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HARMONIC MATERIALS OF
MODERN MUSIC
HARMONIC MATERIALS OF
MODERN MUSIC Resources of the Tempered Scale
Ilowar3™™lfansoir DIRECTOR EASTMAN SCHOOL OF MUSIC UNIVERSITY OF ROCHESTER
New
York
APPLETON-CENTURY-CROFTS,
Inc.
n.
Copyright
©
1960 by
APPLETON-CENTURY-CROFTS,
INC.
610-1
All rights reserved. This hook, or parts thereof, must not he reproduced in any form without permission of the publisher.
Library of Congress Card Number: 58-8138
PRINTED IN THE UNITED STATES OF AMERICA
MUSIC LIBRARY. 'v\t:
H'^
To my dear
who
wife, Peggie,
loves music but does not
entirely approve of the twelve-tone scale, this
book
is
affectionately dedicated.
Preface
This volume represents the results of over a quarter-century of study of the problems of the relationships of tones. The conviction that there
a need for such a basic text has
is
come from the
author's experience as a teacher of composition, an experience
which has extended over a period It
of
more than
thirty-five years.
has developed in an effort to aid gifted young composers grop-
maze
and melodic and searching for an expressive vocabulary which would reach out into new fields and at the same time satisfy their own esthetic desires.
ing in the vast unchartered possibilities,
How
hunting for a
new
of harmonic
"lost chord,"
can the young composer be guided in
far horizons? Historically, the training of the
his search for the
composer has been
and imitation; technic passed on from master to pupil undergoing, for the most part, gradual
largely a matter of apprenticeship
change, expansion, liberation, but, at certain points in history, radical change
and
revolution.
During the more placid days the
apprenticeship philosophy— which
was
is
in effect a study of styles-
and efficient. Today, although still enormously important to the development of musical understanding, it does not, hy itself, give the young composer the help he needs. He might, practical
indeed, learn to write in the styles of Palestrina, Purcell, Bach,
Beethoven, Wagner, Debussy, Schoenberg, and Stravinsky and still
own
have
difficulty in
coming
creative development.
basic,
more concerned with
He
to grips
with the problem of his
more the art and
needs a guidance which
a study of the material of
is
vn
PREFACE less
with the manner of
its
use,
although the two can never
be separated. This universality of concept demands, therefore, an approach in its implications.
The
author has attempted to present here such a technic in the
field
which
radical
is
and even revolutionary
of tonal relationship.
Because of the complexity of the
scope of the work
limited to the study of the relationship of
tones in
melody
or
portant element of
is
harmony without reference rhythm. This is not meant
importance to the rhythmic element.
It
task, the
to the highly im-
to assign a lesser
rather recognizes the
practical necessity of isolating the problems of tonal relationship
and investigating them with the greatest thoroughness composer
is
the
if
to develop a firm grasp of his tonal vocabulary.
hope that this volume may serve the composer in much the same way that a dictionary or thesaurus serves the author. It is I
not possible to bring to the definition of musical sound the same exactness
which one may expect
in the definition of a
word.
It is
possible to explain the derivation of a sonority, to analyze
component
its
and describe its position in the tonal cosmos. young composer may be made more aware of the whole tonal vocabulary; he mav be made more sensitive to the subtleties of tone fusion; more conscious of the tonal alchemy by which a master may, with the addition of one note, transform and illuminate an entire passage. At the same time, it should give to the young composer a greater confidence, a surer grasp of his material and a valid means of self-criticism of the logic and
In this
wav
parts,
the
consistency of his expression.
would not seem necessary to explain that this is not a "method" of composition, and yet in these days of systems it may be wise to emphasize it. The most complete knowledge of tonal material cannot create a composer any more than the memorizing of Webster's dictionary can produce a dramatist or poet. Music is, or should be, a means of communication, a vehicle It
Without that communicate, without— in other
for the expression of the inspiration of the composer.
inspiration, without the viii
need to
PREFACE
words— the creative spirit itself, the greatest knowledge will avail nothing. The creative spirit must, however, have a medium in which to express itself, a vocabulary capable of projecting with the utmost accuracy and sensitivity those feelings which seek expression. It
composer gift
is
is
hope that
in developing his
may express
which
my
the
itself
mark of
this
volume may
own vocabulary
with that simplicity,
all
assist
the young
so that his creative
clarity,
and consistency
great music.
Since this text differs radically from conventional texts on "har-
mony,"
may be
it
helpful to point out the basic differences
together with the reason for those diflFerences. Traditional theory, based on the harmonic technics of the
seventeenth, eighteenth, and nineteenth centuries, has distinct
when
limitations
the
late
applied to the music of the twentieth— or even
nineteenth— century.
Although traditional harmonic
theory recognizes the twelve-tone equally tempered scale as an
underlying basis,
its
fundamental scales are actually the seven-
tone major and minor scales; and the only chords which
it
admits
are those consisting of superimposed thirds within these scales
together with their "chromatic" alterations.
The many other com-
binations of tones that occur in traditional music are accounted for as modifications of these chords
tones,
and no further attempt
is
by means
made
of "non-harmonic"
to analyze or classify
these combinations.
This means that traditional harmony systematizes only a very small proportion of leaves
all
the possibilities of the twelve-tones and
all
the rest in a state of chaos. In contemporary music, on
the other hand,
major and minor
many
other scales are used, in addition to the
scales,
and
intervals other than thirds are
used
in constructing chords. I
have, therefore, attempted to analyze
all of
of the twelve-tone scale as comprehensively
traditional
chords
it
classified
and
the possibilities as
thoroughly as
harmony has analyzed the much smaller number
covers. This vast
and thus reduced
of
and bewildering mass of material is to comprehensible and logical order IX
PREFACE
by four
chiefly
devices: interval analysis, projection, involution,
and complementary Interval analysis
is
scales.
explained in Chapter 2 and applied through-
out. All interval relationship
perfect
reduced to
is
six basic categories
:
the
the minor second, the major second, the minor third,
fifth,
each— except the tritone— conabove and below the initial tone.
the major third, and the tritone, sidered in both
its
relationship
This implies a radical departure from the classic theories of interterminology, and their use in chord and scale construc-
vals, their tion.
Most
of
Western music has
perfect-fifth category.
for centuries
Important as
been based on the
this relationship
has been,
it
should not be assumed that music based on other relationships
cannot be equally valid, as Projection logical
I
believe the examples will show.
means the construction
and consistent process
of scales or chords
of addition
and
by any
repetition. Several
types of projection are employed in different sections of the book. If
a series of specified intervals, arranged in a definite ascending
order,
is
order,
it
compared with a is
similar series arranged in descending
found that there
is
a clear structural relationship
between them. The second series is referred to here as the involution of the first. (The term inversion would seem to be more accurate, since the process is literally the "turning upsidedown" of the original chord or scale. It
might
The
result
was
felt,
however, that confusion
because of the traditional use of the term inversion.
)
any sonority and its and extensively employed later on. Complementary scales refer to the relationship between any series of tones selected from the twelve-tones and the other tones which are omitted from the series. They are discussed in Parts V and VI. This theory, which is perhaps the most important— and also the most radical— contribution of the text, is based on the fact that every combination of tones, from two-tone to six-tone, has its complementary scale composed of similar proportions of the same intervals. If consistency of harmonic-melodic expression is important in musical creation, this theory should bear the most relation of
Chapter
3,
involution
is
discussed in
PREFACE intensive study, for
sets
it
up
a basis for the logical expansion of
tonal ideas once the germinating concept has been decided
mind
in the
The
of the composer.
chart at the end of the text presents graphically the relation-
ship of
all
of the combinations possible in the twelve-tone system,
from two-tone intervals to their complementary ten-tone I
upon
must
my
reiterate
sidered a "method" nor a "system." of harmonic-melodic material. Since
It
is,
rather, a
it is
in his lifetime use
of the material studied.
own
it is
all,
Each composer
those portions which appeal to his
compendium
inclusive of
basic relationships within the twelve-tones,
any composer would
scales.
passionate plea that this text not be con-
all
of the
hardly likely that
or even a large part,
will,
rather, use only
esthetic taste
and which
own creative needs. Complexity is no guarantee and a smaller and simpler vocabulary used with sensitivity and conviction may produce the greatest music. Although this text was written primarily for the composer, my colleagues have felt that it would be useful as a guide to the analysis of contemporary music. If it is used by the student of theory rather than by the composer, I would suggest a different
contribute to his of excellence,
mode I
and
of procedure, namely, that the student study carefully Parts II,
Chapters
exercises— although will enlighten
During the
to 16, without undertaking the creative
I
if
there
is
and inform the first
sufficient
time the creative exercises
theorist as well as the composer.
part of this study he should try to find in the
works of contemporary composers examples of the various hexad formations discussed.
He
will not find
them
in great
abundance,
since contemporary composers have not written compositions
primarily to illustrate the hexad formations of this text! However,
when he
masters the theory of complementary scales, he will have
at his disposal
an analytical technic which will enable him to
analyze factually any passage or phrase written in the twelve-tone equally tempered scale.
H. H. Rochester,
New
York XI
Acknowledgments
The author wishes to Professor
his
deep debt of gratitude
many
his
help-
manuEastman School of Music Wayne Barlow, Allen Irvine McHose, Charles Riker,
and
faculty,
and
for his meticulous reading of a difficult
to his colleagues of the
and Robert Sutton, also
acknowledge
Herbert Inch of Hunter College for
ful suggestions script,
to
for valuable criticism.
His appreciation
is
extended to Clarence Hall for the duplication of the chart,
to Carl A.
Rosenthal for his painstaking reproduction of the
examples, and to their
Mary Louise Creegan and
Janice Daggett for
devoted help in the preparation of the manuscript.
His
warm
generous finally
thanks go to the various music publishers for their
permission
and
especially
to
quote to
from
works
copyrighted
Appleton-Century-Crofts
for
and their
co-operation and for their great patience. Finally,
students
my
devoted thanks go to
who have borne with me
my
hundreds of composition
so loyally all these
many
years.
H. H.
Contents
Preface
vu
1.
Equal Temperament
1
2.
The Analysis of Intervals The Theory of Involution
7
3.
Part
I.
17
THE SIX BASIC TONAL SERIES
4.
Projection of the Perfect Fifth
27
5.
Harmonic-Melodic Material of the Perfect-Fifth Hexad
40
6.
Modal Modulation Key Modulation
56
7.
60
Minor Second
8.
Projection of the
9.
Projection of the Major Second
10.
Projection of the Major Second
11.
Projection of the Minor Third
65 77
Beyond the Six-Tone
Series
90 97
12.
Involution of the Six-Tone Minor-Third Projection
13.
Projection of the
14.
Projection of
15.
Projection of the Major Third
16.
Recapitulation of the Triad Forms
17.
Projection of the Tritone
18.
Projection of the Perfect-Fifth-Tritone Series
19.
The pmn-Tritone
20.
Involution of the pmn-Tritone Projection
158
21.
Recapitulation of the Tetrad Forms
161
Six
110
Minor Third Beyond the Six-Tone the Major Third
Series
Beyond the Six-Tone
Series
118 123 132 136 139
Beyond
Tones
148 151
Projection
xiii
CONTENTS
CONSTRUCTION OF HEXADS BY THE SUPERPOSITION OF TRIAD FORMS Part
II.
pmn
22.
Projection of the Triad
23.
Projection of the Triad pns
24.
Projection of the Triad
177
25.
Projection of the
182
26.
Projection of the Triad nsd
167 172
pmd Triad mnd
187
Part III. SIX-TONE SCALES FORMED BY THE SIMULTANEOUS PROJECTION OF TWO INTERVALS
27.
Simultaneous Projection of the Minor Third and Perfect Fifth 195
28.
Simultaneous Projection of the Minor Third and Major Third 200
29.
Simultaneous Projection of the Minor Third and Major
Second
204
30.
Simultaneous Projection of the Minor Third and Minor
31.
Simultaneous Projection of the Perfect Fifth and Major Third 211
32.
Simultaneous Projection of the Major Third and Minor
Second
207
Second 33.
215
Simultaneous Projection of the Perfect Fifth and Minor
219
Second
Part IV.
PROJECTION BY INVOLUTION AND AT FOREIGN INTERVALS
Projection
225
35.
by Involution Major-Second Hexads with Foreign Tone
36.
Projection of Triads at Foreign Intervals
236
37.
Recapitulation of Pentad Forms
241
34.
Part V.
38. 39.
THE THEORY OF COMPLEMENTARY SONORITIES
The Complementary Hexad The Hexad "Quartets" xiv
232
249 254
CONTENTS
COMPLEMENTARY SCALES
Part VI.
40.
Expansion of the Complementary-Scale Theory
4L 42.
Projection of the Six Basic Series with Their Complementary Sonorities 274 Projection of the Triad Forms with Their Complementary
43.
The pmn-Tritone
44.
Projection of
263
285
Sonorities
with
Projection
Its
Complementary 294
Sonorities
Two
Similar Intervals at a Foreign Interval
with Complementary Sonorities 45.
46. 47.
48.
Simultaneous
Projection
of
298
with
Intervals
Their
303 Complementary Sonorities 314 Projection by Involution with Complementary Sonorities 331 The "Maverick" Sonority Vertical Projection by Involution and Complementary 335
Relationship 49.
Relationship of Tones in Equal
346
50.
Translation of Symbolism into
Temperament Sound
356
Appendix: Symmetrical Twelve-Tone Forms
373
Index
377
The Projection and Equal Temperament Chart:
Interrelation of Sonorities in
inside back cover
XV
HARMONIC MATERIALS OF
MODERN MUSIC
1
Equal Temperament
Since the subject of our study
is
the analysis and relationship
of all of the possible sonorities contained in the twelve tones of
the equally tempered chromatic scale, in both their melodic and
harmonic implications, our
first
task
is
to explain the reasons for
basing our study upon that scale. There are two primary reasons.
The
that a study confined to equal
first is
though complex, a
-finite
retical
within just intonation
A
possibilities
temperament
would be
simple example will illustrate this point.
major
third, E,
G#, above E,
al-
is,
study, whereas a study of the theo-
If
we
infinite.
construct a
above C, and superimpose a second major
we produce
the sonority C-E-G#i
superimpose yet another major third above the tone B#. In equal temperament, however, equivalent of C, and the four-tone sonority
GJj:,
Now we
third,
we
if
reach the
B# is the enharmonic C-E-G#-B# is actually
the three tones C-E-Gfl: with the lower tone, C, duplicated at the octave.
In just intonation, on the contrary,
the equivalent of C.
A
B# would not be
projection of major thirds above
C
would therefore approach infinity. The second reason is a- corollary of the first. Because
in
just intonation
pitches
possible
intonation
ments or
is
for
in
just
intonation
approach
infinity,
the just
not a practical possibility for keyboard instru-
keyed and valve instruments of the woodwind and
brass families. Just intonation
would be possible
for stringed
instruments, voices, and one brass instrument, the slide trom-
bone. However, since
much
of our
music
is
concerted, using
all
HARMONIC MATERIALS OF MODERN MUSIC and since it is unlikely that keyboard, keyed, and valve instruments will be done away with, o£ these resources simultaneously,
at least within the generation of living composers, the system
of equal
temperament
Another
is
advantage
the logical basis for our study.
of
equal
temperament
simplicity possible in the symbolism
is
the
greater
of the pitches involved.
Because enharmonic equivalents indicate the same pitch, possible to concentrate
upon the sound
than upon the complexity of
its
it
of the sonority rather
spelling.
Referring again to the example already cited,
if
we were
find ourselves involved in endless complexity.
to
we
continue to superimpose major thirds in just intonation
would soon
is
The
BJj: would become D double-sharp; the major third above D double-sharp would become F triple-sharp; the next major third, A triple-sharp; and so on. In equal temperament, after the first three tones have been notated— C-E-Gjj:— the G# is considered the equivalent of Aj^ and the succeeding major thirds become C-E-Gfl:-C, merely octave duplicates of the
major third above
first
three.
Example Pure Temperament
"% !
1-1
Equal Temperament
]
ip"
)
This point of view has the advantage of freeing the composer
from certain inhibiting preoccupations with academic symbolization as such.
For the composer, the important matter
sound of the notes, not
G-B-D-F sounds
their "spelling."
like a
is
the
For example, the sonority
dominant seventh chord whether
it
is
G-B-D-F, G-B-D-E#, G-B-CX-E#, G-Cb-C-:^-F, or in some other manner. The equally tempered twelve-tone scale may be conveniently thought of as a circle, and any point on the circumference may spelled
be considered
as representing
any tone and/or
its
octave. This
EQUAL TEMPERAMENT circumference
may
then be divided into twelve equal parts, each
representing a minor second, or half-step. Or, with equal validity,
each of the twelve parts fifth,
since
embraces
all
the
may
represent the interval of a perfect
superposition
of
twelve
perfect
also
fifths
of the twelve tones of the chromatic scale— as in the
familiar "key-circle."
We shall find the latter diagram particularly
useful. Beginning on
C and
superimposing twelve minor seconds
or twelve perfect fifths clockwise around the circle,
the circle at
BJf,
which
in equal
as C. Similarly, the pitch
names
we
complete
temperament has the same pitch of
C# and D^, D# and
Ej^,
and
so forth, are interchangeable.
Example
1-2
GttlAb)
D« (Eb
MK
The term
sonority
of tone relationship,
When we
is
used in
whether
this
book
in terms of
to cover the entire field
melody or of harmony.
speak of G-B-D-F, for example,
we mean
ship of those tones used either as tones of a
harmony. This
may seem
(Bb)
the relation-
melody
to indicate a too easy fusion of
or of a
melody
and harmony, and yet the problems of tone relationship are essentially the same. Most listeners would agree that the sonority in Example l-3a is a dissonant, or "harsh," combination of tones when sounded together. The same efl^ect of dissonance, however, persists in our aural memory if the tones are sounded consecutively, as in Example l-3b:
HARMONIC MATERIALS OF MODERN MUSIC
Example
1-3
(fl)
i The of
first
^
problem
component
its
in the analysis of a sonority
A
parts.
sonority sounds as
it
is
the analysis
does primarily
because of the relative degree of consonance and dissonance of elements, the position and order of those elements in relation
its
to the tones of the clarity in tion,
harmonic
series,
the degree of acoustical
terms of the doubling of tones, timbre of the orchestra-
and the
like.
further affected
It is
which the sonority
is
by the environment
in
placed and by the manner in which
experience has conditioned the ears of the listener.
Of these
factors, the first
would seem
to
be
For example,
basic.
the most important aural fact about the familiar sonority of the
dominant seventh thirds than of
that
any other
of the perfect fifth of the
is
it
contains a greater
number
interval. It contains also the
of
minor
consonances
and the major third and the mild dissonances
minor seventh and the
tritone.
This
is,
so to speak, the
chemical analysis of the sonority.
Example
Minor thirds
It
is
of
f
Perfect
fifth
1-4
Mojor third
paramount importance
to the
Minor seventh
Tritone
composer, since the
composer should both love and understand the beauty of sound.
He
should "savor" sound as the poet savors words and the
painter form and color. Lacking this sensitivity to sound, the
composer
is
not a composer at
a scholar and a craftsman.
all,
even though he
may be both
EQUAL TEMPERAMENT This does not imply a lack of importance of the secondary analyses already referred
The
to.
various styles and periods,
in
tonality
implied— and the
is
analyses strengthen the
historic position of a sonority
its
function in tonality— where
Such multiple
like are important.
young composer's grasp
of his material,
providing always that they do not obscure the fundamental analysis of the
sound as sound.
we
Referring again to the sonority G-B-D-F,
should note
historic position in the counterpoint of the sixteenth century its
harmonic
position
in
the
tonality
eighteenth, and nineteenth centuries, but
observe
its
construction, the elements of
of these analyses are important
the
we
should
and contribute
and
seventeenth,
of
which
its
it is
first
of all
formed. All
to an understand-
ing of harmonic and melodic vocabulary.
As another example of multiple
analysis, let us take the familiar
contains two perfect
chord C-E-G-B.
It
one minor
and one major seventh.
third,
Example
Perfect
It
may be
fifths
fifths,
two major
thirds,
1-5
Mojor thirds
Minor third
*
Major seventh
considered as the combination of two perfect
fifths at
the interval of the major third; two major thirds at the perfect fifth;
or perhaps as the combination of the major triad
and the minor
triad
E-G-B
or the triads*
Example
C-G-B and C-E-B:
1-6
ofijiij^ij ii i
*The word
triad
is
used
to
mean any
C-E-G
three-tone chord.
HARMONIC MATERIALS OF MODERN MUSIC Historically, ties
it
represents one of the important dissonant sonori-
of the baroque and
may be
classic periods. Its function in tonality
subdominant or tonic seventh of the major
scale,
the mediant or submediant seventh of the "natural" minor
scale,
and
as the
so forth.
Using the pattern of analysis employed
and
1-6,
Examples
1-4, 1-5,
analyze as completely as possible the following sonorities
Example 4.
i
in
ji8
ijia^
1%
5.
^
1-7 e.
±fit ift
7.
9.
=^Iia^itftt«^
10.
i
«sp
The Analysis of
Intervals
In order again to reduce a problem of theoretically proportions to a finite problem, an additional device
is
infinite
suggested.
Let us take as an example the intervallic analysis of the major triad
C-E-G:
Example
Perfect fifth
This triad
is
Major
commonly described
combination of a perfect
fifth
analysis
is
incomplete, since
as long as the triad
however, the chord
is is
third
Minor third
in conventional analysis as a
and a major third above the lowest
or "generating" tone of the triad. It
the minor third between
2-1
it
obvious, however, that this
is
omits the concomitant interval of
E and
G. This completes the analysis
in the simple
form represented above.
present in a form in which there are
If,
many
doublings in several octaves, such a complete analysis becomes
more complex. If
we examine
Transfiguration
the scoring of the final chord in Death and
by Richard
Strauss
Example
:i
* *
^m
we 2-2
find a sixteen -tone chord:
HARMONIC MATERIALS OF MODERN MUSIC These sixteen tones combine to form one hundred and twenty
The
different intervals.
between
relationship
C and G
is
repre-
sented not only by the intervals
Example
2-3
eta
<^
-o-
«
a o ^ but also by the intervals
Example
i ^ 5 in
E and E
f^
»^^
-o
»^
*^'*
—©—©—o—
which case we commonly
"inversion" of the
2-4
first.
second relationship the
call the
The same
is
true of the relation of
C
to
to G.
However, the composite impression of the
C
of all of the tones
still
gives the
major triad in spite of the complexity of
doubling. In other words, the interval
C
function in the sonority regardless of the
to
G
performs the same
manner
of the doubling
of voices.
The
similarity of
illustrated
if
an interval and
inversion
may be
further
one refers again to the arrangement of the twelve-
tone scale in the circle of
8
its
fifths:
THE ANALYSIS OF INTERVALS
Example
Here
C
to
it
G
will
and
be seen that
C
C
has two perfect-fifth relationships,
to F; the one,
(ascending) and the other,
C
2-5
C
to G, proceeding clockwise
to F, proceeding counterclockwise
(descending). In the same manner,
C
has two major-second
C to D and C to B^; two major-sixth relationships, C to A and C to E^; two major-third relationships, C to E and C to A\); and two major-seventh relationships, C to B and C to Dt>. It has only one tritone relationship, C up to F#, or C down
relationships,
to G\). It will be helpful in ,our analysis
if
we
use only one
symbol to represent both the interval under consideration and its
inversion. This
is
not meant to imply that the interval and
inversion are the same, but rather that they perform the
its
same
function in a sonority.
Proceeding on
this theory,
we
shall
represent the relationship of the perfect first
tone,
intervals
choose the symbol p to fifth
above or below the
even though when the lower tone of each of the two is
raised an octave the relationship
a perfect fourth:
becomes actually
harmonic materials of modern music
Example
# The symbolization it
is
Perfect
fifth
p
-
2-6
Perfect
Perfect
fifth
fourth
arbitrary, the letter
p being chosen because
connotes the designation "perfect," which apphes to both
intervals.
The major third above nated by the letter m:
or
below the given
Example
tojie will
be desig-
2-7
^ ^^ Major third, m minor sixth)
(or
The minor third above represented by the letter n:
or
below the given tone
Example
i
2-8
B^ Minor third, n (or major sixth)
by
the major second above or below,
Example
s:
2-9
(i'ji)
i
t»to
tib<^
Major second, s (or minor seventh)
the dissonant minor second by d:
Example
2-10
Minor second, d mojor seventh)
(or
10
will
be
THE ANALYSIS OF INTERVALS and the
tritone
by
t:
Example
2-11
M ^ (bo'i
*^
Augmented fourth,^
(or
diminished fifth) (Tritone)
The
letters
pmn,
represent
therefore,
intervals
commonly
considered consonant, whereas the letters sdt represent the intervals
commonly considered
dissonant.
The symbol pmn,
sdt'*
would therefore represent a sonority which contained one perfect fifth or its inversion,
inversion, the
major
sixth;
minor
sixth;
one minor third or
one major second or
one minor second or
augmented fourth symbols at
the perfect fourth; one major third or
or
tiie left of
its its
its
inversion, the
minor seventh;
inversion, the major seventh;
and one
inversion, the diminished fifth; the three
the
comma
representing consonances, those
at the right representing dissonances.
for example,
its
inversion, the
its
by the symbol
A
sonority represented,
sd^, indicating a triad
composed
of
one major second and two minor seconds, would be recognized as a highly dissonant sound, while the
symbol
pmn would indicate
a consonant sound.
The complexity of the analysis will depend, obviously, upon the number of diflFerent tones present in the sonority. A threetone sonority such as C-E-G would contain the three intervals C to E, C to G, and E to G. A four-tone sonority would contain
3+2+1 tervals,
Since task
is
or 6 intervals; a five-tone sonority,
and so
we
or 10 in-
on.
are considering
somewhat
tones in equal temperament, our
all
simplified.
C
the same sound as the interval "
4+3+2+1
to
C
D#, for example, represents
to E^i;
and since the sound
For the sake of uniformity, analyses of sonorities will
list
is
the constituent inter-
vals in this order.
11
HARMONIC MATERIALS OF MODERN MUSIC the same, they would both be represented by the single symbol n.
be
A
table of intervals with their classification would, therefore,
as follows:
C-G (orG-C),B#-G, C-F^K<,etc. C-E (or E-C), B#-E, C-Fb, BJf-Fb, etc. C-Eb (orEb-C),C-Dif, B#-Eb, etc.
C-D
(or D-C), Bif-D, C-Ebb, etc.
C-Db(orDb-C),C-C#,B#-Db,etc. C-F# (or F#-C), C-Gb, B#-Gb, etc.
Example xo
i
i-
-
*0
fv^
»
2-12
efc.
SE
it\^
^°
tf^g'1
£
'^»
r
i^i
m
^^g
Qgyi
¥^
»j|o
t^
etc
efc.
ife
= p = m = n = s = d = t
Ed -XT
bo
*
^^
^^ bo^^'^'
For example, the augmented triad C-E-G# contains the major third C to E; the major third E to G#, and the interval C to G^. Since, however,
C
to
G# sounds
like
C
to Ab, the inversion of
which is Ab to C— also a major third— the designation of the augmented triad would be three major thirds, or m^. A diagram of these three notes in equal temperament quickly illustrates the validity of this analysis.
The
joining of the three notes
C-E-G#
(Ab) forms an equilateral triangle—a triangle having three equal sides and angles:
Example B /-^
'
2-13 -\
CJt
G
12
THE ANALYSIS OF INTERVALS It
of
is,
of course, a figure
which
side
is
used as
which has the same form regardless
its
base:
Example
Similarly the of
which
augmented
of the three tones
G#
•
E C One
is
triad
2-14
sounds the same regardless
the lowest:
B#(C) G#
E
E
G#(Ab)
final illustration will indicate
C the value of this technique
of analysis. Let us consider the following complex-looking sonority in the light of
conventional academic analysis:
Example
2-15
f The chord +1, or 15
contains six notes and therefore has
5+4+3+2
intervals, as follows:
C-D# and Ab-B C-E and G-B C-G and E-B C-Ab and D#-B
augmented seconds major thirds perfect fifths
minor
sixths
13
HARMONIC MATERIALS OF MODERN MUSIC
C-B D#-E and G-A^ Djf-G and E-Aj^
D#-Ab
E-G However,
in the
new
= = = = =
major seventh
minor seconds diminished fourths double-diminished
fifth
minor third
analysis
it
converts
itself into
only four
types of intervals, or their inversions, as follows:
C-G, E-B, and Ab-Eb (Dif ). 6 major thirds: C-E, Eb (D^f )-G, E-G# (Ab), G-B, Ab-C, and B-D#.
3 perfect
fifths:
3 minor thirds:
C-Eb (D#), E-G, and G# (Ab)-B.
3 minor seconds: DJf-E, G-Ab, and B-C.
The
description
is,
therefore, p^m^n^d^.
Example Perfect
i
Mojor thirds
fifths
o
2-16
ii
\fv:w^
Minor thirds
i b^||o)t8
^
b8(tfo)tlit^
^
vObB
l^»
^
Minor seconds
=^a= tfS^^^ Ijl^ ^'^'
A diagram will indicate the essential simplicity of the structure: Example
2-17
G<»(Ab)
14
THE ANALYSIS OF INTERVALS has been
It
who may
my
experience that although the young composer
has been thoroughly grounded in academic terminology at
first
be confused by
own
simplification,
this
embraces the new analysis because
it
he quickly
conforms directly to his
aural impression.
In analyzing intervals, the student will find the habit of "measuring" in half-steps
down),
intervals in terms of the "distance"
all
between the two
for example, will
half-steps
tones.
Seven half-steps (up or
be designated by the symbol
by the symbol m; three
so forth, regardless
practical to form
it
p; four
by the symbol n, and the tones which form
half-steps
of the spelling of
the interval:
V
m n
s
perfect fifth
7 half -Steps
perfect fourth
5
If
II
major third
4
II
II
minor
sixth
8
II
II
minor third
3
II
II
major sixth
9
II
II
major second
2
II
II
10
II
II
II
II
11
II
II
augmented fourth
6
II
II
diminished
6
II
II
minor seventh minor second
d
1
major seventh
t
fifth
Example m
p
9f=^4S=-^c^^
#^
Perfect fifth
2-18
Perfect fourth
44«
n
i."*
Major
Minor
third
sixth
—A?—
4.^
^inor
Major
third
sixth
1
15
HARMONIC MATERIALS OF MODERN MUSIC
=15=
=f^rt
t
^
w
'^XXt
«»i
Minor seventh
Major second
Augmented Diminished
Minor Major second seventh
speaking of sonorities
In
^ ^M
d
s
2T*t°
we
shall
fifth
fourth
make
apparently
little
between tones used successively in a melody and tones used simultaneously in a harmony. It is true that the distinction
addition of the element of rhythm, the indispensable adjunct of its varying degrees of emphasis upon individual by the devices of time length, stress of accent, and the like, creates both great and subtle variance from the sonority played
melody, with notes
as a "block" of sound. Nevertheless, the basic relationship
same. itself
A
melody may grow out
of a sonority or a
is
the
melody may
be a sonority.
Analyze the following sonorities
in the
same manner employed
In Examples 2-15 and 2-16, pages 13 and 14, giving
first
the
conventional interval analysis, and second the simplified analysis:
Example
i^
jt#
#
^S^
I
^i
^
2-19
^^ 3 r^
Repeat the same process with the chords
16
^1^
^S in
Example
w
S3S: ^»S^
'^BT 1-7,
page
6.
The Theory of Involution
Reference has already been made aspect of musical relationship, that
"down"
is,
to
the
two-directional
the relationship "up" and
in terms of pitch, or the relationship in clockwise or
counterclockwise rotation on the circle already referred will
be readily apparent that every sonority
in
to.
It
music has a
counterpart obtained by taking the inverse ratio of the original sonority.
The
projection dovon from the lowest tone of a given
chord, using the
same
in the given chord,
This
chord. original.
intervals in the order of their occurrence
we may
counterpart
is,
the involution of the given
call
so
to
speak,
"mirror"
a
For example, the major triad C-E-G
is
of
the
formed by the
projection of a major third and a perfect fifth above C. However, if
this
same relationship
to
E
has as
C
to
G
its
has as
is
projected downward, the interval
counterpart the interval its
counterpart
[C
|C
to Aj^;
and the interval
to F.
Example
C
3-1
B
17
HARMONIC MATERIALS OF MODERN MUSIC It will
be noted that the involution of a sonority always contains
the same intervals found in the original sonority.
There are three types of involutions: simple, isometric, and enharmonic. In simple involution, the involuted chord differs in sound from the given chord. Let us take, for example, the major triad C-E-G,
which
is
formed by the projection
of a
major third and a perfect
formed by the projection downward and a perfect fifth, is the minor triad '[F-A^-C. The major triad C-E-G and its involution, the minor triad ^F-A^-C, each contain a perfect fifth, a major third, and a
fifth
from
above C.
C
minor
Its involution,
of a major third
third,
and can be represented by the symbols pmn.
Example
i ^m
3-2
^
In the second type of involution, which
we may
call isometric
same kind of sound as For example, the tetrad C-E-G-B has as its
involution, the involuted sonority has the
the original sonority.
involution jDb-F-Ab-C.
Example
3-3
«
18
^
:
THE THEORY OF INVOLUTION
Each of these is a major seventh chord, containing two perfect two major thirds, a minor third, and a major seventh, and can be characterized by the symbols p^irrnd, the exponents in this instance representing two perfect fifths and two major thirds. fifths,
In the third type, enharmonic involution, the invohited sonority
and the
octaves
(except
triad
for
one
common
same tones
tone).
in different
For example, the
C-E-G# involutes to produce the augmented ^F^-Ab-C, F^ and A^ being the equal-temperament equivaof E and G#. Another common example of enharmonic
augmented lents
original sonority contain the
involution
is
triad
the diminished seventh chord
Example
3-4
«
m
All sonorities
sonority with
its
3
.CK.
I
which are formed by the combination
of a
involution are isometric sonorities, since they
have the same order of intervals whether considered "up" or "down," clockwise or counterclockwise. We have already seen
will
that the involution of the triad
C-E-G
is
jC-Ab-F. The two
together produce the sonority F3Ab4C4E3G, which has the same
order of intervals *The numbers
upward
indicate the
or
downward.*
number
of half-steps
between the tones of the
sonority.
19
)
HARMONIC MATERIALS OF MODERN MUSIC the tone
If
E
of the triad
C4E3G
is
used
as the axis of involu-
tion, a diflFerent five-tone sonority will result, since the involution
E3G5C
be J^EsC^gGJ, forming together the sonority GJgCfllsEaGgC. If the tone G is used as the axis of involution, the involution of G5C4E will be J,G5D4Bb, forming together the of
will
sonority Bb4D5G5C4E. These resultant sonorities will to
be isometric
in structure.
(
Example
"^^",
If
44
(3)
'^%j|§ii
two tones are used
be a four-tone isometric
be seen
3-5
(2)
)
all
See Note, page 24.
r
33
.
'
bS-i|-2"3
3
2
as the axes of involution, the result will
sonority:
Example
3-6
5=^
343
313
434
C and G constitute the and E; and in the third E and G. The discussion of involution up to this point does not differ greatly from the "mirror" principle of earlier theorists, whereby "new" chords were formed by "mirroring" a familiar chord and In the
"double
first
of the
axis"; in
above examples,
the second
C
combining the "mirrored" or involuted chord with the
original.
however, we shall expand the principle to the becomes a basic part of our theory. When a major triad is involuted— as in Example 3-2— deriving the minor triad as the "mirrored" image of the major triad seems to place the minor triad in a position of secondary importance, as the reflected image of the major triad. In the principle of involution presented here, no such secondary importance is intended; for if the minor triad is the reflected image of the major triad, it is equally true that the major triad is
At
this point,
point where
20
it
"
THE THEORY OF INVOLUTION image
also the reflected
and the involution
minor
of the
C4E3G
involution of the major triad of the
is
For example, the
triad.
the minor triad |C4Ab3F,
minor triad C3E[74G
is
the major triad
jCaA^F. In order to avoid any implication that the involution speak, a less important sonority, ties
by
construct both the
shall in analyzing the sonori-
sonority
first
and
its
upward
involution
the simple process of reversing the intervallic order. For
example, will
we
so to
is,
if
the
triad
first
C4E3G the
is
involution of this triad
be any triad which has the same order of half-steps
in
reverse, for example F3Ab4C, the comparison being obviously 4-3
versus 3-4.
In this sense, therefore, the involution of a major triad can be considered to be any minor triad whether or not there
an
is
axis
of involution present.
In Example 3-7, therefore, the
F# minor, E^ minor, and
D
B
minor,
minor
B^ minor, G$ minor,
triads are all considered as
C major triad, although there is no When the C major triad is combined with any
possible involutions of the axis of involution.
one of them, the resultant formation
is
a six-tone isometric
sonority.
Example
m Ha
m=^
«o
222 ffi
Wt
Hi^
i^
btxhc^feo>. . ..i'« ti'
i
3
3
bo 2
I
OgP "^^= 3 13 13 i
3-7
12 13
m^
m
b
fc^
13 12
^=j^
M\i
3
2
12
=»=si
"° 2
3
2
3 2
21
2
HARMONIC MATERIALS OF MODERN MUSIC Note that the combination of any sonority with form always produces an isometric sonority, that
which can be arranged
in such a
the same whether thought
intervals
is
the
combination in Example
first
manner
B
to
G
or from
G
or
3-7, if
configuration BiC2D2E2FJj:iG, which
sidered from
up
is
that
its is,
its
involuted a sonority
foraiation of
down. For example,
begun on
B, has the
the same whether con-
to B,
The second combination, C major and B^ minor, must be Bj^ or E to make its isometric character clear:
begun on
BbsCiDbsEiF^G or E,F,GsB\),C,DhThe isometric character of the third combination, C major and G# minor, is clear regardless of the tone with which we begin: C3D#iE3GiG#3B; D^,E,G,GJi^,B,C,
however, for the sake of comparison,
If,
with another major
triad
C
major with
since tion
etc.
major, the resultant formation
a major
is
not isometric,
the same up or down:
CsD^E^FSiG^A;
D^EsFliG^AsC; G2A3C2D2E2F#;
FitiGsAgCsDsE;
There
is
few
are a
we combine
example, the combination of
impossible to arrange these tones so that the configura-
it is
is
D
triad, for
E^FSiG^AsC^D; A.C^D^E^Fi.G.
one more phenomenon which should be noted. There
sonorities
which have the same components but which
are not involutions one of the other, although each has
its
own
Examples are the tetrads C-E-fJ-G and C-F#-G-Bb. Each contains one perfect fifth, one major third, one minor third,
involution.
one major second, one minor second, and one tritone (pmnsdt), but one its
own
We
is
not the involution of the other— although each has
involution.
shall describe
such sonorities, illustrated in Example
as isomeric sonorities.
22
3-8,
the theory of involution
Example ^
IIIVUIUMUIIi Involution'.
3-8
ife
^#
pmnsdt
pmnsdt
Using the lowest tone of each of the following three-tone each
sonorities as the axis of involution, write the involution of
by projecting the
sonority
downward,
Example 2.
i
^5^=
as in
Example
3-9
c^"*
5^
^
i
:^^(*^
4b.
5a.
=S^Q=
I
7o.
Mb.
ft
2b
'
I
^^ 10.
12.
tU^ lOb.
lOo.
'ith^'^
^tbt^^
6b.
^'
9.
U'1%
ytbb-Q^^bt^oo'ro
|bo gboM ^nrgr
>
8.
7b.
llo.
Q
2og
sT-^n
Sn g o*> ^ W 6o.
5b.
7rt o*^
3b.
3o.
2b.
2o.
:x«o=
4o.
3-5.
10
^ -=
12b.
12a.
^^f ^ 5=33= ng»-
Solution: fl/C.
Ib.
i m
^^D»=
SijO-
Zl-oU
-«s^l2
"^^If
The following
^15
scales are all isometric,
formed by the combina-
tion of one of the three-tone sonorities in
involution.
Match the
sonority in
Example
scale in
Example 3-9 with
its
Example 3-10 with the appropriate
3-9.
23
harmonic materials of modern music
Example
^J
>r
J J
U^J JuJ
Ij
jiiJ
3-10
Ji'^
i
j^jjg^J jj i
Ji'-^t
'fiJ^J^rU^jjtJJUjtJJtf^^i^rr^r'Ti^^
t
^^rrrr
IJJ
i
juJJf
J ^ ^ i;ii.JtJbJ
(j^^jiJ^ri|J^^rr Note:
We
ijJjtJ
U|J
i
J
^
^^^
jjJ^^
i^^^^^UJ^tJ^P
Uj jj^^*^ i>J^^ i
''^^rrri>ji>J^^^
have defined an isometric sonority
as
one which
has the same order of intervals regardless of the direction of projection.
The student should note
character of a sonority
is
that
this
bidirectional
not always immediately evident. For
example, the perfect-fifth pentad in the position C2D2E3G2A3(C)
does not at position
apparent.
24
first
glance seem to be isometric.
D2E3G2A3C2(D),
its
isometric
However
character
is
in the
readily
1^
:
PartJ
THE SIX BASIC TONAL SERIES
4
Projection of the Perfect Fifth
We if
have seen that there are
we
types of interval relationship,
consider such relationship both "up" and "down": the
perfect fifth and
and
six
its
the minor sixth;
inversion,
its
inversion, the perfect fourth; the major third
inversion, the major
the minor third and
major second and
sixth; the
minor seventh; the minor second and seventh; and the tritone,— the fifth— which
and
t,
we
its
its
inversion, the
inversion, the major
augmented fourth by the letters,
are symbolizing
its
or diminished p,
m,
n,
s,
d,
respectively.
In a broader sense, the combinations of tones in our system of
equal temperament— whether such sounds consist of two tones or
many— tend
to
group themselves into sounds which have a
preponderance of one of these basic
most
sonorities fall into
fifth types,
There
is
one of the
intervals. In other words,
six great categories: perfect-
major-third types, minor-third types, and so forth.
a smaller
number
in
which two
of the basic intervals
predominate, some in which three intervals predominate, and a
few
in
which four
intervals
have equal strength.
Among
the
six-tone sonorities or scales, for example, there are twenty-six
which one interval predominates, twelve which are dominated equally by two intervals, six in which three intervals have equal strength, and six sonorities which are practically neutral
in
in "color,"
since four of the six basic intervals are of equal
importance.
The
simplest and most direct study of the relationship of tones
27
THE is,
TONAL
SIX BASIC
SERIES
therefore, in terms of the projection of each of the six basic
intervals discussed in
Chapter
By
2.
"projection"
we mean
the
by superimposing a series of similar intervals one above the other. Of these six basic intervals, there are only two which can be projected with complete consistency by superimposing one above the other until all of the tones of the equally tempered scale have been used. These two building of sonorities or scales
consider
first
We
and the minor second.
are, of course, the perfect fifth
shall
the perfect-fifth projection.
we add
Beginning with the tone C,
and then the perfect
fifth,
fifth,
G,
D, to produce the triad C-G-D
or,
first
the perfect
reduced to the compass of an octave, C-D-G- This triad contains,
two fifths, the concomitant may be analyzed as ph.
in addition to the
major second.
It
Example
m
4-1
Perfect Fifth Triad,
^^ 2
The
p^
5
tetrad adds the fifth above D, or A, to produce
contains three perfect
time in
fifths,
this series— a
C-G-D-A,
C-D-G-A. This sonority
or reduced to the compass of the octave,
first
interval of the
two major seconds, and— for the
minor
third,
Example
A
to C,
4-2
Perfect FifthTetrad.p^ns^
m
^^ 2
The analysis is, therefore, p^ns^. The pentad adds the next C-G-D-A-E, or the melodic
5
2
fifth,
E,
forming the sonority
C-D-E-G-A, which
will
be
recognized as the most familiar of the pentatonic scales.
Its
components are four perfect 28
scale
fifths,
three major seconds, two
PROJECTION OF THE PERFECT FIFTH
minor
thirds,
and— for
the
time— a major
first
third.
The
analysis
therefore, p^mnh^.
is,
Example Perfect
4-3
Fifth Pentad,
p^mn^s^
i .
S
^^ o
2
2
The hexad adds
B,
2
3
C-G-D-A-E-B, or melodically, producing
C-D-E-G-A-B,
Example
4-4
Perfect Fifth Hexod,p^nn^n^s^d
m 1 4JJ 2
its
components being
2
3
2
2
five perfect fifths, four
major seconds, three
two major thirds, and— for the first time— the dissonant minor second (or major seventh), p^m^n^s'^d. minor
thirds,
The heptad adds F#:
Example Perfect Fifth
i a
Heptod.p^m^n^s^d^t
^^ '2
2
4-5
2
^ '
•I I
2
2
29
)
THE producing the
first
TONAL
SIX BASIC
scale
which
in
its
SERIES
melodic projection contains
second— in other words, a scale without melodic "gaps." It also employs for the first time the interval of the tritone (augmented fourth or diminished fifth), interval larger than a major
no
C
to
This sonority contains
FJf.
six perfect
seconds, four minor thirds, three major thirds,
and one is
the
tritone: p^m^n'^s^dH. (It will
first
sonority to contain
The octad adds
fifths,
five
major
two minor seconds,
be noted that the heptad
of the six basic intervals.
all
Cfl::
Example
4-6
Perfect Fifth Octod.
p^m^ n ^s^ d^
t^
Am
«5i=
5
12 Its
components are seven perfect
minor
thirds,
12
2
2
fifths, six
major seconds,
five
four major thirds, four minor seconds, and two
tritones: p^m'^n^s^dH^.
The nonad adds G#:
Example
m—
4-7
Nonad, p^m^n^s^d^t^
Perfect Fifth
J^
m
m iff
Its
I
?
components are eight perfect
minor
thirds,
six
major
tritones: p^m^n^s^dH^.
30
thirds,
=
9
fifths,
six
seven major seconds,
six
minor seconds, and three
PROJECTION OF THE PERFECT FIFTH
The decad adds D#:
Example «*!" Perfect
u
-
4-8
Decad, p^m^n^s^d^t^
Fifth
^^
m IT"
Its
I
I
I
I
I
I
components are nine perfect
O I
I
2
eight major seconds, eight
fifths,
and four
minor thirds, eight major thirds, eight minor seconds, tritones: 'p^m^n^s^dH'^.
The undecad adds A#:
Example
4-9
Undecad p'°m'°n'°s'°d'°t^
? s"** Perfect Fifth
,
Isjf
^^
m 1*"^ I
Its
r
components are ten perfect
minor
thirds,
2
I
I
fifths,
II
I
I
ten major seconds, ten
ten major thirds, ten minor seconds, and five
tritones: p^'^m'V^s'Od/'^f^
The duodecad adds the
E#:
last tone,
Example
A^
I s
4-10 I2_l2j2„l2jl2^6
Perfect Fifth Duodecad, p'^m'^n'^s'^d
V^
r
I
r
I
I
I
I
I
I
I
I
31
)
:
THE Its
TONAL
SIX BASIC
components are twelve perfect
SERIES
fifths,
twelve major seconds,
twelve minor thirds, twelve major thirds, twelve minor seconds,
and
six tritones: p'^^m^^n^^s^^d^H^.
The student should observe components of the
intervallic
carefully the progression of the
perfect-fifth projection, since
it
has
important esthetic as well as theoretical implications:
doad:
P
triad:
p^s
tetrad:
p^ns^
pentad:
p^mn^s^
hexad:
p^m^n^s^d
heptad:
p^m^n^sHH
octad:
p'm^nhHH^
nonad:
p^m^n^s^dH^
decad:
p^m^n's^dH''
undecad:
plO^lO^lO^lO^lO^B
duodecad
p'^m^^n'^s^^d'H'
In studying the above projection from the two-tone sonority to the twelve-tone sonority built
should be noted. The
first
is
on perfect
fifths,
several points
the obvious affinity between the
perfect fifth and the major second, since the projection of one perfect fifth
upon another always produces the concomitant
interval of the
whether or not
major second. this
is
(It
interesting to speculate as to
is
a partial explanation of the fact that the
"whole-tone" scale was one of the
first
of the "exotic" scales to
make a strong impact on occidental music. The second thing which should be noted
is
the relatively
greater importance of the minor third over the major third in
the perfect-fifth projection, the late arrival of the dissonant
minor second and,
The its
last of all,
third observation
esthetic implications.
related
by the
sonority, there
32
•
is
the tritone.
of the greatest importance because of
From
the
first
sonority of
interval of the perfect fifth, is
up
two
tones,
to the seven-tone
a steady and regular progression.
Each new
PROJECTION OF THE PERFECT FIFTH
new
tone adds one
interval, in addition to
adding one more to
each of the intervals already present. However, when the projection
beyond seven
carried
is
added. In addition to this
no new
tones,
intervals
any new material, there
loss of
can be is
also
a gradual decrease in the difference of the quantitative formation
same number
of the sonority. In the octad there are the
of major
thirds and minor seconds. In the nonad the number of major thirds,
minor
and minor seconds
thirds,
contains an equal
number
is
the same.
of major thirds,
When
seconds, and minor seconds. sonorities are reached, there
is
no
minor
The decad major
thirds,
the eleven- and twelve-tone
differentiation whatsoever, ex-
number of tritones.* The sound of a sonority— either as harmony or melodydepends not only upon what is present, but equally upon what is absent. The pentatonic scale in the perfect-fifth series sounds as cept in the
it
does not only because
fifths
and because
second or the
On
contains a preponderance of perfect
of the presence of major seconds,
and the major third also because
it
minor
thirds,
in a regularly decreasing progression, but
does not contain either the dissonant minor
it
tritone.
the other hand, as sonorities are projected beyond the
six-tone series they tend to lose their individuality. All seven-tone
example, contain
series, for
difference
in
their
all
of the six basic intervals,
proportion
decreases
as
and the
additional
tones
are added.
This
is
probably the greatest argument against the rigorous
use of the atonal theory in which
all
twelve tones of the chro-
matic scale are used in a single melodic or harmonic pattern, since
such patterns tend to lose their identity, producing a
monochromatic
effect
with
its
accompanying lack of the
essential
element of contrast. All of the perfect-fifth scales are isometric in character, since
any of the projections which * See
page 139 and
we have
if
considered are begun on
140.
33
THE
SIX BASIC
TONAL
SERIES
the final tone of that projection and constructed
same
resultant scale will be the
The seven-tone
scale
as
C2D2E2F#iG2A2B,
the
same
Every scale may have
F+f— and projected
is,
tones: J,F#2E2D2CiB2A2G.
many
as
versions of
The seven-tone
there are tones in the scale.
begun on
for example,
the final tone of the projected fifths— that
downward produces
downward, the
the projection were upward.
if
basic order as
its
example,
scale, for
has seven versions, beginning on C, on D, on E, and so forth.
Example Seven "versions"
i 2
2
Perfect Fifth Heptad
of the
^^ o 2
rtn*
*^
o^^ »
v>
2
2
2
2
(1)
*^ =0^5 O*
2
2
1
2
2
2 (2)
2
2
I
^f
2
2
2
^ 2
(2)
;x4^M
^"*
I
^\ 2
I
=^33 bcsr^ 3s:«i
4-11
(I)
2
2
12
2
(2)
i^
_Ql
:^=KS :&:xsi 2
f^o^
2
(2)
(-C^)
v^g>
#
2 2
2
2
(2)
The student should tion
and the
distinguish carefully
different versions of the
same
between an involu-
An
scale.
involution
is
the same order of progression but in the opposite direction and
is
significant only
if
a
new chord
or scale results.
Referring to page 29, you will see that the perfect-fifth pentatonic scale on C,
C-D-E-G-A, contains a major
minor triad on A. The six-tone triad
on
G
and the minor
nine-, ten-, eleven-
triad
triad
perfect-fifth scale
on
C
adds the major
on E. Analyze the seven-,
and twelve-tone
and a eight-,
scales of the perfect-fifth
and determine where the major, minor, diminished, and augmented triads occur in each. Construct the complete perfect-fifth projection beginning on the tone A. Indicate where the major, minor, diminished, and augmented triads occur in each.
projection
34
PROJECTION OF THE PERFECT FIFTH Since the perfect-fifth projection includes the most famihar scales in occidental music,
The most provocative
innumerable examples are available.
of these
would seem
to
be those which
produce the greatest impact with the smallest amount of tonal
To illustrate the economical use of material, one can no better example than the principal theme of Beethoven's overture, Leonore, No. 3. The first eight measures use only the
material. find
five tones of the perfect-fifth projection:
first
next measure adds
F and
B,
C-D-E-G-A. The
which completes the tonal material
of the theme.
Example
4-12
wm ^^ ^^
Beethoven, Overture, Leonore No.3
*
m
^
i
o
jj i' i
In the same way. Ravel uses the fifth
projection
building to the Suite No.
G-D-A-E-B— or, first
in
^^
first five
tones of the perfect-
melodic form, E-G-A-B-D— in
climax in the opening of Daphnis and Chloe,
2.
Example
4-13
Ravel, Daphnis end Chloe
Permission for reprint granted by Durand et Cie, Paris, France, copyright owners; Elkan-Vogel Co., Inc., Philadelphia, Pa., agents.
The
principal
theme
of the last
movent ent of the Beethoven
Symphony is only slightly less economical in its use of material. The first six measures use only the pentatonic scale Fifth
C-D-E-F-G, and the seventh measure adds Beethoven, Symphony No. 5
Example
A
and
B.
4-14
35
8
THE
SIX BASIC
TONAL
However, even Beethoven with
SERIES
his sense of tonal
economy
extended his tonal material beyond the seven-tone scale without implying modulation. The opening theme of the Eighth Sym-
phony, for example, uses only the
F major
scale in the
first
four measures but reaches
^r an
seven-tone perfect-fifth scale perfect fifth above
E
)
tones F-G-A-B^-C-E of the
six
beyond the
additional tone,
Bt]
(the
in the fifth measure.
Example
4-15
Beethoven, Symphony No.
f T^^r gu i
^
Such chromatic tones are commonly analyzed
as chromatic
passing tones, non-harmonic tones, transient modulations, and the
like,
but the student will find
it
useful also to observe their
position in an "expanded" scale structure.
Study the thematic material of the Beethoven symphonies and determine
how many
of
them
are constructed in the perfect-fifth
projection.
A
useful device of
many contemporary composers
is
to begin
a passage with only a few tones of a particular projection and
then gradually to expand the
medium by adding more
tones of
the same projection. For example, the composer might begin a
phrase in the perfect-fifth projection by using only the
first
four
tones of the projection and then gradually expand the scale
adding the
36
fifth tone,
the sixth tone, and so forth.
by
PROJECTION OF THE PERFECT FIFTH
Examine the opening
of Stravinsky's Petrouchka.
The
first five
measures are formed of the pure four-tone perfect-fifth tetrad
G-D-A-E. The
measure adds
sixth
which forms the
Bt],
perfect-
pentad G-D-A-E-B. The following measure adds a C#,
fifth
forming the hexad G-A-B-Cj|-D-E. This hexad departs momen-
from the pure
tarily
perfect-fifth projection, since
a combina-
and major-second projection— G-D-A-E-B
tion of a perfect-fifth
+
it is
G-A-B-C#.
Measure 11 substitutes a tutes a
which
Bb is
for the
C# and measure
12 substi-
forming the hexad G2A1BI72C2D2E
the involution of the previous hexad G2A2B2C#iD2E.
Measure 13 adds an scale
C
for the previous B,
F, establishing the seven-tone perfect-fifth
Bb-F-C-G-D-A-E.
Continue
type
this
determining
how much
of
analysis
to
of the section
is
rehearsal
number
7,
a part of the perfect-
fifth projection.
Analyze the thematic material of the second movement of the Shostakovitch Fifth Symphony.
How much
of this material con-
forms to the perfect-fifth projection? Excellent examples of the eight-tone perfect-fifth projection are
found
Stravinsky the
first
in
the beginning of
Symphony
all
movements of the movement, for example,
three
in C. In the first
seven measures are built on the tonal material of the
seven-tone perfect-fifth scale on C:
C-G-D-A-E-B-F#. In the
eighth measure, however, the scale
is
expanded one perfect
downward by the addition of the Fki which both F and Ffl: are integral parts of fifth
in the violas, after
the scale. Note the
scale passage in the trumpet:
Example Stravinsky, Symphony
in
C
Copyright 1948 by Schott
&
Co., Ltd.; used
by permission
4-16
of Associated
Music Publishers,
Inc.,
New
37
York.
THE
SIX BASIC
TONAL
SERIES
theme from the first movement of the Sixth Symphony may be analyzed as the expansion of
Similarly, the following ProkofieflF
the perfect-fifth projection to nine tones:
Example Prokotieff,
©
4-17
Symphony No. 6
1949 by Leeds Music Corporation, 322 West 48th
St.,
New
York 36, N.
Y.
Reprinted by permission;
all
rights reserved.
ft-
i m Even when
all of
the tones of the chromatic scale are used, the
formation of individual sonorities frequently indicates a simpler
which the composer had in mind. For example, the first measure of the Lyrische Suite by Alban Berg employs all of the tones of the chromatic scale. Each sonority in the basic structure
measure, however,
is
unmistakably of perfect-fifth construction:
Example
4-18
Albon Berg, Lyrische Suite
Copyright 1927 by Universal Editions, Vienna; renewed 1954 by Helene Berg; used by permission of Associated Music Publishers, Inc.,
38
New York.
PROJECTION OF THE PERFECT FIFTH
Analyze the
first
movement
and determine how much
of the Stravinsky
of
it
is
Symphony
in
C
written in the perfect-fifth
projection.
In
any
analysis,
constructed, that
is,
always try to discover
how much
how
the
work
is
should be analyzed as one frag-
ment of the composition. It will be observed, for example, that some composers will use one scale pattern for long periods of time without change, whereas others will write in a kind of
mosaic pattern, one passage consisting of
many
small
diflPerent patterns.
39
and
Harmonic-Melodic Material
Hexad
of the Perfect-Fifth
Since, as has been previously stated, of the six basic intervals,
all
and
all
seven-tone scales contain
since, as additional tones are
added, the resulting scales become increasingly similar in their
component
different types of tone relationship hes in the six-tone
which
tions,
We
study of
parts, the student's best opportunity for the
offer the greatest
number
combina-
of different scale types.
shall therefore concentrate our attention primarily
upon the
various types of hexads, leaving for later discussion those scales
which contain more than
six tones.
In order to reduce the large amount of material to a manageable quantity, is,
we
we
shall disregard the question of inversions.
shall consider
C-E-G a major
fundamental position— C-E-G; in its
triad
its first
whether
C-D-E-G-A
we
fifths,
its
shall consider
as one type of sonority, that
sonority built of four perfect
That in
inversion— E-G-C; or in
second inversion— G-C-E. In the same way,
the pentad
is
it
regardless of whether
is,
as a
its
form
C-D-E-G-A, D-E-G-A-C, E-G-A-C-D, and so forth. It is also clear that we shall consider all enharmonic equivalents in equal
is
temperament
to
be equally
major triad whether
it is
valid.
We
shall consider
spelled C-E-G, C-F^-G, B#-E-G, or in
some other manner. Examining the harmonic-melodic components fifth
hexad,
These are 1.
The 40
we
find that
C-E-G a
it
of the perfect-
contains six types of triad formation.
in order of their appearance:
basic triad C2D5G, p^s, consisting of
two superimposed
:
1
THE PERFECT-FIFTH HEXAD perfect
with the concomitant major second, which
fifths
is
dupHcated on G, D, and A:
Example Perfect
2
2.
3
2
The
Perfect Fifth Triads
Hexad
Fifth
2
2
2
2
5
.
triads are
duphcated on
Example pns
and
7
The
5
G and on D
5-2
2
7
2
7
2
(involution)
(involution)
3.
2
involutions
,27^
2
5
major second, and a major sixth (or
consists of a perfect fifth, a
minor third ) These
7
2
5
C7G2A, pns, with the involution C2D7A, which
triad
Triad
5-1
(involution)
C4E3G, pmn, with the involution A3C4E, which
triad
consists of a perfect fifth, a
major
third,
and a minor
ing the familiar major and minor triads.
third,
The major
form-
triad
is
duplicated on G, and the minor triad on E:
Example ^
h \
-i J ^
1
J4
4.
The
pmn
Triad
triad
3
5-3
and involu fions
:
1
J
1
• ;i> 4 3
'
—
J
r
r
-
\
4
3
—mJ —Js—r
:
1
3
u
II
4
C7G4B, pmd, with the involution C4E7B,
ing of the perfect
fifth,
consist-
major seventh (minor second), and
major third:
Example Triad
pmd and
5-4 invoiution
* i-H^^t-h-t7
4
4
7
41
— THE 5.
The
triad
SIX BASIC
TONAL
C2D2E, ms^, which
SERIES
which
triad,
is
Triad
2
consists
an isometric
5-5
ms^
^
•
-S-
The
third,
reproduced on G:
Example
6.
two superimposed
consists of
major seconds with the concomitant major
2
2
2
BiCoD, nsd, with the involution A2B1C, which of a minor third, a major second, and a minor second: triad
Example Triad nsd and
I^
5-6 involution
^
,2 (involution) .
I.
The tetrads of the perfect-fifth hexad consist of seven types. The first is the basic tetrad C2DgG2A, p^ns^, aheady discussed duphcated on
in the previous chapter,
Example
i The second
is
2
2
5
2
5
and D:
5-7 p^ns^
Fifth Tetrads
Perfect
G
2
2
5
2
the tetrad C2D2E3G, also duplicated on
G
(G2A2B3D), and the involutions A3C2D2E and E3G2A2B. This
two perfect fifths, two major seconds, one major and one minor third: p^mns^.
tetrad contains third,
Example Tetrads
2 p
223 42
mns
2
*3
5-8
and involutions
22
(involution)
223
,32
2,
(involution)
.
)
THE PERFECT-FIFTH HEXAD one of the most consonant of the tetrads, containing no
It is
strong dissonance and no tritone. Not only does
equal number of perfect the
it
contain an
and major seconds, but
fifths
it is
also
example of the simultaneous projection of two different
first
above the same tone, since
it consists of the two perfect two major seconds above C, that is, C-G-D plus C-D-E, or-above G-G-D-A plus G-A-B. (These
intervals
C
above
fifths
plus the
formations will be discussed in Part
Example
m^ Tetrad
The
p^mns^ as p^+s^
+
fe
2
may
involutions
5-9
i^^
i
s'
III.
2
3
be considered
also
simultaneous projection of two perfect
seconds downward, that
+
is
J J r r
p3
32
to
be formed by the
2
2
and two major J,E-D-C: and J,B-E-A fifths
jB-A-G:
Example
Jj
II ;
The also a
perfect to
I
p2
third
IT =223 I
+
s2
2
p2
i
the tetrad C4E3G2A, duphcated on
is
G
2
3
(G4B3D2E),
predominantly consonant tetrad, which consists of two
C
fifths,
is
G
to
G; the major
p^mnrs. This
E and
5-10
^g^ IT^^ iTt ^ ^^
Involution
E
+
J,E-A-D
?
+
A to E; two minor thirds, A to C and C to E; and the major second, G to A:
and
third,
an isometric tetrad
since,
if
we
begin on the tone
form the same tetrad downward, J^E4C3A2G,
we produce
the identical tones:
Example Tetrads
m
4
3
p^m
2
^
n^
5-11
s.
4
3
J
.11 2
(Isometric involution)
r
r
1
4
3
r
I
^
4
3
2
(isometric involution)
43
:
THE It
may be
TONAL
SIX BASIC
SERIES
considered to be formed of the relationship of two
perfect fifths at the interval of the minor third, indicated
symbol p perfect
@
fifth,
n; or of
indicated
two minor thirds at the by the symbol n @ p:
Example
@
p
It
contains the major triad
n
43
the involution A3C4E;
5-13
m. 34 + involution
C7G2A, pns, with the involution G2A7E
Example
m tetrad,
5-14
J:j
J
7 pns
The fourth
5-12
C4E3G and
)mn
triad
interval of the
il@_P
Example
and the
by the
C4E3G4B,
J
7 + involution
2
2
is
we begin downward, we produce
also isometric, since if
on the tone B and form the same tetrad the identical tones, IB4G3E4C:
Example p^m^n
Tetrad
5-15 d
ij 434 434 r^T^jj^ •'
(isometric involution)
It is
a more dissonant chord than those already discussed, for
contains two perfect
44
fifths,
C
to
G
and E
to B;
two major
it
thirds,
:
:
THE PERFECT-FIFTH HEXAD
C
E and G
to
to B;
one minor
E
third,
C
major seventh (or minor second),
to B: p^m^nd. It
considered to be formed of two perfect relationship of the major third,
C
and the dissonant
to G;
fifths
to G, plus
E
major thirds at the relationship of the perfect
G
to B; or of
C
fifth,
to
E
two plus
to B:
Example
ii @ contains the major triad
5-16
UE @
J
m
p
It
may be
at the interval
P
C4E3G and the
involution, the
minor
E3G4B;
triad
Example
ji^^4 J •^
triad
ij
j
^ 3
3
pmn
and the
5-17
+
r
4'
involution
C7G4B, pmd, and the involution C4E7B
Example
5-18
J j,^r ^i ^4 7 r 7 4 pmd
The fifths,
may
fifth
C
also
to
+
involution
tetrad C2D5G4B, p^mnsd, consists of
G
and
G
to
be considered
fourth above, or
fifth
D, with the dissonance, B. This tetrad
as the
major triad G-B-D with the added
below, G, that
tetrads of this projection
two perfect
is,
C. It
which contains
is
all of
the
first
of the
the intervals of
the parent hexad.
Together with
which
this
consists of the
tetrad
is
found the involution C4E5A2B,
minor triad A-C-E with the perfect
above, or the perfect fourth below, E, namely,
B 45
fifth
:
THE
TONAL
SIX BASIC
Example p^mnsd
iTetrad
5-19
and involution
j^'jiUJ ^
.
*^
4
5
SERIES
4
^
r
2'
5
(i) 9-'
(involution)
The
G2A2B1C, pmns^d, contains one perfect fifth, one minor third, two major seconds, and a
sixth tetrad,
one major
third,
minor second.
We also find the involution B1C2D2E Example Tetrad
*
pmns^d and
2'
2
5-20 involution
2
I
2
(involution)
And which
finally,
we have
the isometric tetrad A2B1C0D, pnh^d,
consists of a perfect fifth,
seconds, and a minor second. It
two minor thirds, two major may be analyzed as the com-
bination of two minor thirds at the interval of the major second, or
two major seconds
at the interval of the
minor
third.
It
contains the triad B1C2D, nsd, and the involution A2B1C; also the triad
D7A2B, pns, and the involution C2D7A:
Example Tetrad
5-21
pn^s^d
L @
The parent hexad
1.
S.
@I}.
Q^
*"
involution
contains three pentad types.
7
2'
pns
The
-^277 + involution
first is
the
basic perfect-fifth pentad C2D2E3G2A, p^mn^s^, also duplicated
on G, G2A2B3D2E:
Example
^
Perfect Fifth Pentads
i^ 46
5-22
p'^nnn^s^
^
^
THE PERFECT-FIFTH HEXAD
The second pentad, C2D0E3G4B, perfect
major fied
fifths,
its
minor
thirds,
more
hke
p^m^n^s^d, predominates in
thirds,
and major seconds.
easily as the superposition of
It
may be
another,
is
with, of course, the triads
Example p^m^n^s^d
Pentad
J i 22
The
J
34 ^
and
r
of
identi-
one major triad upon the
C-E-G + G-B-D; its involution same analysis, and consists projected downward, J^B-G-E plus J,E-C-A:
fifth of
number
parent scale, but has an equal
of
C4E3G0A2B two minor
5-23 involution
Mi^43 ^
j .jTr^L pmn @
^
^
22
p
r
ii J ,^* pmn@ p
pentad consists of the tones G2A2B1C2D, p^mn^s^d.
final
This pentad will be seen to have an equal
number
and major seconds, two minor thirds, one major one minor second. The involution is A2B1C2D2E:
fifths
Example
of perfect third,
and
5-24
Pentad p'^nnn^s^d and involution
i 2
p 2
S
triads
or
pns
#=F
2
I
These pentads
m
may be
2
f J^j^r^irJjJ @ @ pns
s
\
pns
s
m
analyzed further as consisting of two
at the interval of the
major second, projected up
down.
The in this
scales formed of perfect fifths, which have been discussed and the previous chapter, account for a very large segment
of
occidental music.
all
most important of
all
The
five-tone scale in this series
is
the
the pentatonic scales and has served as the
basis of countless folk melodies.
The seven-tone
amination proves to be the most familiar of
all
scale
upon
ex-
occidental scales,
the series which embraces the Gregorian modal scales, including the familiar major scale and the "natural" minor scale.
47
— THE
We
have found
SIX BASIC
TONAL
r
r
SERIES
in the previous chapter that the perfect-fifth
hexad contains two isometric
and ms^, and four
triads, p^s
triads
with involutions, pns, pmn, pmd, and nsd. These triads are
among
the basic words, or perhaps one should say, syllables, of
our musical vocabulary.
They should be studied with the
greatest thoroughness since, unlike words,
it is
necessary not only
them but to hear them. For this reason the young composer might well begin by playing Example 5-25, which contains all of the triad types of the perfect-fifth hexad, over and over again, listening carefully until to "understand"
all
of these sounds are a part of his basic tonal vocabulary. I first measure at least three down, so that he is fully con-
suggest that the student play the times, with the sustaining pedal
scious of the triad's
harmonic
as well as
then proceed with measure two, and so
Example n
^ ^^ '
—
^iiiiM^
^
9
-
m
I
^
-J-
^r^r
f^.
p^s
r
^
5-25
I
rr
J
—^ '
I
I
r
_rr
I
ms^
48
—
J
r ~F~
[_
r
pns
I
i
.
r"Ti
--J-
V
—9
_ ^ r »
I
I
^ ^
—
•
involution
^^^^^ pmn
forth.
, P* r r F^ ^^ -*'- * p r r r r " -^ I
f»
melodic significance; and
involution
nsd
i^^rf"
pnrid
involution
involution
THE
SIX BASIC
TONAL
SERIES
In Example 5-26 play the same triads but as "block" chords, listening carefully to the
sound of each.
Example
When
5-26
the student comes to measures 8 and
may sound
9,
and
10, the triads
"muddy" and unclear in close position. Experiment with these sounds by "spreading" the triads to give them harmonic character, as in Example 5-27. too
Example
^ etc.
5-27
^
^^
etc.
etc.
gpw
^
The sound position
of each of these triads will be affected both by its and by the doubling of its tones. In the Stravinsky
Symphony
of Psalms, familiar sonorities take
on new and some-
times startling character merely by imaginative differences in the
doubling of tones. In Example 5-28, go back over the ten triad forms and experiment with the different character the triad can assume both in different positions and with different doublings.
»"
m
Example
5-28 1*^
£
IT
m ^
etc.
49
THE
TONAL
SIX BASIC
SERIES
In Example 5-29a play the tetrads in arpeggiated form, and in
Example 5-29b play them
as "block" harmonies.
Example
5-29
(«)
IW^P^an Ld"
i
JT^ J^^"^
iJJ-'
^^^^^^^^ !iy^^iiLlal!\JPJ^al}iiil
j
,|j]Tl.r7T3^^l^ »— J * ~
In
» -
-
r
Example 5-30 experiment with
at
—
different
^ 9
positions
and
different doublings of the tones of the tetrads.
Example
^# m
-^ = $
etc.
/if 50
^^-t-
* 5
etc.
etc.
e/c.
/TT ^Nf jJi
-,^l |
5-30
l
f
l
r
ete.
I
jjr
THE PERFECT-FIFTH HEXAD
^^i
^
^^
J=J
^ ^^^^^^ T
In Example 5-31, repeat the same process with the five pentad types.
Example
5-31
(a)
ilTT^SV^ P^^iOUi
SlUr^fTT^n-^oiU ^^
jJJ-iJJJ^^"^
^rr
^rrrJ
r^
iriirrfirr
^mm (b)
e/c.
e/c.
iriMfjii 51
THE
TONAL
SIX BASIC
SERIES
In Example 5-32 repeat the same procedure with the hexad.
Example
5-32
(a)
^!^nJ^ir^P'^ai:!StimJai^ (b)
^
etc.
The student
will find
upon experimentation
basic tetrad seems to keep of
its
much
that although the
same character regardless
of the
position, the remaining tetrads vary considerably in sound
according to the position of the tetrad— particularly with regard
Example 5-30
to the bass tone. Play
which occur
in the
again, noting the changes
sound when different tones of the tetrad are
placed in the lowest part.
Repeat the experiment
in
relation
to
the five pentads in
Example 5-31b and the one hexad in Example 5-32b and notice that as the sonority becomes more complex, the arrangement of the tones of the sonority becomes increasingly important. ( Note especially the complete in the is
in the character of the sonority
second measure of Example 31b when the
shifted from
below
change
position above the
its
C
major
G
major triad
triad to a position
* it. )
The melody perfect-fifth
several times
in
hexad
in
and then
*See Note, page 55.
52
Example 5-33 includes
all
of the triads in the
melodic form. Play the example through finish the analysis.
the perfect-fifth hexad
Example 2
2 PS
#
pmn
p's '
«^^
o o P'S
Q
O
hand
notice
in block
how
«^
pmn
P'3
Example 5-34 harmonizes each left
5-33
harmony. Play
the change of
triad this
harmony
in the left
this
in the
through several times and
we may
melodic line a certain pulse which
Experiment with the changing of
by the same tones
call
hand
gives to the
harmonic rhythm.
harmonic rhythm by
shift-
ing the grouping of the tones in the melody, thereby changing the harmonic accompaniment. (For example, group the eighth, ninth,
and tenth notes
them with an E minor the following this
A
in the
triad
melody together and harmonize
under the melodic tone B, and
minor triad one eighth note earher.
)
shift
Continue
type of change throughout the melody.
Example
''f
T
r
V
JIL^.'
5-34
^r
}'
t'
f
P n f^mmm f
'
i.
i.
L 53
THE
SIX BASIC
Example 5-35 contains hexad of the six-tone
all
TONAL
SERIES
of the tetrads, the pentads,
and the
perfect-fifth scale. Play this exercise several
times in chorale style and listen to each change of harmony.
analyze each sonority on the principle that
we have
Now
discussed in
the previous chapter.
Example
h'UTiiiJ
.-
^r
i
5-35
^ ^W
f f f f
much
you wish of the material which we have been studying, compose a short work in your own manner. Do not, however, use even one tone which is not in the material which we have studied. If you have studied orchestration, it would be desirable to score the composition for string orchestra and if possible have it performed, since only through actual performance can the composer test the results of his tonal thinking. Use all of your ingenuity, all of your knowledge of form and of counterpoint in this exercise. Finally, using as
54
or as
little as
THE PERFECT-FIFTH HEXAD Note:
It is interesting to
speculate
upon
the reason
why two
sonorities containing
tones should sound so differently. The most logical explanation is perhaps that Nature has a great fondness for the major triad and for those sonorities that most closely approximate the overtone series which she has arranged for most sounding bodies with the exception of bells and the like. The human ear seems to agree with Nature and prefers the arrangement of any sonority in the form which most closely approximates the overtone series. In major triads, for example, the case of the combination of the C major and the if C is placed in the bass, the tones D-E-G-B are all found approximated in the If is placed in the bass, however, the first fifteen partials of the tone C. tone C bears no close resemblance to any of the lower partials generated by
identical
—
G
G
the bass tone.
Example
5-36
55
Modal Modulation
Most melodies have some
tonal center, one tone about
the other tones of the melody seem to "revolve." This
only of the classic period with
its
is
which
true not
highly organized key centers,
but also of most melodies from early chants and folk songs to the music of the present
day— with,
of course, the exceptions of
those melodies of the "atonal" school, which deliberately avoid
the repetition of any one tone until
all
twelve have been used.
Even in some of these melodies it is possible to discern evidence of a momentary tonal center.) The advantage of a tonal center would seem to be the greater clarity which a melody derives from being organized around some central tone. Such organization avoids the sense of confusion and frustration which frequently arises when a melody wanders about without any apparent aim or direction. The tonal center, however, is not something which is immutably fixed. It may, in fact, be any one tone of a group of tones which the composer, by melodic and rhythmic emphasis or by the con(
figuration of the melodic line, nominates as the tonal center.
For example, we
may
C-D-E-G-A by having the melody begin on C, depart from it, revolve about it, and return to it. Or we might in the same manner nominate the tone A as the tonal center, using the same tones but in the order A-C-D-E-G. Or, again, we might make either D, E, or G the tonal center of the melody. with
C
One 56
use the pentatonic scale
as the tonal center,
illustration
should
make
this principle clear. If
we
begin
MODAL MODULATION a melody on C, proceed upward to D, return to C, proceed downward to A, return to C, proceed upward to D, then upward to G, down to E, down to A and then back to C, we produce
which obviously centers about C. If, using the same tones, we now take the same general configuration of the melodic line beginning with A, we produce a melody of which A is the tonal center: a melodic line the configuration of
Example
#
»i
O
Finally,
i^t
__ M
ri
O
*^
VI S3I r
we may move from one
6-1
VI
—
-^
%T g.
i
^
*^
o
fc:t
tonal center to another, within
the same tonal group, by changing our emphasis from one tone to another. In other words,
we might
begin a melody which was
centered about C, as above, and then transfer that emphasis to the tone A. Such a transition from one tonal center to another
is
usually called a modulation. Since, however, the term modula-
adding— or more properly, the substiold term and call this tution—of type of modulation modal modulation, since it is the same principle by which it is possible to modulate from one Gregorian tion generally implies the
new
mode
tones,
to another
we may borrow an
without the addition or substitution of
new
tones. (For example, the scale C-D-E-F-G-A-B-C begun on the tone D will be recognized as the Dorian mode; begun on the tone E, as the Phrygian mode. It is therefore possible to "modulate" from the Dorian to the Phrygian mode simply by changing
the melodic line to center about the tone
The
E
rather than D.
six-tone perfect-fifth scale has four consonant triads
which
may serve as natural key centers: two major triads and two minor triads. The perfect-fifth hexad C-D-E-G-A-B, for example, contains the
C
major
and the E minor
triad, the
triad.
We
G
may,
major as
triad, the
we have
seen,
A
minor
triad,
nominate any 57
THE
SIX BASIC
TONAL
SERIES
one of them to be the key center merely by seeing to
it
that the
melodic and harmonic progressions revolve about that particular triad.
We may
modulate from one of these four key centers to
any of the others simply by transferring the tonal seat of govern-
ment from one
to another.
This transferral of attention from one tone as key center to
another in a melody has already been discussed on page 57.
can
assist this transition
We
from one modal tonic to another (har-
monically) by stressing the chord which
we wish
to
make
the
key center both by rhythmic and agogic accent, that is, by fall on a strong rhythmic pulse and by
having the key center having
it
tions will
occupy a longer time value. The simplest of
make
this clear.
illustra-
In the following example, 6-2a, the
three triads seem to emphasize C major as the tonic, while Example 6-2b we make F the key center merely by shifting the accent and changing the relative time values. In the slightly more complicated Example 6-2c, the key center will be seen to be shifted from A minor to E minor merely by shifting the melodic, harmonic, and rhythmic emphasis. first
in
^^ ^
58
(b[
9
3=
MODAL MODULATION
^^ '>'
r
r r f
r
r
r
r
Compose a short sketch in three-part foiin using the hexad C-D-E-G-A-B. Begin with the A minor triad as the key center, modulating after twelve or sixteen measures to the G major triad as the
key center and ending the
Begin the second part with
G
major
first
as the
part in that key.
key center and after
E minor. At the modulate to the key center of C major for a few measures and back to the key of A minor for the beginning of a few measures modulate to the key center of
end
of part two,
the third part. In the third part, pass as rapidly as convenient
from the key center of
A
then to the key center of final
minor
G
to the
key center of
major and back to
A
E
minor,
minor for the
cadence.
In writing this sketch, try to use as
much
of the material
available in the hexad formation as possible. In other words, do
upon the major and minor triads. Since these modulations are all modal modulations, it is clear that the only tones to appear in the sketch will be the tones with which we not rely too heavily
started,
At
G-D-E-G-A-B. glance
first
it
interesting sketch
with only
and the
may seem and
to
difficult or
impossible to write an
make convincing modal modulations
six tones. It is difficult,
discipline of producing
but by no means impossible,
multum
in
parvo will prove
invaluable.
59
7
Key Modulation
In projecting the perfect-fifth relationship,
C
tone
for convenience. It
temperament the
is
we began
with the
obvious, however, that in equal
starting point could
have been any of the
other tones of the chromatic scale. In other words, the pentatonic
C0D2E3G2A may be duplicated on D^,
scale
on D,
as
DoE2F#3A2B; and so
as
Db2Eb2F3Ab2Bb;
forth. It is therefore possible to
use
the familiar device of key modulation to modulate from any scale to an identical scale formation
begun upon a
different tone.
The closeness of relationship of such a modulation depends upon the number of common tones between the scale in the original key and the scale in the key to which the modulation is made. The pentatonic scale C-D-E-G-A, as we have already observed,
the
contains
modulation to the in relationship.
fifth
It
intervals
p*mnV. Therefore
above or to the
fifth
below
is
the closest
have the greatest number of
will
the key
common
tones, for the scale contains four perfect fifths. Since the scale
contains three major seconds, the modulation to the key a major
second above or below lation to the
of
is
the next closest relationship; the
key a minor third above or below
is
modu-
the next order
key relationship; the modulation to the key a major third
above or below
is
next in order; and the last relationship
key a minor second above or below, or original tonic
A
by the
to the
is
to the
key related
to the
interval of the tritone.
practical working-out of these modulations will illustrate
this principle:
60
KEY MODULATION
C-D-E-G-A modulating perfect
fifth
above
"
below
to the:
major second above
"
below minor
third
above
"
below major
third
one new tone
G-A-B-D-E F-G-A-C-D D-E-F#-A-B Bb-C-D-F-G Eb-F-G-Bb-C
gives
If
II
two
"
tones
II
II
II
t>
three
above
E-F#-G#-B-C#
"
four //
Ab-Bb-C-Eb-F
below minor second above
Db-Eb-F-Ab-Bb
"
n
//
A-B-Cif-E-F#
" II
" //
//
all
new
tones
(all
new
tones)
below above
tritone
or
below
gives
V%-G%-A%-C%-D%
Example
7-1
Modulation Perfect Fifth Pentad
to
Perfect Fifth above
Major Second above
to
*
o o
^
Modulation to Perfect Fifth
below
to Major
Second below
-
to Minor Third
i
^ to
i to
*
to
Minor Third below
^
Major Third above
"
b,:
17»-
to
Minor Second obove
|;>
to Major Third below
^^%* ° f'
*
to
1^
^^
'
Minor Second below
i*
>
ff*
Augmented Fourth above It.
*• %- i' ^'
Augmented Fourth below
to
i
above
^
,
_? —
\,-9-
I
L--
"^
!;•
ty
o
*
61
*'
THE
The student should though there
will
TONAL
SIX BASIC
SERIES
learn to distinguish as clearly as possible—
be debatable instances— between,
(1) a modulation from the pentatonic scale
for example,
C-D-E-G-A
to the
pentatonic scale A-B-CJj:-E-Ffl:, and (2) the eight-tone perfectfifth scale,
of
C-C#-D-E-F#-G-A-B, which contains
both pentatonic
In
scales.
the
all
of the tones
former instance, the two
pentatonic scales preserve their identity and there
is
a clear point
which the modulation from one to the other occurs. In the have equal validity in the scale and all are used within the same melodic-harmonic pattern. In the first of the two following examples, 7-2, there is a definite point where the pentatonic scale on C stops and the pentatonic scale on A begins. at
latter case, all of the eight tones
Example
^
^^
7-2
^i^^^ 4 i hJ-
In the second example, 7-3,
all
of the eight tones are
members
of one melodic scale.
Example
I i
7-3
^ti ^^^ r
Although modal modulation
is
the most subtle and delicate
form of modulation, of particular importance poser in an age in which
it
entire tonal palette at the listener, to
to the
young com-
seems to be the fashion to throw the
the tonal fabric. This task
is
it
does not add
new
material
accomplished either by the
"expansion" technic referred to on page 36 or by the familiar device of key modulation.
Key modulation
offers the
advantages of allowing the com-
poser to remain in the same tonal milieu and at the same time to
62
KEY MODULATION
add new tones might— at least major keys and
to the pattern.
in
A
composer of the
classic period
theory— modulate freely to any of the twelve
still
confine himself to one type of tonal material,
that of the major scale.
Such modulations might be performed
deliberately and leisurely— for example, at cadential points in the
made
formal design— or might be
rapidly and restlessly within
the fabric of the structure. In either case, the general impression of a "major key" tonal structure could
This same device
is
equally applicable to any form of the
perfect-fifth projection, or to
The
principle
is
be preserved.
the same.
any of the more exotic scale forms.
The composer may choose the
pattern which he wishes to follow and cling to
he
may
in the process
modulate
it,
tonal
even though
one of the twelve
to every
possible key relationships. It is obvious that the richest and fullest use of modulation would involve both modal modulation and key modulation used
successively or even concurrently.
Write an experimental sketch, using
as
your basic material
the perfect-fifth-pentatonic scale C-D-E-G-A. Begin in the key of C, being careful to use only the five tones of the scale
and
same scale on E (E-F#-G#-B-CJj:). Now moduon F# (F#-Gif-A#-C#-D#) and from F# to Eb (Eb-F-G-Bb-C). Now perform a combined modal and key modulation by going from the pentatonic scale on E^ to the pentatonic scale on B (B-C#-D#-F#-G#), but with G# as the key center. Conclude by modulating to the pentatonic scale on F, with D as the key center ( F-G-A-C-D ) and back to the original
modulate
to the
late to the scale
,
key center of C.
You
will observe that the first
modulation— C
to
E— retains
common tone. The second modulation, from E to F#, retains three common tones. The third, from F# to E^, has two common tones. The fourth, from E^ to B, like the first modulation, has only one common tone. The fifth, from B to F, has no common tones, and the sixth, from F to C, has four common tones. only one
If
you play the key centers
successively,
you
will find that
63
THE
SIX BASIC
TONAL
SERIES
only one transition offers any real problem: the modulation from B, with Gif as the key center, to F, with
require
will
some
ingenuity
on
D
your
as the
part
key center.
to
make
It
this
sound convincing.
Work
out the modulations of the perfect-fifth hexad at the
intervals of the perfect fifth, third,
minor second and
64
major second, minor
tritone, as in
Example
7-1.
third,
major
8
Minor Second
Projection of the
There
is
only one
interval, in addition to the perfect fifth,
which, projected above
itself,
twelve-tone scale. This
is,
gives
of the
all
of course, the
tones of the
minor second, or
its
inversion, the major seventh.
Proceeding, therefore, as in the case of the perfect-fifth projection,
we may
superimpose one minor second upon another,
proceeding from the two-tone to the twelve-tone
Examining the minor-second triad
C-C#-D
C-D:
s(P.
The
series,
we
series.
observe that the basic
contains two minor seconds and the major second
C-C#-D-D#, adds another minor second, another major second, and the minor third: ns^cP. The basic pentad, C-CJ-D-Dif-E, adds another minor second, another major second, another minor third, and a major third:
The
basic tetrad,
basic hexad, C-CJj:-D-D#-E-F, adds another minor second,
another major second, another minor third, another major third,
and a perfect fourth: pm^nh^d^:
Example Minor Second Triad
8-1
Minor Second Tetrad
sd^
ns^d^
^
t^ 2.3^4 mn'^s d
Minor Second Pentad
i
Minor Second Hexad
I
^
yes
"X5 I
pm^n^s^d^
I
I
I
65
THE
SIX BASIC
TONAL
SERIES
The seven-, eight-, nine-, ten-, eleven- and twelve-tone minorsecond scales follow, with the interval analysis of each. The student will notice the same
phenomenon which was observed
in the perfect-fifth projection:
whereas each successive projection adds one new interval,
from the two-tone
to the seven-tone scale
been reached no new interbe added. Furthermore, from the seven-tone to the
after the seven-tone projection has
vals can
eleven-tone projection, the quantitative diff^erence in the propor-
new
tion of intervals also decreases progressively as each is
tone
added.
Example p^^n'^s^d^t
Minor Second Heptad
I
I
I
I
I
r
I
m
I
I
I
I
"^j^o^o o
III
I
Minor Second Undecad p
Octod p'^m^n^s^d^t^
Minor Second Decad
^ v»jtoO^^^»tt« I
Minor Second
I
MinorSeoond Nonad p^m^n^s^d^^
I
8-2
n s d
t
I
I
p^m^n^s^d^t'*
t.^t^^e^f^
III
Minor Second Duodecod p
m
n
s
d
t
^^ojto°"*"°1t°"<'"Lj^v>j)»°"jl"°«°"'' I
i
I
I
I
I
I
I
I
I
I
Proceeding again, as in Chapter
I
I
I
I
I
I
I
sd-,
I
we may now examine
5,
harmonic-melodic material of the minor-second hexad.
have the basic triad C-C#-D,
I
First,
duplicated on the tones
the
we C|:,
D, and D#:
Example )Minor Second Hexad
I
I
The
I
I
I
I
triad CiCJsDJj:, nsd, a
hexad, duplicated on
66
8-3
Minor Second Triads sd^
I
I
I
form observed
C# and D, with
I
I
I
I
in the perfect-fifth
their involutions:
:
projection of the minor second
Example f^
Triads
8-4
nsd and involutions
J|J i J b J Uj J J J J '2 El^ia'^ZI 2 jtJ 2
t
}
2'
I
"^
I
triad
CiC^gE,
J bJ
J
^
12
mnd,
duplicated
J
t|J
2
(involution)
(involution)
The
I
I
I
(involution)
on
C#,
with
their
involutions
Example Triads
mnd
8-5
and involutions
^i|J, ^ J-t^J 3 r 3
I
jti
I
(involution)
The
triad
CiDb4F, pmd, with
J^
J^^lJ ^31 13
(involution)
its
involution C4E1F; which has
already been found in the perfect-fifth hexad:
Example Triad
pmd and
8-6 involution
i>J>U 4 4 I
I
(involution)
The
isometric triad C-D-E, ms^,
which has already occurred
as a part of the perfect-fifth hexad; duplicated
Example
2
8-7
ms^
Triads
i^
on D^;
2
^F^ 2
2
67
THE and the form
triad
also has
SIX BASIC
TONAL
C2D3F, pns, with
been encountered
SERIES
involution, CsE^oF,
its
which
in the perfect-fifth series:
Example
8-8
Triad pns and involution
iJ^XJ,^U> "2
3
3
2
involution
The minor-second hexad ns'^d^,
duplicated on
Cfl:
contains the basic tetrad CiCflliDiDJ):,
and on D:
Example
8-9
Minor Second Tetrads
2
ns d
3
J
i^JJ^j'j^iJjtjJ'JlJ I
The
I
I
I
I
I
tetrad CiCjiDoE, mns-d^, duplicated on C#, with their
respective involutions;
Example
8-10
Tetrads mns^d^and involutions
r'
I
2
2
2
^
(involution
2'
I
I
(involution)
which may be analyzed as the simultaneous projection of two minor seconds and two major seconds above C, or, in its involution, below E:
Example Tetrad
i ^W 68
mns 2h2 d
^
8-11 d2+s2
—
-^
*-
ld2
+
-I
s2
PROJECTION OF THE MINOR SECOND
The
isometric tetrad CjC^aDitiE, mn^sd^, duplicated on C#;
Example mn 2.^2 sd
Tetrads
r
8-12
2
I
*
r
2^*^
I
2
I
I
j>
bj n
nU (S
'
d
mnd and
its
b^Ljt^ d @ n
«
l«^2"
The
^^juM
^
2
I
I
and the involution onCj^, or
involution:
Example
+
nsd
at the relationship
at the relationship
8-13
or as a combination of the triad nsd
the triad
r
(isometric involution)
which may be analyzed as two minor thirds of the minor second, or two minor seconds of the minor third:
Example
2
I
(isometric involution)
2"
8-14
^3^
3
I
mnd
involution
-t-
I
involution
isometric tetrad CiDbgEiF, pm^nd^;
Example
8-15
leiraa pm-ngpm^nd^ * Tetrad
(Ji
Jl J
jl,J J I
3
I
J I
^
3
I
(isometric involution)
which may be analyzed as consisting of two major thirds interval of the minor second, or of two minor seconds
at the at the
interval of the major third;
69
:
THE
TONAL
SIX BASIC
Example
i
hjm d @ m
d
or as a combination of the triad or the triad
pmd, and
mnd, and the involution on D^,
involution:
its
Example
i J 13 mnd
The
J3J
J
1;J
^
tetrad CiCJiDsF, pmnsd",
and
# iitiJ I
I
^
4
involution
involution:
its
8-18
pmnsd^ and
Tetrad
^
|;J
pmd
involution
-t-
8-17
i"14 ^
I
1
Example
The
8-16
^
J
j.
m @
SERIES
involution
UitJ
3
3
I
tetrad CiDbsEb^F, pmns^d, and
I
its
involution,
which has
already been found in the perfect-fifth projection;
Example
8-19
pmns'-o ana and letraa pmns^ ^[Tetrad
12
2
2
involution invc
2
1
and the isometric tetrad CoDiE^oF, pn^s^d, which is also a part of the perfect-fifth hexad, and which may be analyzed as a combination of two minor thirds at the interval of the major second, or of two major seconds at the interval of the minor third
Example
8-20
Tetrad pn^s^d
IJ
J bJ
*
'
^
bJ
J
i 1^
(isometric involution)—
70
^ ^
1
3
@
n
PROJECTION OF THE MINOR SECOND
The student
also
combination of the triad nsd and the involution
be analyzed
as a
on D, or the
triad
pns and
involution:
its
Example
1^ ^"2
12
1
+
risd
8-21
^2
^
3
2
3
pns
involution
involution
Finally, the pentads in the minor-second
basic pentad
C-D-E^-F may
will observe that the tetrad
hexad consist of the
CiCJiDiDJiE, mn^s^d^, duplicated on C#;
Example
8-22
Minor Second Pentads mn^s^d^
ij^i JffJ
III
the
pentad
^
J
'j|i I
I
CiC^iDsEiF,
^J
j
I
I
I
with
pm^n~s~d^,
its
involution,
C,C#2D#iEiF;
Example »
pm
Pentad renraa
2 2 2 3 n s a and invoi involution d ana
12
I
which may be analyzed
8-23
2
1
I
as the relationship of
I
two
triads
mnd,
the interval of the minor second:
Example 2 2 2 3 Pentad pm n s d
i
^^ r
3
I
as
3
8-24
mnd @
I
d
P^W
i involution
71
at
THE
SIX BASIC
TONAL
SERIES
and the pentad CiCJiDiEbsF, pmn^s^d^, with CsDiDJiEiF, which may be analyzed triads nsd, at the interval of the
itr-^itw
and
^ I
I
8-25
involution
J.
lt>
2
^ J * J j J 12 ^
y
The minor-second hexad scale.
For
value. It
reason
may be
J
i
J
i
|
@
it
is,
^
1^
2
1
I
nsd
this
combination of two
major second:
Example Pentad pmn^s d
as the
invokition,
its
I
2
involution
s
quite obviously, a highly dissonant
has perhaps
less
harmonic than melodic
effectively used in two-line or three-line con-
trapuntal passages where the impact of the thick and heavy
dissonance
is
the melodic
somewhat lessened by the rhythmic movement
of
lines.
Example 8-26
constitutes a mild puzzle. It
have the same arithmetic, or perhaps
I
constructed to
should say geometric
Example
relationships, as the melodic line in
is
5-33. It should take
only a short examination to discover what this relationship
Example
is.
8-26
mnd
obo'jjoy o *^oj^o|j^k3'^;_o^"t^l;otlot v3.^ ;
The
six-tone
minor-second scale will be found to be too
limited in compass to give the composer this
restricted
form.
Nevertheless,
it
is
much
opportunity in
valuable to
become
intimately acquainted with the small words and syllables which
72
PROJECTION OF THE MINOR SECOND
go to make up the vocabulary of
this series, since these small
words constitute an important part of the material of some contemporary music. Therefore,
suggest that you play through
I
Example 8-26 slowly and thoughtfully, since the triads of the minor-second hexad. Since these triads in close position, the
melody
is
contains
all
of
have kept
all
of
it
I
even "wormier" than
such melodies need be.
Complete the analysis
of all of the melodic triads
under the
connecting lines and then play through the melody at a more rapid tempo with the phrasing as indicated in Example 8-27.
See
if
you can sing the melody through without the aid
piano and come out on pitch on the
Example
Example 8-28 minor-second
is
of a
final Ej^.
8-27
a four-measure theme constructed in the
hexad.
Continue
its
development
in
two-part
simple counterpoint, allowing one modulation to the "key" of
G-G#-A-A#-B-C— and modulating back again
G—
to the original "key"
of C.
Example
a^ ^i^iJ
8-28
"^^^'CJW^iPrr^^ G=p
^ 73
THE
SIX BASIC
TONAL
SERIES etc.
It is difficult to find
many examples
of the effective use of the
minor-second hexad in any extended form in musical literature
because
of
its
obvious
limitations.
A
charming example
is
"From the Diary of a Fly" from the Mikrokosmos of Bela Bartok. The first nine measures are built on the six-tone scale F-Gb-Gkj-Ab-Ati-Bb. The tenth measure adds the seventh
found
in
tone, C^,
Example
8-29
Bortok, Mikrokosmos
{hi
Lb}^
J ^
JjfjL^^^sWmJ
\i^\\^i^\J^'i
\>^^i\^^
m m ^P |^ ^5 ^pi^r'"^p''t Copyright 1943 by Hawkes
On
& Son (London),
Ltd.
Used by permission
of
Boosey & Hawkes, Inc.
the other hand, examples of the utilization of the entire
chromatic scale within a short passage abound in contemporary music, one of the most imaginative of which can be found in the
first
movement
of the Sixth Quartet of the
Example
same composer:
8-30
Bortok, Sixth Quartet
I^J2^-k '>'
«r '
-
a
Mljl.
Copyright 1941 by Hawkes
74
& Son (London),
Ml Ltd.
Used by permission
SUA-
g of Boosey
& Hawkes,
Inc.
PROJECTION OF THE MINOR SECOND
A is
more obvious example
found
at the
of the use of the minor-second scale
beginning of the second movement of the Bartok
Fourth String Quartet:
Example Bartok, Fourth Quartet, 2
^TT-v
''jT"
^u
l
Copyright 1929 by Universal Editions; renewed 1956. Inc., for the
U.S.A.
8-31
movement ,,
,
_
,-T^
>..
etc.
.
Copyright and renewal assigned to Boosey
& Hawkes,
Used by permission.
A
more subtle example— and one very characteristic of the Hungarian master— is found in the twenty-fifth measure of the first movement of the same quartet. Here the tonal material consists of the seven-tone minor-second scale
E, but divided into
second violin holding the major-second first
violin
and
B^-Btj-C-CJ-D-DJ-
two major-second segments, the triad,
and
B-C#-D#, and the
viola utilizing the major-second tetrad,
Example
cello
B^-C-D-E:
8-32
Bartok, Fourth Quortet
i
*
^ PY^
TTm
^if-^i^y
H-^
ifiw
E
i
Copyright 1929 by Universal Editions; renewed 1956. the U.S.A. Used by permission.
Copyright and renewal assigned to Boosey
& Hawkes,
Inc., for
75
THE PERFECT-FIFTH HEXAD Analyze the determine
first
movement
how much
of
it
is
of the Bartok Sixth Quartet to
constructed in the minor-second
projection.
Modulation of the rninor-second pentad follows the same principle as the perfect-fifth pentad. Modulation at the minor
second produces one tones, at the
new
tone, at the major second
minor third three new tones,
new tones, and Work out all
at the perfect fifth
and
at the
two new
major third four
tritone five
new
of the modulations of the minor-second
tones.
pentad
and hexad.
76
'i
Projection of the
Since the major second
is
Major Second
the concomitant interval resulting
from the projection of either two perfect seconds,
it
would seem
fifths
or of
two minor
be the most logical interval to choose
to
for our next series of projections.*
The
basic, triad of the
major-second series
Example
two major seconds with third: ms^.
We have
C2D2E,
9-1
M Major Second Triad
2
is
"
msf
2
their concomitant interval of the
the perfect-fifth and the minor-second hexads.
The
second produces the tetrad C2D2E2F#, adding the of the tritone,
C
to
major
already observed this triad as a part of both
FJj:.
The
third major
new
analysis of this sonority
interval
becomes
three major seconds, two major thirds, and one tritone: m^sH.
Example
9-2
Major Second Tetrad
^
>.
2
*3
^ 2
= < »
m^^ ^ ^'
• The major second would also seem to follow the perfect fifth and minor second, since it can be projected to a pure six-tone scale, whereas the minor third and the major third can be projected only to four and three tones,
respectively.
77
THE Superimposing
TONAL
SIX BASIC
major
another
SERIES
second
produces
pentad
the
C2D2E2FJj:2G#, which consists of four major seconds; four major thirds,
C
C
F# and
to
D
to E,
D
E
to Ft,
to
to G#,
G# Ab (
and two
tritones,
G#: m^sH\
Example
9-3
Major Second Pentad
2
superposition
rrfs^t^
^^
»-
C5
2
The
to C;
)
2
one more major second produces the
of
"whole-tone" scale C2D2E2F#2GJl:2AJj::
Example
9-4 m^s^t^
Major Second Hexad
i
2
2
2
t"
This scale will be seen to consist of F#,
E
to G#, FJf to A#,
major seconds-C to D,
AS (Bb )
to C;
Its analysis is
D
G#
six
to BJf
to E,
E
It will
^*'
2
major thirds— C to E,
D
(C) and A# (Bb) to D;
to F^,
and three tritones-C m^sH^.
F#
to F#,
to G#,
D
G#
to G#,
to A#,
and E
to six
and
to A#.
be obvious that the scale cannot be
projected beyond the hexad as a pure major-second scale, since
would be BJ, the enharmonic equivalent
the next major second of C.
The major-second hexad only
is
its
up
or
down, but
identical tones. Analyzing different
its
GJf,
its
scale;
not
involution produces the
components,
types of triads:
duplicated on D, E, F#,
78
an enharmonic isometric
form the same whether thought of clockwise or
counterclockwise,
"three
is
we
find that
it
has
the basic triad C2D2E, ms^,
and A#;
—
;
projection of the major second
Example Major Second Triads
jij
w
~St
2
2
ms'^
jitJ
i
j^JtJ
2
2
9-5
i
|tJtJitJ 2
2
2
i
itJ
2
2
2
2
D
the augmented triad C4E4G#, m^, duplicated on
r 2
(since the
remaining four augmented triads are merely inversions of those on C and D )
Example
9-6
Major Third Triads m^
^i '^•'4
ftJ
.1
4
and the
I
J
|J
ii
J
44
i
[jitJiif'rM tJiiJY'r'l4ii
44
triad C2D4FJJ:,
mst, and
i
ir
44
44
|
^^
\
^^ 44
C4E2F#, also
involution,
its
r
duplicated on the other five notes of the scale:
Example Triads
mst and involutions
JitJ Ij
ItJij 4
2
9-7
4
2
Jtl^
;j||J|^ljjJ
4
2
4
basic triad
we have
m^.
The
triad
C2D4F# and
major second, one major
ms^.
third,
C
E
to E,
its
third,
The major-second hexad tetrads: the basic tetrad
i^jj
2
^
i^r^
The augmented
to GJf,
and one
contains
C2D2E2F#,
Example
mst
4
and G#
triad
{A\)) to C,
involution C4E2F#, contain one tritone, mst.
three
different
7n-sH, duplicated
F#, Ab, and Bb;
Major Second Tetrads
:.li|J|^
4
already analyzed as containing two
major seconds and a major contains three major thirds,
2
H'Jjj*
j^jijjtJ«rr':nJiiJ
The
2
9-8
types
of
on D, E,
THE
TONAL
SIX BASIC
SERIES
the isometric tetrad C2D2E4G#, duplicated on D, E, G^, A^, and B\),
containing three major thirds, two major seconds, and one
tritone, rrfsH;
Example
9-9
iBiiuu^ m^s^t III a ^ Tetrads I
4
^24 which may
224 also
224
2
2
224
4
224
be considered to be formed by the simultaneous
projection of two major seconds and
Example
i iJ
thirds;
9-10
i
^
[f'
two major
J
fr
and the isometric tetrad C4E2F#4AJf, duplicated on D and E, which contains two major thirds, two major seconds, and two tritones, m^sH^:
Example I
9-11
Tetrads letraas m^s^t^ m^s t~
^
f^ '
r'^'i Jii-'J '^T' 9>i4 JitJi^tJ
This
may
also
be analyzed
as
i
i
^^^
jJi tJrr it>.ii'r'^'ri>('fr'nii i
424
two major
424
424
thirds at the interval of
the tritone; as two tritones, at the interval of the major third; as
two major seconds
at the interval of the tritone, or as
tones at the interval of the major second.
Example
m
80
@
t
t
@ m
9-12
s
@
t
t
@
s
two
tri-
:
PROJECTION OF THE MAJOR SECOND highly
This
Poeme de
particularly in the
There
is
remaining
five
m'^s
JuJiiTnY
r 222
2
Its
material. It
made up
is
I
jttJiiJiJ
2222
2222
n Y2222 *ni r
examination of
weakness.
9-13
f
I
I*
I"
show both
this series will
strength
lies
in the
one of the most homogeneous
very homogeneity
is
also
its
strength and
of all scales, since
exclusively of major thirds, major seconds,
mary dissonances (the minor second Its
its
complete consistency of
only mildly dissonant in character, since
It is
Scriabine,
pentads are merely transpositions
ij|j J j|jjtJji^ %2222 2222
An
of
first
Major Second Pentads
i
favorite
TExtase.
Example
(|
a
only one type of pentad in the six-tone major-second
scale, since the
of the
was
sonority
isometric
it
and
its
it is
tritones.
contains no pri-
or major seventh).
weakness, for the absence of
contrasting tonal combinations gives, in prolonged use, a feeling
monotony. Also, the absence of the perfect
of
scale of
any consonant "resting-place," or
gressions sound vague, lacking in contrast,
Nevertheless, in the
it is
effective use
thirty
deprives the
tonic, so that its pro-
and without
direction.
an important part of the tonal vocabulary and,
hands of a genius, adds a valuable color
which should not be Its
fifth
is
lightly discarded
to the tonal palette
by the young composer.
illustrated in Debussy's "Voiles," the first
measures of the
first
section of
which are written
entirely
in the whole-tone scale.
The same composer's "La Mer"
contains extended use of the
81
THE
same
SIX BASIC
TONAL
SERIES
scale in the excerpt below:
Example
9-14
Debussy, "lo Men"
^m. fVff^rp ayrt-tjj
*=*.:
'/hh
P^
^^^=tF
S'
-
'
iji^g f^ff :
^
^E
^ ^ 'e-
#^ ^
fi^^ bi
^
^
rt ^m
^
^
Z
^
^^
Permission for reprint granted by Durand et Cie, Paris, France, copyright owners; Elkan-Vogel Co.,
Inc., Phila-
delphia, Pa., agents.
An example expected
82
is
of the whole-tone scale
where
it
might not be
found in the opening of an early song, "Nacht," of
—
:
PROJECTION OF THE MAJOR SECOND
Alban Berg, the
two forms
of the
measures of which are in one of the
first five
whole tone
scale
Example
Alban Berg, Nacht
9-15 J0_
Copyright 1928 by Universal Editions, Vienna; renewed 1956 by Helene Berg; used by permission of Associated
Music Publishers,
Inc.
o
*J
It will
second
o
o
o
be observed that whereas the
series
may be
o
perfect-fifth
and minor-
transposed to eleven different pitches, giv-
ing ample opportunity for modulation, there
is
only one effective
modulation for the whole-tone scale— the modulation to the whole-tone scale a half-tone above or below C-D-E-F#-G#-AJj:
scale
modulation E,
etc., all
is'
the
to
scale
it,
that
is,
from the
Db-Eb-F-G-A-B.
Modal
impractical, since the whole-tone scales on C, D,
have the same configuration:
Example
9-16
^
The two Major Second Hexads
i
a — 2
tt..
:
2
^^ ^"
'-"^
^
^*.^
tjo
1
%T
(2)
o
(2)
In the introduction to Pelleas et Melisande Debussy begins
with the material of the perfect-fifth pentad for the
measures— C-D-E-G-A, changes to the pure whole-tone
first
four
scale for
83
— THE the
fifth, sixth,
fifth-series in
SIX BASIC
TONAL
and seventh measures, and returns
to the perfect
measures 8 to 11:
Example Debussy,"Pel leas and Melisonde
bi
SERIES
Jtr
9-17
*
J.f^
i
J^''
^J*"|r
>T
Permission for reprint granted by Durand et Cie, Paris, France, copyright owners; Elkan-Vogel Co.,
Inc., Phila-
delphia, Pa., agents.
From
the same opera
we
find interesting examples of the use
of whole-tone patterns within the twelve-tone scale
by
alternat-
ing rapidly between the two whole-tone systems:
Example
9-18
Permission for reprint granted by Durand et Cie, Paris, France, copyright owners; Elkan-Vogel Co., Inc., Philadelphia, Pa., agents.
Whereas the minor-second hexad may not be 84
as
bad
as
it
PROJECTION OF THE MAJOR SECOND sounds, the careless use of the whole-tone scale frequently makes it
sound worse than
it
is,
particularly
when used by
improvisors. Because of the homogeneity of
its
casual
material,
it
is
often used in the most obvious manner, which destroys the subtle nuances of
which
capable and substitutes a "glob" of
it is
"tone color."
The author tone scale in this scale
is
its
not making a plea for the return of the whole-
unadulterated form, but
it
must be
said that
has qualities that should not be too lightly cast aside.
Example 9- 19a gives the triads; 19b the tetrads, 19c the pentad, and 19d the hexad, which are found in the six-tone scale. Play them carefully, analyze each, and note their tonal characteristics in the di£Ferent positions or inversions.
Example
9-19
(«)
(b)
^3=- = ii=;,^^^% = bEE-^^ =
^^^tb^
^
liPPjyftjj;^
(c)
(d)
i>jJWii^iW^W*^^ir¥[lS
hrrr^ 4
.
85
THE
SIX BASIC
TONAL
SERIES
Play the triad types in block form as in Example 9-20a. Repeat the same process for the tetrad types in 20b; for the pentad type in 20c;
and
for the
hexad in 20d.
Example
9-20
(a) etc.
i ^ ^r
'^/^^ (h)
(c) etc.
titijt.
^4
^^r
In Example 9-2 la, experiment with the triad types in various
Repeat the same process for the
positions.
tetrads, as in 21b; for
the pentad, as in 21c; for the hexad, as in 21d.
Example
9-21
(a)
i
i
r
F
m ^^^ 86
K
4
J
^^ f^
^
PROJECTION OF THE MAJOR SECOND (b)
(hi i
i\l
'}
f^f
f
«hi
i \
i
J
/h^^^
^ ^^
"F
^fe
(c)
i
ii ^i
'>t
ile
itJ
tit
(|iiiij|g|i
'>'.^^tp
i
u^ i
iJ|itdiii
f#«f»f
:
Experiment with different doublings and positions of above
sonorities, as in
Example
Example
Have
of the
9-22
^
m
all
9-22.
i
the material of Example 9-21 played for you in different
order and take
it
down from
dictation, trying to
reproduce not
only the notes but their exact position.
Analyze in detail the
first
section of Debussy's "Voiles"
and note 87
THE
TONAL
SIX BASIC
SERIES
not only his use of the widest resources of the scale but also his
employment
of the devices of
In detailed analysis
it
change of position and doubling.
seems generally wise to analyze every
note in a passage regardless of
its
relative importance, rather
than dismissing certain notes as "nonharmonic" or "unessential" tones, for all tones in a passage are important,
may be
even though they
only appoggiaturas or some other form of ornamentation.
Occasionally, however, the exclusion of such "unessential" tones
seems obvious. The oflFers
thirty-first
measure of Debussy's "Voiles"
an excellent example of such an occasion. Every note
in
every measure preceding and following this measure in the first
section of the composition
scale, Ab-B^-C-D-E-FJI:,
and D^
in
measure
is
in the six-tone major-second
with the exception of the two notes
31, Since
both of these notes were quite
obviously conceived as passing tones, to analyze
them
G
it
would seem
as integral parts of the tonal
Example
unrealistic
complex.
9-23
Debussy, "Voiles"
4 ^0^-
^
Permission for reprint granted by
—
^i^r
Durand
—
^^ ^^
*
et Cie, Paris, France, copyright owners;
Elkan-Vogel Co.,
Inc.,
Phik-
delphia. Pa., agents.
In using any of the tonal material presented in these chapters,
one all-important principle should be followed: that the composer should train himself to hear the sounds which he uses is reason to fear that some young composers— and some not so young— have been tempted at times to use tonal relationships which are too complex for their own
before he writes them. There
aural comprehension. This of
is
comparable
to the use
by a writer
words which he does not himself understand— an extremely
hazardous practice!
88
PROJECTION OF THE MAJOR SECOND
When
you
feel confident of
write a short sketch
your understanding of the material,
which begins with the use
of the major-
second hexad on C, modulates to the major-second hexad on G,
and returns at the end to the original hexad on C. See to it that you do not mix the two scales, so that the sketch consists entirely of major-second material.
89
10
Major Second
Projection of the
Beyond We
the Six-Tone Series
have already observed that the major-second
pure form cannot be extended beyond
scale in
six tones, since
major second duphcates the starting tone.
We
its
the sixth
can, however,
produce a seven-tone scale which consists of the six-tone majorsecond scale with a foreign tone added, and then proceed to
superimpose major seconds above
this
from any of the tones which
select this foreign tone arbitrarily
we
are not in the original whole-tone scale. If
the perfect
fifth
above
C
We may
foreign tone.
take, for example,
as the foreign tone to
be added,
we
produce the seven-tone scale CoDoEoF#iGi*G#2A#(Co). (The foreign tone
is
indicated by an asterisk to the right of the letter
name.) This again proves to be an isometric scale having the same configuration of half-steps
on the tone of
D
downward, 2221122;
and form the
scale
we
whole- and half-steps,
since
downward with
shall
if
we
begin
the same order
produce the same
scale,
jD,aBb2AbiGiF#,E,2,(D):
Example
10-1
Major Second Heptad p4n n
I ' It
Jti»
2
2
should be noted that the choice of
2
G
'
^^
s
d^
^'
112
as the
added foreign tone
is
arbitrary.
The
addition of any other foreign tone would produce only a different version of the same scale; for
90
example, CiC#iD2E2F#2G#2A#,2)(C).
:
:
FURTHER PROJECTION OF THE MAJOR SECOND
We may now
form the eight-tone scale by adding a major
second above G, that
is,
CJD^EM.C'GtjA.^'A^^^AC):
A:
Example
10-2
Major Second Octod p'^m^n'^s^d'^t'^
$ The nine-tone
^^ 2
2
«
'
tfo
2
I
I
scale becomes, then, the
major second above
tf'
above scale with the
A added, that is, B
C2D2E2FiG,*G#,A,*A#iB,„*(C):
Example
10-3
pmnsdt
Major Second Nonod
^^^^^^^
I The ten-tone
2
2
2
scale adds the
major second above B, namely, C#,
CiC#,*D,E,F#,Gi*G#iA,*A#,B(,, * ( C )
Example
10-4
Major Second Decod p
*
r
The eleven-tone namely,
Dfl:,
#
^..
J 1
scale
2
8
9 8 4 m 8 n 8 sdt
^o 'i^
2
I
I
^ I
adds the major second above C#,
C,C#i*DiDJfi*E2F#iGi*G#iA/A#iBa,*(C):
Example ^ Major c Second^ n Undecad ».
•
,
^1^
ojj.
O
10-5
10
p
It"
m
10 10
n
s
10,10.5 d t
'i" 'J'"
III 211
•
91
THE
The twelve-tone is,
TONAL
SIX BASIC
SERIES
major second above
scale adds the
DJ;, that
E#, and merges with the chromatic scale,
Example ..
r^.. DuodecadJ
.
,-
•
Major Second
^
^^
v>
ij,
o
tf,
10-6
12
p
12 .12,6
m12 n 12 s
d
^^
^>
t
'
fi-* I
I
If
we diagram
perfect-fifth series,
the
first
projection in terms of the twelve-tone
this
we
find that
we have produced two
hexagons,
and the second employ first all of
consisting of the tones C-D-E-F#-G#-Ajj:,
consisting of the tones G-A-B-Cij:-D#-Efl:.
the tones of the
first
hexagon a perfect
fifth
We
hexagon, and then
above the
first
move
to the
and again proceed
second to
add
the six tones found in that hexagon.
Example
10-7
A-'
The
following table gives the complete projection of the
major-second scale with the intervallic analysis of each:
92
FURTHER PROJECTION OF THE MAJOR SECOND
C D C D E C D E F#
CD CD CD CD CD
E E E E E
s
ms^ rn^sH
mHH^
C C# D C C# D C C# D
F# G# F# G# A# F# G G# A# F# G Gt A A# F# G G# A A# B E F# G Gif A A# B D# E F# G G# A A# B D# E E# F# G G# A
We have
already observed that the six-tone major-second scale
m^sH^
fm^nhHH^ fm^n's'dH^ p^ni^n^s^dH^
p'm'n's'dH^ ^10^10^10^10^10^5
HB
^12^12^12^12^12^6
contains only the intervals of the major third, the major second,
and the
tritone.
The addition
of the tone
G
to the six-tone scale
preserves the preponderance of these intervals but adds the
new
C to G and G to D; the minor and the minor seconds, F# to G and
intervals of the perfect fifth, thirds,
E
to
G
and
G
to B^;
GtoAb. It
adds the isometric triad ph, C2D5G; the triad pns, G7D2E,
and the involution Bb2C7G; the triad pmn, C4E3G, and the pmd, G7D4F#, and the involution Ab4C7G; the triad mnd, EsGiAt), and the involution F^iGsBb; the triad nsd, GiAb2Bb, and the involution E2FJt:iG; the two isometric triads, sd^, FJiGiAb, and nH, EgGsBb; and the triad pdt, involution GsB^^D; the triad
CeF^iG, with the involution GiAfjeD. The addition of these triad forms to the three which are a
and mst, gives this types which are possible in the
part of the major-second hexad, ms^,
seven-tone scale
all of
the triad
rrf,
twelve-tone scale.
Example £!i
pns
and
10-8
involution
pmn
and
involution
THE pmd
*J
and
Sd
47
2
^iiiJ
J
l
J
l
J^r
1
3
3
I
J 'e
ond i
SERIES nsd
involution
?
^'l
^
pdt
. I
I
and
' 3
n^t
^
^
mnd
involution
4
7
TONAL
SIX BASIC
and
I
involution
'
2^
f
I
involution
J;Ji-J f=
|J
16
I
The seven-tone impure major-second
scale therefore has cer-
tain advantages over the pure six-tone form, since
it
preserves
the general characteristic of the preponderance of major seconds,
major
thirds,
and
tritones
but adds a wide variety of
new
tonal material.
For the reasons given
earlier,
we
shall
spend most of our time
experimenting with various types of six-tone projections, since
we
find in the six-tone scales the
variety.
We
shall
make an
maximum
of individuality
and
exception in the case of the major-
second projection, however, and write one sketch in the seventone major-second scale, since the addition of the foreign tone to the major-second scale without at the It is
hexad adds variety
same time
to this too
entirely destroying
homogeneous its
character.
a fascinating scale, having some of the characteristics of a
"major" scale, some of the characteristics of a "minor" scale, and all
of the characteristics of a whole-tone scale.
Begin by playing Example
10-9,
which contains all of the each triad and then com-
triads of the scale. Listen carefully to
plete the analysis.
Example ^?
10-9
fi<^\ iiiwiwi^i j^j :NiiJiJ,it^it^j ^'H»^".
^^^
u^ g
s
Example 10-10 contains
all
m
^^
of the tetrad types, but in
no
regular order. Play the example tRrough several times as sensi-
94
— FURTHER PROJECTION OF THE MAJOR SECOND tively as possible, perhaps
with a crescendo in the third and
fourth measures to the
beat of the
fifth measure, and then Note the strong harmonic accent between the last chord of the fifth measure and the first chord of the sixth measure, even though the tones of the two chords first
a diminuendo to the end.
are identical.
Have another student play accurately from dictation.
the example for you and write
Now
analyze
all
it
of the chords as to
formation including the sonorities formed by passing tones.
—— —
EXAIV[PLE 10-10
r-
Tfc^*
\
\
h
\
i-j
=f=r^
-f
J
t*y
^
#r^
^^ ^ m ^
\/
h
£^
r
f
\
r
1
1
^^
J
F
gT
i
[j*
r
bJ^
The following measure from Debussy's offers a
r
f p .
1
3
^r^ fi^
^.Lt^Ul
'
bp
1—
I
^J
r
r
^—
I
C///77.|
rthh
r
J
TIT r ^E^ T
-
r r
'
y.
Pelleas et Melisande
simple illustration of the seven-tone major-second scale,
the foreign tone, E^, merely serving as a passing tone:
Example Debussy
,"
Pelleas
ond Melisande"
±^jij ^ .i^rribj'^ fofv ^^fr^
10-11
t
>^tl:^^b*)b^
w^ 2
2
2
I
Permission for reprint granied by Diirand et Cie, Paris, France, copyright owners; Elkan-Vogel Co.,
Inc., Phila-
delphia, Pa., agents.
95
THE
SIX BASIC
TONAL
A somewhat
SERIES
more complicated illustration Alban Berg song, "Nacht," already referred to the pure whole-tone scale:
Example
found in the
is
as
beginning in
10-12
Albon Berg, "Nacht
Copyright 1928 by Universal Editions, Vienna; renewed 1956 by Helene Berg; used by permission of Associated Music Publishers, Inc.
m
,i'°,^'
The student should now be ready
ii
",b»(it.^
to write a free improvisatory
sketch employing the materials of this scale (Example 10-1). will notice that the scale has
C
major and one on
G
two natural resting
He
points,
one on
G
minor,
minor. Begin the sketch in
modulate modally to C, establish C as the key center, and then modulate back to the original key center of G. See that only the tones
much
C-D-E-Ff-G-Ab-Bb are employed
in this sketch, but get as
variety as possible from the harmonic-melodic material
of the scale.
96
11
Projection of the
The next
Minor Third
which we
series of projections
shall consider
is
the
C we
projection of the minor third. Beginning with the tone
superimpose the minor third E^, then the minor third G^, forming the diminished triad CgEbsGb, which consists of two minor thirds
and the concomitant
tritone,
from
C
to G^.
Upon this we we shall call
superimpose the minor third above G^, B^^, which
by
its
enharmonic equivalent, A, forming the familiar tetrad of
the "diminished seventh," consisting of four minor thirds:
Eb,
Eb
to Gb,
Gb and Eb
Gb
to
Bbb (A), and
A
to C;
and two
tritones:
C C
to
to
to A; symbol, nH^\
Example
11-1
Minor Third Tetrad
i
i.
o
^o
u^\^
^^^C^-)
3
As
in the case of the
major-second
projected in pure form beyond
scale,
six tones,
which could not be so the minor third
cannot be projected in pure form beyond four tones, since the next minor third above
wish to extend
A
duplicates the starting tone, C. If
this projection
beyond four tones we must,
introduce an arbitrary foreign tone, such as the perfect
and begin a new
series
of minor-third projections
we
again,
fifth,
G,
upon the
foreign tone.* ** The choice of the foreign tone is not important, since the addition of any foreign tone would produce either a different version, or the involution, of the
same
scale.
97
THE
SIX BASIC
The minor-third pentad,
TONAL
therefore,
Example
SERIES
becomes C3Eb3GbiGt]2A:
11-2
pmn^sdt^
Minor Third Pentad
>obo "h*
jJt^^tjJ 3
3
It contains, in
^
12
addition to the four minor thirds and two tritones
aheady noted, the perfect the major second,
G
analysis of the scale
fifth,
C
to G; the
therefore, pmn^sdt^.
is,
major
third,
and the minor second, G^
to A;
preponderance of minor thirds and
The
tritones,
E^
to G;
to G.
scale
still
The
has a
but also contains
the remaining intervals as well.
The
minor third above the foreign tone G, that is, Bb, the melodic scale now becoming C3Eb3GbiG2AiBb. The new tone, Bj^, adds another minor third, from G to Bj^; a six-tone scale adds a
perfect
fifth,
from E^ to
Bj^;
a major third, from
major second, from B^ to C, and the minor second,
G^
A
to B^; a
to B^, the
analysis being p^m^n^s^dH^:
Example Minor Third Hexad
>o^» The component
--
11-3
p^m^n^s^d^t^ ^'^'
jbjbJtiJ ^^r
triads of the six-tone minor-third scale are the
basic diminished triad CgE^gGb, nH,
which
is
also duplicated
on
Eb, Gb, and A;
Example Minor Third Triads
98
t
C3Eb4G and Eb3Gb4Bb, pmn, with the one the major triad Eb4G3Bb, which are characteristic of
the minor triads involution,
n
11-4
PROJECTION OF THE MINOR THIRD the perfect-fifth series;
Example pmn
Triads
the triads
6^)2^70;
C7G0A and
found
in
and
4
3
11-5 involution
4
4
3
Ej^yBl^aC, pns,
72
series;
11-6
and
pns
Triads
with the one involution
and minor-second
the perfect-fifth
Example
3
involution
27
72
the triads Gt)iGk]2A and AiB^aC, nsd, with the one involution
GsAiBb, which we have also met as parts of the perfect-fifth and minor-second projection;
Example nsd
Triads
and
I the triads
we have
G(;)4B|72C,
involution
2
I
Eb4G2A and
11-7
2
I
mst, with no involution, which
encountered as part of the major-second hexad;
Example
11-8
Triads mst
i
4
2
4
2
the triads E^aGbiGt] and Gb3AiBb, mnd, with the one involution
GbiGtjsBb; which
is
a part of the minor-second hexad;
99
:
the
tonal
six basic
Example Triads
mnd
series
11-9
and
involution
U J^f ^jJl^J 3 3 l
ibJ^J^f 3
1
I
I
and the triads CeG^iG and E^eAiBb, fdt, without which are new in hexad formations
Example Triads
11-10
pdt
6
The student should study
I
series
introduces.
doubtedly, be thoroughly familiar with the
diminished
triad,
but he will probably be
triad ipdt. Since, as I
first
He
and
will,
less familiar
with the
have tried to emphasize before, sound
"new" sounds, experimenting with
un-
of these, the
all-important aspect of music, the student should play to these
new
carefully the sound of the
which the minor-third
triads
involution,
and
is
the
listen
diflFerent inversions
different doublings of tones until these sounds
have become
a part of his tonal vocabulary.
The
tetrads of the six-tone minor-third scale consist of the
basic tetrad CgE^gGbgA, the familiar diminished seventh chord, consisting of four minor thirds
and two
tritones, nH^, already
discussed;
Example
11-11
Minor Third Tetrad
^
4 2 n
t
the isometric tetrads C^¥.\)4GzB\), p^mn^s, and GsAiBbsC, pn'^s^d,
both of which fifth
we have
already met as a part of the perfect-
hexad, the latter also in the minor-second hexad;
100
projection of the minor third
Example p^mn
Tetrad
4
3
four
one
new
tetrad types,
11-12 Tetrad pn
s
2
3
s
d
2
I
consisting of a diminished triad plus
all
tone: C3Eb3Gb4Bt) and A3C3Et)4G, pmn^st; and C3Eb3GbxG4 Eb3Gb3AiBb, pmnHt; GbiGt^^AgC and AiBbsCg Eb, pn^sdt; Eb3GbiG^2A and Gb3AiBb2C, mnhdt;
"foreign"
Example pmn
Tetrads
^J
ji,j 3
11-13
pmn
st
t
kfA r
^
I
3
r-[
4
3
3
J
mn J
^J 2
the
tetrads
r
*
I
I
2
'
C6GbiGtl3Bb,
analysis pmnsdt, the
first
3
I
J^p
j^j 3
3
I
sdt
-^
3
^^
!
i^J 3
I
2
and Eb4G2AiBb, both having the appearance in any hexad of the twin
tetrads referred to in Chapter 3,
Example
Example
jbJuJ^r 6
3-8;
11-14
pmnsdt
Tetrads
^
^jfej
hj^J^j
r
r
"r
3
dt
J 3
11^
pn^sdt
2
^
ibJ
^'^r
4
3
I
2
I
and the two isometric tetrads EbsGbiGtisBb, prn^n^d, which will to consist of two major thirds at the interval of the minor third, or two minor thirds at the relationship of the major third;
be seen
Example pm^n^d
Tetrad
(j
I,
J
i'^
3
11-15
t^ I
^r 3
hi^^ ^if^ \
ni
@ —
a.
@j]i
101
:
THE
TONAL
SIX BASIC
SERIES
and GbiGtisAiBb, mn^sd^, which consists of two minor thirds at the interval relationship of the minor second, or two minor seconds
minor third
at the interval of the
Example mn'^sd
Tetrad
2
I
The pentads
11-16
*" n.
1
@d
d_
@ji
consist of the basic pentads C3Et)3GbiGfcj2A,
and
EbaGbsAiBbsC, pmn^sdt^;
Example
11-17
Minor Third Pentads pmn'^sdt^
li^J'J^^^ 3
I
12
3
^J^^^ 12 3
3
the pentad CgE^gGbiGtisBb, p^m^nhdt, which
may
also
be ana-
lyzed as a combination of two minor triads at the interval of the
minor
third;
Example
11-18
Pentad p^m^n^sdt
liU^V^I^i^ 13 mn @ 3
3
p
n^
the pentad C3Et)4G2AiBb, p^mn^s^dt, which as
two
triads
pns
at the interval of the
Example
ti
-J-
''• I
3
102
4
11-19
3^2,
.2
2
1
-il-
pns
may
minor
P*"
@£
also
third;
be analyzed
PROJECTION OF THE MINOR THIRD the pentad E^aGbiGtioAiBb, pm~n^sdH, which
lyzed as the combination of two triads
minor
mnd
may
also
be ana-
at the interval of the
third;
Example
11-20
Pentad pm^n^sd^t ffl [
.JbJ^J
and
the
pentad
Ji^f
12
3
l^jJ^JbJJ^f 3
3
1
mnd
1
@
1
ji
pmrfs^dH,
GbiGl:]2AiBb2C,
which may be
analyzed as the combination of two triads nsd at the interval of the
minor third;
Example
11-21
Pentad pmn^s^d^t
12 12 The
contrast
between the
is
_n_
and the be immediately apparent. Whereas
limited to various combinations of major thirds,
major seconds, and of harmonic
'2
I
@
six-tone major-second scale
six-tone minor-third scale will
the former
2
I'
nsd
tritones, the latter contains a
and melodic
possibilities.
of course, in the interval of the
The
wide variety
scale predominates,
minor third and the
tritone,
but
contains also a rich assortment of related sonorities.
Subtle
examples
of
the
minor-third
Debussy's Pelleas et Melisande, such
Example
hexad are found
in
as:
11-22
Debussy, "Pelleas and h^elisande"
Permission for reprint granted by Durand et Cie, Paris, France, copyright owners; Elkan-Vogel Co., Inc., Philadelphia, Pa., agents.
103
THE
SIX BASIC
TONAL
SERIES
Play each of the triads in the minor-third hexad in each of three versions, as indicated in
Example
11-23. Play
several times slowly, with the sustaining pedal held. If sufficient pianistic technic,
play
all
hands in octaves, otherwise the one each
its
each measure
you have
of the exercises with both
line will suffice.
Now
analyze
triad.
Example
11-23
rn. ^jjii.mi^irmi^LJ ^^LjLLJ 1;
-
i>
1
^
^ p^r
''
fp ^^dripi"^LJ j
'
^-
^'u^Lii i^
i^^^
i^n^^^^dlifj^alLLS^^^'iLlL bm
i I)
1,
iff ^^^ JJ
104
I
t-i^^
[^
;mr
\
^cU 1^^^
bf^
M
k^ ^LL
PROJECTION OF THE MINOR THIRD
Repeat the same process with the tetrads of the
Example
scale:
11-24
|jP^.mc:tfLtfr jy..^clJ^Lffl i
,,f,
jw n^^^a!J\^.P^i^ ^ci^ ^
(liP^i-^crJcdJ
F
Lph
JJ^^ft^^MJ^^^yrJ^cfT bp
^f^F ^f^f-j^
p^^crtfrdT^cttri^ffl^^ciLrigj
^^
k-.^
b>f-
^
b*r^i
b[B
bet?
^a^^'LlU^^.W^W (|
i'i?^^r£jc!lin''cll
Repeat the same process with the
Example
^
jM
^
M^'^^ r^r
six
pentads and the hexad
11-25
r
r'l
^
^r r'll [/'tT^ 105
THE
#
SIX BASIC
TONAL
SERIES
J^JJ^^^^^^ypJ
bJ^JjJ^
vH
I
jjt-^t>''
(|
jn7i:^,jT3T:^cxUlrciiir
''
r^r
Y
r 't
I,
^
r'TT ^c_r
r'T
b.
1,
-Vlir,
i
rWr
'cmrftc^^rrT c'TrTT yrT_l
^^^^^^
V4
y^kr
^^^^^^^^^^^^ ^H One
of the
degree
of
'te--±^
fe^
most important attributes of any sonority
its
is
consonance or dissonance, because the "tension"
induced by the dissonance of one sonority
may be
reduced, or released by the sonority to which interesting
^rk^'t'''
f-
and important study, therefore,
is
it
increased,
An
progresses.
the analysis of the
relative degrees of dissonance of diiferent sonorities.
At
first
glance, this
may seem
to
be an easy matter. The
vals of the perfect octave; the perfect fifth
perfect fourth; the major third and
and the minor third and generally sonority.
106
its
its
its
inter-
inversion, the
inversion, the
inversion,
considered to perform
The major second and
its
and
minor
sixth;
the major sixth,
are
a consonant
function
inversion, the
minor seventh;
in
a
PROJECTION OF THE MINOR THIRD the minor second and
and the
inversion, the major seventh;
its
(augmented fourth or diminished
fifth)
are generally
considered to perform a dissonant function.
When
these intervals
tritone
are
mixed together, however, the comparative degree
sonance in different sonorities
is
indeed, cannot be answered with
We may
Some
not always clear.
of dis-
questions,
finality.
assume that the dissonance of the major seventh and minor second is greater than the dissonance of the safely
minor seventh, major second, or however, there
listeners,
is
tritone.
much
not
To
the ears of
difference
many
between the
dissonance of the minor seventh and the tritone.
Another problem
arises
when we compare
the relative con-
sonance or dissonance of two sonorities containing a different
number since
of tones.
C-E-F#-G
conclude that the
is
the tritone, whereas the
minor second, the might
contains three dissonances— the
first
tritone,
and the major second. However,
it
be argued that whereas the sonority C-E-F#-G con-
also
tains a larger
number
of dissonant intervals, C-FJf-G contains a
greater proportion of dissonance. is
we might
For example,
more dissonant than the sonority C-F#-G, the second contains two dissonances— the minor second and
sonority
The
analysis of the
first
sonority
pmnsdt—one-hali of the intervals being dissonant; whereas the
analysis of the second sonority
is
pcff— two-thirds of the intervals
being dissonant:
Example Tetrad
pmnsdt
m' i i Finally,
it
11-26 Triad pdt
»'
would seem
ii^i^U
I
fe°
i
v^-
d
that the presence of one primary dis-
sonance, such as the minor second, renders the sonority more dissonant than the presence of several mild dissonances such as
the tritone or minor seventh. For example, the sonority C-D#-E-
G, with only one dissonant interval, the minor second, sounds
107
— THE
SIX BASIC
more dissonant than the
TONAL
SERIES
tetrad C-E-Bt>-D,
which contains four
mild dissonances:
Example Tetrad
pm 2 n 2 d
11-27
m 2s 3
Tetrad
With the above
t
theories in mind, I have tried to arrange
all
of the sonorities of the minor-third hexad in order of their relative dissonance, beginning with the three
most consonant
triads— major and minor— and moving progressively to the indissonant
creasingly
sonorities.
Play through Example
11-28
carefully, listening for the increasing tension in successive sonorities.
Note where the degree of "tension" seems
approximately the same. Analyze
you agree with the order
all
to
of the sonorities
of dissonance in
them. Have someone play the example
for
which
I
remain
and see
if
have placed
you and take
it
down
from dictation:
Example
'^
r^
J
tl-~"
—
3
r
108
r
3i
1
iittii.-.
1-^
11-28
LJJiW=^
\rh 'i
J
J
^
hN i4
p
J
f^Tw ffi 3
PROJECTION OF THE MINOR THIRD
Reread Chapters 6 and 7 on modal and key modulation.
it is
hexad
minor-third
the
Since
has
the
p^m^n^s^dH^,
analysis
evident that the closest modulatory relationship will be at
the interval of the minor third; the next closest will be at the
and the third order
interval of the tritone;*
be at major
the
interval
minor
or
third,
of the minor third
four
common
will
perfect
the
of
Modulation
second.
have
common
five
Example
f^
^
Minor Third Hexad
@
Modulation i:
^
..k J^"^*
i @—
!?•
-0
lj
7- bo^'
'1'
'
p
m
n
s
^
@m
^
the
interval
at
two common
tones.
d t^
^ n^
^rt^<
M'^'^' @P
@1
^
^
^^^^
rWV4^
v\ •
S
l
second,
11-29
^^^ ^53
pr^^^
major
tones; at the tritone,
tones; at the other intervals
Modulation of
of relationship will
fifth,
\
P^
^
,
}m
bot
|<
^
k^b *
Write a sketch using the material of the minor-third hexad. Begin with C as the key center and modulate modally to E^ as the key center, and back to C.
Now perform
a key modulation to
the minor-third hexad a minor third below
modulate
to the
key a
fifth
C
(that
above (E), and then back
is.
key of C. See Chapter
17,
A);
to the
pages 139 and 140.
109
12
Involution of the Six-Tone
Minor-Third Projection
The
first
three series of
projections, the perfect fifth,
second, and major second, have
For example, the
all
produced isometric
perfect-fifth six-tone scale
minor scales.
C2D2E3G2A2B, begun
on B and constructed downward, produces the identical
B2A2G3E2D2C. This tion. If
The same
we
is
scale,
not true of the six-tone minor-third projec-
projection
downward produces
a different scale.
take the six-tone minor-third scale discussed in the
previous chapter, C3Eb3GbiGti2AiBb, and begin
it
on the
final
note reached in the minor-third projection, namely, B^, and
we add first the minor third below B\), or G; the minor third below G, or E; and the minor third below E, or Cjj:. produce the same scale downward,
Example
12-1
Mi nor Third Tetrad
^
downward
at^
We
then introduce, as in the previous chapter, the foreign tone
a perfect fifth below
B\),
or E\), producing the five-tone scale
BbsGsEkiiEbsCJ:
Example Minor Third Pentad
110
12-2
INVOLUTION OF THE MINOR-THIRD PROJECTION
By adding another minor
third
below E^, or C, we produce the
six-tone involution BbgGsEtiiEboCjfiCfc]:
Example
12-3
Minor Third Hexad
*
b.
t-
A
simpler
method would be
7
^^
^
J
^^
to take the configuration of the
minor third hexad, 3 3 121, beginning on C, but in reverse, 1213 3, which produces the same tones, CiCJsEbiEtjs original
GsBb:
Example Minor Third Hexad upward [.o
bo
12-4
Involution
^
t?o
t^ If
to
we examine
be the same
involution.
the components of this scale
We
4
|
;)
we
shall find
them
conceived upward but in
as those of the scale
The
p^m^n^s^dH^.
£
bo
analysis of the scale
is,
of course, the same:
again, the four basic diminished triads
find,
C^gEsG, EgGsBb, G3Bb3Db(C#), and A#(Bb)3C#3E;
Example
12-5
Minor Third Triads
4V
i-jl
\,^{tr)iHf^
the major triads— (where before
and Eb4G3Bb, with the one
we had minor
involution, the
Example Triads
t
n^^t
12-6
pmn and
H
triads)— C4E3G
minor triad C3Eb4G;
involution
''i>i
111
THE the
triads
TONAL
SIX BASIC
SERIES
BbsCyG and Db(Cfl:)2Eb7Bb,
pns,
with the one
involution, Eb7Bt)2C;
Example and
Triads pns
4
12-7
^ 2i 7
involution
pjur
•>
Y-~
'bJ 7
7
2
the triads Bl:)2CiDb(C#) and CJfaDJiE,
T2r
nscZ,
together with the
one involution CiDb2Eb;
Example Triads
and
nsd
iJ'aiU 2
12-8 involution
lijJ^J " 2
1
I
^^ I
2
the triads Bb2C4E and Db2Eb4G, mst;
Example
12-9
Triads mst
ITlTg
^
2
4
the triads CiCJgE and DJiEgG, mnc/, with the one involution
CsDfiE;
Example Triads
mnd
12-10
and
involution
4 ^,t^3 ^^A 13 ^
'^jt
3
and the
triads CiCJfeG
and D^iEgBb, pdt:
Example Triads
i 112
12-11
pdt
WFWf \v
6
I
6
INVOLUTION OF THE MINOR-THIRD PROJECTION
The first
Bb,
tetrads consist of the
same isometric
tetrads found in the
minor- third scale: the diminished-seventh tetrad, CifgEgGa the
nH^,
other
isometric
pmVc/,
CsDJiEsG,
C3Eb4G3Bb,
tetrads,
and
pnVc/,
BbsCiDbsEb,
jrmn^s,
CiDbsEbiEl^,
rmnrsd^;
Example ^Tetrad
12-12
1343
333
212
313
mn^d
Tetrad
Tetrad p^mn^s Tetrad pm^n^d Tetrad pn^s^d
n^^t^
121
four tetrads consisting of a diminished triad and one foreign tone, each of
which
tetrad
similar
Eb4G3Bb3Db,
GgBbsCiDb
be discovered
will
the
in
first
to
minor-third
be the involution of a scale:
C4E3G3Bb and
pmnht; CiCJfsEsG and D^iEgGsBb, pmnHtand Bb3Db2EbiEl^, pnhdt; and Bb2CiC#3E and
C^sDftiEsG, mn^sdt;
Example
12-13
pmn^st
Tetrads
A 4
^ 3
Tetrads
A
y^ 3
y,
and the
'
Tetrads
p 2
y
\?m I
I
"twins",
3
3
t^p
\?m
^
^
yk
Tetrads pn^sdt
3
pmn ^dt
'7 2
3
mn^
I
3
sdt
^
vi-y^
I
^
\-ii^ 2
3
^ I
CgD^iEsBb and CiDb2Eb4G, pmnsdt, the
involu-
tions of similar tetrads discussed in the previous chapter:
Example
12-14
Isomeric Tetrads
J
jt^
3
I
J 't 6
pmnsdt
Ui-J I
1^ 2
3
4
113
THE
The pentads
SIX BASIC
TONAL
SERIES
CJaDJiEgGsBb and
consist of the basic pentads
Bb2CiC|:3E3G, pmn'^sdt^ (the involutions of the basic pentads in the previous chapter);
Example
12-15
pmn
Minor Third Pentads
sdt^
^^^^^^ 13
2
3
13
2
the pentad CgEj^iEtjsGgBb, p^m^n^sdt, a
which may be analyzed
combination of two major triads
minor
3
at
the
Pentod
I
the
12-16
p^m^n^sdt
J ^ V i 3^J 13 3
i0
•'
^
ii
amn
@
n_
may be
the pentad CiDb2Eb4G3Bb, p^mn^s^dt, which
minor
of
third;
Example
as the
interval
as
combination of two
analyzed
pns, at the interval of the
triads,
third;
Example Pentad
12-17
p^mn's^dt
l-iJ^J^YLi^J^Jt 12 4 3 2
^2
7
7
may be
the pentad CiDb2EbiEti3G, pm^n^sdH, which the
combination of two
minor
triads,
rand,
at
the
third;
Example g,
Pentod CIIIUW pm^n^sd^t ^111 3U I
II
12 13 114
12-18 I
I
3
mnd
_
@
3
I
_n_
analyzed as
interval
of the
INVOLUTION OF THE MINOR-THIRD PROJECTION
and the pentad BboCiDboEbiEt], pmn^s^dH, which may be analyzed as the combination of two triads, nsd, at the interval of the minor third:
Example
12-19
Pentad pmn^s^d^t
^^^^^m 12
2
2'
1
All of the above pentads will
2
1
nsd
@
1
_n_
be seen
to
be involutions of
similar pentads discussed in the previous chapter.
many examples of the involution of the minor-third hexad we may choose two, first from page 13 of the vocal score From
the
of Debussy's Pelleas et Melisande;
Example Debussy, "Pelleas and
U]ITi\'^^
^y-
12-20
Melisande"^
____^
m--^-m--r-0-r-
p?
^Jt
p
m
"3:
Permission for reprint granted by Durand et Cie, Paris, France, copyright owners; Elkan-Vogel Co., Inc., Philadelphia, Pa., agents.
and from the second movement of Benjamin and string orchestra:
Britten's Illumina-
tions for voice
Example Benjomin Britten, "les 772./
12-21
Illummations"
espress.
espress. e sost. Copyright 1944 by Hawkes
& Son (London),
Ltd.
Used by permission
of
Boosey & Hawkes,
115
Inc.
THE
SIX BASIC
TONAL
SERIES
Analyze the following two measures which come
at the
end
a section of Debussy's "Les fees sont d'exquises danseuses." of the notes of the of
one
scale,
two measures
we have
of
If all
are considered as integral parts
the rather complex scale iC-Cb-B^-A-Ab-
G-Gb-F-E^-D composed of the two minor-third tetrads, jC-A-GbE^ and iF-D-C^-Ab, plus the minor third, Bt)-G (forming the ten-tone minor-third projection).
A
closer— and also simpler— analysis, however, shows that the
measure
first
contains
the
notes
of
pattern transposed a perfect
fifth,
minor-third
the
|F-D-Cb-Ab-Bb-G, and the second measure to begin
is
hexad
the identical scale
on C, I C-A-Gb-E^-
F-D. This simpler analysis posers,
whose
desire
is
is
to
much
to be preferred, for most comcommunicate to their listeners rather
than to befuddle them, tend to think in the simplest vocabulary
commensurate with
their needs.
Example
12-22
Debussy, "Les fl es sont d'exauises danseuses"
Permission for reprint granted by
Durand
et Cie, Paris, France, copyright owners;
Elkan-Vogel Co.,
Inc., Phila-
delphia, Pa., agents.
a\^>-^
'
* A
i^U^be»
;t,k.^=^
[
detailed
comparison of the material of the minor-third
hexad discussed Chapter 12 116
_:H-^^»t>o p^=*^
in
Chapter 11 with that of the material
will indicate that the isometric material of the
in
two
INVOLUTION OF THE MINOR-THIRD PROJECTION hexads tions,
is
identical,
but that where the sonorities have involu-
each sonority of one scale
is
the involution o£ a similar
sonority in the other. For example, the minor-third hexad dis-
cussed in Chapter 11 contains two minor triads and one major triad,
whereas the involution of the hexad contains two major
triads
and one minor triad. The involution does not, therefore, add any new types of sonorities, but merely
strictly speaking,
substitutes involutions of those sonorities.
117
13
Minor Third
Projection of the
Beyond
the Six-Tone Series
We
produced the six-tone minor-third scale in Chapter 11 by beginning on any given tone, superimposing three minor thirds above that tone, adding the foreign tone of the perfect fifth, and
superimposing another minor third above that tone.
We may now minor
thirds,
complete the
series
by superimposing two more
thereby completing a second diminished-seventh
chord, then adding a second foreign tone a perfect fifth above
the
first
foreign tone, and superimposing three
more minor
thirds,
thereby completing the third diminished-seventh chord. For the student
who
is
"eye-minded"
following diagram
may be
as
Example
118
well
helpful:
13-1
as
"ear-minded," the
FURTHER PROJECTION OF THE MINOR THIRD
Here
it
be seen that the minor-third projection divides the
will
twelve points in the circle into three squares, the
We
on C, the second on G, and the third on D.
A
begin by super-
imposing
E\),
G[},
imposing
B\),
D\),
and F^ (E), and then adding
The
and
A\),
and superand super-
(B):
Cj^
produced, with their respective analyses,
thus
scales
G D
above C, then adding
and
imposing F,
beginning
first
become:
Example p^m^n^s'^d^t^
Minor Third Heptad
i
"
^g*
l>o
13-2
^^
^*
i J
4 4 8 4
Minor Third Octad
t;o
bo
m
p
"
n
s
d
3
p^
m^
n^ s® d^
b« ^*
2
I
If.
ij =
r
I
S
^^
p' ^m'^n'^
Minor Third Undecod
k-
.
,
11111112
iT^b-*-
=S
the
seven-tone
I
Jg 1
^^^S^^ I
I
I
I
I
I
I
p'^nn'^n'^ s '.^d'^t^
^^b»b»
-K;:b^ b,l;i'
U
iJbJ^J^J^J^r ^^Il^ ibJ^J^J Jl I
these
^r 2
I
^
I
Minor Third Duodecod
of
2
I
I I
J
^J
i^J
^jgiJllJbJ^J
s'^d'^t
*?:
All
I
p^m^n^s^d^ f^
Minor Third Decad
tl'
2
I
t"^
•*-
"
I
|^^^^§^^ ^
iP^bo^°
^
t
2
^*
2
I
4 4
tl>
Minor Third Nonad
^^
^J ^^ 2
I
scales scale,
are
isometric
the
involution
I
I
l I
I
I
I
I
I
^
I
with the exception of
which produces
different scale:
119
of
a
:
THE
TONAL
SIX BASIC
Example Minor Third Heptad
These
scales
SERIES
13-3
Involution
with their rich variety of tonal material and their
generally "exotic" quality have
made them
the favorites of
many
contemporary composers.
A
beautiful example of the eight-tone minor-third scale will be
found
in the first
Example this scale,
13-4,
movement
where the
of Stravinsky's
first
of Psalms,
seven measures are consistently in
EiF2GiG#2A#iB2C#iD
Example
2 Strovinsky, /1AIT03 AlTos
Symphony
2
1
Symphony
2
I
of
I
13-4
(2)
Psalms
I
J
J
^'
Ex
1-1.
I
>>;j.^^X
J
'^m
J
J
ro -
tl
I
-
J
J -
nem
J3 iri
mm
J J me -am,
J7J
rp'rrmr
^^m ^^ ^m "^m Copyright by Edition Russe de Musique. by permission of Boosey & Hawkes, Inc.
120
Revised version copyright 1948 by Boosey
& Hawkes,
Inc.
Used
i
—
:
FURTHER PROJECTION OF THE MINOR THIRD
A
completely consistent use of the involution of the seven-tone
minor-third scale will be found in the
same composer's Sijmphony
number
rehearsal
in
first
movement
of the
Three Movements, beginning
and continuing without deviation
7,
at
for
twenty-three measures
Example Symphony
Stravinsky,
in
13-5
Three Movements
^^"
,
^•^mr
marcato
V
mf I-
-^-»-
'i^'r^
*""*^°
Dizz
i
^
^^T-Vi
pizz.
m
f
f
^
b
l?Qj>
j}
*/
*/
g
g
•OS?
^
orco
3_^ -»-y-
i
—
pizz.
^•^
pocosj 06p
J-
^m p
pizz.
a Jj^»^i ^4?
^J5?
pizz.
orco I
5
s
-1~ t»
n^y
a s^
W
^
pizz.
vT pocosjz
J
J
I
r
p
J
r
mf
^ ^
^ ^ pizz.
Copyright 1946 by Associated Music Publishers,
^^-i
12
3
Inc.,
12
New
York; used by permission.
I
121
THE
SIX BASIC
TONAL SERIES
Another interesting example of the eight-tone minor-third scale
found
is
siaen's
at the
opening of the third movement of Mes-
VAscension: Example
Messloen
13-6
,"l' Ascension"
Vif
^M\h
>
"/g^^^
-.
%
i-
itJiitfe
Reproduced with the permission of Alphonse Leduc, music publisher, 175 rue Saint-Honore, right by Alphonse Leduc.
I
4-
^
*
fl"0
=
I
2
I
2
I
2
I
aS
Paris.
Copy-
(21
Analyze further the Stravinsky Symphony of Psalms and try to find additional examples of the minor-third projection.
122
14
Major Third
Projection of the
We
have observed
that there are only
two
intervals
which can
be projected consistently through the twelve tones, the perfect
and the minor second. The major second may be projected
fifth
through a six-tone
and then must
series
resort to the interjection
of a "foreign" tone to continue the projection, while the
form through only four
third can be projected in pure
We
come now
to the
major
third,
which can be projected only
the major third, E, and the second major third,
C-E-G# consisting G#, and G# to B# (C), m^:
ing the augmented triad
C
to E,
E
to
Example
I ^ project the major third
add the foreign tone
of the three major
14-1
Major Third Triad
To
we superimpose E to G#, produc-
Beginning again with the tone C,
to three tones.
thirds,
minor
tones.
mj
^°'"-'
°
tf°
beyond these three
Gtj*, a perfect fifth
tones,
we
again
above G, producing the
basic major-third tetrad G4E,oGiGJj: having, in addition to the
three major thirds already enumerated, a perfect to G; a
GtoG# •
minor
from
E
to
from
C
G; and a minor second from
{k\));pnv'nd:
Here the choice
A# with
third,
fifth,
their
of the foreign tone
is
more important,
since the addition of D, F|, or
superimposed major thirds would duplicate the major-second hexad. The
addition of any other foreign tone to the augmented triad produces the same tetrad in a different version, or in involution.
123
the
tonal
six basic
Example
To produce
14-2
pm^nd
Major Third Tetrad
^.
t^
series
^g
J4J
3
we superimpose
the pentad,
I
a major third above
G, or B, forming the scale C4E3GiG#3B, and producing, in addi-
G
tion to the major third,
minor
G#
third,
to B;
and the minor second, B
Example
to B; the
to C; p^m^n^d^:
14-3
p^m^n^d^
.Major Third Pentad
To produce
E
to B, the perfect fifth,
we add
the six-tone major-third scale,
the major
third above B, or D^, giving the scale CgDJiE.sGiGJfsB.
D^,
tone,
in addition to
an additional major another perfect
fifth,
and a minor second,
forming the major
third,
G#
DJj:
from
Hexod
i If
minor
we proceed six-tone
augmented scale,
to G.
It
C
to
third,
The new DJj:,
adds
also DJj:
adds
(E^);
p^m^n'^d^
iitJ to analyze the
which
C and on
is
G.
^
^
«^
r
melodic-harmonic components of
major-third scale,
triad,
m^, on
to
14-4
*
this
B
to E; p^m'^n^(P.
Example Major Third
(El^)
Dfl:
to DJf; a
third,
we
find
that
it
contains
the
the basic triad of the major-third It
contains also the major triads
C4E3G, E^GifsB and G#4B#3(C)D#, pmn, with their involutions, the minor triads C3Eb4(D#)G, E3G4B, and
124
Gjj^,B,Djj^;
projection of the major third
Example Triads
14-5
pmn
and
involutions
C,G4B, E,B4D#, and Ab(G#),Eb(D#),G, pmd, together with their invohitions C^E^B, E4G#7D# and Ah(G#)4
and the
triads
C^G:
Example
14-6
pmd
Triads
and
Finally,
74
74
74 it
involutions
47
47
47
contains the triads CJD^-JE, EgGiGJ, and GJyBiC,
mnd, with the involutions BiCsDfl:, DJiEgG, and GiG^sB, which have already been seen as parts of the minor-second and minorthird scales but which would seem to be characteristic of the major-third projection:
Example Triads
mnd
id ^
I
The
J
14-7 and
^^^
J 3
i^r 3
I
involutions
ji^j 13
r I
tetrads consist of the basic tetrads,
tfjj^^
13
new
^m 13
to the
hexad
series, C4E4G#3B, E4G#4B#3(C)D#, and Ab(G#)4C4E3G, which are a combination of the augmented triad and the major triad,
pm^nd, together with
and
their involutions
G#3B4Dfl:4F-)<(-(Gt:]),
which
augmented triad and a minor
pm^nd
Etj3G4B4D]|:,
consist of the combination of the
triad;
Example Major Third Tetrads
C3Eb4G4B,
14-8 and involutions
THE
^
—
—
SIX BASIC
TONAL
1
SERIES
the isometric tetrads C4E3G4B, E^G];^^Bj:>g and (DJj:)G, p~m~nd, which
we
Ab4(G#)C3Eb4
observed in the perfect-fifth
first
projection;
Example •Tetrads p
2
14-9
m 2 nd
^"%34
434
-54
4
the isometric tetrads CgDSiEaG, EsGiGJsB,
we have encountered
pm^n-d, which
as
and GI^BiCsDJ,
parts of the minor-
third series;
Example .Tetrads
pm n^d
13
3
14-10
13
3
3
13
and the isometric tetrads B^C^Dj^^E, DJiEgGiGJ, and GiGJgBiC, pmrnd^, which can be analyzed as two major thirds at the interval of the
minor second, or two minor seconds
at the interval of
the major third, previously observed in the minor-second series:
Example ^Tetrads
7'
1
\-rr-\
•J
J(t^
1
-^r:iitJ I
14-11
pm^nd^
3
^
i^ I
I
The pentads
3
I
— = ^JjtJ r
!
—
w—
1
\-r,
I
3
r-^
^ Ni— m @
I
^
d
consist only of the basic pentads
'
^ d
—
\
^Jm (g
"
C4E3GiG#3B,
E4G#3BiC3D#, and Ab4(Gt)C3D#iE3G^ p-m^n'd^ together with their involutions C3DtfiE3G4B, E3GiG#3B4D#, and Ab3(G#)Bi C3Eb4(D#)Gti.
Example Major Third Pentads
p^m^n^d^
14-12 and
involutions
PROJECTION OF THE MAJOR THIRD
From
this analysis
scale has
it
will
be seen that the six-tone major-third
something of the same homogeneity of material that
The
characteristic of the six-tone major-second scale.
is
includes only the intervals of the perfect
fifth,
scale
the major third,
the minor third, and the minor second, or their inversions.
does not contain either the major second or the tritone.
however, a more striking scale than the whole-tone
It
scale, for
It is,
it
contains a greater variety of material and varies in consonance
from the consonant perfect
The
we
fifth to
six-tone major-third scale
is
the dissonant minor second.
an isometric
scale,
because
begin the scale CgDSiEgGiGJgB on B, and project
reverse, the order of the intervals remains the same.
therefore,
A
clear
sixth
no involution
as
was the case
it
in
There
is,
in the minor-third scale.
example of the major-third hexad
may be found
in the
Bartok string quartet:
Bartok, Sixth Quartet
Example
14-13
Vivacissimo
Copyright 1941 by Hawkes
& Son (London),
Ltd.
P^
Used by permission
of Boosey
& Hawkes,
Inc.
(b«^ 3
if
13 13 127
— THE
TONAL
SIX BASIC
An harmonic example
of the
same
SERIES
scale
is
illustrated
by the
following example from Stravinsky's Petrouchka:
Example
Stravinsky, "Petrouchko"
i
g
VIos.
j'^^bS
j!
[b^§
^
P
m
^^
p
cresc.
_
^% ^s l
^^
14-14
J
jiJ 3
^
b*-!
^r t r 13 13
Copyright by Edition Russe de Musique. by permission of Boosey & Hawkes, Inc.
#
V-l
^^
]
Revised version copyright 1958 by Boosey
& Hawkes,
Inc.
Used
A purely consonant use of this hexad may be found in the opening of the author's Fifth Symphony, Sinfonia Sacra: Example
14-15
Honson, Symphony No. 5
Bossesby-
b^S-
.
tt
!
W
yM^ H.
^.
Copyright
A
charming use of
Prokofieff's Peter
this scale
©
is
1957 by Eastman School of Music, Rochester, N.
Y.
the flute-violin passage from
and the Wolf:
Example
14-16
Prokofieff, "Peter and the Wolf" Fl.
Copyright by Edition Russe de Musique; used by permission.
128
PROJECTION OF THE MAJOR THIRD
P Play the
b^N
\^A
SE
'r
i
r
r
TT
13 13
3
triads, tetrads,
'
pentads, and the hexad in Example
14-17 which constitute the material of the major-third hexad.
Play each measure slowly and listen carefully to the fusion of tones in each sonority:
Example
14-17
^
m
^^ rPi^ Vrnmi lU jJm^-"^'l^ ^ i
'
Is ^'
^^JbJ\J
^^
(|j7;pja^i-^ jjr, [Trpi^ i
.
129
:
.
THE
SIX BASIC
TONAL
SERIES
Experiment with different positions and doublings of the characteristic sonorities of this scale, as in
Example
W (j
^ d
d
u,
i
Hi
P
i^
^=H
^H
^i
i^
^«» etc.
etc.
i
ii
The following
etc.
etc.
%
n J
14-18:
14-18
etc.
etc
/
Example
^
'
T
H
exercise contains all of the sonorities of the
major-third hexad. Play
it
through several times and analyze
each sonority. Have someone play through the exercise for you
and take
it
down from
dictation
Example
^^
^« ^^ ^^ ^^^ff #^^ ^^
m
130
14-19
^m
^
PROJECTION OF THE MAJOR THIRD
"^
(|^
n'-JlJ
Ji.^ Lnj
4
d
^
S
liti
^
tfc^
w
^
¥
*
Write a short sketch Hmited to the material of the major-third
hexad on C.
Example 14-20 scale.
illustrates the
modulatory
possibilities of this
Modulations at the interval of the major
up
third,
or
down,
produce no new tones; modulations at the interval of the perfect fifth, minor third, and minor second, up or down, produce three new tones; modulations at the interval of the major second and the tritone produce
all
new
tones.
Example
14-20
p^m^n^d^
og»
oflo S 3
13 13
Modulation
n-e-
#@ 7-
,j|.
olt'""'
n
>^°'«°' @d
@m
^S
Modulation
^
@
p
^3
^^ ^ ^^^ @1
.»^."'«" ^
^^
Write a short sketch which modulates from the majors-third hexad on C to the major-third hexad on D, but do not "mix" the two keys.
131
15
Projection of the
Beyond
If
we
the Six-Tone Series
we
refer to the diagram below
points in the circle first
Major Third
may be connected
to
see that the twelve
form four
triangles: the
consisting of the tones C-E-Gif; the second of the tones
Gt]-B-D#; the third of the tones Dt^-F#-A#; and the fourth of the tones Ati-C#-E#:
Example
We tones scale.
15-1
may, therefore, project the major third beyond the six by continuing the process by which we formed the six-tone Beginning on C we form the augmented triad C-E-G#;
132
:
FURTHER PROJECTION OF THE MAJOR-THIRD
add the foreign tone, Gt|, and superimpose the augmented triad G-B-DJj:; add the fifth above the foreign tone G, that is, Dt], and superimpose the augmented triad D-F#-AJ|:; and, finally, add the fifth above the foreign tone D, or At], and superimpose the augmented triad A-Cj-E^f. Rearranged melodically, we find the following projections
Seven tone: C-E-G# p^m^n^s^dH, with
+
CsDiDJiEaGiG^gB, involution CaDJiEgGiGJiAaB:
its
Example Major Third Heptad
^
Eight
G#3B,
tone-.
15-2 and
p'''m®n'*s^d'*t
+
C-E-G#
+
Gt^-B-DJ
fm'nhHH^, with
ISline
tone:
p^m^ n
3^
^1^119 13 12 3
Dtj-FJ
^ s'^d ^ t^
and
+
Gti-B-D#
involution
j|j JJ ^11^^ I
113
2
+
CaDiDJiEaFJiGi
15-3
iJiJjit^^«-'r
C-E-G#
=
1
involution CgD^iEiFaGiGJiAaB:
its
Example
^^
involution
2T13I3
=
Major Third Octad
=
G-B-D# +• D^
3
I
I
2
=
Dt^-FJf-AJ,
12
I
CaDiDJiEs
F#iGiG#2A#iB, p^m^n^s^dH^:
Example Major Third Nonod
li «^
(This
is
^
M
steps,
-H
an isometric
proceed downward,
p^m^n^s^d^t'
^H
-»-
15-4
1
1
1
1
J
,
=
^2
scale, for if
we have
the
r
we
1
2
1
H
1
J
,
itJ
2
1
1
begin the scale on
same order
of
A# and
whole and half
21121121.)
133
))
THE
Ten
tone: C-E-Gif
TONAL
SIX BASIC
+
+
G\\-B-Dj^
SERIES
Dt^-Ff-AJ -f
QDiDfti
Al^
E2F#iGiG#iAiA#iB, fm^nhHH'-.
Example Major Third
Decad
p^m^n^s^d^t"*
^^ (This scale
^
j J^J Jj[J-'tfJ^1 II II 2 112 1
also isometric, for
is
15-5
we have
progress downward,
if
we
begin the scale on F# and
the same order of whole and
half-steps.
Eleven tone: C-E-G#
+
+
Gti-B-D#
CiC#iDiD lE^FSiG.GJi AiAliB, f'w}'n''s''fH'
Major Third Undecod
^
=
15-6
p'^ m'^n'^s'^d'S ^
^^^^^^
-*-
Twelve tone: C-E-Gif
Al^-Cft
:
Jf
Example
+
Dt;-F#-Afl
11
+
12
I
Gtj-B-Dft
+
1
I
I
+
Dt^F#-A#
I
I
Al^-C#-Et
C,C#,D,D#,E,E#,F#,G,G#,AiAif,B,p^WW^c/^-T:
Example Major Third Duodecac
^^ (
The
eleven-
12
p
m
15-7
I2„I2,I2 .12.6 n s d t
rff^i
and twelve-tone
r
I
I
I
I
I
III
scales are, of course, also isometric
formations.
The student
will observe that the seven-tone scale
adds the
formerly missing intervals of the major second and the tritone,
while
still
maintaining a preponderance of major thirds and a
proportionately greater
number
and minor seconds. The
scale gradually loses
istic as
of
additional tones are
of perfect fifths,
added but
its
134
thirds,
basic character-
retains the
major thirds through the ten-tone projection.
minor
preponderance
FURTHER PROJECTION OF THE MAJOR-THIRD
The following measure from La siaen, fourth
movement, page
Nativite
du Seigneur by Mes-
2, illustrates a
use of the nine-tone
major-third scale:
Example Messiaen^La Nativite
^
i
f
15-8
du Seigneur"
f
i
i^ p
f ^'f
Reproduced with the permission of Alphonse Leduc, music publisher, 175 rue Saint-Honore, right by Alphonse Lediic.
Sr m i/ii
il
8
The long melodic of the
same
line
VAscension
Copy-
iJ^J^JitJJi'^^r^^ 2
composer's
Paris.
I
I
2
I
I
2
I
(I)
from the second movement of the same is
a striking
example of the melodic use
scale:
Example
15-9
Mes3ioen,"L'A scension"
^^i^\r[^ >^-^^^
(
\iIiJ?\^-} a
\
Reproduced with the permission right by Alphonse Leduc.
of
Alphonse Leduc, music publisher, 175 rue Saint-Honore,
2
I
I
2
11
2
I
and
try to find other
Copy-
(I)
Analyze further the second movement of Messiaen's sion
Paris.
VAscen-
examples of the major-third projection. 135
16
Recapitulation of the Triad
Inasmuch as the projections
that
we have
Forms
discussed contain
of the triads possible in twelve-tone equal temperament,
and
if
if
ment
all
may
summarize them here. There are only twelve types we include both the triad and its involution as one form,
be helpful in all
it
we
to
consider inversions to be merely a different arrange-
of the
same
There are composition:
two perfect
triad.
five triads
which contam the perfect
fifth in their
(1) the basic perfect-fifth triad p^s, consisting of
and the concomitant major second; (2) the triad pns, consisting of a perfect fifth, a minor third, and a major second, with its involution; (3) the major triad pmn, consisting of a perfect fifth, major third, and minor third, with its involution, the minor triad; (4) the triad fmd, consisting of a perfect fifth, a major third, and a major seventh with its involution; and (5) the triad pc?f, in which the tritone is the characteristic interval, consisting of the perfect fifth, minor second, and tritone with its involution. Here they are with their involutions: fifths
Example 2
I.
i 1/
p s
•#-
2.
2
m 4.
pmd
136
and involution
psn
=f 7
5
2
2
and involution
r
^
J
J
16-1
r
5.
I
pdt
J
^[J
3.
pmn 4
7
ond involution
m
I
6
and
3
involution
3
4
RECAPITULATION OF THE TRIAD FORMS
The
has appeared in the perfect-fifth hexad.
p^s,
first,
The second,
pns, has appeared in the perfect-fifth, minor-second,
The
third hexads.
third,
pmn,
is
found
and major-third hexads. The
third,
encountered in the
The
hexads.
fifth,
perfect-fifth,
and minor-
in the perfect-fifth,
pind,
fourth,
minor-
has
been
minor-second, and major-third
pdt, has appeared only in the minor-third
hexad, but will be found as the characteristic triad in the projection to
be considered
There triads,
in the next chapter.
are, in addition to the perfect-fifth triad p^s, four other
each characteristic of a basic
Example
2
2
The
triad
also
found in the
ms^
is
nH, m^, and sd~:
16-2
4
4
3
3
series: ms^,
It*
I
the basic triad of the major-second scale, but perfect-fifth
is
and minor-second hexads. The
The triad m^ has been found only in the major-second and major-third hexads. The triad sd^ is the basic triad of the minor-second projection and is found in none of the other hexads which have triad nH, has occurred only in the minor-third hexad.
been examined. There remain three other
triad types:
Example 10.
mnd
and involution
triad
nsd
and
r3l2
31 The
II.
mnd
is
found
hexads.
The
is
and mst:
nsd,
16-3 involution
21
12 .mst
24
and involution
42
in the major-third, minor-third,
minor-second hexads. The triad nsd
hexad and
mnd,
is
and
a part of the minor-second
also found in the perfect-fifth and minor-third
twelfth, mst, has occurred in the major-second
minor-third hexads.
137
and
THE
TONAL
SIX BASIC
SERIES
Since these twelve triad types are the basic vocabulary of
young composer should study them them in various inversions and with various doublings, and absorb them as a part of his tonal vocabulary. If we "spell" all of these triads and their involutions above and below C, instead of relating them to any of the particular series which we have discussed, we have the triads and their involutions as shown in the next example. Notice again that the
musical expression,
the
carefully, listen to
first
five triads— basic triads of the perfect-fifth,
minor-second,
major-second, minor-third, and major-third series— are
same "shape" as the all have involutions.
metric, the involution having the
The remaining seven
triad.
triads
Example p^s
ijj ri
25 r -r
24
:
r
16-4
n^t
m'
pdt and
i
05 22
^J
V 24
and involution
I
involution
i
44A
-^X 33
and involution
r
fe^^ ^
mnd
Rl 61
A.
pmn
43
iso-
original
iJ.i >'^-MitJ«^ ?i'r:r
and involution
pmd
#
i
ifll
m^ mst
r
ms^
s^d
all
43
and involution
pns
1
7 nsd
61
^ ^^^ and
involution
'
'
1
^^
2
7
2
and involution
^
74
138
74
31
31
12
12
17
Projection of the Tritone
The student will have observed, in examining the five series which we have discussed, the strategic importance of the tritone. Three of the six-tone series have contained no tritones— the perfect-fifth,
minor-second, and major-third series— while in the
other two series, the major-second and minor-third series, the tritone
is
It will
a highly important part of the complex.
be observed,
further, that the tritone in itself
when one
is
not use-
superimposed
ful as a unit of projection,
because
upon another, the
the enharmonic octave of the
tone.
For example,
result if
we
is
is
place an augmented fourth above
first
C we
have the tone F#, and superimposing another augmented fourth
we have
above F#
BJf,
the enharmonic equivalent of C:
Example
^^ For
this
17-1
t^^^ may be said to have An example will illusscale contains, as we have
very reason, however, the tritone
twice the valency of the other intervals. trate this.
The complete chromatic
seen, twelve perfect fifths, twelve
minor seconds, twelve major
seconds, twelve minor thirds, and twelve major thirds. It contains,
however, only
to A, Ft] to At],
Bb, and
Aij:,
six tritones:
and F
C
to F#,
D^
to G, Dt] to G#,
to B, since the tritones
Bti are duplications of the
above
first six. It is
F|:,
E^
G, A^,
necessary,
139
THE
TONAL
SIX BASIC
SERIES
therefore, in judging the relative importance of the tritone in scale to multiply the
number
of tritones
by two.
we found
In the whole-tone scale, for example, thirds, six is
the
major seconds, and three
maximum number
tone sonority, and since
major thirds which can that this scale
is
exist in
major
which can
maximum
any six-tone
any
exist in
six-
of major seconds or
sonority,
we may
say
saturated with major seconds, major thirds, and
and that the three
tritones;
the
six
tritones. Since three tritones
of tritones
six is
any
tritones
have the same valency
as
the six major seconds and six major thirds. Since the tritone cannot be projected upon
itself to produce a must be formed by superimposing the tritone upon those scales or sonorities which do not themselves contain tritones. We may begin, therefore, by super-
scale, the tritone projection
imposing tritones on the tones of the perfect-fifth Starting with the tone C,
the perfect and,
finally,
tritone
G#,
fifth
we add
the tritone
series.
Fif;
we
then add
above C, or G, and superimpose the tritone C#; the fifth above G, or D, and superimpose the
we add
forming the projection C-F#-G-C#-D-G#,
which
arranged melodically produces the six-tone scale CiC^iD^Fj^i
GxG#:.
Example
17-2
^
Tritone- Perfect Fifth Hexad p'*m^s^d'*t'
i^iU
tf^""
I
I
4
I
I
This scale will be seen to consist of four perfect seconds, two major thirds,
fifths,
four minor
two major seconds, and three
p'^m^s^dH^. Multiplying the
that this scale predominates
number
of tritones
in tritones,
tritones:
by two, we
find
with the intervals of the
perfect fifth and the minor second next in importance, and with
no minor
thirds. This
is
an isometric
scale, since the
same order
of intervals reversed, 11411, produces the identical scale. If
we
superimpose the tritones above the minor-second projec-
140
PROJECTION OF THE TRITONE tion
we produce
the same scale:
C
to
Ffl:,
D^
to Gt], Dk] to
G#, or
arranged melodically, CiDbiDl:]4F#iGiGJj::
Example -
Tritone
17-3
Minor Second Hexad p^m^s^d^t^
I
The components and
I
of this perfect-fifth— tritone projection are the
characteristic triads
pdt,
I
I
CeF^iG, CJeGiGJ, FJeCiCS, and GeC^iD, CiCJsG, CJiDgGiJ:, FJiGgCJ, and
involutions
their
GiGifeD, which, though they have been encountered in the minor-third scale, are more characteristic of this projection;
Example
^6
end
pdt
Triads
I
the triads
«^6
17-4
I
6
C2D5G and
6
1
1
involutions
^% 16
*
16c .
16c
16c
,
.
FJaGfgCjj:, p^s, the characteristic triads of
the perfect-fifth projection;
Example iiiuu:> M Triads
2
the triads
CiC#iD and
17-5
p^s p :>
5
FJfiGiGJj:,
2
5
5
the characteristic triads
of the minor-second projection;
Example
17-6
Triads sd^
r
I
II 141
:
THE
TONAL
SIX BASIC
SERIES
the triads C#,G#4B#(C) and G^D^Fj^, pmd, with the involutions Ab4C7G and D4F#7C#, which have been found in the six-tone perfect-fifth,
minor-second, and major-third projections;
Example Triads
^
-?
pmd
17-7
and involutions
A.
A
A.
-t
A
-I
-I
and the tions
triads C2D4F# and Y%.Q%^%(^C), mst, with the involuD4F#2G# and Ab(G#)4eoD, which have been met in the
major-second and minor-third hexads:
Example Triads mst
J 2 j^j 4 The
and
in
involutions
^J^J^t^'r'Nlt^^t^ 2
4
4
series contains five
appeared
17-8
new forms
''^(^ 4
2
of tetrads
any of the other hexads so
^ 2
which have not
far discussed:
1. The characteristic isometric tetrads of the series, CiC^gF^iG and GjfiDgGiGJ, p^cPf, which contain the maximum number of tritones possible in a tetrad, and which also contain two perfect fifths and two minor seconds. These tetrads may also be considered to be formed of two perfect fifths at the interval of the tritone, of two tritones at the interval of the perfect fifth, of two minor seconds at the interval of the tritone, or of two tritones
at the interval of the
minor second
Example
17-9
Tetrads p^d^t^
151 2.
The 142
151
isometric tetrads
P@t
t@p
d@t
t@d
CiC^iDsG and F#iGiG#5C#, p^sdH,
:
:
PROJECTION OF THE TRITONE
which
also contain
two perfect
which contain only one tetrads
may be
and one major second. These
tritone
fifths
and two minor seconds
Example p^sd
Tetrads
r
.tf
3.
The
lie 115
c 5
, I
MZ*f
_2 p2 + .
d'
and F^iGeCjfiD, p^mdH, two minor seconds, one major
isometric tetrads CiCflieGiGJ
and one
relationships of the
17-10
t
which contain two perfect third
and two minor seconds, but
considered to be formed by the simultaneous
two perfect
projection of
fifths
:
tritone;
fifths,
and which
will
be seen to embrace two
the relationship of two perfect
fifths at
the interval
minor second, and the relationship of two minor seconds
at the interval of the perfect fifth
Example Tetrads
17-11
2_^2.
p'^md'^t
@ 4.
The
Tetrads pmsd
ijijjit^
The
tetrads
their involutions
@
p
pmsdH, with
and FJeCiCJfiD:
Example
5.
d
CiCtiDeGit and FJiGiGSgD,
tetrads
their involutions CgF^iGiGiJ:
116
d
17-12
and involutions
t
tfJ^ii^r
116
ijiiJ JitJ 6
11
^Jfg^ 6
I
I
CsD^GiGJ, and FtsG^gCifiD, p^msdt, with
CiC#5F#oG# and F^iGgCoD: 143
the
tonal
six basic
Example
oRi 2 5
involutions
o*;! 2 5
i+i*io r 5 2
I
I
The remaining
tetrad
the
is
we have aheady
m^sH^, which
17-13
and
p'^msdt
Tetrads
series
isometric
I*;5
2
I
tetrad
C2D4F#2G#,
discussed as an important part of
the major-second projection:
Example Tetrad
17-14
m^s^t^
^^P 4
2
The
2
two new pentad forms and
series contains
their involu-
the characteristic pentads CiC#iD4F#iG, p^msdH^, and
tions:
with
F}fiGiGif4CiC#,
the
involutions
C#iD4F#iGiG#
and
GiG#4CiC#iD;
Example Pentads p'msd^t^
iijj«J^ I
I
4
I
17-15
and involutions
iiJ^»^r"ri^tiJ)iJJiJ 114 14 11 1
and ClCil:lD4F#2G#,.p2mVc^2f^ and which also predominate in tritones:
Example
its
m
k^^
14 11
involution C2D4F#iGiG#,
17-16
Pentad p^m^s^d^t^and involution
ii^j.ji^i^ 'ijjt^ ^0^ r 4 2 2 4 11 I
The
characteristics of the
dominance 144
hexad
will
be seen to be a pre-
of tritones, with the perfect fifths
and minor seconds
PROJECTION OF THE TRITONE
and with the major third and the It will be noted, furthermore, that the six-tone scale contains no minor thirds. Listening to this scale as a whole, and to its component parts, of secondary importance,
major second of tertiary importance.
student will find that
the
tonally interesting material. of
it
contains
highly dissonant but
The unison theme near the beginning
the Bartok sixth quartet dramatically outlines this scale:
^W ^ ? ^^ ^ ^ ^
Bartok Sixth Quartet ,
Example
17-17
T^^afBl
i
F
^ ri
i
'
-
r
T^r-^r
Copyright 1941 by Hawkes
i
& Son (London),
tssz
Ltd.
Used by permission
of Boosey
& Hawkes,
Inc.
ti"
k i^
See also the beginning of the
fifth
movement
fourth quartet for the use of the same scale in
its
of the Bartok
five-tone form.
Play several times the triad, tetrad, pentad, and hexad material of this scale as outlined in
Example
Example
hii^^^i^ii^^ (|i
J
17-18.
17-18
ij,j'\iJ
JjtJ7^ij^i;3itJ Jl^
i'ijj J7^
i
in
ii^LiF
Jpp^cJ
iiJiiJ)tJltJ«^ijitJ«J|'LL/
145
J
THE
TONAL
SIX BASIC
j^
JJlt^^iJ^tl^ JJJ
^
^i^
l
SERIES
^fJ^I u-
^W
I isometric \ involution/
I
new tetrad forms, two new pentad forms, new hexad form. Experiment with these new
This scale adds five and, of course, one
Example
sonorities as in
17-19,
changing the spacing, position,
and doublings of the tones of each
Example
^
m
§
^ % ^ etc.
etc.
*
'> F
146
)
i
^^ l|ii
i
i"F
17-19
^
^i $
sonority.
"{^
i^ P^^ etc.
^
fct.
p
i
.j
H
^p ^^
PROJECTION OF THE TRITONE
I
^^ Now
SeH
4i I^MPt
if^
I
i
3—i*-|%
f
write a short sketch based on the material of the perfect-
fifth— tritone hexad.
Example 17-20
indicates the modulatory possibilities of the
perfect-fifth— tritone hexad. Write a short sketch employing
one of the
Example ^-S^>
.. iU.
o
Modulation
1^ ^^o @
d
*^
|o o^
@t
«
iM«=?=
*
jo
j
ftr ^
1
'/'
17-20
OflO ^^^ f^^
^^ 114
i
any
modulations, up or down.
five possible
»3
f^
@
P
'
^^ ^
3^^
|^
» '
^"^t
"^''
@_n
@_m !... [,,
b»
•
o i*
b>
^^" b>b---"*^
^^
147
18
Projection of the Perfect-Fifth-
Beyond Six Tones
Tritone Series
Beginning with the six-tone perfect-fifth— tritone scale CiC#iD4 Ffl:iGiG#, we may now form the remaining scales by continuing the process of superimposing tritones above the remaining tones
The order
of the perfect-fifth scale. fore,
be
C
G
to FJ,
to C#,
D
of the projection will, there-
to G#,
Example
A
to
E
D#,
to A#,
W
E#:
^^ £
331
i|i
JjiJ
JiJn
Seven tone: CiC|iD4F#iGiG#iA, fm^nhHH^, with tion
to
18-1
ayp
O go
B
its
involu-
CiC#xDiD#4GiG#iA:
Example Perfect Fifth - Tritone
Heptad
18-2
p^n
3
d^t
and
1
^h^
iU 14bt-^11td
^'°
involuti on
J.J J^ ^i^ ia w 11)411 ,
•
J
J
Eight tone: CiC#iDiD#3F#iGiG#iA (isometric), fm^n^s^dH^:
Example Octad
18-3
p^m'^n'^s'^ d^l^*
*^^
•ffc*
Ties
148
r
IT"
I I
113 I
I
^ I
I
—
FURTHER PROJECTION OF THE TRITONE Nine tone: C.CiJD.DJl^.E^FJl^.G.Gj^.A, p'm^nhHH\ with
its
involution CiC#iDiD#2FtiiFitiGiG#iA:
Example .,
m6 n6
7
.
Nonad
p
6 .7.4 s
d
and involution
t
tlH '-
4-
Ten
18-4
CiC#iDiD#iE2F#iGiG#iAiA#
tone:
(isometric),
p^m^n^s^dH^'.
Example p^m^n^s^d f t^
Decad
.
i
18-5
I
.
iS^Js
iU r
Eleven
tone:
g^fe
,,. '°
12
I
1
ii^
(isometric),
1
1
hn
1
jjjj Jtt^
Duodecod
p
^ m
m
n.
s
d
fl'°'
s
(ti
^ti''
"^^^^
i
18-7
t
*°
j^
jjiJJitJ^-'^it^^i'^r I
I
I
I
I
I
I
i
I
The melodic of the
r
pi2^i2^i25i2
Twelve tone: CiC#iDiD#iEiE#iF#iGiG#iAiA#iB,
Example
I
18-6
^^
^^-
»9
^ I
1
CiC#iDiD#iE2F#iGiG#iAiA#iB
Example
P
-^
JmJ J|J I
I
line in the violins in measures 60 to 62 of the first Schonberg Five Orchestral Pieces, is an excellent example
of the eight-tone perfect-fifth— tritone projection:
149
1
the
tonal
six basic
Example
series
18-8
SchOnberg, Five Orchestral Pieces, No.
113 By permission
of C. F. Peters Corporation,
Measures 3 and 4 of the Stravinsky Concertino r
r
1
I
'
music publishers.
for string
quartet are a striking example of the seven-tone perfect-fifth— tritone projection in involution:
Example Stravinsky, Concertino sfz
p
^rt -
^J-i
^
»F Afp
lA "^
,^
^F^
18-9
mIz
m ^i
^
f ^P
:ot
A3. |
'\
I
*g
>
t
pizz.
1<
is
By permission
18-10
B^""-"^^ r^^\c#
<^^
Xg»
/
150
I
I
of the publishers.
a graphic representation of the
Example
M^^~^~~~
k"y
I
«^l
perfect-fifth— tritone projection.
f
4
'
Copyright 1923, 1951, 1953 by Wilhelm Hansen, Copenhagen.
The following diagram
I
V
G^^~^__ _^.^E«
\
D«
yA«
19
The pmn-Tritone Projection
There are nine triads which contain no already described by the symbols p^s, sd^, pmd, mnd, and nsd.
Example P^s
i
i
*^l?f^l
5
2
and
pns
i It
i
^
involution
i ^72 w27 nsd
pmn
^2
2
pmd
pmn,
ms^, m^,
pns,
19-1
ms
sd'
tritones, the triads
^4
tt^
I
and involution
4
3
mnd
4 7 47
7 4 74
J
J
J
4
and involution
^ 3
and
3'1 3^
i
I
4
involution
m
\V
and involution
bJ
fc'J
^
i2
would seem,
duce a
^ I
therefore, logical to
assume that we might pro-
six- tone
tritone projection using each of these triads.
we
use each of the above triads as a basis for the
However,
if
projection of the tritone,
we
find that only
one new scale
is
produced. The projection of tritones upon the triads p^s and sd^,
as
we have
CiCjj^iD^F^iGiGjf^.
already
The
seen,
produces
the
same
projection of tritones on the triad
151
scale,
pmd
J
THE also
SIX BASIC
produces the same
TONAL
C-G-B
scale,
SERIES
+
F#-CJj;-EJj:
=
B^CiCjl^J^Jl^i
F#iG:
Example pmd
The
19-2
+ tritones
and m^ proF#-G#-A# = C2D2E2Ffl:2 F#-A#-C>^-(D) = C2D2E2F#2G#2A#:
projection of tritones above the triads ms^
duces the major-second
+
G#2A#; and C-E-G#
scale,
Example ms^
+
fl»
ItU
m^
tritones
'
+
C-D-E
19-3
+ tritones
fe
fo'
^
^
o o ^" 2
The
tl°
fl'
2
projection of tritones above the major triad, however,
new
(Example 19-4a). The projection and nsd produces the involution of the same scale, that is, two minor triads, C-E^-G and F-jf-A-C^j:, at the interval of the tritone ( Example 19-4??, c ) The projection of the tritone above the triad mnd also produces the involution of the first scale: two minor triads, A-C-E and DJj:-F#-A||:, at the interval of the tritone (Example 19-4
produces a
six-tone scale
of tritones above the triads pns
.
Example o)
pmn it»
+
tritones
'°
b)
i^
c)^nsd + tritones
i
i
3
I
J|JitJ»^ I
152
2
3
tritones
I
d)
mnd
+
(3)
3
1
2(3)
tritones
flJJiiJir'H"
"f'ita-jtU't; itf
2
J ^'ri|[^4
j^itJ^J 12
(2)
M '
'Jl^ig-tf'J
pns +
JB^r'i ii»""*-°
JjtJ
3 2
19-4
3
I
2 3
I
(2)
i}«M
1
THE pmn-TRITONE PROJECTION Beginning with the major triad C-E-G, we project a tritone above each of the tones of the triad: C to F#; E to A#, and G to C#, producing the six-tone scale CiCJaE.FifiGaAJ. This scale has two perfect
major
two major thirds, four minor thirds, two two minor seconds, and three tritones:
fifths,
seconds,
p^m^n^sWf.
contains a large
remaining
predominates, therefore, in tritones, but also
It
number
of
minor thirds and only two each of the
intervals. Its sound,
is,
somewhat similar which predominates
therefore,
that of the six-tone minor-third scale
minor thirds but
two of the possible three this scale are the two major
also has
The components of and F#4A#3C#, pmn; the diminished
to
in
tritones.
triads
C4E3G
triads CJaEgG, E3G3Bb(A#), G3Bb(AiJ:)3Db(C#), and A#3C#3E, nH; the triads (A#)Bb2C,G
and EsF^^Cj, pns; the
CiC#3E and F#iG3A#, mnd; the EsFJ^G and A#oCiC#, nsd; the triads EoFJ^AJ and Bb2(A#)G4E, mst, with the involutions FJ^AJfoC and C4E2F#; and triads CgFJiG and FJfeCiGJf, pdt, with their involutions CiC^gG and FJiGeCfl:; all of which we have already met: triads
triads
Example prnn
-
Triads
p^m^n^s^d^t^
tritone
pns
Ljj
J w^ 'r
2727
jt
Triads lUU 9 I
I
pmn
mnd
Triads
^
mst mo
and UIIU
I
I
19-5
i^j J 13
involutions IIIVUIUII\.'IIO
iiJ
J
i
Triads
n^t
Triads
nsd
tJ
J
I
13
jiJ
^
J
21
Triads IIIUUO pdt ^J\J l
Ul and lU
21
involutions IV\Jt U III^IIO II
^>rr;itJji>iriJiJUi[JJit^ri'riu^ § jttJit^^r< 24 24 4242 61 61 1^+6 i'
It contains
i^^'t
16
the isometric tetrads CjfsEsGaAfl:, nH^, CiC#5F#aG,
p^dH^ (which will be recalled as the characteristic tetrad of the previous
projection),
and
C^EzFJl^iAjf;,
m^sH^;
the
tetrads
C4E3G3Bb(A#) and F#4A#3Cif3E, pmnht; CxC#3E3G and FJ^Gs 153
THE
SIX BASIC
TONAL
SERIES
A#3C#, pmnHt; C^sEsFJiG and GsAJsCiCt, pnhdt; and EoF^i GgAfl: and AfaCiCJfgE, mn^sdt (which will be recalled as forming important parts of the six-tone minor-third scale); and the two
pmnsdt, C4E2Fij:iG and F|:4AiJ:2CiCfl;, and and CiCJgEsFJ FJfiGsAJfaC, both of which have the same analysis, but neither of which is the involution of the other. None of these tetrads is a new form, as all have been encountered in pairs
"twins,"
of
previous chapters.
Example Tetrads
n'^t^
3
^
4 2
1
4
4 3
4
3
3
3
mn^sdt
pn^sdt
133^321 13 3
33
3 3
213
321
213
pmnsdt
Tetrads
4
15
3
pmnsdt
Tetrads
imn'-st
j-Jti^^iiJif^^^iiJ^t^^'^UJ^^^^^rr
i
^r-is
3
m2s2t2
fJ^d^t^
iKi^^'^^^
19-6
2
4
1
Finally,
we
2
13
3 2
I*
1
2
find the characteristic pentads CiCJfgEoFfiG
fmnhdH\
F#iG3A#2CiC#,
and and C4E2FtiG3A# and FJf^Ait^CiCJg
and the characteristic pentads of the minor-third CiC#3E3G3A# and F#iG3A#3C}f3E, pmn'sdt^:
E, pm-nh^dt^; scale,
Example p^mn^sd^t^
Pentads
r
3
2
I
19-7 pm^n^s^dt^
13
2
1
13
3
3
4 2
13
4
2
13
iPentads pmn'''sdt^
Of these pentads, only the 154
first
two are new forms, the
third
THE pmn-TRITONE PROJECTION having already appeared as part of the minor-third projection. This projection has been a favorite of contemporary composers since early Stravinsky, particularly observable in Petrouchka. Strovinsky, Petrouchko ^ Rs.,Obs., EH.
Example
19-8
z
i tH CIS.
"t^ Bsns.
P*^
^
^l^^b
S
^^^i
Horns
Tpts.,
Comets
0t
3
i Piano, Strings
Copyright by Edition Russe de Musique. by permission of Boosey & Hawkes, Inc.
A Boris
striking earlier use
Revised version copyright 1958 by Boosey
is
& Hawkes,
Example I,
Used
found in the coronation scene from
Goudonov by Moussorgsky:
Moussorgsky, "Boris Godounov", Act
Inc.
19-9
Scene 2
155
THE
A more
SIX BASIC
TONAL
SERIES
may be found
recent example
Les Illuminations, the entire
in
movement
first
Benjamin
Britten's
which
written
of
is
in this scale:
Example
19-10
Benjamin Britten, Les Illuminations, Fanfare 1Vlns.i
^
VIOS. 3. .
'Cellos
Bosses
m ^
1¥
p
^
o " ^^botlQ°jlfc(g
13
=^
13
2
^ i^ ^
^
i^L
Pr
/
Copyright 1944 by Hawkes
& Son (London),
Ltd.
Used by permission
Play over several times Examples 19-5,
6,
& Hawkes,
of Boosey
and
7;
Inc.
then play
the entire six-tone scale until you have the sound of the scale firmly established.
Play the two characteristic pentads and their involutions, and the six-tone scale, in block harmony, experimenting with spacing, position,
and doubling
as in
Example
Example
0* ^
19-11
tt|l^^
lii
tt
etc.
^i^i H^»"^
ii^
19-11.
f
etc.
w^
4i>;J
i^^-^
tt'if
n^
Write a short sketch using the material of the six-tone pmntritone projection.
Example 19-12 scale. It will
indicates
possible
be noted that the modulation
no tones; modulation
two
the
at the
minor
third,
modulations of at the tritone
up
or
this
changes
down, changes
tones; modulation at the perfect fifth, major third, major
second, and minor second changes four of the
156
six tones.
THE pmn-TRITONE PROJECTION
Example rm
p
I >>;
n
s
g
>
13 ..
It..
T
tt
^v%
*^
2
»
13
lt°
#
'
^ :«=«
@ £ !>^
.
1
,
19-12
- »tf'
ot
^
' i
Modulotion
@
@
t^
bo " *s^ bo »
^^
il
""' -it..it"«-'
i
^^^= ^^^^3
@m
@S
5^
^^ N^
@ ,
I
'
fj
d.
*
t
Write a short sketch employing any one of the possible modulations.
Analyze the third movement of Messiaen's
VAscension for the
projection of the major triad at the interval of the tritone.
157
20
Involution of the
pmn-Tritone Projection
If,
instead of taking the major triad C-E-G,
tion, the
minor triad
|G
E^i C,
tone of the triad— G to C#,
E^
and project a to A,
C
we
take
tritone
its
involu-
below each produce
to F|;— we will
the six-tone scale J,GiF#3E[^2C#iCt]3A(2)(G)
having the same
intervallic analysis, p^m^n'^s^dH^.
This scale will be seen to be the involution of the major triadtritone scale of the previous chapter.
Example
13 pmn
Minor Triad .
,Q
r^
2
20-1
I
+ tritones
17-0...
:h."
r «r
^ ^m
t
i^r
(2)
The components
of
components of the major
this
scale
are
the
involutions
triad-tritone projection.
They
of
the
consist of
C3Eb4G and F^^A^Cj^, pmn; the diminished triads C3Eb3Gb(F#), D#3(Eb)F#3A, FJt3A3C and AaCsEb, nH; the triads C7G2A and F#7C#2D}f(Eb), pns; the triads EbsFJiG and A3CiC#, mnd; the triads CiDb2Eb and FJiGgA, nsd; the triads Eb4G2A and A4CJj:2DJj:(Eb), mst, with the involutions G2A4C# and Db(C#)2Eb4G; and the triads CiC#eG and FJiGeCif, the two minor triads
pdt, with their involutions CgFJfiG
158
and FifsGiCJ.
involution of the pmn-tritone projection
Example pmn
.Triads
Triads n
20-2 Triads pns
t
mnd
Triads
3
12
24
24
4 2
4
2
and involutions
pdt
Triads
2
mst
Triads
3
I
and involutions
It
nsd
Triads
7
2
7
61
61
16
l-ffe
contains the isometric tetrads CgEbsFjIgA, nH^, CiCifgFJiG,
D#3(Eb)F#3A4C}f and and FJsAsCiCJ, pmnHt; A3C3Eb4G, pmn^st; CsEbsFJfiG and FJiGoAsC, pnhdt; EbsFftiGsA and CiCjj:2DlJ:(Eb)3Fi|: AaCiC#2D#(Eb), mn^sdt (all of which will be seen to be p^d'f, and Eb4G2A4C#,
mVf^; the
tetrads
involutions of the tetrads in the major triad-tritone projection);
and the involutions of the two pairs of the "twins," CiCfl:2Eb4G and F#iG2A4Cif, and C#2D#(Eb)3F#iG and GaAsCiCif, pmnsdt.
Example
n^
Tetrad
Tetrad _^dftf Tetrad m^£t5
JibJi J ^IjJI t^ l
M letrods
3
3
pmn
3
.Tetrads
'
5
r'
5
3 3
1
m 1^*
2
pmn^st
^Ti^JibijitJ^^r 4
I
Tetrads
2
4
3
^1
3 4
3
3
3
4
Tetrads lerraas mn^sai mn^dt
Tetrads letrads pn sdt
''* dt
1
12
3
3
12
3
12
pmnsdt
F24 Finally,
J
t
3
3
20-3
124
we have
**23l
231
the characteristic pentads CiCJoEbsFSiG and
FitiG2A3CiC#, p^mnhdH^; and Eb3FJfiG2A4C# and A3CiC#2Eb4G,
159
THE
SIX BASIC
TONAL
SERIES
pm^nh^dt^; and the characteristic pentads of the minor- third
and AsCgEbsFJiG, pmn^sdf,
scale, EbsFifgAsCiCij;
are
involutions
the
of
pentads
of
the
all
major
of
which
triad-tritone
projection:
Example Pentads
,2 .2*2 p'^mrrsd^r
w
20-4
pm^n^s^dt^
jj^JttJJ
t,i
I
I
^rY'^ 3
12
4
sdt
„j J r 3
3
i
I
tr
Jiif''^ 3
3
3
1
Since the triad has only three tones,
it is
clear that the resultant
formed by adding tritones above the original triad cannot be projected beyond six tones. The complementary scales beyond scale
the six-tone projection will be discussed in a later chapter. Write a short exercise, without modulation, employing the involution of the pmn- tritone hexad.
160
:
21
Forms
Recapitulation of the Tetrad
We
have now encountered
all
of the tetrad forms possible in
the twelve-tone scale, twenty-nine in
all,
with their respective
The young composer should review them carefully, them in various inversions, experiment with different
involutions. listen to
types of doubling and spacing of tones, until they gradually
become a part of his tonal material. The six-tone perfect-fifth projection introduces the following tetrad types with their involutions
(where the tetrad
not
is
isometric )
Example
21-1
p^mn^s
p^ns^
3 j[Jrrr] 252 ^432 34
i^
.
J
i
3
p^mns^
223 The
1
and involution p^mnsd and
322
Jf j j 434
involution
452
254
pmns
ns^d'^
III
mn
sd
121
pm
ncr
131
Jrrr 212
and involution
122
221
six-tone minor-second projection adds five
Example
d
i
new tetrad types:
21-2
fpn^s d
1
212
mns^d
and
involutic
Tl22 161
I
I
THE pmnsd^
SIX BASIC
and
involution
TONAL [
SERIES
pmns^d
and
J ^
r The
I
3"
3
2
I
I
2
'^
12
i
new
2
tetrad types:
21-3
m 3^2. s
m2^3. m s t
'il^Jt^
1
six-tone major-second scale adds three
Example
involution!
r«2c2t2 m s t
t
Ji-^Uj J«^ii^«^* iJ 222 424 24 2.
The
six-tone minor-third scale adds eight
Example 4*2
pmn^st
n'^t
333 pn^sdt
and
pmnsdt
4
The
2
1
involution
involution
12
4
3
13
pm'^nd
162
new
tetrad;
21-5 and involution
r J 4 jit^ 4 3
tritone-perfect-fifth scale
one
:
adds
i^J 3
five
^ 4
new
r33
pmnsdt and
2„2, pm'^n'^d
Example
The
and Involution
331
and involution
six-tone major- third scale adds
#
pmn^dt
433
mn^sdt
tetrad types;
21-4
and involution
334 and
new
4
tetrads:
Involution
recapitulation of the tetrad forms
Example
V
5
r
1
p^msdt
and
iJ^t^ 2 5 we
build
5
I
6
I
I
involution
guit^it 1^ 5
The pmn-tritone If
21-6
all
2
new
projection adds no
of the tetrads on the tone
tetrads.
C and
construct their
involutions— where the tetrads are not isometric— below C,
have the sonorities
as
in
Example
arranged in the following order: fifth
21-7.
The
sonorities
we are
those in which the perfect
first,
predominates, then those in which the minor second pre-
dominates, then the major second, minor third,* major third, and
which the tritone predominates. These are followed by the tetrads which are the result of the simultaneous projection of two intervals: the perfect-fifth and major second; the major second and minor second; two perfect fifths plus the tritone; two minor seconds plus the tritone; and finally the simultaneous projection of two perfect fifths and two minor seconds. These are followed by the tetrads which consist of two those
finally,
in
similar intervals related at a foreign interval.
EXAMPLE p^
i
p^mnsd
ns''
-O-KSI
5
2
m2^t :xs
2
^ ^^ pmns
5
2 2
2 2
4
2 5
pm nsd^
44
*
I
I
I
,4*2
^tet^=c=^ 2~"2 r
^
3
3 3
I. ,
3
3
I
pmn'^st
n^t
l7o-o-,f
I
^^
^^^^^g ^^^-
331
d
^35
2
ns^d^
EC»I 2
21-7
r
I
3
,
.beg:
3
4
3
^^
3
4
* In the case of the minor-third tetrads it would be more accurate to say that they are dominated equally by the minor third and the tritone because of the latter 's double valency.
163
THE pmn^dt
TONAL
SIX BASIC
mn^sdt
pn^sdt
^ ^^
'SSl.
4'^"0°
^^
tec
3
3
3
3
I
pm^nd
I
2
I
2
3
4
4
2
33l
IXS
4
2
2
3
^^^S
4
I
2
6
I
2
2
I
xx:
4
I
13
^^
6
I
3
2
I
bo
(>
c^
2
I
,
P^
|'>^
4
2
I
3
iij^
2
2
3
t
4 ^'^^
*
p^m^nd
tf
'^^ii
2 2
251* 116
251
» ogo "
rro-
pmsd^
p^msdt
fe^eeO:?^=ec I
^^M
3
p^mns^
fe°*^ mns^d
i.
2
I
^
e
fc^
pmnsdt
4
3
3
£Vt2
^
4
SERIES
i
pm^n^d
i
fi
116R i
i
pm^nd^
w
Play the tetrads of Example 21-7 as indicated in previous chapters, listening to each carefully different positions
164
and doublings.
and experimenting with
Part
11
CONSTRUCTION OF HEXADS BY THE SUPERPOSITION OF TRIAD FORMS
22
pmn
Projection of the Triad
Having exhausted the single intervals
of projection in terms of
possibilities
we may now
turn to the formation of sonorities
—or scales— by the superposition of triad forms. For reasons which will later become apparent, we shall not project these triads
beyond
six-tone chords or scales, leaving the discussion
of the scales involving
We have
found that there are
will
nsd.
six
tones to a later section.
five triads
and which exclude the
different intervals
mnd, and
more than
Each
which
tritone
of these triads projected
produce a distinctive six-tone scale
:
consist of three
pmn,
upon
in
its
pns,
pmd,
own
tones
which the three
intervals of the original triad predominate.
Beginning with the projection of the major
C— C-E-G— and
major triad upon triad
upon
its
fifth,
triad,
we form
the
superimpose another major
producing the second major
triad,
G-B-D.
This gives the pentad C2D2E3G4B, p^m^n^s^d, which has already
appeared in Chapter
page
5,
47, as a part of the perfect-fifth
projection:
Example .Pentad
i i pmn «
*
The symbol pmn
@
the interval of the perfect
p^m^n^s^d
i
f
@
22-1
p
=
p should be
2
-'
J
J 2
3
g 4
translated as "the triad
pmn
projected at
fifth."
167
— :
:
SUPERPOSITION OF TRIAD FORMS
We
then superimpose a major triad on the major third of the
original triad, that
the
first triad,
E-G#-B, producing
is,
aheady observed
Pentad
Hi pmn @ m
on
E and
we have
)
22-2
p^m'^n^d^
fc triad
combination with
as a part of the major-third projection
Example
The
in
the pentad C4E3GiG}t:3B, p^m^n^(P (which
jj|j
J
J =
the triad on
EgGiGJfsBaD, p^m^n^sdt (which
G
^ 3
I
together form the pentad
we have
observed as a part of
the minor-third projection )
Example
22-3
Pentad p^m^n^sdt
4 The combined triad projection
@
pmn
triads
=
n
G
on C, E, and
3
form the six-tone major-
CsDoEsGiGJsB, p^m^n^s^dH:
Example pmn Hexod
22-4
p^m'^n^s^d^f
^
—a 2
2
The
3
I
chief characteristic of this scale
maximum number
is
third, the scale as a
from the
whole
perfect-fifth, major-third,
and has a preponderance third, and minor third. 168
it
contains the
of major triads. Since these triads are related
at the intervals of the perfect fifth, the
minor
that
is
major
third,
and the
a mixture of the materials
and minor-third projections
of intervals of the perfect
fifth,
major
pmU
PROJECTION OF THE TRIAD
The
new
major-triad projection adds no
triads or tetrads. It
aheady mentioned (comtwo major triads at the intervals of the perfect fifth, major third, and minor third, respectively), three new pentads: the pentad C2D2E3GiG#, p^m^ns^dt, which may be analyzed as the simultaneous projection of two perfect fifths and two contains, in addition to the pentads
binations of
major
thirds;
Example
p^m^ns^dt
Pentad
^^
22-5
j
^ «^
J2 J 2 J
3
p2
1
the pentad C2D2E4GiJ:3B, pm^n^s^dt,
^i ^2
^
which may be analyzed
as
the simultaneous projection of two major thirds and two minor thirds
above
G# (Ab); Example Pentad
I
i
pm^n^s^dt J
J 2
22-6
2
ti^
4
r
W
'f
3
and the pentad CsDgGiGJsB, p^m^n^sdH, which may be anafifths and two
lyzed as the simultaneous projection of two perfect
minor
thirds,
downward:
Example Pentad
i tj
The
involution
p^m^n^sd^t «^
•L
2
of
22-7
5
the
1
r 3
i}^
\i I
p2 +
projection
of
n2i
the
major
triad
C2D2E3GiGiJ:3B will be the same order of half-steps in reverse, that
is,
31322, producing the scale C3EbiEt]3G2A2B:
169
:
SUPERPOSITION OF TRIAD FORMS
Example pmn Hexod -
Involution
o o
—
2
3
2
22-8
i
bo
o
^i
*:^
This will seem to be the same formation as that of the previous chapter,
if
begun on the tone B and constructed downward:
Example
i we
*
upward
22-9
n
i4
downward, it becomes the projection of three minor triads: A-C-E, C-E^-G, and Etj-G-B. The scale contains six pentads, the first three of which are If
think the scale
formed of two minor major
third,
and minor
rather than
triads at the interval of the perfect fifth, third, respectively
Example 22-10
|j ^ r^^JjN pmn @p
i
2 3 4
2
=
I
The remaining pentads
r^J^jjibi^ ^JJjJ
bi
pmn @m^
4 3
=
3
1
i
£2
+
rn2l
=
3
133
22-11
|r^^j^j|4 M^^^i/'f 2
pmn @n
are:
Example
2
13
2 2 4 3
i
ym^ r^j^i j I 13 i
n^ +
2
t p2 +
5
nf
f
All of these will be seen to be involutions of the pentads
discussed in the
A
first
part of this chapter.
short but clear exposition of the mixture of
at the interval of the perfect fifth
Symphony 170
of Psalms:
may be found
two
triads
pmn
in Stravinsky's
PROJECTION OF THE TRIAD
Example
*
Imn
StravinsKy, "Symphony of
Lou
^
do
-
V^
O
p
n
I
J
@
22-12
Psalms'
Sop.
;i
m
pmn
-
'r
te
Boss
i^ do
Lou
te
Si ©'
'
Copyright by Edition Russe de Musique. by permission of Boosey & Hawkes, Inc.
The
Revised version copyright 1948 by Boosey
short trumpet fanfare from Respighi's Pines of
movement,
constitutes another very clear
tion of the triad
pmn:
Respighi "Pines of
Rome"
Example Tpts.
»
ll
iii
M
k kf f
Rome,
example of the projec-
wrijitiiiiiig
w
#|*H
By permission
t
is
found
in the
^iH
of G. Ricordi
exposition of the complete projection of the triad
involution
first
3
^ i
Used
Inc.
22-13
,
ff
An
& Hawkes,
&
Co., Inc.
pmn
in
opening of the seventh movement,
Neptune, from Gustav Hoist's
suite.
The
Planets:
Example 22-14 Gustov Hoist, "Neptune" from "The Planets" Flute
i
^^ *
Bossflute
J.
By permission of Curwen & Sons, Ltd.
171
23
pns
Projection of the Triad
To PROJECT THE TRIAD pus, wc may begin with the triad on C— C-G-A— and superimpose similar triads on G and A. We produce first the pentad C7G2A + G7D2E, or C2D2E3G2A, p^mn^s^, which we recognize as the perfect-fifth pentad: Example
23-1
Pentad p'^'mn^s^
I i@ Next
we
i
T-
pns
J
J 2
p
2
^ 2
3
superimpose upon C7G2A the triad A7E2F#, producing
the pentad C4E2F#iG2A, p^mn^s^dt;
Example Pentad
<|
jpns
23-2
p^mn^^dt
7 @ n
j JjJ 4
=
and, finally, the pentad formed
2
^
^
12
by the combination
and A7E2F#, or G2A5D2E2F#, p^mnhH:
Example Pentad
23-3
p'mn^s^
=1 pns
172
@
_3
2
5
2
2
2
2
12
of
G7D2E
— PROJECTION OF THE TRIAD pUS
Together with the original triad C-G-A, they produce the tone scale C2D2E2F#iG2A,
p'^m^n^s'^dt.
equally logical analyses.
may be
major triads
+
It
D-FJf-A; and
it
may
also
tone, that
is,
+
C-G-D-A-(E)
C-D-E-F#
—a 2
It is
pmn
pastoral quality from
its
=
C-D-E-FS-G-A:
^
* 2
a graceful scale in
C-E-G
23-4
p^m^n^s^dt
Hexad
is,
and three major seconds above the
Example pns
considered to consist of two
be formed by the simultaneous pro-
jection of three perfect fifths first
This scale has two other
major second, that
at the interval o£ the
six-
which
@
3
to write, deriving a certain fifths
and
one strong
dis-
equal combination of perfect
major seconds and having among
its
sonance of the minor second, and one
intervals
tritone.
This scale contains, in addition to the pentads already discussed, three 1.
The
more pentads, none
of
which has appeared
before.
isometric pentad C2D2E2F#3A, p^m^n^sH, formed
by
the projection of two major seconds above and two minor thirds
below C, which
we
shall consider in a later chapter:
Example
23-5
p^m^n^s^t
^m 2
2.
2
2
The pentad C2D2E2F#iG,
3
w
fjjj(»^) 32
p^m^ns^dt, which
as the simultaneous projection of
two perfect
may be fifths
analyzed
and three
major seconds: 173
—
"
:
SUPERPOSITION OF TRIAD FORMS
Example
23-6
p^m^ns^dt
J|J.^lj ^liJ 2 2 2 1
The pentad
3.
,
^P 3
2
which may be analyzed two (or three) perfect fifths above and two
C2D4FJj:iG2A, p^mn^s^dt,
as the projection of
minor thirds below
C Example
23-7
p3 mn^s^dt
f
The
?*=
+
•
n^i
involution of the projection C2D2E2FiJ:iG2A, pns, will have
the same order of half-steps in reverse, 21222, forming the scale
C2DiEb2F2G2A:
Example pns
i
Hexad
Involution
—»
o
^-o 2
2
23-8
—a—
*»
>^
:
2
2
o
»- ^
I
This scale will be seen to be the same formation as the original
pns hexad
if
begun on the tone
A
and constructed downward:
Example
23-9
2
The
2 2
scale contains six pentads, all of
12
which are involutions of first and fourth
those found in the original hexad, except, the
pentad, which are isometric.
174
The
first
pentad contains the involu-
PROJECTION OF THE TRIAD pUS
two
tion of
triads
pns at the interval of the perfect
second at the interval of the major
sixth;
fifth;
and the third
the
at the
interval of the major second:
Example J»l I
pns
^'''JJ.
@p
2 2 3 2
" 'bJ J J 4 2 12
b ^^-t
' I
i_pns
23-10
@n
^"'* bJ"''!'!
1
i
pns
l*'"'t; J
@£ 2522
I
2212
This scale contains, in addition to the pentads already discussed, three
more pentad forms,
all
of
which
will
involutions of the pentads discussed in the
be found
to
be
part of this
first
chapter: 1.
The
two minor
as the projection of
thirds above, A:
Example
^ 2.
The
which may two major seconds below, and
isometric pentad AoGoFsE^gC, p^mrn^sH,
be analyzed
pentad
2
2
2
3^
23-11
*s2
A2G2F2EbiD,
'7
L^t
p^m^ns^dt,
which
analyzed as the simultaneous projection of two perfect
may be fifths
and
three major seconds below A:
Example
2
2
2
1
*^1
3.
The pentad A2G4EbiD2C,
23-12
p" + s^4
p^mn^s^dt, which
lyzed as the projection of two perfect
fifths
may be
below
A
ana-
and two
minor thirds above A: 175
SUPERPOSITION OF TRIAD FORMS
Example
23-13
mW
M i
p2
n^t
+
The smooth, pastoral quahties of this scale are beautifully by the following excerpt from Vaughn-Williams' The
illustrated
Shepherds of the Delectable Mountains:
Example
^=?
m
^^
pns
or
b.o
pmn
|b Jijuij see
'
o
^
i
fc^
@^
Voughn -Williams "The Shepherds
of the
J
ev'-ry day
23-14
flowers
Delectable Mountoins"
J op
l
-
.^ }^i the (and
^ peer
in
zzf Copyright 1925 Oxford University Press; quoted by permission.
The
involution of this scale
is
clearly projected in the
from the Shostakovich Fifth Symphony,
first
theme
movement:
Example 23-15 Shostokovic h,
Sym phony
No. 5
m
h^
If
m
hi
i
ricaca
-T^^s^-j^r)
pmn @ 3
^^JJJj 2 2 2
12
'
b^-b^-h>^-h Copyright
MCMVL
by permission.
176
by Leeds Music Corporation, 322 West 48th
All rights reserved.
Street,
New
York 36, N.
Y.
Reprinted
24
Projection of the Triad
The projection of frnd @ p, C,G4B +
the triad C-G-B,
fmd, produces the pentad, C2D4FJfiG4B, fm^nsdH;
G^D^Fjj^, or
Example ,
pmd
24-1
Pentad p^m^nsd^t
pmd
the pentad,
pmd
@
@£
C7G4B
d,
^^^ 14
4
2
=
+
B^Fj^^Ag or CeFJfiGsAftiB,
p^m^nsdH;
Example Pentad
^
pmd
and the pentad,
pmd
p^m^nsd't
Jf @
24-2
d
m
ii^ ^"^
r =
6
3
I
r
1
@ m, G7D4F# + B7F#4A#, or G3A#iB3D4FiJ:,
p^m^n^d^;
Example Pentad
24-3
p^m^n^d^
pmd @£L
=
5
'
3
4
177
:
:
SUPERPOSITION OF TRIAD FORMS
which we have aheady observed characteristic
and
pentad of the major-third
two projections
the
the
as
Example
series.
form
together
involution
the
The
of
triad
six-tone
the
pmd scale
24-4
pmd Hexad p^m'^n^s^d^t
^^3
o
^^
—2 CT 4
:
I
In addition to the three pentads already described, the prnd projection contains three other pentads 1.
The pentad CoDiFJiGgAJ, p^m^ns^dt,
the projection of two
perfect fifths and two major thirds below D, already found in
the involution of the projection of the triad
Example
iHJflJ 2. Afl:,
24-5
p^m'ns^dt
Pentad
2
pmn:
4
Uhj Wi
13
^p^ +
m**
The pentad C2D4F#4A#iB, pm^ns^dH, which,
may be
thirds
if
begun on
analyzed as the simultaneous projection of two major
and two minor seconds above
Example
AJf
(
or
B^ )
24-6
Pentad pm-^ ns^d^t
2
3.
4
4
1
m2
The pentad CoDgGsAJfiB,
+
d^
p^rn^n^s-d^,
lyzed as the projection of two perfect
minor seconds below C: 178
which may be anaabove C and two
fifths
:
pmd
PROJECTION OF THE TRIAD
Example
24-7
p^m^n^s^d^
Pentad
s
fc
"r 3
5
" i
"i
^ji^ Wi dS
tp2 +
I
This scale has one major and two minor triads which serve as key centers
if
the scale
G
begun on
is
may
or on B. It bears
the closest affinity to the major-third scale but contains both
major seconds and a
The
involution of
which the major-third scale lacks. the projection pmd will have the same order tritone,
Since the order of the original
of half-steps in reverse.
was 24131, the order CiDb3EiF4A2B:
projection or
of the involution will
Example pmd Hexad
24-8 Involution
S 2
—
o
P
4
3
I
If we begin on B and project we produce the same scale
the original triad
Example
If The
J ^^rr^-
p
2
4
scale contains six pentads, the
the
of
perfect
pmd downward,
24-9
I
3
I
three of
first
formed by the relationship of the involution of vals
pmd
be 13142,
major
fifth,
seventh,
pmd
which are
at the inter-
and major
third,
respectively;
Example 24-10
Jf
f
pmd
@^
I
^^ Ti if i
2
4
14
pmd
J'^iVrri^J @
d
6
13
1
I
pmd
@m
I' l
3
r^ 13
179
4
:
SUPERPOSITION OF TRIAD FORMS the pentad B2A4FiE3Db, p^m^ns^dt, the projection of two perfect fifths
and two major
above A, aheady found
thirds
in
the
major-triad projection;
Example
24-11
^
3 r^^jj 4 13
*
p'
2
m'
+
the pentad B2A4F4DbiC, pm^ns^dH, which, if begun on D^, may be analyzed as the simultaneous projection of two major thirds
and two minor seconds downward;
Example
to
w?
i f=r=' 2
24-12
4
4
1
and the pentad BsAgEsD^iC, p^m^n^s^d^, which may be analyzed as the projection of two perfect fifths below B and two minor seconds above B:
Example
r^J^Ji 2
5
3
1
24-13
rrt'
'
li p2
i
+
d2
t
All of the above pentads will be observed to be involutions of the pentads in the
An
first
part of this chapter.
illustration of the use of the triad
the perfect
fifth,
at the interval of
used as harmonic background, in the Danse
Sacrale from Stravinsky's
Le Sacre du Printemps,
Example 24-14
pmd @
180
pmd
p
follows
—
—
pmd
PROJECTION OF THE TRIAD "Danse Sacrale"
Stravinsky,
fh^ •w
:
^— — fH^
f»-
a
y
•7
i r
— P— -^— '-y
Vm• • g y
p. » i
p
1.
TlHu •
.
p. »
w
p.
V
*
7
1
ft
^^#
Copyright by Associated Music Publishers,
k
Inc.,
1
—
0— •^^v ^
r
r
1 r^r— T I-
|»
f-
•
J
r r
r F
1
^
r
r New
*
York; used by permission.
All of the above pentads will be observed to be involutions of the pentads in the
An
first
part of this chapter.
illustration of the use of the triad
the perfect
fifth,
used as
Sacrale from Stravinsky's
pmd
at the interval of
harmonic background,
Le Sacre du Printemps,
in the
Danse
follows
181
25
Projection of the Triad
The projection
mnd
@
which,
if
mnd
CgD^iE, mnd, forms the pentad
of the triad
+
EgGiGS, or CgDJiEgGiGJ, fm^nH^, begun on G#, or Aj^, will be seen to be the characteristic m, CsDj^jE
pentad of the major-third
series;
Example p^m'^n^d^
Pentad
W ^
mnd
the pentad
mnd
@
IHi jJJ
J I
3
„ (S
1
rp
+
C-Dif-E
n,
25-1
D#-F#-G, or CgDJiEsFJiG,
pm^n^sdH;
Example
25-2
Pentod pm^n^sd^t
3". 3
I
mnd
the pentad
mnd
@
d,
@
3
"^
^
I
\
2
n
DJsFSiG
+
EgGiGJf, or Dj^.E^F^.G.Gl
pm^n^s^d^:
Example .Pentad
^ ^ 182
3
mnd
25-3
pm^ n^ s^ d^
U
(g
d
3
1
12 11
:
PROJECTION OF THE TRIAD mtld
Together
they
form
the
six-tone
Example
scale
CgDJiEsFJiGiGJj:,
25-4
mnd Hexad p^m^n^s^d^t
i
^ #Q
^%
:f^
I
The
remaining
pm^n^s^dt, which of
two major
pentads
may be
thirds
CgDJiEsFJaGJ,
and which has already
thirds,
pmn projection; Example
3
pentad
the
are
analyzed as the simultaneous projection
and two minor
appeared as a part of the
I
12
25-5
2
the pentad C4E2F#iGiGfl:, pm^ns^dH; which has already been
observed as a part of the
pmd
projection,
and which may be
analyzed as the combination of two major thirds and two minor seconds below
G|f;
Example
25-6
pm 3ns2d2t
ilH^ 11Ji|J|||j 4
2
4
t)JiiJ|tJ
m^
+
d^
II
I
and the new pentad CaDJfgFJiGiGfl:, p^m^rfsdH, which may be analyzed as a combination of two minor seconds above, and two minor thirds below F#
Example
25-7
-Pentad p^m^n^sd^t
3
3
11
t d'
+
n';
183
:
SUPERPOSITION OF TRIAD FORMS This hexad has a close affinity to the six-tone major-third scale
C-D#-E-G-G#-B. The presence of the
and two major the major-third hexad but
seconds destroys the homogeneity of
tritone
produces a greater variety of material. Since the projection of the triad
mnd
has the order 31211,
the involution of the projection will have the same order in reverse, 11213, or
CiCJiDsEiFgAb:
Example mnd
Hexad
i
25-8
Involution
o—^t'^— ^Fo 12
3
If
1^
1
we begin with the tone A^ and project we obtain the same results
the triad
mnd down-
ward,
Example
3
This scale has
six
,-rm
-|^H
'3-,H
I
25-9
pentads, the
first
three of which are formed
combinations of the involution of the triad of the
major
third, the
minor
third,
mnd
5
W
I
@ m
mnd
at the intervals
and the minor second:
Example 25-10
^
by
'{'{Mh.^@ 3
I
mnd
1
^^ W 3
1
3
I
2
Ji
^^^^^^ @
mnd
The
d
Others are the pentad A^gFiEsDaC, pm^nh^dt,
be analyzed 184
as the
which may
simultaneous projection of two major thirds
:
PROJECTION OF THE TRIAD
and two minor
below
thirds
A\) (or
Example 4j
mnd
G#);
25-11
Pentad pm~n~5 reniuu pm^^n^s^dt ar
2
2
i
m2
t
n2|
the pentad Ab4EoDiC#iCt|, pm^ns^dH, which as the simultaneous projection of
may be
analyzed
two minor seconds and two
major thirds above C;
Example
25-12
f and the pentad AbaFsDiCjfiC, p^m^n^sdH, which may be analyzed as being composed of two minor seconds below and two minor thirds above D
Example Pentad
25-13
p^m^n^sd^t
^ 3
A
3
I
nineteenth-century example of the involution of this scale
may be found
in the following phrase
from Wagner's King des
Nibelungen:
Example 25-14 Wagner,
Iw
o po 3
I
^^
2^
I
3^ 185
I
or
:
SUPERPOSITION OF TRIAD FORMS
Another simple but
effective
example of the involution of
this
projection from Debussy's Pelleas et Melisande follows
Example Debussy,
'
Pelleas and
Melisande"
i^i
i=i^
i
25-15
kit
s^
m
:
~gT
P.p. p\P
it^^'PvCs p
pp
^
^^h^
i
^
h
Permission for reprint granted by Durand et Cie, Paris, France, copyright owners; Elkan-Vogel Co., Inc., Philadelphia, Pa., agents.
3
186
12(1
:
26
Projection of the Triad nsd
Finally,
we come
to the last of the triad projections, the projec-
tion of the triad nsd. Beginning with the triad
C-Db-E^, we form
the three pentads: 1.
is
CiDbsEb
+
DbiD^sE^
== CiDbiDfc^iEbiEti,
mnhH\
which
the basic pentad of the minor second series
Example I
Pentad mn^s^d'^
@
nsd
2,
The pentad nsd
CiDb2Eb
+
26-1
d.
@ n,
EbiFb2Gb
=
CiDb2EbiFb2Gb, pmn'sWt:
Example Pentad
26-2
pmn^s^d^t
s
s
bJ^'^t'^ Jl,JbJ ^ ibJbJ ^12+ 2 + 122 ==^l122 12 nsd @ n I
3,
The pentad nsd
DbiDt^^E
+
@
EbxFb^Gb
I
I
s,
=
DbiDtiiEbiFb2Gb,
pmnhH': 187
:
:
:
SUPERPOSITION OF TRIAD FORMS
Example M.
Pentad pmn remaa pmn^s^d^ s a
^
V Z
+
@
nsd
The
26-3
2
I
f
;
I
I
2
2
together produce the scale CiD^iDtiiEbiFbsGb'
three
may
pm^n^s^dH, which
also
be analyzed
projection of three minor seconds
C; or as two triads
mnd
as
the simultaneous
and three major seconds above
at the interval of the
Example
major second:
26-4
nsd Hexod pm^n^s^d^t
b^ Ho bo k^ ^^
ljj^iJjtJt,J[)J Jjt^
I
\)'^^ mnd
@
_s
This scale contains three other pentads 1.
which may be analyzed as the two major seconds above D and two minor seconds
CiCJiDtjsEsFfl;, pm^ns^dH,
projection of
below D; or
as
simultaneous projection of three major
the
seconds and two minor seconds above
Example Pentad
2.
C
26-5
pm^ns'^d^t
The pentad
CJD\)-J)[\iE\)sG[),
pmn^s^dH, which
may be
analyzed as the simultaneous projection of two minor thirds and
two minor seconds above C
Example Pentad
I
188
r
26-6
pmn^s^d^t
I
3
n2
+ d2
:
:
PROJECTION OF THE TRIAD nsd 3.
The
which may two minor thirds
isometric pentad CsDiEbiFboGb, m^n~s^dH,
be analyzed
as the simultaneous projection of
and two major seconds above
C
Example
m^n^s^d^t
Pentad
112
2
26-7
n2
+
^^ s^
This hexad will be seen to have a strong affinity to the minor
second six-tone
scale.
does, however,
It
have somewhat more
variety with the addition of the tritone.
Since the projection of the triad nsd has the order 11112, the involution of the projection will have the reverse:
same order
may be
21111, or CaDiE^iEtiiFiGb. This hexad
in
ana-
lyzed as the simultaneous projection of three minor seconds and three major seconds below
G^
(FJj:),
or as
two
triads
mnd
at the
interval of the major second
Example nsd Hexad pm^n^s^d^t
26-8
Involution
we begin with the tone G^ and ward, we obtain the same result: If
Example
4
1 1
This scale has
O
2
I I
six
" '
O
2
1 I
project the triad nsd
26-9
O^ 2"'
II III I
o^ 12^ I
pentads, three of which are formed by
combinations of the involution of the triad of the
down-
minor second, minor
third,
7^sd at the interval
and major second: 189
;
superposition of triad forms
Example
12
•^
@
nsd
llll
=
'
J,J,J
It
r22 I
@
nsd
26-10
^
1212
12'*=
122 I
@
nsd
d
n
P
UIJ^ /- U J 1112 l
_3
contains also the pentad GbiFiEoDoC, pmrns^dH,
be analyzed
which may
the projection of two major seconds
as
E and two minor
below
seconds above E; or as the projection of three
major seconds and two minor seconds below G^ (F# )
Example
i
^^ ^ J
112
Jiijj Ji 2
I s2
+
26-11
J
J^^Ti|j(U)jj
d2
t
the pentad GbiFiEiEbgC, pmn^s^dH, which
J
jtj^j +
is-
may be
d2 \
analyzed as
the simultaneous projection of two minor thirds and two minor
seconds below G^;
Example 26-12
fe
13
|n2
d^i
and the isometric pentad GbsFbiEbiDoC, m^n^s^dH, which may be analyzed as two minor thirds and two major seconds below Gb(Fif):
Example
i))»JJ|,JJ 112 2
190
l
l
l
26-13
Hi I
nZ
1
^ s2*
:
PROJECTION OF THE TRIAD Usd All of these pentads are, again, involutions of the pentads dis-
cussed in the
first
The remaining
part of this chapter.
add no further possibilities. The superand 5
triads
position of the triads p~s, ms^,
major-second, and minor-second scales, already discussed.
The
superposition of the
tones duplicates
augmented
triad, nv\
upon
its
own
itself:
Example 26-14
(|
The one
ij i^i
ij«-ti
superposition of the diminished triad, nH, produces only
new
tone:
Example
26-15
^^^^^P The
projection of the triad mst merges with the five-tone
major-second scale:
Example
ij.j^^ The
J
26-16
J»^ m-ii^^r
JttS iJ 2 2 2 2
projection of the triad pdt merges with the five-tone
tone— perfect-fifth projection
Example 26-17
jij
J
|jir<'r
J^ri'
j^^jw I"
I
4
I
191
tri-
:
SUPERPOSITION OF TRIAD FORMS
An its
excellent example of the projection of the triad nsd, with
whole
characteristic combination of four half-steps plus a
step,
is
found in the
quartet where the
first
first
movement
and second
of the fourth Bartok string
violins project the scale
with
a stretto imitation at the major ninth below in the viola and cello
Example 26-18 Bortok, Fourth Quartet
^
1^ i
^ #
the U.S.A.
h^'»'^
j*
Copyright 1929 by Universal Editions; renewed 1956. Inc., for
'
^^'^^$^
i
4 h^
^'
Copyright and renewal assigned to Boosey
& Hawkes,
Used by permission.
iuj^j
i^jj|jjJi^^jjjsiJ I
I
I
I
J 2
W
J^^
I'^'^iJ I
I
I
I
2
Review the material of the projections of the triads pmn, pns, pmd, mnd, and nsd. Choose the one which seems best suited to your taste and write a short sketch based exclusively on the six tones of the scale which you select.
192
'
|
.
^
..
..
x 3,rL
SIX-TONE SCALES
JJ^X-^
FORMED
BY THE SIMULTANEOUS PROJECTION OF
TWO INTERVALS
— 27
Simultaneous Projection of the
Minor Third and We
have already seen that some
by the projection
of triads
(see
Perfect Fifth
of the six-tone scales
formed
Example 23-4) may
also
be
explained as the result of the simultaneous projection of two intervals.
difiFerent
We may now
method
explore further this
of scale structure.
We
shall
begin with the consideration of the simultaneous pro-
jection of the
minor third with each of the other basic
intervals,
since these combinations offer the greatest variety of possibilities.
Let us consider perfect If
first
the combination of the minor third and
fifth.
we
project three perfect fifths above C,
C-G-D-A. Three minor thirds above
C
we form
the tetrad
produce the tetrad
C-E^-Gb-A. Combining the two, we form the isometric hexad, CsDiEbaGbiGtiaA, fm~n^s-dH^:
Example
27-1
Hexad p^m^n'^s^d^ t^
^& _p'
bo "^^
«i:
+
11-3)
perfect
its
^1
=21312 «i:
n3
This scale, with fifths, is
bo
cr
t^^
bo
bo
tv
predominance of minor thirds and perfect hexad (see Example
closely related to the minor-third
except
for
the
relatively
greater
importance
of
fifth.
195
the
,
:
SIMULTANEOUS PROJECTION OF TWO INTERVALS It contains three
pentads, each with
Example
i 3U which are the
^^
^^
3
'^
p^rr?
i
^
J 2
characteristic pentads of the
bJ ^^ J 2 J 13
^^
4
b^i
p'
+ n'
1
^^
bJ 13
^ 3
minor third
scale;
and
27-3
and
n^sd^t
involution
involution
1
Example Pentad
own
27-2
and
Minor Third Pentad
its
involution
N 13
bJ ^^
^
li^
12
^
\i^H
V
+ n^
which we have already encountered as a part of the pmn projection (Chapter 22); and which is formed by the simultaneous projection of two perfect fifths and two minor thirds; and
Example p^mn^s^dt
Pentad
and
i^±
J
J ttJU'^H' 4
27-4
r
I
involution
^m
ttti
tp2+
n^
m 4
2
n2t
ip'
*
which v/e have met as a part of the pns projection ( Chapter 23 ) and which is formed by the projection of two perfect fifths above and two minor thirds below C,
One
interesting fact that should be pointed out here
is
that
every isometric six-tone scale formed by the simultaneous projection
of
two
intervals
has an isomeric "twin" having the
identical intervallic analysis.
imposing three perfect
we form
fifths
For example,
instead of super-
and three minor
thirds
above C,
the relationship of two minor thirds at the interval of
the perfect
fifth
we
derive the scale
or CiDbsEbsGbiGl^aBb, p'm~n'sWt^:
196
if,
C-E^-Gb
+
Gl:]-Bb-Db,
:
minor third and perfect fifth
Example p3m2n4s2d2t2
Hexod
Analyzing to G,
Eb
^^
^
n2
@
fg
we
to
and Gb
find
3
1
to contain three perfect fifths,
it
to D^i;
^^
3
2
I
t'g^
bo
1,^
p
this scale B\),
27-5
two major
thirds,
E^
to G,
C
and G^
C to E^, E^ to G^, G[\ to B^, and Bb to Db; two major seconds, D^ to E^, and B^ to C; two minor seconds, C to D^ and G\) to Gt|; and two tritones, C to G^, and to B\); four
minor
thirds,
same
D\) to Gt^; p^m^n^s^dH^, the
existed in the scale
perfect fifths
neither scale
and three minor is
interval combinations that
formed by simultaneous projection of three thirds. It will
be observed that
the involution of the other.
This scale also contains three pentads and their involutions
Example ^neiiiuu Pentad
and uiiu
p^m^n^sdt III 3UI yj
II
pmn
which were found
@
27-6 luii involution iiiYuiu 1
?mn
n
in the projection of the triad
pmn
@
ji^
as the
com-
bination of two major or two minor triads at the interval of the
minor
third;
and
Example Rented
27-7
and involution
p^mn^s^dt
pns
@
n
pns
@ 197
n_
SIMULTANEOUS PROJECTION OF TWO INTERVALS
which were found and
in the projection of the triad
pns
at the
minor
third;
Example Pentad
p^ mn^sd^t^
U^J J^iWsi
12
3
tJ
r
+
rt
\>h^^r
^
pmn
Involution
^*=^
Ji f*f
^i
1
27-8
and
13
2(1-3)
2
1'^
1
^u
pmn
\_
+
2(1-5)
t'
which was found in the pmn tritone projection (Chapter 19), as a major or minor triad with added tritones above the root and the
fifth.
An example
formed by the simultaneous
of the six-tone scale
projection of three perfect fifths and three minor thirds in the following excerpt
can, of course, also
is
found
from Stravinsky's Petrouchka, which
be analyzed
as a
dominant ninth
in
C# minor
followed by the tonic:
Example Stravinsky,
Petrouchka
Bsn.
I
27-9
»i^
ItJJJj VIn.pizz.
Copyright by Edition Russe de Musique. by pennission of Boosey & Hawkes, Inc.
^ ^
^
Revised version copyright 1958 by Boosey
& Hawkes,
Inc.
Used
"twin" sonority, formed of two minor thirds at the interval
Its
by the excerpt from Gustav where the sonority is divided into two pmn, one major and one minor, at the interval of the
of the perfect Hoist's triads
Hymn
tritone:
198
fifth,
is
of Jesus,
illustrated
minor third and perfect fifth
Example 27-10 Hoist,
Hymn
m (|
4
of Jesus
r Oi
ij
ir -
vine
a t»
r
r
Grace
is
done
^
ing
Wff ^t^tff W^ ^ m^ '
T^
^
m
By permission
,JUJ _n2
@
Jfi^y p
f
of
Galaxy Music Corporation, publishers.
l,JtJUJ^«^
12
3
1
3
199
28
Simultaneous Projection of the
Minor Third and Major Third C and two
Projecting three minor thirds above
above C,
we form
C-Et^-GJf, or CgEbiEt^oGboGftiA,
major thirds
C-E^-Gb-A
the isometric six-tone scale
+
having the analysis p^m^n's^dH\
This scale bears a close relationship to the minor-third series but
with a greater number of major thirds:
Example Hexad
p^m^n'^s^d^t^
^ ^ +
m"
This scale contains two
^
new
J 3 bJ
t'Q
12
^ 2
1
28-2
ii
tiJ 1
4
which is formed of a major below C, tm~n^; and ^
t
third
Example
m'
and a minor third above and ^^ ^ 28-3
p^mn^s^d^t
|^J(a^JjW,^ 200
3
tjo
p^m^n^d^t
Pentad
Pentad
bo
isometric pentads:
Example
i
28-1
j^n^U
i
^
:
MINOR THIRD AND MAJOR THIRD which below
formed of a minor third and a major second above and Fjl; and two pentads with their involutions, is
Example 4
Minor Third Pen tod pmn sdt J
fj!
'T
r 3
3
"r 12
*r
which are the basic pentads
which
is
may be
n 2
involution
^^ 3
I
3
and
of the minor-third series;
Pentad pm^n^s^dt
o 2
2
11.
Example
^ 4A
28-4
28-5
and involution
i_2 4m^
I 1
a part of the
.
+
A
T«
_2i
ipvfin
^J ^ * 24 2
I I
mnd
and the
t^^
I'^^nyit
tm2+n2t
projection,
analyzed as the simultaneous projection of
and which two major
and two minor thirds. If we now project two minor thirds at the interval of the major third, we form the isomeric twin having the same intervallic thirds
analysis, p^m^n^s^dH^:
Example
28-6
p^m^n^s^d^t^
to _n_2
|,o
^fg @ _m.
jo
I*"
I
^
This scale contains three pentads, each with
Example Pentad
i iP
p^m^n\dt
jbJiiJ ^'T 3
1
3
3
ki
^' @ n
involution:
28-7 and
pmn
its
involution
^jbJ^^^^V 3
3
13
hi
pmn
d
@
201
n
: i
SIMULTANEOUS PROJECTION OF TWO INTERVALS
which has already appeared in the pmn projection pmn, at the interval of the minor third; and
Example
involution
which has already appeared in the projection mnd at the interval of the minor third, and
Example Pentad
pm^n^s^dt^
^4213
as
two
triads
28-9
3124
I
which has already been found
Two
mnd
^
and involution
r
triads,
28-8
and
pm^n'sd^t
Pentad
two
as
f^ '
in the tritone-pinn projection.
quotations from Debussy's Pelleas et Melisande illustrate
the use of the two hexads.
The
first
by the
uses the scale formed
simultaneous projection of minor thirds and major thirds:
Example
V-
Mel son de"
and
Debussy, "Pelleos
28-10 .
rrrr
J-
HP
^^b^ Permission for reprint granted by Durand et Cie, Paris, France, copyright owners; Elkan-Vogel Co.,
Inc., Phila-
delphia, Pa., agents.
Jlo-
bo
<^
*' _n^
+
The second employs
m^
3
12
2
1
the hexad formed of two minor thirds at
the interval of the major third
202
"
bo
minor third and major third
Example
28-11
ibid.
ij^
*>••
p i i i 1 i i
p p
ugQ
,,j{o^«^ I
@
n'
m^
^^ 12 13
«.^BO
3
J2J0
•^•-
muuiuu
The following found
in the
interesting
example of the second hexad
second of Schonberg's Five Orchestral Pieces:
Example Schonberg
,
Five Orchestral
28-12
^2
Pieces, No.
n.,E.H.
'ueiio,^ Cello,
aa.,BsWP By permission
t;vi
tio
@
ot C. F. Peters Corporation, music publishers.
i^
JL
3
12
13
203
is
29
Simultaneous Projection of the
Minor Third and Major Second Projecting three minor thirds and three major seconds above C,
we form
the
six-tone
+
C-E^-Gb-A
scale
C-D-El^-Ffl,
or
C2DiEbiEt|2F#3A, with the analysis phn^n^s^dH^:
Example Hexad p^m^n'^s^d^t
i
bo
^°
^
'^
+
which
will
be seen
29-1
^
V5
\^
^2
tf"
33
to
be similar
.^
112^
bo
t}o
:
3
to the minor-third series,
but
with a greater number of major seconds. This scale contains two isometric pentads:
Example Pentad
p^m^n^s^t
^^ 2
2
29-2
2
3
,
\
„2 P
„2 £L
which has appeared in the projection pns (see Example 23-5), and may also be considered as the projection of a perfect fifth and a minor third above and below A; and
204
:
minor third and major second
Example
m^n^s^d^t
Pentad
iy j
29-3
which has been found
t^g.i^o
bo
JbJtiJjiJ
112
2
n}
J
in the projection
d^
nsd and
may
also
be con-
sidered as the projection of a minor third and minor second
above and below
There are
E^).
also
two pentads, each with
its
involution
Example Pentad
IVlinorThird
pmn^sdj^ ^
involution
a
^2133
12
3
3
.
29-4
which are basic pentads
of the
minor third
Example .Pentad
p^mn^s^d^
29-5 Involution
# ^^ r
I
tt»
P'
+
3'
and
series;
2
3
2
P'
-^
11
^
J|.JI| + d'
3*
g ^
d^
and may be analyzed as the simultaneous projection of two perfect fifths, two major seconds, and two minor seconds above D or below E. If we now project two minor thirds at the interval of the major second, we produce the isomeric twin C-E^-Gb + D-F-Aj^, or CsDiEbaFiGbsAb, with the same analysis, p^m^n^s^dH^:
which appears here
for the first time
Example
#
an^
"0
b i'
@
3
CF 2
29-6
bo
''*
be
^^
I
205
SIMULTANEOUS PROJECTION OF TWO INTERVALS This scale contains three pentads, each with
Example Pentad
p^mn^^dt
29-7 Involution
12
2
3
pns
which has already appeared in the pns projection pns at the interval of the minor third; and
Example Pentad
pmn^s^d^t
212
as
n-M nsd
@
ji
two
triads
29-8
12O
O 2
n n^
fn\
interval of the
Pentad pm^n^s^dt^
...^ nsd
I
I
1
@ /^
n
nsd as a combination of
in the projection
Example
minor
third;
and
29-9
Involution
ti.
i4± 13
@
Involution
which has appeared two triads nsd at the
2
involution:
its
2
2
3
12
which has appeared in the p^nn-tritone projection. The climactic section of the author's Cherubic Hymn begins with the projection of two minor thirds at the interval of the major second and gradually expands to the eight-tone minorthird scale:
Example 29-10 Hanson/TVie Cherubic Hymn"
^i ^j\ rr >-z
%
ri
^
rg bhj n^
rr
Copyright
206
r r
©
r
r
@
s
r^
1950 by Carl Fischer,
Inc.,
New York,
N. Y.
30
Simultaneous Projection of the
Minor Third and Minor Second Projecting three minor thirds and three minor seconds above C, we form the six-tone scale C-Eb-Gb-A C-Db-Dt|-Eb, or
+
CiDbiD^iEb3Gb3A, with the
analysis fm^n'^s^dH^:
Example Hexad
30-1
p^nnVS^d^t? r
I
This scale
is,
3
I
3
again, similar to the minor-third series, but with
greater emphasis on the minor second.
This scale contains three pentads, each with
Example Min^yji^d Pentad
i which
is
3
3
its
involution:
30-2
involution
12
-
2
3
I
3
the basic pentad o£ the minor third series; and
Example Pentad
p^m^n^sd^t
30-3 Involution
3 4
I
I
Td^'
* +^n2 t
207
SIMULTANEOUS PROJECTION OF TWO INTERVALS
which has occurred
in the projection
mnd and
appears here as
the projection of two minor seconds above and two minor thirds
below C;
or, in involution, as
two minor seconds below and two
minor thirds above D#; and
Example Pentad
pmn^s^d^t
30-4 Involution
y jj^juu^ ij^jjj^^^rTrriTrT'' \>m tip
1113
d2
which has occurred
+•
111
3
n2
\\'F
C or below E^. two minor thirds
i?|»
I
^
^P . J n2 ^
4d2
may be
in the projection nsd. This
as the simultaneous projection o£
•
analyzed
two minor seconds and two
minor thirds above If
we
second,
project
we produce
the isomeric twin C-E^-Gb
CiCJfsEbiEtisGbiGtl, with the
i
n}
@
same
analysis,
Example
30-5
^ j^
t^^
i^a
g^ b^^i-
at the interval of the
I
This scale contains three pentads, each with
Pentad ^ reniau
pm^n^sd^t pm su ri
^»
t^"
2
I
Example
CJ-Eti-Gti, or
fm^n^sHH^:
tio
±
+
minor
its
involution:
30-6 Involution mvumiiuii
i
3
mnd
1
@
3 n
1
12 13 ,
r,
i
>.
.3 r^
mnd
@
13 n
which has appeared in the projection mnd as a combination two triads mnd at the interval of the minor third; and 208
of
minor third and minor second
Example Pentad pmn ^s^d^t
Wyi^'^' which has appeared two triads nsd at the
W @
nsd
Involution
interval of the
nsd
minor
@
1
J2
combination of
as a third;
and
30-8
Involution
t^
pmn
which has aheady occurred
A
2
I
nsd
n^
in the projection
p^mn^sd^
M^ W
^^H^'ȴ:^^^f
Example Pentad
30-7
@
j;^
in the pmn-tritone projection.
review of Chapters 27 to 30, which have presented the
simultaneous projection of the minor third with the intervals of the perfect
fifth,
major
third,
show
that
respectively, will
major second, and minor second all
of the hexads so
naturally into the minor-third series, since
all
of
formed
preponderance of minor thirds with their concomitant
The
fall
them contain a tritones.
short recitative from Debussy's Pelleas et Melisande ade-
quately illustrates the hexad formed by the simultaneous projection of
minor thirds and minor seconds:
Example
30-9
Debussy, Pelleas and Melisande
j)i
#
^'
^'
'/
g'j^^jT I'/pipp^^'
^
n-
Permission for reprint granted by
"^ bo Durand
^
J^
)iM)i\^
^
i
et Cie, Paris, France, copyright owners;
Elkan-Vogel Co.,
Inc., Phila-
delphia, Pa., agents.
The quotation from example
Stravinsky's Petrouchka
of the projection of
is
an excellent
two minor thirds of the interval of
the minor second:
209
simultaneous projection of two intervals
Example
30-10
Stravinsky, Petrouchko
Harp Copyright by Edition Russe de Musique. by permission of Boosey & Havifkes, Inc.
Review the projections
Revised version copyright 1958 by Boosey
& Hawkes,
Inc.
Used
of Chapters 27 to 30, inclusive. Select
the hexad which most appeals to you and write a short sketch
based exclusively on the material of the scale which you
210
select.
:
31
Simultaneous Projection of the
and Major Third
Perfect Fifth
If
we
project three perfect
above C, C-G-D-A, and two
fifths
we produce
major thirds above C,
C-E-Gfl:,
scale CsDsEsGiGifiA,
fm^nhHH: Example
1 p3
+
the six-tone isometric
31-1
«s
m^
2
2-
I
I
It
bears a close relationship to the perfect-fifth series because
it
is
the perfect-fifth pentad above
C
with the addition of the
chromatic tone G#. It contains
two isometric pentads
Example
31-2
Perfect Fifth Pentad
P^mn2s3 I
2
J
J
J
.1
2
2
3
already described as the basic perfect-fifth pentad; and
Example Pentad
*J
-0-
4 J.
31-3
p^m^n^d^
1.
3
\ 1
I
M.
I
t
~
9 2
m^
.9
aZ 1'
211
SIMULTANEOUS PROJECTION OF TWO INTERVALS
which
a
is
new
which may be analyzed as and a minor second above and
isometric pentad, and
the formation of a major third
below G#,
Wd\
contains two pentads, each with
It also
Example p^mns ^d^t W-
-ZgL
involution:
31-4
^m 115 ^m Involution
W
f-f^
11
5
2
its
-^
2
p3+ d2
»^it'
t p3 + d2
which may be analyzed as the simultaneous projection of three perfect fifths and two minor seconds, and which has not before been encountered; and
Example p^m^ns^dt
Involution
i^\h^ 2
2
3
nj.
«i
p' +
1
which we have met before
^ r
iiJ
:
r
13
m'
2
2
^ms i
p2
+ m^
by the simultaneous projection major
two perfect
of
fifths
both the
is
formed
and two
thirds.
we now
major
I
as a part of the projection of
'pmn (Chapter 22) and 'pmd (Chapter 24) and
triads
If
31-5
third,
two perfect
project
we form
fifths at
intervallic analysis as the previous scale,
involution of the
the interval of the
another isomeric twin having the same
first scale.
The
but not constituting an
scale thus
formed
is
C-G-D
+
E-B-F#, or C2D2E2F#iG4B, which also has the intervallic formation p^m^nrs^dH:
Example
i p?
212
@
m
31-6
a
^€i^
2
2
PERFECT FIFTH AND MAJOR THIRD This scale will be seen also to have a close resemblance to the perfect-fifth series, for perfect-fifth scale It
it
consists of the tones of the seven-tone
with the tone
A
omitted.
contains three pentads, each with
Example n^s^d
p-^m2
2
i
pmn
@
which has already occurred
^
p3m2nsd2
pmn
31-7
h^n'r 4
p
Example
31-8
m
Involution
pmd
4
@p
14
2
M @
p
as the relation-
pmd
as the projection of
and
two
@
p
triads
pmd
and
Example
31-9
p^m^ns^dt
Involution
^^
fe=*
2
simultaneous
striking
2
I
^2 +
in the projection of the triad
projection
major seconds.
2 2
p« + s^
1
which we have met'
of
two
perfect
fifths
^=m s3
i
pns
as the
and
three
,
example of the projection of two perfect
the interval of a major third Stravinsky
i
pmn
2
s^
ii
at the interval of the perfect fifth;
A
2
pmn projection
in the
which has already occurred
2
3
at the interval of the perfect fifth;
t
^^
2
involution:
Involution
J.J^r 2 3 4
ship of two triads
its
Symphony
is
fifths at
found in the opening of the
in C:
213
simultaneous projection of two intervals
Example Symphony
Strovinsky,
#^ m
in
31-10
C
e ^ ^ n Jf iSj,
Strgs., Hns.,
*'"^-
him
m
¥
i^
Winds
p2@nn-
^--.^v.^
rimp. Copyright 1948 by Schott
An
&
Co., Ltd.; used
by permission
of Associated
Music Publishers,
Inc.,
New
York.
perfect fifths
example of the simultaneous projection of two and two major thirds, giving the pentatonic scale
CDEG
may be found
excellent
Ab,
in Copland's
Example
A
Lincoln Portrait:
31-11
Copiond,"A Lincoln Portrait"
^m
Hns.
nnti
^
^Sr
4u iuDa,Tro., bo ,Trb.,' cellos, Cellos, Basses
r
r
Copyright 1943 by Hawkes
214
r
& Son (London),
r Ltd.
Used by permission
of
Boosey & Hawkes,
Inc.
:
32
Simultaneous Projection of the
Major Third and Minor Second Projecting major thirds and minor seconds simultaneously,
form the six-tone scale C-E-G#
+
we
C-Ci|:-D-D#, or CiCJfiDiDJi
E4G#, with the analysis p^m^n^s^dH. This scale
is
very similar to
the six-tone minor-second series with the exception of the addition of the tritone
and greater emphasis on the major
Example Hexad
^'
third:
32-1
p^m^n^s^d^ t
%.T3ft^ " "
Ss
^J-
tt
I
I
This scale contains two isometric pentads
Example Pentad
P which is formed below G#; and
32-2
p^m^n ^sd^
2^
4
I
t
of a perfect fifth
Example
,»^
™2
32-3
mn^s^d^
^'
^ I
^2
and a major third above and
Minor Second Pentad
#^
'^
I
215
SIMULTANEOUS PROJECTION OF TWO INTERVALS
which
the basic minor-second pentad. There are two additional
is
pentads, each with
its
involution:
Example pm^ns^d^t
.Pentad
1^
I
2
4
32-4
Involution
tm2 + d
2
-*
I
I
4m2
,+
d2
which has been found as a part of the projection pmd and mnd, and is analyzed as the simultaneous projection of two major thirds and two minor seconds; and
Example Pentad p^mns^d^t
32-5 Involution
aJ ltiJ|
!+
I
which consists of the simultaneous projection of two perfect fifths and three minor seconds, and which appears here for the first
time.
we project two minor seconds at the interval third, we form the isomeric twin C-C#-D + If
CiCifiDaEiFiFfl:, having the
same
Example
of the major E-F-Ffl:,
or
analysis, p^m^n^s^dH:
32-6
Hexod p^m^n^s^d'^t
^#^." @
o
"
^
=°=#^
^
r
This scale contains three pentads, each with
Example Pentad
216
pm^n^s^d^
32-7 Involution
I
I
its
I
involution:
MAJOR THIRD AND MINOR SECOND which
mnd
is
a part of the projection
at the interval of the
mnd, being formed
Example Pentad
p^m^nsd^t
(|j,,iJ/iri
triads
is
r'ff @
:J3J, J|,
Iff
which
is
||
d
at the interval of the
Example pm^ns^d^t
Pentad
I
2
2
triads
32-8
a part of the projection
pmd
two
Involution
pmd
which
of
minor second; and
iJ
i
pmd
@
lJ d[
pmd, being formed
of
two
minor second; and 32-9 Involution
+
s3
d^
2 2
11
a part of the nsd projection and
I
+
s3
may be
d2
considered as
the simultaneous projection of three major seconds and two
minor seconds. Copland's
A
Lincoln Portrait contains the following example
of the projection of
two minor seconds and two major
thirds,
producing the pentad J,At)-G-F}f-E-C:
Example
32-10
Copland, "Lincoln Portrait"
Copyright 1943 by Hawlces
& Son (London),
Ltd.
Used by permission
of Boosey
& Hawkes,
217
Inc.
SIMULTANEOUS PROJECTION OF TWO INTERVALS
An example
of
hexad
the
formed
by the
simultaneous
projection of three minor seconds and major thirds will be found
beginning of Le Tour de Passe-Passe from Stravinsky's
at the
Petrouchka:
Example
^
32-11
Stravinsky,"Petrouchkgl_^
^ ^ BasSj
^"^r-[jir
C.Bsn,
^
Bab
Revised version copyright 1958 by Boosey & Havv-kes, Inc.
Copyright by Edition Russe de Musique. by permission of Boosey & Haw'xes, Inc.
An
S m 53l^
Bsns^p-r-[Jr
Used
unusual example of the projection of two minor seconds
at the interval of the
end of the
first
major third
is
found
cadence
at the
of the Five Orchestral Pieces of Schonberg:
Example Schonberg, "Five Orchestral
32-12
Pieces'
By permission
218
in the
of C. F. Peters Corporation,
music publishers.
33
Simultaneous Projection of the
and Minor Second
Perfect Fifth
The simultaneous projection
and three C-C#-D-D#, or be analyzed as
of three perfect fifths
+
C-D-G-A
minor seconds produces the scale
CiCJiDiDJfiGaA, p^m^n^s^dH^, which may also the triad pdt at the interval of the major second:
Example p^m^n^s^d^
Hexad
33-1
t^
11142
j'+d'*"
pdt
&_
3
This does not form an isometric six-tone scale but a more
complex pattern, a scale which has has
its
own
its
isomeric "twin" which in turn has
its
involution
own
and
also
involution. This
type of formation will be discussed in detail in Chapter 39. If
we
second,
project
two perfect
we form
fifths at
the six-tone scale
the interval of the minor
C-G-D
+
D^-Ab-Eb, or
CiDbiDtiiEb4GiAb, with the analysis p^m^ns^dH^:
Example Hexad
p^mSnsSd^fS
i
b p^
This scale
is
33-2
@
o 6_
i>o
.^I
most closely related
tj I
o
P
'
I
to the projection of the tritone
discussed in Chapter 17.
219
SIMULTANEOUS PROJECTION OF TWO INTERVALS It contains three
pentads, each with
Example ^Pentad
p
4=1
msd 3t2
^
^ #dkJ 114 which
1
«l
-IT*
L W
2dt
1
involution:
33-3 Involution
f—
\r2 17
its
^i'^^ ^v W .-.'--JJ !
@
-tt'l
^ -^=^^^'— f-^P
14 11
p
\k-^
^t @
p
a part of the tritone-perfect-fifth projection and
is
analyzed as the triad fdt at the interval of the perfect
Example Pentad p^m^ nsd^t
iU
^
i,J
2
1'^
14
4
r
and
33-4
Involution
H @tp pmd
^j J 4
Q^
which has appeared previously of the perfect fifth; and
^-i
r
14
p^mns^d^t
y
^
if
2
@
pmd
pmd
as the triad
Example Pentod
may be
fifth;
p
at the interval
33-5 Involution
which may be analyzed as the simultaneous projection of two perfect fifths and three minor seconds. If we now reverse the projection and form two minor seconds at the interval of the perfect fifth,
G-G#-A,
or
CiC^iDgGiGJfiA,
Example Hexed
we form having
the scale
same
the
p'^m^ns^d^tg
220
@
"
jto I*'
I
5
I
+
analysis,
33-6
&^I^ ft dz
C-C#-D
I
PERFECT FIFTH AND MINOR SECOND This scale contains three pentads, each with
Example p^ msd^t^
Pentad
tt^l
11
5
its
involution:
33-7
Involution
pdt
®
I"
p
5
I
^
I
'
@
pdt
which
p
a part of the tritone-perfect-fifth projection, being a
is
combination of two triads pdt at the interval of the perfect fifth;
and
Example ^ Pentod
^
#
p'^m'^ nsd'^t
^ifi^ie It* 6 I
I
nmd pmd
ra
33-8 Involution
T*
*^*6II
d
'
@
pmd
d
which has occurred in the projection 'pmd as the combination of two triads fmd at the interval of the major seventh; and
Example ,3„„^2 p^mns^
Pentad
r
I
^2* d'^t
5 2
33-9
Involution
)3
2
+ d?^
5
11
i|p3
+
d2
which may be analyzed as the simultaneous projection perfect fifths and two minor seconds.
The
first
dominance
of the hexads discussed in this chapter has a pre-
all
and third have an equal and minor seconds. This means
of tritones, while the second
strength of tritones, perfect that
of three
fifths,
three scales have a close resemblance to the tritone-
perfect-fifth projection.
The following measure from
the Stravin-
sky Concertino illustrates the simultaneous projection of three
minor seconds and three perfect variant
of
ample
18-9.
the
illustration
of
fifths.
the
It will
tritone
be seen
projection
to
of
221
be a Ex-
simultaneous projection of two intervals
Example Stravinsky. Concertino
33-10
.
pizz. Copyright 1923, 1951, 1953 by Wilhelm Hansen, Copenhagen.
By permission
of the publishers.
This concludes the discussion of the simultaneous projection of
two
intervals, since the only pair
of the major second
and the major
remaining
is
the combination
third, the projection of
which
forms the major-second pentad.
Review the hexads of Chapters 31 to 33, inclusive. Select one and write a short sketch confined entirely to the material of the scale you select.
222
Part
lY
PROJECTION BY INVOLUTION AND AT FOREIGN INTERVALS
34
Projection by Involution
If
we examine
again the perfect-fifth pentad C-D-E-G-A,
C-G-D-A-E, we
shall
observe that this combination may be formed with equal by beginning with the tone D and projecting two perfect above and below the starting tone:
fifths
formed of the four superimposed
Example
i
-^
o
All such sonorities will obviously
Using
this principle,
fifths,
34-1
*>
be isometric.
we can form
a
number
of characteristic
pentads by superimposing two intervals above the also projecting the
same two
intervals
below the
Referring again to the twelve-tone circle of
we have
six
logic
fifths,
tone and
first
starting tone.
we
note that
tones clockwise from C: G-D-A-E-B-Fjj:, and six
F-B^-Eb-Ab-Db-Gb, the G^ duplicating the F}. The following visual arrangement may be
tones
counterclockwise from
C:
12
3
4
5
D
A
E
B
of aid:
G C
F#(Gb) F
G
and F form the perfect
Bb fifth
Eb Ab Db above and below C;
D
and Bb
225
INVOLUTION AND FOREIGN INTERVALS
form the major second above and below C;
A
and E^ form the
E and A\) form the major third and B and D^ form the major seventh above
major sixth above and below C;
above and below C;
and below C. Taking the combination perfect-fifth pentad: ^ ^
of 1
Example
of 1
23-5):
$p^s^,
2,
we
2 2 3 2
and 3 forms the pentad |5^n^|
G A tp^n%
F Eb or,
arranged melodically C3Eb2F2G2A, p^m^nhH:
Example p
$
i^^
^
The combination
b<
2«2 p''n
m
34-3 n
^ 3
ST
iS 2
2
2
and 4 forms the pentad
of 1
G E C
,
tp^rn^ or
F Ab C4EiF2GiAb, p''m^nh(P:
Example p
4=^ -r*^ u" \jvs
'g>
%
226
duplicate the
34-2
2 3 2 3 C
$ p2s2
The combination
and
p2m2
m
34-4 n^sd^
=^ 12 —
4
—
-r.
•- vu 1
(
Example
PROJECTION BY INVOLUTION
The combination
and 5 forms the pentad
of 1
G B C
tp'd',
,
F Db or
CiDb4F2G4B, p^mhHH^:
Example
34-5
t
* ^"
bo
j-bJ
14
p2d2
The combination
^
^ 2
r 4
and 3 forms the pentad
of 2
D A C
ts^n^
,
Bb Eb
p^mnhHH:
or CsDiEbeAiBb,
Example
34-6
p^ mn^s^d^t
iJj^T 16
bo X
The combination
2
s^n'
1
and 4 duphcates the major-second pentad
of 2
D E C
Xs^rn\
,
Bb Ab or C2D2E4Ab2Bb, m^sH^:
Example
#
^'t^e %
«2m2
\
M
i2J 2 J
1'^
4
34-7
^r 2
^
(''^
2
1
2
2
2
227
INVOLUTION AND FOREIGN INTERVALS
The combination
of 2
and 5 duplicates the minor-second pentad
D
Bti
C
ts^d%
,
Bb Db or CiDbiDtisBbiBti, mnh'd^:
Example
# =^©:
iv^rr iJW^r^r 116
=F^
t
34-8
s2d2
I
1
The combination
of 3
I
^ s I
I
and 4 forms the pentad
A E C
tn^m\
,
Eb Ab or C3EbiEl^4AbiAl^, p'^m^nHH:
Example y
Ml
I
The combination
n£m2
of 3
I
II
3
34-9 I
I
\J
14
I
1
and 5 forms the pentad
A B C
,
XnH\
Eb Db or CiDb2Eb6A2B, m^n^s^dH,
Example 26-7 minor
as the projection of
thirds, A-B-Cjj:
228
which has
+ A-Ct^-Eb:
also
been analyzed
in
two major seconds and two
projection by involution
Example m
n
s
d
34-10
1
fej^
5 12\,j^r 6
And
finally,
[^r
2
2
rV 112
the combination of 4 and 5 forms the pentad
E
B
C
tm^d^
,
Ab Db or CiDb3E4Ab3B, p^m^n^sd^:
Example
* m bo t
The only way
in
J.^ 13
m^d^
(or Gb). For example,
and add the six-tone scale
g 1^ 4 3
which an isometric
formed from the above pentads
F#
34-11
tritone
if
is
we
six-tone scale can
by the addition
take the
first
above and below C,
of the tritone
of these pentads
we produce
C2D3FiF#(Gb)iGli3Bb, p^m^nhHH-.
Example
#
n
\}Q
34-12
*
t p2s2t
i 2J
The remaining pentads with the
J 3
tt^ 1
tritone
be
mT ^
13
added become
C3Eb2FiF#iG2A, fm^n^sHH^: 229
the
INVOLUTION AND FOREIGN INTERVALS
Example
34-13
p2m2n^s3d2t2
^»^
li^J Jp2n2t
^
112
2
3
C4EiFiF#iGiAb, p^rrfnhHH:
Example 34-14 p2m3n2s3d^t
I$p2ri5J 4 J
'-^
^
h\^ I
I
I
I
CiDb4FiF#iG4B, p'mhH'f:
Example
34-15
P^m2s2d^t3
1 p^d^t
I
4
I
4
I
CaDiEbsFJsAiBb, p^m^n's^dH^:
Example 34-16 p2m^n^s2d2 9 u |/
Jl
S C2D2E2F#2Ab2Bb,
111
,2n2T ?
I I
I
f
t2 I
3
I
m«s«^3.
Example 34-17
}
230
s2m;1f 2
TCT "^
€»^
2
^' 2^
2
2
)
PROJECTION BY INVOLUTION
CiDbiD^4F#4BbiB^,
fm^nhHH;
sHH
(duplicating 34-14)
n^mH
(duplicating 34-16)
t
CgEbiEoFJ.AbiAl^, fm^n's-dH^- %
CiDbsEbsFSsA.B, p^^Vs^cZ^^^; t nHH (duplicating 34-13) CiDbsEsFSsAbsB, fm^nhHH; mHH ( duplicating 34-12 Since
all
of the six-tone scales
produced by the addition
of
the tritone have already been discussed in previous chapters,
we need not
analyze them further.
231
35
Major-Second Hexads with Foreign Tone
Examining the seven-tone major-second Bb,
we
find that
it
contains the whole-tone scale C-D-E-F#-Gfl:-
A#: and three other six-tone
scales,
Example m
p
* 1.
n
s
d
o 2
CaDsEsFifiCsBb
©- ff" 2
2
with
the
MS U ~ p2m4n2s4dt2 III
\)
2
which may
I
2
each with
its
involution:
35-1
t
Example M
scale C-D-E-Fjf-G-Ab-
2
bo
;cH
"
t
11 involution
EgGiAbaBbaCaD,
35-2 Involution IIIVUIUIIUII
12
3
2
2
be considered to be formed of four major
also
seconds above, and two minor thirds below
B\) or, in involution,
four major seconds below and two minor thirds above E;
Example
35-3
m ^m^^m ^g i i=F
2
232
2
2
2
1
ts''
HI*
is.'*
+
n^ t
MAJOR-SECOND HEXADS WITH FOREIGN TONE 2.
with
CoD.EoFJfiGiAb
the
involution
Example p'^m^ns^d^t'
'2
2
F^iGiAbsBbsCsD,
35-4
Involution
12
2
2
2
which may also be considered as the projection of four major seconds and two perfect fifths above C, or below D;
Example
+
3.
C4E2F#iGiAb2Bb
p2
with
35-5
the
Example
+
s*
I
involution
p2 I
E2F#iGiAb2Bb4D,
35-6 nvolution
*^
*-4
2
I
12
112
2
4
which may also be considered as the projection of four major seconds and two minor thirds above E, or below B^:
Example
#
12
2
s"
35-7
+
n2
\s'
The theory of involution provides an even simpler analysis. Example 35-2 becomes the projection of two major thirds and two major seconds above and below D, and one perfect fifth below D; and the involution becomes two major thirds and two major 233
INVOLUTION AND FOREIGN INTERVALS seconds above and below C, and one perfect that
is
|mVp|.
X'^^s^pi or
Xm^s^n^
Example
:|)mVn|.
or
Example
^m
1 l* ^m 1 £
^
5
*J!?
fifth
C—
above
Example 35-4 becomes 35-6 becomes t:mV
Similarly,
11^
35-8
*i!?
i
il^
^^
i
1*
*2!
1
£
impure major-second scales will be seen to have the characteristic predominance of the major second, major All of these
third,
and
tritone.
impure major-second scale of Example 35-6, where one might not expect to find it, will be seen in the following excerpt from Stravinsky's Symphony of Psalms:
A
striking use of the
Example StrovinsKy,"
Symphony
ot
Copyright by Edition Russe de Musique. by permission of Boosey & Havifkes, Inc.
i An
4
35-9
Psalms"
^ 11
Revised version copyright 1948 by Boosey
ki
Inc.
Used
b«
2
earlier use of the scale illustrated in
234
& Hawkes,
Example 35-2
will
be
.
MAJOR-SECOND HEXADS WITH FOREIGN TONE found
in the excerpt
from Scriabine's Prometheus:
Example
35-10
Scriobine, "Prometheus"
7'
m
if
Hns
m
'
4«^ '^
2
2
i^
^^
2
fep
^
it'
ttf-
A
more
m
familiar example
is
found
same composer's Le Poeme de TExtase
Example Scriobine, "Le
beginning of the
at the
:
35-11
^m
Poeme de TExtose" 5
i
Q
\i^y b«
2
2
o 2
at i
13
i Write a short sketch using the material of the hexads of Examples 35-2, 35-4, or 35-6.
235
.
:
36
Projection of Triads
at Foreign Intervals
we
In Part II vals
fmn
discussed the projection of triads upon the inter-
which were a part
@
p,
fmn
@
ra,
own
of their
pmn
@
n,
composition, for example,
each of which forms a pentad,
and the three together forming the six-tone pmn projection. It is obvious that we may form a six-tone scale directly from a triad by projecting it at a foreign interval, that is, at an interval which is
pmn
not in the original triad. For example,
at the interval of
the major second produces the six-tone scale which
already
discussed
Chapter
in
C-E-G
23,
+
we have
D-F#-A
=
C2D2E2F#iG2A, which has been analyzed both as the projection of the triad pns and as the simultaneous projection of three perfect fifths
and three major seconds
Example
i pmn
We
have noticed,
of
two
triads
iJ
tj @ ^
2
also, that
projection of the triad nsd
36-1
J
2
12
the six-tone scale formed
may be
The
triad
the scale
236
by the
analyzed as the relationship
mnd at the major second
(
see
Example 26-4 )
Certain of these projections, however, form
have not so
i
J»J 2
new hexads which
far appeared.
pmd
C-G-B
at the interval of the
+
major second produces
D-A-C#, or CiCJiDsGsAsB, p^m^nh^dH, with
PROJECTION OF TRIADS AT FOREIGN INTERVALS its
involution CsDsEgAiBbiBtl:
Example p3m2n2s^d3t
I pmd i
itiJ r 5
"J @ s
Involution
^r 2
I
36-2
^
J
J
J 2
2
2
5
^^r^r 11
j
^i
pmd
@s
The same triad pmd at the interval of the minor third forms scale C-G-B + Eb-Bb-D, or C2DiEb4G3BbiB^, with its
the
fm^nhH^:
involution CiC#3E4G#iA2B,
^ pmd
The
@
Example p^m^n^s^d'
36-3
Involution
m
i ju ^^r 14 3 2
n_
r
1
triad ins^ at the interval of the
isometric
six- tone scale,
p^m^n^s'^d^,
may be
C-D-E
which predominates
+ in
12
4
3
Eb-F-G, or CsDiEbiEtjiFaG,
major seconds, but which also
F (F-C-G-D
Example
in
minor third forms the new
analyzed as a projection of three perfect
three minor seconds below
@
pmd
+
fifths
above, and
F-E-Eb-D):
36-4
p3nn2n3s4(j3
@
s^
n
2
I
I
d'i
tp^
2
I
The triad mst at the interval of the perfect fifth forms the C2D4F# + G2A4G#, or CiC#iD4F#iG2A with its involution
scale
C2DiEb4GiAbiAI::], p'^m^n^s^dH^,
which
is
most closely related
the tritone-perfect-fifth series:
Example p4m2n2 s2d3t2
mst
@
p
I
2
36-5 Involution
2
I
4
I
I
mst
@ 237
p
to
INVOLUTION AND FOREIGN INTERVALS
The same
triad, mst, at the interval of the
+
minor second forms
=
CiDbiD^iEbsFJiG, with its involution CiDbsEiFiFJiG, p^m^n^s^dH^, which also resembles the scale C^D^Fif
Db2Eb4G
the tritone-perfect-fifth projection:
Example p3m2n2s2d4^2
@
mst
There
Involution
d
@
mst
d^
are, finally, eight projections of triads at foreign tones,
which the what similar in
They
36-6
scales
and
their involutions follow a pattern some-
to the projections discussed in Chapters
27 to
33.
should, for the sake of completeness be mentioned here,
but will be discussed in detail in a later chapter. They are:
The
projection of the triad
pvm
at the interval of the
second, which forms the scale CiDbsEiFaGiAb, with
its
minor
involu-
CiDb2EbiFb3GiAb, p^m'^n^sdH; the triad pns at the major third, CiCJfgEgGaAaB, with its involution CaDoEaGsBbiBt^, tion
p^m^n^s^dH; the triad pns at the minor second, CiDbeGiAbiAtji Bb, with
its
involution CiDbiDtjiEbeAiBb, p^m^n^s^dH; the triad
pdt at the major second, C2D4F}t:iGiGJ|:iA, with
CiDbiDt|iEb4G2A, p^m^rrs^dH^, which
may
involution
its
also
be analyzed
as the simultaneous projection of three perfect fifths
minor seconds; the triad pdt
third,
C4E2F#iG3A#iB,
involution CiDb3EiF2G4B, p^m^n^s^dH^; the triad nsd at
with
its
the
perfect
fifth,
C2DiEb4G2AiBb, third,
major
at the
and three
CiDb2Eb4GiAb2Bb,
p^m^n^s^dH;
CiDb2EbiE^iF2G, with
p^m^n^s^dH; and the triad
A#iB, with
238
its
the its
mnd
triad
with
nsd
its
at
involution
the
major
involution C2DiEbiEti2F#iG,
at the perfect fifth,
involution CiDbsEgGjAbsB, p^m^n^sdH.
CgD^iEaGg
projection of triads at foreign intervals
Example p^m'*n^sd'^t
^^^
it J
J
J 3
I
2
I
Involution
I
6
I
2233
2 4
I
I
@ m
pns
@
I
I
I
r
42
I
p3m3n2s2(j3t2
Involution
p^m^n'^s^d^t
Involution
)3+d5
@ m
pdt
@
pdt
12
4
12
2
14
@
S
i mnd
@2
Of the four
_m^
may
trated in
I
I
I
^313 S
2
2
p^m^n^sd^t
3
new hexads
be explained
Example
@£
Involution
I
2
I
I
@
nsd
thirteen also
2
n_
2 nsd
p^m^n^s^d^t
nsd
d.
Involution
ilp3+<^|
I
pns
Involution
p3rTi2n2s3d3j2
_pdt@£
d^
Involution
p^na^n^s^d^t
I
@
1
P3322
@m_
M
^''^^JJUtJ -l"^ hl 12 13 pmn
^3m3n3c:3r|2 p-^m-^n'^S'^d'^t
pns
36-7
Tji
Involution
r
UJ
J
J 3
bJ
r
^''^r
3
13
^ iJ
by
'J^
innd@p
discussed in this chapter,
as projection
I
all
but
involution, as illus-
36-8.
239
involution and foreign intervals
Example pmd
@n
i h jt» t
»2 m'
^2 d;
I
„U n't
T: ^*> »2
tp'
pdT dt@m
d2
~.2 m'
^^ I
^1 d"
t
^
bl-g
m'i
tn2
pns@d
«" ^"
^p'=
H
iH
I
*
i,„ s}
^_
t
n
^
T
p^l
tji2
d2
I
tt^»
E* *£
mU
H
ra jm^
The four new hexads which cannot be arranged manner are: pmd @ s, s^ @ n, mst @ p, and mst @ d.
240
g
^
1
nnnd@_g_
it^ I
pdt@s
t^^e i^» ^^^
nsd@m
nsd@p
^ t£2
pn8@ m
pmn@ d ^"^
36-8
j^:2
Ti 2'
*
in similar
*
37
Forms
Recapitulation of Pentad
We
now encountered
have
the pentad forms which are
all
found in the twelve-tone equally tempered
It
them and
carefully, play
listen to
them
in all of their
and experiment with them, both melodically and
inversions
harmonically. All of the pentads are projected above
comparison and, where the pentad is
projected
downward from
Pentads numbered
number 6 seconds.
wise,
is
summarize them here. The student should review
therefore, to
them
scale.
1 to
is
for
C.
5 predominate in perfect
contains an equal
number
Pentads numbered 7 to
fifths,
while
and major
of perfect fifths
11
predominate in minor
number 12 containing an equal number
seconds, with
C
not isometric, the involution
of
minor
seconds and major seconds. Pentad number 13 has major thirds,
major seconds, and tritones in equal strength. Pentads numbered 14 to 17 predominate in major seconds. Pentad number 18 pre-
dominates in minor thirds and
predominate major
in
thirds.
minor
The
thirds.
tritone,
dominates pentads 30 to
34 to
33,
numbers 19
tritones;
22
to
Pentads 23 to 29 predominate in considering
its
double
valency,
and the remaining pentads, numbers
38, are neutral in character.
Example p^mn^s^
i=E ^— 2
2
2.
37-1
p^m^^s^d
Involution
=§ 3
2
2
2
3
4
mn pmn
@ (S)
n p
2.
2.
3
4 j
pmn
@
241
p
INVOLUTION AND FOREIGN INTERVALS
p^m^nsd^t
3
Involution
i ^2414 14 4
2
/^^
Involution
4
? ^i +
t p2
2
I
4
^^W b^ pmd
j
5^
^!^ 2
2
nX, @ ra ^n
p^ mn^s^ dt
.
14 2414
^
r'Tt'J
i
2
n^'^*.
pf
+ _n2
pns
@
i
p3mns2(j2t
2
2
13
6
I
2
s_
III IISU P^rn^nsd^ p
8 g
r'
I
@
pns
mn^s^d" 5 U* M 7 J mil
1^
10.
2^
tI
^mnd
pmd
1
@
F
I
2
I
6
d
13
lVt2
d2*^
nsd 14
^rr^r^jjJ
+ n2
2
242
2
2
pmd
@ d^
mnd
@ ^
@
lejtjmi^
1113
,h2
„2
^
3
nsd
@
_s^
p^m^n^s^t
ig=i.
iJJltJil^ 2
j,
j
J.
I3.J
^^
1
12 11
d^
JJi^J-^^^j
3
I
@
_s
Involution
itiJU^^ I
lilVUIUIIUII nvolution
Ll
pmn^^d^t
I**
2
Involution
I
I
2
5
I
pm^n^s^d^
I
t
Involution
2
5
p
te i
12
4
@
UbJj-l^ 3
2 2
Jr.
2 J
p2„2
:^j.i^jJ or
2 2 2
3
^^ f^Z
+
^c^
RECAPITULATION OF THE PENTAD FORMS p^m^ns^dt
15
Involution
^
jjJtJ^ ^ " 16.
2
2
pm
^ns^'d^t
2
JJJ I ^ i + p2
2
£3
1
f 2
2
I
4*^'
13^^
1
Involution
J r
2
2
+ d2^
3^
pl^plJj-^'^4j[;J
II 112
is '
2
ptlpb|
. d?
m^n^s^d^t 17 m-n-5-o-i ^17 1
12 IQ
6
2
1^
t IL^
pmn^sdt^
Involution
^ ^33! bJ
J
ijj
3
I
i
pmn @
3
n
20 p mn'^s^dt
r
2
4^
i,''j'
.:^
@
pns
I
^
I
3
2
mnd @
I
24.
nsd
@
2
1 3
3
J
J
2
I
^^
@
n
ly^ @
; pns
12
1
I
12 12
r^
13 itJ
I
1
r
P^rn^ns^dt
2
'i pmn
n
mnd
@
n
4 nsd
@
n
Involution
J
J
4 /I
4
3
n
p^m^n^d^
i
J
Involution
12 12
•-
3
Involution
pmn^s^d^t
23.
i
{
i
13
^ J
r
n
2i.pm^n3sd^t
3*
^
Involution
S
3
3
Involution
iH,^^s-'/i-
iF^f
^^^ 12
^
r
'
2
p^m^n^sdt
ig
^^
or
^
r
''^
4
A.
3
Tf.
^ 13
nvolution
1
p"^
+
m''
2
2
3
1
I
p^
+
243
m'
J INVOLUTION AND FOREIGN INTERVALS 3n2c2, 25 prtT^n^s^dt
ft /''\
12
3
1
26.
IIIVUIUIIUII nvolution
1-
m!
2
pm^ns^d^t
d^
29
2
I
m2
+
^
S
p^2
J
i^J^Jl-^'l^
^2
I
m^
P^^n2d2t
28.
\ CT
+
id''
14
3
13 bJ
.
S
J
.
i
I'-^^
n^
m^2
t
1
4
3
°
Jif.
;
It-
p2mn2sd^t2
ijJiJ 3 2
^
^^
Ig^
I).
4
e
°
lis
:rl'^^i' Jb^J 4
^^ 2
^^^^
2
r
4
13
tw.-tf »
^>
* J
d2
p_'
involution
iJi,j^^^^
ii p2
1
b^i +
n2
p^m^n^s^d^
'
^1 rV 2 13 V 1
t^
t
1
i^p2
^-
n2
Involution
+
244
^
^-i^ir'r'^^^j^^ -^ 13 2
34 p^m^n^sd^t
13
-1 1
nvolution
33.p!m!sVt2
2
i'^
1
I
J|J Jil-' ^"4213
.
T T 4
II
Involution
32 pm^n^s^dt^
i
r
4
I
^
n^
%
Involution
J I't'
"8
l.»
30 p^msd^t^
i
n'
+
m-^
i
p^m^n^sd^
i
31
2
Involution
^ 27 p^m~n^sd
4
12
3
+J1^
p2 + d2
I
4
p2+ d2t
RECAPITULATION OF THE PENTAD FORMS
36
p^m^rr^sd^t
Involution
J
tri^ rrJ
p2+32 :^fl
i
4 d2
I
p2+ s^
+ d<
p^mn^s^d^t J 2
I
6
¥^Ins'
t'*
^"it
2
I
t.a
«2
245
|^^Part,.,_Y,.
THE THEORY OF COMPLEMENTARY SONORITIES
38
The Complementary Hexad
We
come now,
logically, to the rather
complicated but highly
important theory of complementary sonorities. that
the projection of five perfect
fifths
We
have seen
above the tone
C
produces the hexad C-G-D-A-E-B.
Example
Referring to our twelve-tone
form a figure having to B.
We
note,
also,
five
38-1
circle,
we
note that these
six
tones
equal sides and the baseline from
C
that the remaining tones form a com-
plementary pattern beginning with
F and proceeding
counter-
249
THE THEORY OF COMPLEMENTARY SONORITIES clockwise
formation as
complementary hexad has the same counterpart and, of course, the same intervallic This
G^.
to
its
analysis.
Example
38-2
pSm^n^s^d
p^m^n's^d
#
f^
m
k^z bo
2
m Q
«
gy. tJ
2
''"
-
^"
I'o
==
2
3
u> i^
2
Since the hexad |F-Eb-Db-Bb-At)-Gb
the isometric involution
is
of the original, it will be clear that the formation is the same whether we proceed clockwise or counterclockwise. That is, if instead of beginning at F and proceeding counterclockwise, we
begin at
G^ and proceed
note, also, that the
clockwise, the result
is
complementary hexad on G^
the same.
We
merely the
is
transposition of the original hexad on C:
Example
38-3
A more complicated example of complementary hexads occurs where the original hexad is not isometric. If we consider, for example, the hexad composed of major triads we find an important difference. Taking the major triad C-E-G,
second major triad on
G— G-B-D,
on
E— the
triad
hexad CsD^EsGiGifsB:
Example
^ 250
^
m
38-4
i2
.
J 2
•' 1
a
we produce
and a third major
E-GJf-B. Rearranging these tones melodically,
we form
ti^
r
THE COMPLEMENTARY HEXAD
we now diagram
If
this
hexad,
in the following example, the
by
solid lines
we produce
the pattern indicated
major triad hexad being indicated
and the complementary hexad by dotted
Example
lines:
38-5
G^(Ab)
(Bbb)A
i Here
it
j
will
;^Eb
a
g^
^g
il^^ ^ g
^k
be observed that the complementary hexad
F-Bb-Eb-Db-Gb-A
(Bbb)
is
not
the
transposition
involution of the original, and that the pattern of the
be duplicated only
in reverse, that
is,
this
the
first
can
F and statement may
by beginning
proceeding counterclockwise. The validity of
but
at
be tested by rotating the pattern of the complementary hexad within the circle and attempting to find a position in which the
second form exactly duplicates the
original.
discovered that the two patterns cannot be in this
manner. They will conform only
upon C and the second pattern
is
if
It
will then
made
the point
to
F
be
conform is
placed
turned ouer— similar to the
turning over of a page. In this "mirrored" position, the two patterns will conform.
Transferring the above to musical notation,
we
observe again
251
THE THEORY OF COMPLEMENTARY SONORITIES that the
complementary hexad
|F2EboDb3BbiBbb3Gb.
involution,
the
It will
triads,
as
triads
the tones of a major triad, so the second hexad
combination of three minor
is its
be noted further that
hexad was produced by the imposition of major
first
upon
hexad CoDoEgGiGJoB
to the
is
a
the minor triad being the
involution of the major triad:
Example
38-6 p^m'^n^s^d^t
p3m''n3s2d2t
2 2 3
13
As might be expected, the sonorities
identical:
is
2
2
intervallic
three perfect
analysis
fifths,
3
of
3
1
the
two
four major thirds,
three minor thirds, two major seconds, two minor seconds, and
one
tritone, p^m'^n^s^dH.
The
and most complicated, type of complementary
third,
hexad occurs when the remaining
six
tones form neither a trans-
position nor an involution of the original hexad but an entirely
new
hexad, yet having the same intervallic analysis. For example,
the triad
C-E-G
+
hexad C-E-G hexad
minor second forms the
at the interval of the
Db-F-A^, or CiDbsEiFoGiAb.
Its
complementary
consists of the remaining tones, DiEbsFJsAiBt^iBti.
Both
hexads have the same intervallic analysis, p^m'^n^sdH but, as will
be observed similarity
in
one
Example
38-7,
the two scales bear no other
to the other.
Example
j^-j pmn
252
@
d
I
3
J
J I
2
38-7
"^ I
:
^^rp
j^JjjJ I
3
3
I
I
THE COMPLEMENTARY HEXAD Ftt
THE COMPLEMENTARY HEXAD
A III,
fourth type includes the "isomeric twins" discussed in Part
Chapters 27 to 32,
If,
for example,
we
superimpose three
perfect fifths and three minor thirds above C we produce the hexad C-G-D-A plus C-EbGbA, or C2-Di-Eb3-Gbi-Gti2-A. The remaining tones, C#3EtiiE#3G#2AJfiB, will be seen to consist of
two minor plus
thirds at the interval of the perfect
A#-C#-Et^
E#-G#-B.
Example ,
fifth,
p3m2n4s2d2t2
i
bo^°'^ pi +
38-8
p3m2n4s2(j2
t2
-ijg^^t^^ :^tJ|j«Jj^ 2
13 12
3
13
2
1
^
W n^ @
253
p
39
??
The Hexad
We
are
'Quartets
now ready
to consider the
more complex formations which are
resulting from the projection of triads at intervals
own
foreign to their
construction.
We
have already noted
in the
previous chapter that every six-tone scale has a complementary scale consisting in each case of the remaining six tones of the
twelve-tone scale.
We
have
also
noted that these complementary scales vary in
their formation. In certain cases, as in the
tone— perfect-fifth projection cited plementary scale
is
example of the
Example
six
38-3, the
com-
simply a transposition of the original
scale.
in
In other cases, as in the major-triad projection referred to in
Example
38-5, the
original
scale.
complementary scale is the involution of the However, in fifteen cases the complementary
scale has an entirely diflFerent order, although the
same
inter-
vallic analysis.
We
have already observed in Part
formation of what
we have
III,
Chapters 27 to 33, the
called the isomeric twins— seven pairs
of isometric hexads with identical intervallic analysis.
A
still
more complex formation occurs where the original hexad is not isometric, for here the original scale and the complementary "twin" will each have
its
own
involution. In other words, these
formations result in eight quartets of hexads: the original scale, the involution of the original scale, the complementary scale,
and the involution of the complementary scale. The first of these is the scale formed by two major 254
triads
pmn
:
THE HEXAD at
QUARTETS
the interval of the minor second, already referred
involution will have the order,
Its
to.
12131, or CiD^oEbiFbaGiAb,
having the same analysis and consisting of two minor triads
The complementary
the interval of the minor second.
at
scale of the
original will consist of the tones DiEbsF^aAiBbiBti, also with the
analysis p^m^n^sdH.
+
fifth.
same
Begun on
two
FJsAiBb, or
B,
may be
it
mnd
triads
analyzed as B3D1EI9
at the interval of the perfect
This scale will in turn have
its
involution, having again the
analysis:
Example p^m^n^sd^t
39-1 Complementary
Involution
Involution
He xad,
@
mnd
The
triad
six-tone
pns
C-G-A
scale
p^m^n^s^dH.
at the interval of the
or CiDbeGiAbiA^iBb, becomes involution CiDbiD^iEbeAiBbi- The
Its
complementary
+
minor second forms the
Db-Ab-B^,
scale of the original
is
DiDifiEiFiFJfgB, with
its
involution
Example p^m^n^s^d^t
*^
PM
The
©" d
triad
six-tone scale
I
6
pns
I
I
39-2 Comp. Hexed
Involution
'
I
I
!•
I
6
I
at the interval of the
C-G-A
+
r
I
I
I
5
Involution
5 II
I
I
major third forms the
E-B-C#, or CiCJgEgGaAsB, p^m^nh^dH.
255
THE THEORY OF COMPLEMENTARY SONORITIES Its
becomes CaDaEgGaBbiBti. The complementary
involution
scale
DiE|:)2FiGb2Ab2Bb, with
is
its
Example
involution:
39-3 Comp. Hexad
p'm^n^s^d^t Involution
12 12
The
+
C-F#-G
scale
p^m^n^s^dH^.
involution
Its
plementary scale
is
D-G|-A,
The
DboEbiEl^iFgBbiBt], with
39-4 Comp. Hexad
triad pdt at the interval of the
C-F#-G is
C4E2F#iG3A#iB
is
+ E-A#-B,
3
nsd
scale
p^m^nh^dH.
Its
2
its
involution:
39-5 Comp. Hexod
112
124
3
1
at the interval of the perfect fifth
C-Db-Eb
+
G-Ab-Bb,
or
is
Involution
I
3 2
I
I
forms the
CiDb2Eb4GiAb2Bb,
involution becomes C2DiEb4G2AiBb.
plementary hexad of C-Db-Eb-G-Ab-Bb
256
I
I
or C4E2FJfiG3A#iB, p^m^nh^dH^.
Involution
@ m
triad
5
major third forms the
DbiDl^iEboFgAbiAti, with
p^m^s^d^t^
six-tone
Involution
CiDt)3EiFoG4B. The complementary scale of
Example
The
involution:
I
involution
pdt
its
_s_
six-tone scale Its
CoD4F#iGiG#iA,
or
p^m^n^s^d^t^ Involution
@
1
becomes CiC#iDl:iiEb4G2A. The com-
Example
pdt
12
2 2
pdt at the interval of the major second forms the
triad
six-tone
2
Involution
The com-
D2EaFiFjj:3A2B, with
THE HEXAD its
These hexads, with
involution.
and secondary strength
fifths
QUARTETS
in
preponderance of perfect
their
major seconds and minor
thirds,
are most closely related to the perfect-fifth series:
Example
nsd
triad
six-tone
scale
p^m^nh^dH.
Its
Comp.Hexod
Involution
p*m^n^s^d^t
The
39-6
+
Et|-F-G,
or
CiDbsEbiEt^iFoG,
involution becomes C2DiEbiEtl2FJt:iG.
plementary hexad of C-Db-Eb-Et^-F-G its
major third forms the
at the interval of the
C-Db-Eb
is
involution. This quartet of hexads
with an equal strength of major
thirds,
Involution
The com-
D4FiJ:2G#iAiAifiB, with
is
neutral in character,
minor
thirds,
major
sec-
onds, and minor seconds:
Example p
@
nsd
m
n s d
t
39-7 Comp.Hexod
Involution
Involution
_nn
The last of these quartets of six-tone isomeric scales is somewhat of a maverick, formed from the combination of the intervals of the perfect fifth, the
major second, and the minor Second.
C and project simultaneously two two major seconds, and two minor seconds, we form the pentad C-G-D + C-D-E + C-C#-D, or GiC#iD2E3G, If
we
begin with the tone
perfect
fifths,
with
involution C3Eb2FiFJ|:iG, p^^mnrs^dH:
its
Example
39-8
Pentad p^mn^sTl
p2 f S2
If
we now form
+
t
Involution
d^
a six-tone scale
by adding
first
a fifth below C,
257
THE THEORY OF COMPLEMENTARY SONORITIES
we
form
the
with
CiCifiD2EiF2G,
scale
involution
its
C^DiEbsFiFJiG:
Example p^m^n^s^d^t
iff
If
we add
I
?
I
39-9
Involution
2
2
122 I
the minor second below C,
CiCfliDsEgG^B, with
1 I
I
we form
the six-tone scale
involution C4E3G2AiA#iB:
its
Example 39-10 p^m^n^s^d^t
Involution
43211
['"1234 Upon examining that they
all
have the same
also discover in
we
find
intervallic analysis, p^m^n^s^dH.
We
these four scales, Examples 9 and 10,
Example 11 that the complementary hexad of
Example 9 is the same Example 10:
scale as the involution of the scale in
Example Hexod
Original
39-11 Transposition above
Comp. Hexed
C
I
12 12 (If
we
4
3
11
2
4
3
2
11
take the third possibility and add a major second
below C, we form the six-tone scale CiC^iDaEsGsBb, which is an isometric scale with the analysis p^m^n^s^dH^, already discussed in Chapter 29.
It will
be noted that
both the pentad CiCjfiDsEsG and
Example p2m2n4s3d2t2
258
its
this scale contains
involution jDiCJfiCl^sBbsG.
39-12 Involution
THE HEXAD
The complementary hexads
may
all
be analyzed
of
QUARTETS
Examples 39-1
as projection
by
to 39-7, inclusive,
involution, as illustrated in
Example 39-13:
Example
^
mW^ ih
^
_
3.
^
u ° t
^g
^-
m2
p^
s'
*
°tj
{t^
fs
JB
O d^
p' t
J
s^
d^
p'
i
4
^
^^^^^^
^
tp2
39-13
m^
s'
t
$ n^
S'^
TTl'
I
7. :|,
tm^
n^
s't
Jm^
n^
s' t
tp^
s^
d' t
Jp^
s^
d'i
J
m^
d^
l
^fi^8^«
s'Um^d^s't
259
^Part^
COMPLEMENTARY SCALES
40
Expansion of the Complementary- Scale Theory We
have noted
six-tone scale consisting of all of the notes in the original scale, vallic analysis.
tone
scale
An
will
complementary
that every six-tone scale has a
and that these
scales
which are not present have the same
inter-
analysis of all of the sonorities of the twelve-
reveal
the
every
that
fact
has
sonority
a
complementary sonority composed of the remaining tones of the twelve-tone scale and that the complementary scale will always
have
same
the
type
of
intervallic
predominance of the same interval or
analysis,
that
the
is,
intervals. In other words,
every two-tone interval has a complementary ten-tone
scale,
every triad has a complementary nine-tone scale, every tetrad has
a
complementary eight-tone
complementary seven-tone
For example, the major triad scale as triads
its
make
be found
which
is
intervallic analysis has a
intervals of the perfect fifth,
major
projection of the major triad, since
third,
to
it is
allows
the
have a nine-tone
saturated with major
predominance of the
and minor third which
we
shall call the
in fact the expansion or
projection of the triad to the nine-tone order.
it
six-tone scale has
up the major triad. This nine-tone scale
this principle to the
a
scale.
will
counterpart, a scale
and whose
and every
scale,
another complementary six-tone
every pentad has
scale,
The importance
of
composer can hardly be overestimated, since
composer
to
expand any tonal relation with
complete consistency.
263
COMPLEMENTARY SCALES
The is
process of arriving at such an expansion of tonal resources
not an entirely simple one, and
carefully, step
by
we
shall therefore
step, until the general principle
is
examine clear.
it
The
major triad C-E-G has a complementary nine-tone scale consisting of the remaining nine tones of the chromatic scale, the tones
C#-D-D#-F-F#-G#-A-A# and this scale that
it
and three tritones— that
shall observe in analyzing
it
six
major seconds,
six
minor seconds,
predominates in the same three
which form the major
triad.
we
again revert to our circle and plot the major triad C-E-G,
find,
proceeding counterclockwise, the complementary figure
If
we
We
has seven perfect htths, seven major thirds, and
seven minor thirds, but only
intervals
B.
E#-A#-D#-G#-C#-F#-B-A and D:
Since, as has already
been noticed, clockwise rotation implies
proceeding "upward" in perfect rotation implies proceeding
may
transfer the above
264
fifths
"downward"
diagram
to
and counterclockwise in perfect fifths,
we
musical notation as follows:
:
:
expansion of the complementary-scale theory
Example
^
Triad
i
pmn
If
we
it
consists
40-2
Complementary Nonad
V
*r I I
^'r 1
m
p
1112 j^^j 1'^
tt^
r
O
n^s^d^t^
1
I
I
analyze the complementary nine-tone scale,
GJf,
/IV"
(ir
we
find that
downward from E#,
a nine-tone projection
of
upward from
O
I
not of the major triad, but of
its
or
involution,
the minor triad
Example
40-3
I '' I
If
^
'V^^j^il ii|j.,ji;^
we now form
by and whole
the involution of the nine-tone sonority
constructing a scale
which has the same order
tones proceeding in the opposite direction,
of half
we
construct the
following scale:
Example Involution of
i Analyzing
this
J 2
ttie
Complementary Nonad
jj II
we
scale,
40-4
.
J 1
find
ii^
it
II to
^
t^
consist
r 2
I
of
(I)
the nine-tone
projection of the major triad
Example
40-5
We may therefore state the general principle that the nine-tone the involution of
complementary
projection of a triad
is
We
same principle applies and pentads.
shall find, later, that this
projection of tetrads
The tone which
is
used as the
initial
its
scale.
also to the
tone of the descending
265
COMPLEMENTARY SCALES complementary scale— in converting tone.
this case
perfect fifths or minor seconds
superimpose
E#
(or Ft])— we shall call the
choice in the case of the superposition of
Its
twelve
perfect
simple. For example,
is
fifths
above
the
C,
if
final
we
tone
is F, which becomes the initial tone of the descending complementary scale. The complementary heptad of the perfectfifth pentad C-D-E-G-A becomes the scale FsE^oDbaCbiBboAba Gb(i)(F). The seven-tone projection of C-D-E-G-A becomes therefore the complementary heptad projected upward from C,
reached
or
C2DoE2F#iG2A2B(i)(C) (See Ex. 41-1, lines 4 and 6.) The converting tone of any triad is almost equally simple
determine being the of the original triads
final
triad.
upon the tones
tone arrived at in the upward projection
For example, of the
if
we
superimpose major
major triad C-E-G and continue
superimposing major triads upon each resultant all
to
twelve tones have been employed, the
final
new
tone until
tone arrived at
will be the converting tone for the complementary scales of that
we form the and (G)-B-D, giving the new tones G#, B, and D. Superimposing major triads above G#, B, and D, we form the triads (G#)-(B#)-D#, (B)-D#-F|t and (D) F# A giving the new tones D#, Fjj:, and A. Again superimposing major triads on DJ, F#, and A, we form the triads (DJf )-(F-^ )-A#, (F|;)-A#-C#, and formation. Beginning with the major triad C-E-G,
triads (E)-GJ|:-B
(A)-CJj:-(E), giving the Finally,
form the final
that
new
tones
C}f.
triads (A#)-(C>x< )-E#
A# and
Cf,
we
and (C#)-E#-(G#), giving the
twelfth tone E#. This tone becomes the converting tone, is,
the initial tone of the descending complementary scale.
Example j>
>/
A# and
superimposing major triads above
40-6
^jiy.;tlly
»l
266
l*P
»
ii
t
% ^1^
tii
l
f/yl|JH¥^
)
EXPANSION OF THE COMPLEMENTARY-SCALE THEORY
The complementary heptad major triads
pentad composed of two
of the
at the perfect fifth,
C-E-G
+
G-B-D, or C2D2E3G4B,
becomes, therefore, iFaEbsD^gBbiAiAbsGb. The projection of
C-D-E-G-B 2 and In
therefore CsDsEsGiG^iAaB.
is
(See Ex. 42-1, lines
5.
many
other cases, however, the choice of the convert-
ing tone must be quite arbitrary. For example, in the case
any
of
choice
composed
sonority entirely
is
for example,
arbitrary.
entirely
major
of
seconds,
the
The whole-tone hexad above C,
C2D2E2F#2G#2AJ1:. Since this scale form super-
is
imposed on the original tones produces no new tones but merely octave duplications,
is
it
obvious that the converting tone of
the scale C2D2E2F#2G#2A# will be B-A-G-F-Eb or D^, giving
complementary
the
scales
J,B2A2G2F2Et)2Db'
iA2G2F2Eb2
DbsB, iG2F2Eb2Db2B2A, jF2Eb2Db2B2A2G, jEb2Db2B2A2G2F, or
iDb2B2A2G2F2Eb. The choice of
Example 40-7
is
I
I
^ ti"
I
22222 jf-'
ff""
40-7
Complementary Hexads
Major Second Hexod
J
as the converting tone in
therefore entirely arbitrary.
Example
J
F
I I
"
r r 'I r r
't b^
22^22 I
I
i
I
—22222
|rr'rhvJir"rh- .Jji'rVr^JJ
^
22222
Take,
again,
22222 |
the
CsDJiEaGiGfligB. If
upon each
of the
CsDJtiEeGiGSsB;
22222
major-third
we superimpose tones
hexad
in
i
^rp^^ 22222 Example
of the hexad,
we form
F#,
A#,
(G)3A#,(B)3D,(D#)3F#;
and
D
40-8,
the hexads
iE),(G),{G^), (G#)3(B),(C)3
(D#)i(E)3(Gti); and (B),D,(Dj)^)sFUG)sAl giving the tones
T
this intervallic order, 31313,
(D'i^)sFUG)sAUB),D^;
(B)i(C)3(D#);
r L
r
and producing the nine-tone 267
new scale
)
COMPLEMENTARY SCALES C2DiD#iE2F#iGiG#2A#iB(i)(C.) The remaining tones, F, Db,
and A, are therefore
equally the result of further superposition and are
all
possible converting tones, giving the descending
all
complementary
iFgDiDbsBbiAsGb, iDbsBbiAsGbiFsDl^, and lAaGbiFgDiDbsBb- Our choice of F is therefore an arbitrary
choice from
scales
among
three possibilities.
Example
^mo
40-8
^
"
«
^"*'^" av^^ ^ '
' ^'
13 13
3
-^fr
13 13
3
Complementary Hexads
13 13
3
One
final
13 13
3
example may
suffice.
The
tritone
3
13 13
hexad of Example
40-9 contains the tones CiC#iDjFJfiGiG#. This scale form superimposed upon the original tones gives the hexads CiCj^J^^Fjj^iGi
G#;
(C#),(D),D#4(G),(G#)iA;
(G),(G#),(C)i(C#),(D);
(D),DS,E,(G#)AAJf;
(F#),
(G),(GJf)iA4(C#),(D),D#;
and
(G#)iAiA#4(D)iD#iE, with the new tones D#, E,
A
and A#,
producing the ten-tone scale CiC#iDiD#iE2F#iGiG#iAiA#,2) (C).
The remaining
tones,
F and
B, are therefore both possible con-
verting tones giving the descending complementary scale of the
hexad CiCjf.D^FJfiGiGJf
Our choice possibilities.
268
of (
F
is
as
iF,EiEb4BiBbiA or jBiBbiA^FiEiEb-
therefore an arbitrary choice between
See the Appendix.
two
5
expansion of the complementary-scale theory
Example
"^"
V- ..|t» o
I
^
'<^
itI
I
4
I
40-9
^ ^ I
I
I
1l--«t°
-^4*^
ft. I
4
I
I
I
I
4
I
I
i^-^=^ 4
I
I
I
i"^:
°fr
I
Complementary Hexads
i
I
4
I
114
11
11
In certain cases where sonorities are built-up from tetrads or
pentads through connecting hexads to the projection of the
complementary octads or heptads respectively, the converting tone of the connecting hexad
An
is
used.
*
understanding of the theory of complementary scales
especially helpful in analyzing contemporary music,
shows that complex passages
by an examination
ejffectively
in
the
passage.
Let us
may be for
it
analyzed accurately and
of the tones
take,
since
is
which are not used
example,
the
moderately
complex tonal material of the opening of the Shostakovitch
Symphony:
Fifth
Example
40-10
Shostakovitch, Symphony No.
Moderoto
S^
/ Copyright
MCMVL
by permission.
*
A
by Leeds Music Corporation, 322 West 48th
"connecting hexad"
pentad and
Street,
New
York 36, N. Y Reprinted '
All rights reserved.
is
is defined as any hexad which contains also a part of that pentad's seven-tone projection.
a specific
269
COMPLEMENTARY SCALES
r_yJP
io i An
o 12
tfv>
iiJ
j^
o
12 >
^
^r
I
jij
omitted tones
^
-P= 2
^^f
I
^
examination of the opening theme shows not only the
presence of the tones D-D#-E-F#-G-A-B[;,-C-CJf, but the absence
and B. Since F-G#-B is the basic minor third becomes immediately apparent that the complementary nine-tone theme must be the basic nine-tone minor third scale. of the tones F, G#, triad,
it
A re-examination of the of
two diminished
scale confirms the fact that
it is
composed
tetrads at the interval of the perfect fifth plus
a second foreign tone a
fifth
above the
first
foreign tone— the
formation of the minor-third nonad as described in Chapter 13. This type of "analysis by omission" must, however, be used
with caution,
lest
a degree of complexity be imputed which was
never in the mind of the composer. The opening of the Third
Symphony which, at it
actually
of
Roy
first
Harris offers a fascinating example of music
glance, might
seem much more complex than
is.
Example Harris,
Symphony No. 3
270
40-11 Vios.i
_
^^
K
^J^-J).
EXPANSION OF THE COMPLEMENTARY-SCALE THEORY
^ 1
V^
=. «*-«»
^>,,
-
|Trr [rrrri^r^rr^ir
By permission
«r^
of G. Schirmer, Inc., copyright owner.
we examine the first twenty-seven measures of symphony, we shall find that the composer in one If
this
long
melodic line makes use of the tones G-Ab-Ati-Bb-Bfc|-C-C#-D-D#E-F-F#; in other words,
Upon
all
of the tones of the chromatic scale.
closer examination, however,
we
find that this long line
organized into a number of expertly contrived sections, together to form a homogeneous whole. consist
the
of
perfect-fifth
The
first
all
linked
seven measures
C-G-D-A-E-B,
projection
is
or
melodically, G-A-B-C-D-E, a perfect-fifth hexad with the tonality
apparently centering about G.
The next
phrase, measures 8 to 12, drops the tone
C
and adds
the tone B^. This proves to be another essentially perfect-fifth projection:
the perfect-fifth pentad G-D-A-E-B
with an added
minor
third.
B^i,
(G-A-B-D-E)
producing a hexad with both a major and
(See Example 39-6, Chapter 39, complementary
hexad.) Measure 15 adds a
momentary
A[)
which may be ana-
lyzed as a lowered passing tone or as a part of the minor-second tetrad G-Ab-Atj-B^. Measures 16 to 18 establish a cadence consist-
ing of two major triads at the relationship of the major third—
Bb-D-F plus D-F#-A (D-F-F#-A-Bb-Example
22-2).
271
'
COMPLEMENTARY SCALES Measures 19
22 establish a
to
new
perfect-fifth
D-A-E-B-F#-C# (D-E-F#-A-B-C#), which transposition of the original hexad of the
measure 23 the modulation
to a
B
D—
seven measures. In
first
tonality
hexad on
be seen to be a
will
is
accomplished by
the involution of the process used in measures 16 to 18, that
two minor
triads at the relationship of the
Measures 24
G-D-A-E-B-F#, tion of the
to
AJf
F#
DJf
B
major
is,
third:
27 return to the pure-fifth hexad projection
in the
melodic form B-D-E-F#-G-A, a transposi-
hexad which introduced the theme.
The student may well ask whether any such detailed analysis went on in the mind of the composer as he was writing the passage. The answer is probably, "consciously— no, subconsciously—yes." Even the composer himself could not answer the question with
finality,
for
even he
is
not conscious of the
workings of the subconscious during creation.
happens
is
that the
composer uses both
What
actually
and his homogeneous
his intuition
conscious knowledge in selecting material which
is
and which accurately expresses his desires. A somewhat more complicated example may be cited from the opening of the Walter Piston First Symphony: in character
Example Piston,
Symphony No 'Cellos,
'-n
40-12
:
Bosses pizz
j]^
^'v/jt
j
\i
O \i>r^ By permission
Here the
first
I
j;^
..rjj
of G. Schirmer, Inc., copyright owner.
three measures, over a pedal tone, G, in the
tympani, employ the tones G-G#-A-Bb-Bti-C-C#(Db)-D-E, of the tones except F, scale
Ffl:,
might be considered 272
and D#, to
in
all
which case the nine-tone
be a projection of the triad nsd.
EXPANSION OF THE COMPLEMENTARY-SCALE THEORY
Such an analysis might, indeed, be justified. However, a simpler analysis would be that the first five beats are composed of two similar tetrads,
perfect
two
fifth;
CiDboEgG and GiG^gBgD,
at the interval of the
and that the remainder of the passage consists of and GiGj^iA^Cjl^, at the interval
similar tetrads, B[)iBl:|iC4E
of the
minor
third.
Both analyses are factually correct and
supplement one another.
273
41 Projection of the
Six Basic Series with Their
Complementary Sonorities We may now
begin the study of the projection of
all
sonorities
with the simplest and most easily understood of the projections,
Here the relationship of the involution of complementary seven-, eight-, nine-, and ten-tone scales to their five-, four-, three-, and two-tone counterparts will of the
that
be
perfect-fifth
series.
easily seen, since all perfect-fifth scales are isometric.
Referring to Chapter scale
contains
the
5,
tones
we
find that the ten-tone perfect-fifth
C-G-D-A-E-B-F||-C#-Gif-Dif
or,
ar-
ranged melodically, C-C#-D-Dit-E-FJf-G-Git-A-B. We will observe that the remaining tones of the twelve-tone scale are the tones
F and B^. If we now examine the nine-tone-perfect-fifth scale, we find that it contains the tones C-G-D-A-E-B-F#-C#-G# or, arranged melodically, C-C#-D-E-F#-G-G#-A-B.
We
observe that
the remaining tones are the tones F, B^, and E^. If
we is
we now
build up the entire perfect-fifth projection above C,
find that the
complementary
the perfect fifth beginning on
interval to the ten-tone scale
F and
constructed downward;
the complementary three-tone chord to the nine-tone scale con-
two perfect fifths beginning on F and formed downward, F-B^-E^; the complementary four-tone chord to the eight-tone scale consists of three perfect fifths below F, F-Bb-E^-Ab; and the complementary five-tone scale to the seven-tone scale consists of four perfect fifths below F, F-Bb-Eb-Ab-DbThe first line of Example 41-1 gives the perfect fifth with its complementary decad. The projection of the doad of line 1 is sists
of
274
PROJECTION OF THE SIX BASIC SERIES
which
therefore the decad of Hne
9,
complementary decad of
1.
line
the involution of the
is
Line 2 gives the perfect-fifth triad
complementary
v^ith its
nonad. The projection of the triad becomes the nonad, line
8,
the involution of the complementary nonad of line
2.
vi^hich
is
Compare, therefore, 4a with
6,
and
with
5<2
8a the involution of 4,
line la
2,
5.
Note
with line
2a with
9,
also that 9a
is
7a the involution of
and 5a the involution
of
7,
the involution of
1,
6a the involution of
3,
5.
Example Perfect Fifth
3a with
8,
Doad
41-1 lo.
p
Complementary Decad '
^r
7
II
^ 2
r^r
^
I
Ji^Ji^ I
I
I
2
(5)
Perfect Fifth Triad
i
I
^°-
p^s
Complementary Nonad
r
^S
Perfect Fifth Tetrad
2 5 2
r'T^r'f r I
5(5)
3a.
p^ns^
I
r
2 2
^'^^^J 2 (I)
I
I
Complementary Octad
Z
r'T^rir^r JbJ
r
i
I
I
(3)
2 2
'
\'2 12
(I 2 {0
4a. Perfect
2 2
Pentod p^mn^s^
^m
Lompiementq omplementqry Heptad
3 2
2 2 2
Fiftti
(3)
Perfect Fifth Hexad p
$
"^^
^2
^P
2 3 2 2
m
i
-^
2 2 2
I
"
{I (1^
Complementary Hexad
2
(I)
Heptad p
2 2
2
fi
^^
5t
2
^^^
n^s d
60. Perfect Fifth
1' I
m
n s
d^
2 3 2
^
2
(I)
Complementary Pentad
2
(I)
m
2 3 2(3)
7a.
il^i
i
Perfect Fifth Octad
p^m^n^s^d^t^
Complementary Tetrad
^
r
I
2 2
I
2 2
(I)
2
5 2
(3)
275
(I)
COMPLEMENTARY SCALES 8a.
8.
j^
4
Nonad pfim^n^s^d^t^
Perfect Fifth
*
^
m
12
i^
1<'^I
r
^
2 5 (5) 9a. Complementary Doad Decod p^m^nQs^d^t^
2
Perfect Fifth
jl -e-
Complementary Triad
I
1
I
2
I
I
2
(I)
2
I
I
7
(1)
(5)
lOo.
10.
Q »" Jul*'} r^i
#^
^m 4t^
I
2
I
I
T
The minor-second
I
I
I
I
I
Duodecad
Perfect Fifth
r
pjVWW*
^^^
Perfect Fifth Undecod
fej;
I
I
r
I
(I)
I
p'^m'^n'^s'^d'^t^
I
I
(I)
I
shows the same relationship between
series
the two-tone interval and the ten-tone scale; between the triad
and the nine-tone
scale; the tetrad
and the eight-tone
the five-tone and the seven-tone scale. Line 9
is
scale,
and
the involution
of la; line 8 of 2a; line 7 of 3a, line 6 of 4a,
and hne 5 of
Conversely, line 9a
8a the involution
is
the involution of
of 2, line 7a of 3, line 6a of 4,
and
line
Example Minor Second Dead
d
Minor Second Triad
sd^
^
i
276
5a of
5.
41-2 IQ
2a.
Complementary Decad
Complementary Nonad
^^^^ I
I
I
I
I
I
(10)
Minor Second Tetrad ns^d^
r
line
1,
^^
Complementary Octad
I I
I
(9)
f^
I
I
S I
I
I
(5)
5a.
PROJECTION OF THE SIX BASIC SERIES mn^s^d^
Minor Second Pentad
4
m
i ^^^ 5.
Co,mplenr|entary Heptod
4o
£^ •
P-f-
I
pm^n^s^d^
Minor Second Hexed
1^
I
I
I
r
r
r
I
^
Complementary Lpmpiementa ^W ^bi^
^IHJ I
I
I-
r' r
I I
^
7^
I
I
I
I
I
I
ii
I
I
mp
(9)
^ ^^
p^m®n^s^d®t'
'^
^
8q.
Complementary Triad ^
^
I
I
J Minor o SecondJ II Undecod .
I
(loy
(4)
I
I
...
iO/»
^^
(8)'
(5)
I
II III
I I
I
Pentad
Complementary Tetrad
I
I
I
I
I
^
Minor Second Decad £^nri^n^s^d2t'*9Q
9
II
I
s4^4„ 5,6^7*2 p^m^n^s^dM*
Minor Second Nonod
r
(6)'
(6)
8./I
iTi
I
'^
s^
|J J
Minor Second Octod
7
I
(7)
60.
*^
f^
I
I
*^^ ^^ -km ^^£#^ m
Minor Second Heptad p^m^^s^d^t
i "^
,&,
Complementary Hexod
5q.
i itr 'l+*'l
,
(8)
^
'^
f»-bf-
I
Cqmpiementary Doad
(3)
pmnsdt
10 10 10 .10.5 loa.
10
Minor Second Duodecod p'^m'^n'^s'^d'^t^
I
I
I
I
I
I
I
The major-second though
it is
in line 9
nonad 8
is
is
I
I
I
(I)
projection follows the
same
pattern, even
not a "pure" scale form. Note again that the decad the involution of the complementary decad, la; the the involution of the complementary nonad 2a; and
so forth. Note also that 9a tion of 2,
I
and so
is
the involution of
1,
8a the involu-
forth.
277
complementary scales
Example £
^ Major Second Dood
I.
2 2.
/I
41-3 la.
Complementary Decad
2o.
Complementary Nonad
(10)
I
ms ^
Major Second Triad
2
2
2
(8)
m^s^t
3.^ Major Second Tetrad
3a.
2
2
5y,
2
2 2
1^
5a.
m^s^t^
2
(2)
2
2
2
^^«
I
CO
m
n
d
6o.
t
2
2
2
2 7o.
I
8.^ Major Second
2 9Jj
2 2
I
I
I
^1^
flJ
2 2
I
I
^^1^
I
I
9a.
r
I
r
I
jj[j 2
2
(I)
^jt^ ^ii^ I
I
I
I
r I
(2)
2 (2)
^^ 55 2 2 (2)
'f^^ (I)
Major Second Duodecad p'^m'^n'^s'^d'^t^
|q^
2
bp J
2
2
^
(4)
^bphp :ti^ 2 2 (6)
^
2 (8)
Complementary Doad
'f
r I
I
I
Complementary Triad
2
(I) (1)
Major Second Undecad p'Om'Qn'Qs'Qd'QtS
Is lly,
I
Ba.
Major Second Decad p^m^n^s^d^t^
m 10/. 10/}
2
I
Nonad p^m^n^s^d^t^
I
I
Complementary Tetrad
f
Jti^^it^r' 11 (2)
2
I
Complementary Pentad
l
jjjjj
2
p
p\n^nV^d^^
7y, Major Second Octad
(I)
^
Ji^
I
r
2 2
'^pt^
2 (2)
I
^
1
c o ^ s
I
I
Complementary Hexad
2 Q
i ^^
T^r
2 2
'
Second Heptad p
6.^ Major
l
r'T^ r
^
I
I
Complementary Heptad
2
(4)
Major Second Hexad
i2 J 2 J^2rt2
4a.
II
Complementary Octad
2
(6)
^ Second Pentad m^s^t
4yi Major
2
2
I
1
11
2 2
r'Tt^f
njJi[Jr
2
2
i
m (10)
(i)
PROJECTION OF THE SIX BASIC SERIES
The minor-third projection follows the same pattern, with the exception that the minor-third scale forms are not all isometric. It should be noted that while the three-, four-, eight-, and nine-tone formations are isometric, the
each has
scale
and seven-tone
five-, six-,
involution. (See Chapters 11 through 13.)
its
The student should examine with
particular care the eighttone minor-third scale, noting the characteristic alternation of a
and whole step associated with so much
half-step
of
con-
temporary music.
Example I-
^
i
^
Minor Third Doad
3
41-4 la
n
II
(9)
I
2.M Minor Third Triad
3
(6)
I
3.^ Minor Third Tetrad
n
4 2 t
3o.
P^p 3
3
I
pmn^sdt^
4q
j3bJ 3^Jft^^r' 2 (3)
Hexod p^m^n^s^d^t^
5a.
i^M^^Yr' 6-
31
2
(2)
I
MinorThird Heptad
^p 2
I
1
2
2
p'm^ n^s^d^t^
6o.
1
2
I.
2
T (?r
2
1
2
1
7a.
3
2
I
T
(2)
Complementary Hexad
3
I
2
T
(2)
Complementary Pentad
^
3
p%i^ n^s'^d'^t'^
-tt-o\-^)
Complementary Heptad
?
(2)
1
7.^ Minor Third Octad
1
Nonad
m ^^ m s ^W 3
jtj J^rr> 3
1
Complementary Octad
I
3
2
I
I
19
I
I
i
4 yi Minor Third Pentad
5.^ Minor Third
I
J ? ^f2 '^Jj 12 12
3 (3)
1
I
I
I
2a. Complementary
n^t
^^ 3
Complementary Decod
3
12
m (^f")
Complementary Tetrad
(2)
279
m/
COMPLEMENTARY SCALES QM
Minor Third Nonod
,6w,6„8e6H6t4 p°m°n°s°d°t^
Complementary Triad
8a.
3
3
Decad p^m^n^s^d^t'^
9.^ Minor Third
I
I
I
I
I
I
2
:
i
t^ ,^ 10
.
MinorThird Duodecod p'^m'^n'^s'^d'^t®
I
The six-,
I
I
I
I
I
I
I
and
I
I
I
(I)
major-third projection forms isometric types at the three-,
and nine-tone
projections;
eight-tone projections
all
the four-,
step followed
by two
its
whole
half-steps, or vice-versa.
^ Major Third Doad m^
41-5 io.
m
A
especially the nine-tone
characteristic progression of a
Example
i
and
seven-,
five-,
having involutions, (See Chapters 14
The student should examine
15.)
major-third scale with
I.
Complementary Doad
9o.
(2)
I
Undecod —p'°m'°n'°s'°d'°
,„_ Minor Third 10^
11.^
I
(6)^
Complementary Decad
w
^T^r^fii'^l. 2
(8)
ii Major Third i^A 2 ihird Triad
m^
2a.
I
I
2
I
I
I
I
I
(I)
$
Complementary Nonad
r^V^fJ JJ^ j 4 4
2
(4)
3/5 Major Third Tetrad
pm^nd
jJ^tt^Tg 4 3
4.^ Major Third Pentad
I
2
I
I
2
Complementary Octad
S 2
p^
m^n^d^
I
I
2
11
3
iF=i=^
280
1
3
(I)
(I)
4a. Complementary Heptad
rr^ry-^^JJ
jj^ti^rT 4 3
I
rrr'ri^r^a
(4)
I
3o.
I
2
1
I
3'
I
3
(I)
(I)
PROJECTION OF THE SIX BASIC SERIES
p^m^n^d^
Hexad
5y. MajorThird
^ ^m
5a.
Complementary Hexad
te
r^r^^hi
it-*-
3
3
I
6y, MajorThird
3
I
3
r
3
I
I
7-
Major Ttiird Octad
8^
Major
» 9.
Ttiird
6a.
p'^m^n^s^d^t
Heptad
2
A Mojor
p^nn^nSs^d^t^
F^*r 2
I
Third
^ 2 r
2
I
Decad
2
I
I
I
I
I
I
I
I
2
I
I
.1
I
I
3
(I)
Complementary Triad
9a
,
sw 4
(41
Complementare Doad
w 4
(I)
I
I
8a.
(I)
pOm^nQs^d^H
3
Complementary Tetrad
4
MajorThird Undecod p'^m'^n'^s'^d'^t^
I
^
(I)
7a.
jiJjtfJ^ii^r'r^ I
3
^¥^
Nonod p°m n°s d°t^
I
1
Complementary Pentod
4
3(1)
jj.jJiiJJttJjt^rt' (
3
fe
f
j-jjiJ J ^i^ r 2
I
(1)
(8)
IOq
(I)
12 12 .12,6 u K, A r^ ^ A '2 rti12ii n a iniru Duodecad major T^uuuuBt;uu s u d t p "y) yj MajorThird
m
r
The
I
I
I
I
I
I
I
i
I
I
I
(I)
upon the perfect-fifth series which predominate in tritones—
projection of the tritone
produces a
series
of
scales
remembering the double valency
of the tritone discussed in
previous chapters. All of the scales follow the general pattern of
the triad pdt, with a preponderance of tritones and secondary
importance of the perfect six-,
five-,
fifth and minor second. The four-, and eight-tone forms are isometric, whereas the three-, seven-, and nine-tone forms have involutions.
281
complementary scales
Example
^
I.^Tritone
6
2. -
t
lo
-
^m
^^^^^ I
Mn
2
I
2a.
pdt
Tritone Triad
I
i
^
p^d^t^ u p
3o.
i
I
I
I
I
I
o' 2
i
r~
I
I
g
i~^ I I
I
1 3
'
1'! I
I
i
T I
(3)
(5)
I
Pentad p^msd^t^
#^ M
4a.
Hexad p'^m^s^d'^t'
5o.
4
I
r i~i
Complementary Octod
IJiiJ J^
r
I
Complementary Heptad
r r r^^3*^5 r Ji'^ ^
I
4
I
(5)
I
r
I
I
(3)
Complementary Hexad
J rTT Tl'r^fH^ i
ifl
6.^
4. 4
I I
I""
I 1
t
\ I
I
Heptad p^m^n^s^d^t^
6a.
i^^iJitJ^tiJ^if r 7.i5
4
I
I
3
I
I
I
I
7a.
III 1211
r
I
122
I I
Undecad
I
11./, 1.^
I
I
I
I I
9o.
I I
I I
I
11 I
I
lOa.
I
I
P I
(I)
Duodecad p uuoaecaa p'^m'^n'^s'^d'^t^ m n s a t
I
II
282
I
I
I
I
I
sm
5
I
(5)
I
I
I
(I)
I
(5)
Complementary Doad
6
(2)
J«^ ^^^
JtlJ
122
(4)
mplementary Tetrad
3
6
(3)
p'Om'On'Os'Od'QtS
JJJ^J r
I 1
I
Complementary Triad
Decad p^m^n^s^d^t^
1^
'Oj^
I
Complementary Pentad
I
(3)
^g Nonad p^m^nQsQd^t'*
9yi
4
I
(3)
^^
I
I
rT'Tl'r^r^ II 4 (5)
pgm^^nMd^t^
Octad
I
I
I
I
Complementary Nonad
(5)
I
r 5
5^,
Complementary Decad
i
Perfect Fifth
leiruu 3.A 3.^ Tetrad
4./I
.
(6)
6
1^
41-6
(6)
(3)
(i (^)
)
PROJECTION OF THE SIX BASIC SERIES
An
excellent example of the gradual expansion of the projec-
be found
tion of perfect fifths will for Violin
two and
The
in
Bernard Rogers' "Portrait"
and Orchestra (Theodore Presser Company). The
D-E-F
a half measures consist of the tones
and
third, fourth,
fifth
first
(triad nsd).
measures add, successively, the tones
G, A, and C, forming the perfect-fifth hexad, D-E-F-G-A-C
(F-C-G-D-A-E). This material suffices until the fifteenth measure which adds the next perfect
fifth,
The seventeenth measure adds
B.
Cfl:,
the
nineteenth measure adds F#, and the twenty-first measure adds
G#, forming the perfect-fifth decad, Ft|-C-G-D-A-E-B-F#-C#-G#. In the twenty-third measure this material
is
exchanged
in favor
of a completely consistent modulation to another perfect-fifth projection, the At^-Et^.
nonad composed
This material
is
of the tones
Ab-Eb-B^-F-C-G-D-
then used consistently for the next
twenty-four measures. In the forty-seventh measure, however, the perfect-fifth projection
is
suddenly abandoned for the harmonic basis F#-G-A-
Cp, the sombre, mysterious pmnsdt tetrad, rapidly expanding to
pmnsdt tetrad on A (A-B^-C-E), and again to a similar tetrad on C# (CJf-Dti-E-Gf), as harmonic background. The opening of the following Allegro di molto makes a similara
ly
similar
logical
projection,
beginning
again
with
the
triad
nsd
(F-G^-A^) and expanding to the nine-tone projection of the triad nsd, E^-Etj-F-Gb-Gti-AI^-Akj-Bb-C, in the
The
first
four measures.
projection of the most complex of the basic series, the
by a passage which has been the by theorists, the phrase at the Wagner's Tristan and Isolde. If we analyze the
tritone, is beautifully illustrated
subject of countless analyses
beginning of
opening passage
as
one unified harmonic-melodic conception,
it
proves to be an eight-tone projection of the tritone-perfect-fifth relationship,
that
is,
listening to this passage,
AiA#iB3DiD}t:iEiF3G#(i)(A).
even without
Sensitive
analysis, should convince
the student of the complete dominance of this music tritone relationship.
(
See Example 41-6, line
by the
7.
283
COMPLEMENTARY SCALES This consistency of expression
is,
I
believe, generally charac-
master craftsmen, and an examination of the works of
teristic of
Stravinsky, Bartok, Debussy, Sibelius,
name but
a
few— will
and Vaughn-Williams— to
reveal countless examples of a similar ex-
pansion of melodic-harmonic material.
The keenly
analytical student will also find that although
composer confines himself
to only
one type of material,
no
many
composers show a strong predilection for certain kinds of tonal material— a predilection which It
might
to a
in
many
composer
cases be
may change
more
during his lifetime.
analytically descriptive to refer
as essentially a "perfect-fifth
composer," a "major-
and the like himself exclusively to one vocabu-
third composer," a "minor-second-tritone composer,"
—although no composer
limits
lary—rather than as an "impressionist," "neoclassicist," or other similar classifications.
284
42 Projection of the
Triad Forms with Their
Complementary Sonorities Before beginning the study
of the
complementary
sonorities or
scales of the triad projections, the student should review Part II,
Chapters 22 to 26 inclusive.
fmn,
pns,
pmd, mnd, and
We
have seen that any of the
tones or intervals, produces a pentad. all
three of
its
upon one
nsd, projected
The
The seven-tone
triads
its
triad projected
tones produces a hexad which
the original triad form.
of
is
own upon
"saturated" with
have the same and the nine-tone
scales
characteristics as their five-tone counterparts,
scale follows the pattern of the original triad.
now examine Example
Let us
42-1,
which presents the
projec-
pmn. Since the projection of the triads pmd, mnd, and nsd follow the same principle, the careful study of one should serve them all. tion of the major triad
pns,
Example ,
pmn
42-1 Complementary Nonad
Triad
I
iF* Z.A
pmn
^^ A 4
@
p
2
^
Pentad p^m^n^s^d
2 3-/1
r'Tr^r'fr^iJj 1 3
2
3
@m
Pentad p^m'^n^^
f
ti
jj.ifl.ir
l
4
3
13
1'!
I
2
Complementary Heptod
r'T^r^f
4
pmn
112
2
2
^ 31
Ji-J
*
12
Complementary Heptad
:
U)
(2)
I'Tr^rh-AJ 2
113
285
COMPLEMENTARY SCALES pmn@
kf.^l^lMIl^ 4-ji
m
-
p \J
Hexad IIC^VJVJ
III
2
2
^^
\^\./iii(^n^iii\*iiiv«ijr Hexad i\*^*.jvi Complementary
p^rri^n^s^d^t Ml M o VJ \J
I
3
3
1
>
2
(I)
1
2
2 3
np5m4n4s4d3t
com p. Heptad
6/1 Involution of
I
i iJMJ.. 2
r
'
'
.-
i
i 2
I
I
^ Ar:f 2 '
I
(I)
I
3
pmn@n
+
*
*^
I'yj
1-33
4
*P
3
3
r
I I
2'
Involutionof comp.Heptad p'^m'^n^s^d^t^
i^P"3 ^'2^
m^ I
of comp.iNonaa or comp.Nonod
2
p^mVs^d^t^ p^mrrs^^ElL
I
^P I
2M
I
3
I
21
Complementary Hexad
3
13
*=*
15 ^ 12
^
Complementary Pentad
m 13 3
2'
12
2
Complementary Heptad
I
ml Hexad p^m^n^s^d^t
b* ^3
I2y) mvoiuTion '^y^ Involution
^ I
Pentad p^m^n^sdt
n
3 10^
^
Complementary Triad
flf^ 4
@
"
\^
13
3
Complementary Nonad
pnn Triad S/i pmn 3^
9/1 P'T'"
4
comp.Nonod p^m^n^sQd^^
7.A Involution of
^^m
4
f^^i'^Y^
13
3
I
2 3
II)
Complementary Pentad'(2)
(2)
iJitJ J^tf^rP"""^"^'-^"' 2 f
13
3
'T^rtT^J 2
p'
I
I
2
Complementary Pentad
Involution of comp.Heptad(l)
5./I
^
«M
3
Complementary ^.Ajinpu
Triad
i I I
The
? 2
I I
9 2
I I
I I
O 2
I I
Ml (I)
4 3
Example 42-1 shows the major triad C-E-G by a dotted line, its complementary nonad— the remaining tones of the chromatic scale begun on F and projected downward. The second line shows the pentad formed by the first
line of
and, separated
superposition of a second major triad, on G, again with
its
complementary scale. The third line shows the second pentad formed by the superposition of a major triad upon the tone E with its complementary scale. 286
PROJECTION OF THE TRIAD FORMS
The
fourth hne shows the hexad formed by the combination three major triads, on C, on G, and on E, with
of the
complementary hexad.
same
scale has the
its
be noted that the complementary
It will
relationship in involution— in other words,
the similar projection of three minor triads.
The fifth line shows the projection of the first pentad, line 2, by taking the order of intervals in the complementary heptad (second part of line 2) and projecting them upward. Its complementary pentad (second part of involution of the pentad of line
line 5) in turn
2,
downward and
half-steps— 2234— but projected
becomes the
having the same order of therefore repre-
senting the relationship of two minor triads at the perfect
The
sixth
(line 3)
heptad Its
line
shows the projection of the second heptad
by taking the order in the
fifth.
of half-steps in the com.plementary
second part of
line 3
and projecting
complementary pentad (second part of
line 6)
upward.
it
becomes
in
turn the involution of the pentad of line 3 and presents, therefore, the relationship
major
of
two minor
triads
at
the interval of the
third.
Line seven half-steps
in
is
formed by the projection upward of the order
complementary
the
(second part of line 1).
Its
scale
complementary
involution of the original triad of line
1,
the
of
that
original
triad in turn is,
Note the consistency of interval analysis
of
triad is
the
the minor triad.
as the projection
progresses from the three-tone to the six-tone to the nine-tone
formation:
three
p'^m'^n's^dH^.
In
tone— pmn, six-tone— p^m^n^s^dH; nine-tone—
all
of the intervals p,
them we m, and n. of
In examining the hexad
see the characteristic domination
we
additional relationship, that of
discover the presence of one
two major
triads
at the
con-
comitant interval of the minor third— E-GJf-B and Gt|-B-D. Lines 8 to 12 explore this relationship by transposing third so that the basic triad
major triad C-E-G with
its
is
again
C
it
down
a major
major. Line 8 gives the
complementary nonad begun on 287
A
.
COMPLEMENTARY SCALES and projected downward (A being the converting tone of the connecting hexad of hne 10 ) Line 9 gives the pentad formed by the relationship of two major triads at the interval of the minor third, with its complementary heptad. Line 10 is the transposition of line 4, beginning the original hexad of line 4 on E and transposing it a major third to C, the order of half-steps becoming 313 (1)22; with its accompanying complementary hexad which is al-
down so
its
involution.
Line 11
is
the projection of the order of half-steps of the
complementary heptad (second part complementary pentad will be seen line 9, or the relationship of
of
two minor
line
upward.
9)
Its
be the involution of
to
triads at the interval of
the minor third.
Line 12 gives the projection upward of the order of half-steps of the complementary nonad (second part of line 8), its
complementary sonority being the minor triad D-F-A, which is the involution of the major triad of line 8. It should be observed that the nonads of lines 7 and 12 are the same scale, line 12 having the same order of half-steps as line 7, nonad of line 12 on E, a major third above C.
Study the relationships within the
pmn
if
we
begin the
projection carefully
and then proceed to the study of the projection of the triad pns (Example 42-2), the triad pmd (Example 42-3), the triad mnd
(Example 42-4), and the
triad
nsd (Example 42-5).
Example
Complementary Nonad
1^ pns Triad
'7
?A pns@p^ 19^
\
iT
^A pns|
n
2
I
Pentod(i)
p^mn^s^
WJJ 232 "2"
Pentod
m 4
288
42-2
2
(2 )p^mn^s^dt
I
I
I
2
I
2
Complementary Heptad
(l)
rr'fbpbJu
222122
Complementary Heptad
-hnfi
rrVr^r'^^ 12
2
2
112
1
(2)
PROJECTION OF THE TRIAD FORMS
2
2
12
2
i iJilH^T P^m^n^^s^d^t 2
2 2
^1^ Involution of
comp.Heptad
^3
I 8-/}
(2)
i
p^i^^/ri
7/5
S
Complementary Pentad
4
^
compNonodp^rn^nV^d^B Complementary
f
*ff*
itt^
r
I
¥ 12
2
7
I
I
p^n^s^d
Pentad
Bn§@s
^P
j-
pns@s
ij^ 2 5 2
'2
'2i«
comp.Heptad p^m^'S^d^
of
12
comp. Nonad
I
2
2
I
p^mVVd^^
^
pmd
Jj 2
I
I
t?
^^
(»
3
2
*
2
2
'[
;»
I
1
Complementary Hexad
Complementary Pentad
Complementary Triad
I
Example I-
i
2
I
2522
32211
Involution of
r
2
.4„2„3r4 Hexad p^m^n^s^dt
+ pt
"y* Involution
^
2
I
Complementary Heptad >
10.
Triad
2
'Tl'f'r
i
(2)
Complementary Nonad
pns Triad
7 2
9.^
J*
^^
r^r r^r irJ
^ ^^ 2122 Involution of
I" 17b
2
2
'
p^m^n^s^d^t^
p
6^^==!'
9~V9 3' 2
~~9
2
2
1
12
2
2
Complementary Pentad(l)
Involution of comp.Heptod(l)
5-/5
2
42-3 Complementary Nonod
Triad
7 4
I
I
3
I
I
2
I
I
289
J
COMPLEMENTARY SCALES 2>.
2
pmd
m
pmd
Pentad
(a p
(3 d_
Pentad
(2)
13
I
"
Hexad p3m4n2s2d3t
comp. Heptad
m
JttJ jg^ ^_ „•','. "
o-ij 6-/5
1
Involution ot of
I""
4
I
7/) 7.^ involution Involution ot of
r i^'
m 8,/,
pmd
"^
3
I I
13
r
I
1
3
I
(2)
I
13
4
n
i
.III' 1
1
Complementary miiiury Pentad retiiuu
^g14 2
p4m4n^s^d^t ^
id) ivi /
4
Complementary Kentod complementary Pentad
(2)
13
6
^\d] ,(2)
^
1
I
p' i\onad p^m^n^^d^t' comp.Nonad comp
9 2
I I
I I
4
I
1
2
1
1
3
13
Complementary Hexad Hexac
(I)
comp. He Heptad
1
'
pSm^n 3s3d4t2
'
3
(I)
Complementary Heptad
1
2
5/5 Involution of
2
p^m^nsd^t
jj'r'PiJii'i 4 13 3
s^^g
14
2 4
prnd@p+^
2
Complementary Heptad
iwr 6
^
p^m^nsd^t
(I)
I I
Complementary Triad
^
7
I I
Triad
i
4
Complementary Nonad
^^^^P
t^
«
7
9.pmd@rn
4
Pentad
p^m^n^d^
jjr
3
@m
pmd "^wnvu^^^T + pi 10^ yjj,
11
3
13
1111
2
Complementary Heptad
'^r^r''^
4
Hexad p-in^n'^s'^t p^mVs^d^t
I
I
2
I
3
Complementary Hexad
ji
if'l4i4jjJ-ir-^rt'MJjt"jj"P P^ 311244 *3II24 112 4 3
3
'75 Involution of l^lnvolut
I
comp. Heptad p'^m^n^s^d^t
^
2
I
^ 3
3
I
*^^ Involution of comp. No[;iad
g^ iiJ J J ^fJ ^
I
290
I
2
I
I
II
p^m^n^s^d^t^ r
1
I
2
Comolementary Pentad
13* I
VW
A 4
Complementary Triad i
^1
7
4
i
projection of the triad forms
Example ••/I
mnd
Complementary Nonad
Triad
M
i t>j. 2/1
3
mnd@
n
^«Ltti
r
j
3
^
3
m
(I
j
Ui) 12)
13
2
I
comp. Heptad hff.
3
I
2
I
I
p^m'^n^d^ p m n d
3^
^^ 3
^•
I
2
I
I
I
(I)
II
1
12
(2)
13
1
Complementary Hexad
3 I
I
2
I
I
I
II)
(2)
Complementary Pentad
>
m
\;
m
Jj
(I)
^ ^ j|
12
3
1
Complementary Pentad (2)
? 13
3
tff^
J
itrl^r
comp. Npnad p^m^n^s^d^t^
Involution of
I
Complementary Hep Heptad
3
^p^mSn^ S'=^d^
3
I
12
1
I
comp. Heptad
Involution of
Jg
J
i
^^m
3
^I'^ii^ 9^ j^j J;iJ p3m4n5s3d4t2
^A
I
I
Complementary Heptad
1
Hexad _pVnJs^dft
3 o.A In 5./ Involution volution of
pm^n^sd^t
12
Pentad
''y5mnd@n +
J
I'i'p
2
I
I
Jit.
m
mnd @ mnd(g
tif
J
Pentad
3 3-^ ^f^
42-4
1
Complementary Triad
(
I
°"^
mnd
2
I
I
I
II Complementary Nonad
Triad
III IQ5 ncmd
@d
+ ji
i
Hexad
p^m^nVd^t
p^^n^s'^d^t comp.Heptad ept
14
Complementary Hexad
I
"/5 Involution of
I
2
I
I
4
Complernentary
Pentad ,
i^
I
r
I
14
1211 291
COMPLEMENTARY SCALES 12.^
Involution of comp. Nonaid
p^m^n^s^^t^
Example nsd Triad
2
I
nsd
@
d
Pentad (I) mn^s^d^
^A risd 7^ nsd
@
n n_
Pentad
^
@
nsd
+
I
pmn3^d^t
comp.Heptad
iiHV^^?V.' B.A Involution of
i
r
I
I
8y( nsd
Triad
^ " nsd
@
I
I
I
Ji Jb J ;
s^
l
,
Jl' J
-
^
r^r^;N 2
I
(2)
-"-"^-^^J
2
Complementary Triad
2
Pentad pmn^s^d ^
dl i^jJi|J
I
I
2
3
Complementary Heptad
1112
12
Hexad pm^n^s^d^t
Complementary Hexad
^W 12 I
292
•QJiJiJJ
I
Complementary Nonad
_s
@
I
(I)
Complementary Pentad
(2)
III
'°-^nsd
(2)
3
JiJ J
r:r I
I
12
I
ri>r ^
comp Nonad p^m^n^s^d^t^
I
I
Complementary Pentad
(I)
JHH^f.^Vp3.3...,e Involution of
I
1112
'pe.3n.s5.s,
comp.Heptad
I
(I)
Complementary Hexac
III 5/) Involution of
I
I
II
12
Hexad_gnA^|s^d^
ji
1112
I
I
Lomplemen Complementary Heptad
(2) pmn'-y-d^t l^l
12
I
Complementary Heptad
I
I
^/5
42-5 Complementary Nonad
I
2fl
Complementary Triad
I'
I
2 6
5
1112^
1
r
PROJECTION OF THE TRIAD FORMS Complementary Pentad
"yj Involution of comp.Heptad fAn^n'^s^dSf
112
I
5
1
comp.Nonad p^m
Involution of
12-5
m ww^
J^pi^r
¥W=*
I
I
I
I
'
^
Complementary Triad
^ '^^^ '
2
I
3
2
I
VsVf
I
Since the triad mst cannot be projected to the hexad by the
superposition,
counterpart
is
and proceed
simplest
to consider
it
Example
as in
method
Example
4
Nonad
da
r
(6)
Involution of
nine-tone
42-6 Complementary
^^ 2
its
major-second hexad,
42-6:
mst Triad
I
forming
of
as a part of the
comp.Nonad
pmnsdt
r
T
^r ^r ^r
112
2
m
J li I
I
(2)
Complementary Triad
:^ IT
I
The p^s,
9
?
I
I
I
I
f?)
4
2
(6)
projection of the triad forms of the six basic series—
sd^,
nH, m^, and pdt— were shown in Chapter 41.
ms^,
The opening
of the author's Elegy in
Memory
of Serge Kous-
sevitzky illustrates the projection of the minor triad first six
third,
pmn. The
notes outline the minor triad at the interval of the major
C-Eb-G
+
Et]-G-B.
The addition
of
D
and fourth measures forms the seven-tone A-B, the projection of the pentad
pmn
@
and
A
in the
second
scale C-D-E^-Eti-Gp.
The
later addition
of Ab and F# produces the scale C-D-Eb-E^-FJf-G-Ab-A^-B, whioh proves to be the projection of the major triad pmn.
(See Ex. 42-1, line 7.)
293
43 The pmn-Tritone Projection with Complementary Sonorities
Its
We may
combine the study
of the projection of the triad
mst
with the study of the pmn-tritone projection, since the triad mst is
the most characteristic triad of this projection. Line 1 in
Example 43-1
gives the pmn-tritone hexad with
ary hexad. Line 2 gives the triad mst with
nonad, begun on
A
its
its
complement-
complementary
and projected downward.
Lines 3 and 4 give the two characteristic tetrads pmnsdt,
with their respective complementary octads. Lines 5 and 6 give the
two
characteristic pentads with their
complementary heptads,
and line 7 gives the hexad with its complementary involution, two minor triads at the interval of the tritone. Line 8 forms the heptad which is the projection of the pentad in line 5 by the usual process of taking the order of half-steps of the complementary heptad (second part of line 5) and projecting that order upward. Its complementary pentad ( second part of line 8) will be seen to be the involution of the pentad in line 5.
Line 9 forms the second heptad by taking the complementary
heptad of
upward.
line
Its
6 and projecting the same order of half-steps
complementary pentad becomes the involution of
the pentad in line
6.
Line 10 forms the
eight-tone projection
first
by taking the
second part of line 3 ) and projecting the same order of half-steps upward. Its complementary octad first
is
complementary octad
(
the involution of the tetrad of line
3.
Line 11 forms the second eight-tone projection in the same
manner, by taking the complementary octad of 294
line
4 and
THE pmn-TRITONE PROJECTION
same order
projecting the
of half-steps
upward.
complement-
Its
ary tetrad becomes the involution of the tetrad of line Finally, line 12
derived from the complementary nonad of
is
2 projected upward,
line
its
complementary
involution of the triad mst of line
@
P'^'^
-^P^ 13 2 13 2^
13
^^ '4
2
i 1
1
J id
\
4
I
2
I
I
p^mn^sd^t^
2
2
^
I
I
3
1^' 3
2
I
3
'
pT
°y) Involution of comp.Heptads p
i
Iff
3
:^
? 2
Si ^ 13
m
r
I
^^i^ 12'
1
Complementary Complement ary Hexad
3
4 3 4 3
a
y
r
3
I
^
p^m^nVd^t^ pmnsdt
Hexad
r
^^
^ 2
2
13 2^
'l^
3
Complementary Heptads
pm^n^'s 2„2.2d^2
7-^
1
I
112
2
^m
i 3
2
3
I
I
3
I
^75 Pentads
or
^
->
2
3
3
Complementary Octads
pmnsdt
1^
I
2 2 ^%"^H,^,^^M^
I
I
pmnsdt
3y) Tetrads
2
Complementary Lpmpie Nonad
mst
Triad
43-1 Complementary Hexad
Hexad
^
being the
triad
2.
Example '•/I
4.
i4
3
n s d^t
2
3'
I
^
"
I
T
f-j^
Complementary Pentads
^3 3
1
*i 2
^^ i *rt r
I
p^mVs'^d^tS
r
3
2
112
4
2
i
¥^;^ 295
COMPLEMENTARY SCALES p^m^n^s^d^t ^
10^ Involution of comp.Octads
pi ^
^^ 3
11
2
p^m^n^s^d^t^
^^
Ji^ ^"^
J
|v
2
2
I
r
^
13
I
comp.Nonad p^m^n^s^d^t^
'2« Involution of
2
I
3
I
^
Complementary Tetrads
2
I
m
2
Complementary Triad
2
I
This projection offers possibilities of great tonal beauty to
composers clearly
who
allied
are intrigued with the sound of the tritone. It to
the
minor-third
but
projection
is
is
actually
saturated with tritones, the minor thirds being, in this case, incidental to the tritone formation. Notice the consistency of
the projection, particularly the fact that the triad and the nonad,
the two tetrads and the two octads, and the two pentads and the
two heptads keep the same pattern
The opening
of interval
of the Sibelius Fourth
sixteen measures (discussed in Chapter
C
Symphony— after the first 45)— shows many aspects
The twentieth measure
of the pmn-tritone relationship.
a clear juxtaposition of the
dominance.
major and G^ major
contains
triads,
and
the climax comes in the twenty-fifth measure in the tetrad C-E-FJf-G, pmnsdt, which with the addition of
C#
in
measures
twenty-seven and twenty-eight becomes C-Cfli-E-Fif-G, the major triad with a tritone added below the root and fifth.
The student
will profit
symphony, since earlier
it
from a detailed analysis of
exhibits
a
fascinating
C
this entire
variation
between
nineteenth-century melodic-harmonic relationships and
contemporary material.
The opening
many
trates
Symphony No. 2, Romantic, projection. The opening chord is
of the author's
aspects of this
major triad with a tritone below the root and with a
illus-
a
D^
third, alternating
G major triad with a tritone below its third
and
fifth.
Later
the principal theme employs the complete material of the projection of the pentad
296
Db-F-G-A^-B, that
is,
Db-Dt^-F-G-Ab-Al^-B.
THE pmn-TRITONE PROJECTION However,
it
is
not necessary to examine only contemporary
music or music of the exotic scale forms.
late nineteenth
The strange and
century for examples of
beautiful transition from the
scherzo to the finale of the Beethoven Fifth nificent
tones
Symphony
is
a
mag-
example of the same projection. Beginning with the
h.\)
and C, the melody
first
outlines the
configuration
Ab-C-Eb-D-FJj:, a major triad, A^-C-Eb, with tritones above the
and third— D and Ffl:. It then rapidly expands, by the addiand then E, to the scale Ab-A^-C-D-Eb-E^-F#-G which is the eight-tone counterpart of A^-C-D-Eb, pmnsdt, a root
tion of G, A,
characteristic tetrad of the pmn-tritone projection.
This projection
is
but the relationship
essentially is
melodic rather than harmonic,
as readily
apparent as
if
the tones were
sounded simultaneously.
297
)
44
Two
Projection of
Similar Intervals
at a Foreign Interval
with Complementary Sonorities
The next projection
be considered
to
which are composed
tetrads
relationship
of
two
similar intervals
We
a foreign interval.
of
the projection of those
is
at
begin with the
shall
examination of the tetrad C-E-G-B, formed of two perfect at the interval of the
interval of the perfect
Line
1,
major fifth.
(
third, or of
two major
See Examples 5-15 and
fifths
thirds at the 16.
@
m with its comp Line 2 gives the hexad formed by the
Example
44-1, gives the tetrad
plementary octad.
@
@
m) m, with complementary hexad. Line 3 forms the eight-tone projection
projection of this tetrad at the major third— (p its
the
of the original tetrad
upward the order
by the now
familiar process of projecting
complementary octad (second part
of the
of
line 1).
Since
all
of these sonorities are isometric in character, there
are no involutions to
be considered.
Example 1
75
p
@m
^m Tetrad
4
2j^p^@rn@m
Hexod
H
id 3
298
p^
m^
nd
Complementary Octad
p^m^n^d ^
3
11
3
1
Complementary Hexad
r^ry
Jtf^r M'
J
JiJ^
^ri^
r 2
4
3
I
44-1
3
I
3
J^
12
TWO
PROJECTION OF 3yi
i
Involution of
comp.Octad p
m
d
n s
SIMILAR INTERVALS Complementary Tetrad
t
W
'^' ' J i 2 JiiJ 113 12
^m
4
4
3
1
The remaining tetrads Example 44-2 presents the
projected
are
similar
in
manner:
minor third
interval of the
at the
relationship of the perfect fifth:
Example
i
£ @ £ s=
p^mn^s
Tetrad
4
^S
f\
Involution of
2
3
p^mfn^s^d^^
comp.Octad
t
It""
i.Jjt 4 2!
2- m@t_@ m
^ m
Tetrad
or
§
s
2
i^j
m
n
^
J "Tpfs
3ft^
44-3
r 'r
Hexad m^s^t^
p
1
Complementary Tetrad
4
compOctod
2
I
Complementary Octad
t
fer^^'^t^^ i J Jtf^*> 2 2222 3y) Involution of
J 12
at the tritone;
Example m @
;J
^D^itJ Z
34
There follows the major third
Vj
l
1
Complementary Hexad
^2121121
'
2
^^12 ^^ 2
M^ 12
i|J
r
3
Hexad p^m^n^s^d
2ji«_n@£@p
^
Complementary Octad
jj ^T 3
44-2
s
d'^t
2
I
I
J
^r 'r
r
2
^ I
I
Complementary Hexad ;
r
22222N 'r ^r
i^r^
Complementary Tetrad
^
i 2
1
12
2
4 1
I
2
'
4
299
COMPLEMENTARY SCALES the minor third at the interval of the major third;
Example @ m
n
I.
4 b^
pm
Tetrad
ib^
^8
3
2'/5n@m@m
2
2
n
Complementary Octad
d
r
3
I
3
13 13
J^bJ
|(J
3
it
2
I
I
^
^
I
I
I
2
^li
13 13
Complementary Tetrad
^^ 13 ^
?= 2
I
I
^r^r
r
^
3
p^m^n^s^d^t^
3y, Involution of comp.Octad
2
I
Complementary Hexad
gti°ibJ^jT^r
t®<
^^^
^r 't ^r
r 3
p^m^n^d^
Hexad
44-4
3
the major third at the interval of the minor second;
Example ''/5
m @
d
J&.
b is
pm
Tetrad
2
=^^ 3S
1;.€U
:#^
i^ 3
3.^ Involution of comp.Octad
iP
ibJ I
bJ
IjJ I
I
3
I
I
I
3
'
^^^ J
^r
r
I
3
3
I
I
p^m^n^s'^d^t^ Complementary Tetrad
ft^p
^ i
t|J I
II
Complementary Hexad
J
W-g^ I
I
i
II
p^m^n^d^
Hexad
^g^
^^ ^ Complementary Octad
I
I
^- m@d.@iTi
nd
44-5
2
3
I
13
jU 1
the minor third at the interval of the major second;
Example '•^Jl
@
_s
pn^s^d
Tetrad
2
I
2
44-6 Complementary Octad
2
II
1
112
Complementary Hexad
2
300
I
I
I
2
I
TWO
PROJECTION OF ^yjlnvolution of
SIMILAR INTERVALS
compOctod p^m n^s^d
Complementary Tetrad
t
9f
W.2
I
I
I
jtJ
12
2
2
I
I
ji
f
the minor third at the interval of the minor second;
Example '
n
/I
@
Tetrad
d
!+ 1^*
fl
Q
? 2
^&S
I I
2 3 4 5 n
,
.
,
,
comp. Octad p
I I
m
n s d
and the perfect
@
izf^ iJ I
'" ^
m 2 s 2 d4
iu r^J U ^^1
l"
I
The
m)
I I
3
•» 3
I I
I
r'T r'T^^ 1113 113
4
I
I
-X
:
I I
i
It
r
I'
r
Hexad
I
p^m^n^s^d^t^ Complementary
113 I
2
I
^~^
itJ
I
C^mplerr^entary
t
11
14 Involution of comp.Octod
I
44-8
I
4
I I
Complementary Octad
-"'^
Hexad p
l^@jd^@j^
3
I
^^
p^md^t
6
I I
the interval of the minor second.
fifth at
Tetrad
d
I I
I
Example p
I I
Complementary Tetrad
t
I
I
I I
I
III 13
1^1 r
I I
Complementary Hexad
d
s
[III .ft
Involution of
Complementary Octad
I I
Hexad pm
'^ji@d@d^
^S^
mn2sd2
44-7
C 6
I
Tetrad
«Ce:
I I
@ m) @ p; (n @ p) @. n; n @ @ s) @ n; and (n@ d) @ n are not
reverse relationship of (p
@ n;
(m
@ d) @ d;
(n
used as connecting hexads in Examples 44-1, respectively because they
all
(
2, 4, 5, 6,
and 7
belong to the family of "twins" or 301
COMPLEMENTARY SCALES The
"quartets" discussed in Chapters 27-33, 39.
relationships of
@ p; and (p @ d) @ d; are not used as connecting {p @ hexads for the same reason. The reverse relationship of Example 44-3, (m @ t) @ t, is not used because it reproduces itself end)
harmonically.
In the second the
first
movement
of the Sibelius
Fourth Symphony,
nineteen measures are a straightforward presentation of
the perfect-fifth heptad on F, expanded to an eight-tone perfectfifth scale
by the addition
of a
B^
the Beethoven example. Chapter
Measures
twenty-five
of
the
twenty-eight
present
the
heptad
@
opening being built on the expansion of the tetrad
C-E-Gb-Bb to (See Example
302
to
measure twenty. (Compare Example 15).
in
n pentad. Measures twenty-nine to however, depart from the more conservative material
counterpart of the thirty-six,
pmn
4,
its
eight-tone counterpart C-D-E-F-Gb-Ab-Bb-Bt^.
44-3.)
45 Simultaneous Projection of Intervals with Their
Complementary Sonorities We
come now
to the projection of those sonorities
formed by
the simultaneous projection of different intervals. As
we
shall
which may be projected to their eight-tone counterparts, whereas others form pentads which may be projected to their seven- tone counterparts. In Example 45-1 we begin with the simultaneous projection of the perfect fifth and the major second. Line 1 gives the projection of two perfect fifths and two major seconds above C, resulting in the tetrad C-D-E-G with its complementary octad. Line 2 increases the projection to three perfect fifths and two major see,
some
of these projections result in tetrads
seconds, producing the familiar perfect-fifth pentad, with
its
complementary heptad; while
line 3 gives the pentad formed two perfect fifths and three major seconds, with its complementary heptad. Line 6 gives the heptad formed by projecting upward the order of the complementary heptad in line 2, with its own complementary pentad— which will be seen to be the isometric
by the
projection of
involution of the pentad of line gives the heptad
which
plementary heptad of
is
line 3.
the
2.
Line seven, in similar manner,
upward
projection of the com-
Line 8 becomes the octad projection
of the original tetrad.
Lines 4 and 5 are the hexads which connect the pentads of lines
3 and 4 with the heptads of lines 6 and 7 respectively.
There
is
which is not hexad projection.
a third connecting hexad, C-D-E-G-A-B,
included because
it
duplicates the perfect-fifth
303
COMPLEMENTARY SCALES
Example
^
p^mns^
Tetrod
2
2
-
-^
2./)
p2
3./)
+ s^
Pentad
5/^
I f^ ^'^
^2 J 2
i
J
2
.1
jiJ
2
S r'T
i
^^
12
2
2
V Y ^r 11 2
r 'T 2 2
r "r ^r
2
2
2
I
2
Connecting Hexad (?)p2rT (2)p2m^s^d^t2
2
^f
'f
(1)
iJ
2
Complementary Hexad
It 'y ^r
'T
J 2 J 2 Ji2 11
I
''^ I
^
^^12
(2)
^
Complementary Hexad
&
(I)
V'T^r''^^^ 2
Complementary Heptad
I
2
I
Complementary Heptad
p2m2ns3dt
J
11
2
Connecting Hexad (l)p'*m^n^s^dt
(2)
J
l
comp.Heptad
6.^ Inv.of
^
(2)
2
2
3
7 ^2232
^ 4.ij
Complementary Octad
PentadlDp^mn^s^
-^
-0-
45-1
J2 J2
(I)
2 p
jj|J
J
2
12
nrr
n s d^t
fpplementqry Pen Pentad Complementory
2
2
2
^r
B.^ inv. Inv.of ot 8./5
2
11
2
2
I
comp. Octad p comp.uctad p^i^n^sV\^ i n s d t
2
2
11
2
Example 45-2
I
2
2
(I)
^2
3
Complementary Pentad ^r
2
2
1^
^1" comp.Heptad ^2) (2)p^m^n^s^d^t^
Inv.of
2
2
(2)
k
i^F
^
1
Complementary Tetrad Lompiementi
2
2
3
^
gives the projection of the minor second
and
the major second which parallels in every respect the projection just discussed:
Example '/)
d^
+
A
I*'
304
mnsd
Tetrad
I
2
45-2 Complementary Octad
II
III 12
SIMULTANEOUS PROJECTION OF INTERVALS
^
^fi
Pentad
s^
+
(i)
r d^
^fi
*
F 1*1
I
r
|1f
1^
r
I
I
I
12 6y5
(I)
A
7. 7.1^
Inv.of
B.ij 8.<5
I
2
I
Iff
The
I
Jiij i~
I
I
is
(2)
2 (!)
J I
I
Pentad Complementary Pe
112
(2)
2
Complenrientary Tetrad u
IP
third illustration
2
I
2
2
(I)
^—d
J [J
bp
p
JjtJfl^ I
(2)
9
Complementary Pentad
2
J I
9
I
I
II 2™,3„4^5^6. pfnrrTs^d^
I
jii
I
Complementary Hexad
t^ Inv.of pVn^s^d^t^ m^n^s nv. of comp.Octad p
j^
I
I
I
*
112
I
I
^m
(2)p^m^n^s^
r
I
Complementary lenrary Hexad
II
I
I
(1)
III 22
I I
2
2
I"
I
9 2
9 2
I
Complementary Heptad
pm^ns^d^t
pm^n^s^d^t
iiJ. V«J I*'
II
I
p^m^ns'^d^t^ (2) p'^m^ns^d'^t'^ t2)
comp. Heptad
Inv.of
Complementary Heptad
o 2
I
Connecting Hexad
5-^
(2)
I I
Connecting Hexad U)
4/«
I
I
s^ Pentad
+
mn^s^d ^
^
J
br
r I
J
=
I
arranged somewhat differently, as
concerns a phenomenon which
we
encounter for the
it
time.
first
In referring back to the simultaneous projection of the perfect fifth
and the major second, we
two pentads
of
Example
shall see that
45-1, line 2,
if
we combine
formed of three perfect
fifths
plus two major seconds, and line
fifths
and three major seconds, we produce the hexad of
which Line
3,
formed of two perfect
a part of both of the heptads of lines 6 and
is
1
of
Example 45-3
gives
its
fifths
line
4
7.
the tetrad formed
simultaneous projection of two perfect seconds, together with
the
by the
and two minor
complementary octad. Line 2 gives
the pentad formed by the addition of a third perfect fifth— three
305
COMPLEMENTARY SCALES perfect fifths and
two minor seconds— with
complementary
its
heptad. Line 5 forms the heptad by projecting
complementary heptad of
the involution of the pentad of line
2.
of line 6 will
of the original tetrad of line
i mF-^
^ -^.H r
3.^ Connecting
iff l«
I
I
Inv.of Inv.ot
6/1 Inv.of
I
4
?
I
^^ 2
I
'
I
2
2
^ ^r
r
i
J
2
'
I
Complementary Hexad
5 2
o 2
R~1 5
i I
'
Complementary Pentad
?
j|j jtfJ ^ i J j 14 11 12
I
'
4
I
I
comp.Octad p^m'^n'^s^d^t^
4
r »r I
r Iff
I
Complementary Hexad
r
comp. Heptad p^nn^ p^m^n^ n^s^d^ t^
4
I
2
2
P
Complementary Heptad
2
I
j
b|> j>f I
2
Hexad p^m^n^^d^t
*s^r^ r 5
Complementary Octad
5
5
^
The
45-3
r
Hexad p'^m^n^s^d^t^
4
I
4.^ Connecting
^* >*/
I
1.
be seen to be the involution
m
^ ,
Pentad p^mns^d^t
2/1 -^
i
p^sd^t
Tetrad
of line
1.
Example P^d2
is
Line 6 forms the octad
by projecting upward the complementary octad complementary tetrad
upward the
complementary pentad
line 2. Its
I
ST
h. 5
I
^2
i,J
'
Tetrad Complementary ipieme
r
r
T ^p
Iff
Example 45-4 line
2
is
is
the same as 45-3, except that the pentad of
formed by the addition of a minor second— that
perfect fifths and three minor seconds— with in line 5,
306
and the two connecting hexads
its
is,
two
projected heptad
of lines 3
and
4.
simultaneous projection of intervals
Example +d
P
i/l
Tetrad p^sd^t
^
*
Complementary Octad
S
-
|tr
5
I
w
^
Connecting Hexad
i 5^5
I
I
:^
I
J
p ^^ I
I
p-^m^n^s^d^t^
^s^ mv. 6,/j 6,|^ Inv.
of
4^3„3.,4j5*2 p^m^n^s^d^t
J ^J
i I
^
it
2
I
1
I
J
'^
I
3
I..
I
^^ ^1114
^ ^^
comp. Octad p^m'^n^s^d^t^
*^
2
2
Complementary Pentad
^
i^jt i*
bJ
I
^^
i7J I
^
I
12 11
I
Complementary Hexad
1
of comp. Heptod
Inv.
2
I
I
hI*
} ^
I
Complementary Hexad
2
2
I
4
I
J
2
I
Complementary Heptod
p^m^n^s^d^t
Connecting Hexad
^•/^
I
I
^J
'-^
^
^r I
Pentad p^mns^d^t
I^T
45-4
Complementary Tetrad
I
I
Example 45-1, we shall observe an interesting difference. If we combine the two pentads in 45-1 formed by the projection of p^ + 5^ and p^ -f s^, we form the connecting hexad of line 4, C-D-E-F#-G-A, which If
we compare Examples
consists
of
three
perfect
45-3 and 4 with
fifths,
However,
seconds, C-D-E-Ffl:.
if
C-D-G-A, plus three major we combine the pentads of
by the projection of p^ + d^ and f + d^, we form the hexad C-C#-D-G-A + C-Cft-D-Eb-G, or C-C#-D-Eb-G-A, which is not a connecting hexad for either Examples 45-3 and 45-4, formed
projection.
The reason the
for this
is
isomeric "quartets"
curious
hexad C-CJf-D-Eb-G-A is one of discussed in Chapter 39. It is the
that the
propertv both of the "twins" and the "quartets" of
hexads, as
we have
hexads are not their
already observed, that their complementary
own
involutions as
is
the case with
all
307
other
:
COMPLEMENTARY SCALES hexad forms. This type of hexad, therefore, does not serve as a connecting scale between a pentad and its heptad projection.
Example 45-5 gives the pentad formed by the projection of two perfect fifths upward and two minor seconds downward, with its projected heptad and connecting hexads:
Example ta^ + d^l
\.jt
Pentad
5
2
^
Ii
^
^
J 5
2
^^
p^m^n^ s^d^
2
T
2
1
*
tJ
r
"r
2
1
5
u 2
J
11
i„
Complementary Pentad
12
4
(I)
Complementary Hexad(2)
CO mp. Heptad p'^m'^n'^s'^d^t
"^ij Inv.ot
13
4
2
1
5/5 Connecting Hexa,d (2)
11
Complementary Hexad
m ^
3
12
4
2
1
i
4
2
3
45-5 Complementary Heptad
U)p^mVs^d^
2y, Connecting Hexad
i
p^m^n^s^d ^
j
+ d^ t
p^
Example 45-6 gives the projection of two major seconds and two major thirds from the tetrad to the octad which is its counterpart, using the whole-tone scale as the connecting hexad
Example '
^
+
-S
nr
iJJtii J 2 J 2^s^m^@s
«iw 'i*
Inv.
of
m s^
Tetrad
2
45-6 jrnpiem Conjipleqfientary
jM 4
r ^r 2
Hexad m^s^t ^
2
2
Octod
V II ^r
^
Complementary Hexad
"ry^^
k
i2J
comp. Octad
Jj 2
^2JttJ^J 2 2 i
p'^m^n^s^d^t^
i
r 2
Comp
2
2
Tetrad
2
2 '
«^
^'
^ 2
I
' *
SIMULTANEOUS PROJECTION OF INTERVALS
Example 45-7
and
gives the projection of the perfect fifth
major third:
Example [.M
m 4
2./J
p^m
Pentad
P^+iH^
dt
-I
i2i
Connecting Hexad
ns
^^
J
2
(I)
3
45-7 Complementary Heptad
T
I'
Connecting Hexad
2
2 4./( Inv.of
2
2
2
/)
+ m^
d5
Pentad pm^ns^d^t
^
*
*
^ I* —
2
I
m
^
Iff-
2
I
3/5 Connecting
ff*^ iTT-
Inv.of
I I
3
I
(2)
13
3
Complementary Pentad
and
^
Hexad
(2)
? 2
2
p^m^ns'^d^t^
s
.
Example 45-9 third;
downward
Complementary Heptad
^
J
J
^
l>J
2
^^ ^ ^^ ^112 I
I
I
Complementary Hexad
I
2
ll)
3
I
Complementary Hexad
(2)
*
2
2
II
45-8
I
comp. Heptad p^m5n^^d^l2 III 9 u
? «^ minor
2
2
(r
4 .2_4„3.2^3 2A Connecting Hexad lOpfnvjTfsfd^t 1^
4^
(I)
gives the projection of the minor second
^^
5*it
I^
^.i 3
I
third:
Example I.
'
13
1
Example 45-8
2
^
hI
Conriplementary Hexad
3
comp. Heptad p^m^n^s^d^t^
2
major
p^m'^n^s^d^t
(2)
'T 2
Complementary Hexad
p^m^ns^d^t^
2
3/5
^r 2
2
1
I
2
Complementary v^v.riiipidiici iiui Pentad CI iiuu MHHHM jr
112
I
4
M'
+
gives the projection of the perfect fifth
with the second interval in both
its
m'
and
upward and
projection:
309
COMPLEMENTARY SCALES
Example P^lg plm^nZsdZt Comp.Heptod
45-9 Comp. Heptad
pf-if*g|"r^°n2S2dt,
12
I
^^'^S&.^fz" 1^
I
I
3
13 13
2
^
I
I
I
r r r T'-r '
3
^^
3.^(2)p3m4n3s2d2t
ffa'miL&^fe"^ ^ CompHexodsU)
Conr,p.He«odsU)
I
r
I
a/^^X^"^
i Example 45-10 minor
I
'
/)
d'^
.
^ + n^pmn^s^d^t Comp. Heptad
III
3
I
I
pm2n3s4d4t
I
D.jt Kci p-'m'- n*- s'Ki^T'^ 3.^l2)p3m2n2s2d4t2
UjJbV 13
<^\
310
I
I
(2) [.£.)
I 1
p3m3n4s4d5t2
I
2
I
:r^
Inv.ofComp. Heptad
r
>•
^ r
I
m il)
I
4 3
I
I
I
I
and
I
3
I
2
I
r^pb[>iJ
I
^ * ^ Comp. Pentad
1113
4'
Connecting Hexads r«r«r, uovnHcn) Comp. Hexads U) p4m2n2s2d3t2
114 12
2
^§ I
I
(I)
i^^^iJJ
J
4/1
3
^ Pentad j^i p^m^n^sd^t Comp. Heptad
„ td^^
2
r«n«r, u«v«He Comp.Hexads
1112 ^1
45-10
-Pentad
Connecting Hexads
^l^(l)
^y
I
n^
third:
r „
2412
gives the projection of the minor second
Example _
C°"'l''^"'°''
Comp.Triod
"^ 2
I
12
2 2 2
12212
°
2
I
'-i
;i"frir^r^J
'g'5°n!5'??"s4'gi':i''
Iff
i
2
I
(2)
Nonod p^m^nQs^d^t^
=
4
I
i
^r^ toto|t»
lit
I
2 2 2 12
2131
Heptads
5.A Combination of
2
I
iJJJ
*fi
13 13
CompPentod
II3I3
4
I
(2)p4m2n3s4dt
(2)
2
12
2
(2)p2m4n3s2d5t
(2)
m
|ff
14
3
1
I
Comp.Heptad
Inv.of
p4m4n4s3d4t2
I*
I
4
I
2
I
^m I
4
3
I
^ * j Pentod Comp. o
114
3
SIMULTANEOUS PROJECTION OF INTERVALS ^^5 Combination of Heptads
Example 45-11 minor
Nonod p^m^nQs^d^t"*
=
2./,
(
s2+I?m^l^s5d2t
Wn3s4?4r''
Comp.Hexads(l)
I
I
II I
(2)pmVs^d^2 ^ v^'i
1124
comp. Heptad 4.^ p2m4n4s5c|4tf nv. of
5
,5
21124
^ ^ ^ Pentad Comp. o.
I
I
?
I
I
Combination of Heptads
Example 45-12 minor
CI
^6
122
(I)p2m4n3s2d.3t
3
I
2
I
I
(2) p2n1»n2s4dt2
^^
2223
^ d * ^ Comp. Pentad
^^
2
23
2'
Comp.Triad n2t
^
^
3
3
f and
gives the projection of the major third
Comp. Heptad
3l2'l 12
3 Connecting Hexads „
i JtJnJ^''H 3
^^
third:
m^-*-!? pm3n2s2dt
,
o
f 1212
I
Example
i
^
comp. Heptad
p^m^n^s^d^t^
fO [jo DO 13 t>o
r
S s^ 22231
222121'
Nonad
:i
(2)
p4m4n4s5d2t2
I
=
22212
p2mVs^2
Inv.of
2112
9 4 11124
rff 1"^
(2)
(!)
^
-0~ '2
^
'r
^
Comp.Hexads
^ ^5 2212
2
I
Comp. Heptod
n2£lf^n2s5f
^rfJM^T''
(2)
|
JtJ^^r
I
I
45-11 5.2+
m
9
I
3 3./)
2
and
gives the projection of the majoi second
Comp. Heptad
I*"
I
n^t
third:
Example I./1
Comp. Triad
I
? 2
?
...
,.
Comp.Hexads (I)
3
1^ I
2
I
I
m2^j^2| pm5n2s2dt Comp. Heptad
"^4 ^ 2 d d 2 jf jt *"jf Connecting Hexads (i) p3m4n3s2d2t
J
12
2
2
13
1 \
\
:>
_
d. 2 2 iL
,,
1 \
.
Comp.Hexads
,
^fl^-3 o 3 2
9 2
I
1
(2)pm4n2s4d2t2
(2)
3
45-12
"^^
2 2
13
2 2
(2)
4 2 2 '
I
I
1
311
1
\
,,< (!)
COMPLEMENTARY SCALES Inv.of
comp.Heptad
1
Connbination of Heptads
5./I
)H*" °l"
projected both
is
pentad
°*^ o
^^ I
in Examples up and down,
be noted that
It will
third
results.
will
It
^ 422
32211
!*
4 2 2
Comp. Triad
Nonad p°m°n°s°d°t'
=
_ „ . ^ Comp. Pentad
p3m5n4s4d3t2
3122
T,o.,9 12 3
connp.Heptad
Inv.of
^ r^ . ^ Comp. Pentad
p3m5n4s4d3t2
4./.
^ 10,
9,
and
the minor
11,
since in each case a
be observed that
also
3
3
I
1
n'^t
in
new
of these
all
examples the combination of the heptads produces a minorthird nonad. In first
heptad
Example
12,
results since the
however, only the involution of the
augmented
triad
is
the same whether
constructed up or down.
Example 45-13 shows the pentad formed by the simultaneous projection of two perfect fifths, two major seconds, and two minor seconds, with its seven-tone projection and Finally,
connecting hexads.
Example p
^
+
-t-
^
Pentod p^mn^s^d^t
1^I
2
/)
Connecting Hexad
(I)
J
I
Connecting Hexad
r
i
Inv.of 4y^ '*fj
I*
J
J
J I
2
I
'^
r 4
2
I
tl)
g
>J
t-^
4
3
^^
'^^
^
4'
2
I
Complementary Hexad
I
^ 2
''^
112
:^r
r
2
'
Complementary Hexod
4
comp.Heptad p p'^m^n'^s'^d'^t^ m^^n^s d t*^
9'iii
'
^^r
p3m2n3s3d3t
3
2
J
J
J
J I
(2)
Complementary Heptad
3
p^m^n^s^d^t
i ^ f* 12 12 3/}
2
45-13
(2)
^ I
t'
2
Pentad Complementary Penta Complementory
<^r
112 '\M ^
''^
3
The hexads of Example 45-13 have already been discussed in Chapter 39, Examples 39-8, 9, 10, and 11. It will be noted again 312
SIMULTANEOUS PROJECTION OF INTERVALS that the of
complementary hexad
of
hexad
(
1
)
is
the involution
hexad (2), and vice-versa. Note: The projections p^
the former latter
is
is
+
s~l
the involution of p^
and d^
+
s^
+
5^| are not
used since
(Ex. 45-1, line 3), and the
the involution of d^-\-s^ (Ex. 45-2, line
3).
Projections at
m^ are obviously the same whether projected up or down. The opening of the first movement of the Sibelius Fourth Symphony, already referred to, furnishes a fine example of the projection illustrated in Example 45-1. The first six measures utilize the major-second pentad C-D-E-F#-G|j:. The seventh to the eleventh measures add the tones A, G, and B, forming the scale C-D-E-F#-G-G#-A-B, the projection of the tetrad C-D-E-G.
313
46 Projection by
Involution with
Complementary Sonorities In chapter 34
we
observed
how
isometric triads and pentads
could be formed by simultaneous projection and below a given axis. From this observation
apparent that an isometric perfect
fifth,
series,
above becomes equally
of intervals it
such as the projection of the
can be analyzed as a bidirectional projection as well
as a superposition of intervals.
Example 46-1 to
make
illustrates this observation graphically. In
the illustration as clear as possible
out" the circle to F|; at
the other.
make an
Now
proceeding one perfect C,
its
complementary
if
ellipse,
placing
C
we have at
order
"stretched
one extreme and
we form a triad of perfect fifths by above C and one perfect fifth below
fifth
scale will
be the nine-tone scale formed by
the projection of the remaining tones above and below FJf at the other extreme of the ellipse.
Example
314
46-1
PROJECTION BY INVOLUTION
Example 46-2 proceeds
to illustrate the principle further
forming the entire perfect-fifth
series
above and below the
by axis
C, the complementary scale in each case being the remaining
tones above and below the axis of
Fj|.
Example
ESS3 flu *-
=m:
i *J
p2
46-2
ap^s
*s= p3
in
bQ
p3
^
S^
£_^
:g=m -^V^
*
¥
b«
''
£^
^°=^^ if
«
^
*
^^^ ^
i :=^^
p5
It will
be obvious that
equally well
by the
and below the
The
this principle
may
also
be
illustrated
projection of the minor-second scale above
starting tone.
projection of the basic series of the perfect fifth or the
minor second by involution rather than by superposition does not, of course,
another
projection different
add any new tonal
explanation is
of
material, but merely gives
the same material.
However,
if
the
based upon the simultaneous involution of two
intervals,
new and
interesting
sonorities
and
scales
315
COMPLEMENTARY SCALES Example 46-3a shows the simultaneous projection by
result.
involution of the intervals of the perfect fifth and the major
and below C.
third above
The fifth
first
line gives the perfect-fifth triad
above and below C, with
its
arranged in the form of two perfect
minor
formed of a perfect
complementary nine-tone fifths,
two major
thirds,
and two major seconds above and below
thirds,
second line adds the major third above C, with
its
scale
two
The
FJf.
complementary
octad arranged in a similar manner, and the third line shows a perfect fifth above and below C, with a major third below
C—
the two tetrads being, of course, involutions of each other.
The fourth
line gives the
and two major
thirds
pentad formed of two perfect
above and below C, with
its
fifths
complementary
heptad. Line 7 forms the projection of line 4 by the usual process of projecting
upward the order
of the
being arranged as two perfect
line 4, the tones of this scale
two major
fifths,
C.
The
complementary heptad of
and two minor thirds above and below
thirds,
right half of line 7 presents
arranged as two perfect
its
complementary pentad
and two major
fifths
thirds
above and
below F#. Lines 5 and 6 give the connecting hexads between lines
4 and
7.
Lines 8 and 9 form the octad projection by pro-
jecting
upward the order
and
their
3,
of the
complementary tetrads being the involutions
original tetrads of lines 2 is
complementary octads of
and
3.
lines
2
of the
Line 10 forms the nonad which
the prototype of the original triad by projecting
upward the
The complementary
triad of this
complementary nonad of
line
nonad
same formation
line
is,
of course, the
1.
1.
Example
316
46-3a
as the original triad of
.
PROJECTION BY INVOLUTION 2.^
}
p""
m'tTetrods
m
p^
3
2 p
I
jjji'^ J
P^
mnsd
[r
12
4
1
2
I
I
I
I
m^
n^
2
n? Pentad p^m^n^sd^
p2m2 f^s[i
^^^^ 12
11
1
I
Complementary Heptad
p2
|
m2
ri2
i 12
4
3
1
Hexods p,3„4„3„2^3 m n s d
5y5 Connecting
II
2
2
I
I
I
*r riir r nJ ii, 12 11 (3) i
(4)
I
I
j,Lompiementary Complementary Hexads
>JtJJ>^T-' 3
p^m^n^s^d^
jjjMJf ^"4
2
I
I
^w^p 3
(3)
I
Iny.of Comp.Heptad mv.oTUDmp.i-iepToa
7.^
JJ J>J 12 11
3
J
i
I
i
iJUp
il»
J]
I
(4)
1
4
12
o J^*^
•
m
^
1
1
Complementary Tetrads
2 \
W
na;re.:Vr«rr
p
mi I
.
I
2
11
12
4
II
I
p6m5n5s5d5t2
2
P
2'
I
iTf^rr^^
,2_2„2„l t
I
Complementary Pentad
4
P^m^ nSs^d^t'
j ,.^.
o
i:i."ii'"
Comp.Octads Lomp.uct
Inv.of
9/1
l
1
o
J^^n? It
|
J^ .
3
8-/,
o
p^m^n^s^d^t
im
J
I
2
2m2 n2 eh p'^m'^n^s't
5 1
2
1
J
p2 m'
J
p2
j
4 12
I
Comp.Nonad p8m6n6s7(j6|3 Inv.of
10. 'fi
2
11
12
1
s' T
I
t
fr iJ
I
5 2
p^
}
^^^^f^
2
I
o
i \
Complementary Octads
m
4
3.^ J
p'^mnsd
II
^
p
mr np sf
Complementary Triad
5
2
317
COMPLEMENTARY SCALES Example 46-3b forms the projection example
of the previous
the perfect
fifth
rather than
same two intervals two major thirds plus
of the
in reverse, that
is,
two perfect
fifths
plus the major
The pentad, heptad and connecting hexads
third.
the same, but the tetrads and octads are
are, of course,
diflFerent.
Example 46-3b Jm^
Triad
iS ^
m
2.tm £T 1
.1!
J jJ'' 4 3
3. .
Jm ^"'p
I
|l^»
1
^
13
Pentad
1
p^m^n^sd^
Hexads p
^ 112m I
2
I
m
n s d
2
I
2
I
I
I
2
I
112
1
318
12 11
f
p2
n2
r7i2
*^
2
^
2 ^^2 ^2
JbMJ ^^'T^
I
Complementary Hexads
I
2
I
I
^ruf^r^r 3 ?
? ? m'^IL
I
I
r"^ 2
I
Complementary Pentad
4
12
1
(jl
^
ii"M»:g "^ ~j|-C5f«-
p^m'^n^^d^
t£
'
I
I
1
tes rrr^^ 3
I
«
XJ-
2
Complementary Heptad
4
I
Z^g^m^n'^s^d^t
I
^fTr^
Comp. Heptad
Inv.of
3
3
1
P^
f4
I
V^r^iirritJiiJ
1
6./J
2
I
^^^^^s
4 3
l
3
n^ d^
tp2 m2 n2
jJ ^ jj 4 12
5.^ Connecting
I
"rV'Trii^^^ I
^ ^m
^
2
Complementory Octads
pm^nd
4
tm^
pm^nd
I
'
s 4
1
Tetrads
m^
'rftirriiJiiJJ,|i^MH
4
4
^^
Complementary Nonad
m'^
^^2 ^2
^fy
1
d' t
PROJECTION BY INVOLUTION Comp.Octads
Inv.of
8.^
2
I
2
I
I
J
I
2
I
I
p^m^r^
* t» ^ Complementary Tetrad /-
,
d't
$
1^
1
tm2
13
4
Comp.Nonod p6m9n6s6(j6t3
.2
^^Z^ d'*
2
I
,
4 3
I
I
9^ £^rn[n^s^d£j2
3
o o
o
p5m7n5s4d5t2
4 3
pi t
1
Inv.of
12
1
2
I
I
{p^rr^ji^d^
Complementory Triad ^
^z
2
I
Example 46-4a continues the same process and the minor third;
for the relationship
of the perfect fifth
Example I
^
^£
p^s
Triad
5
i
iS 3
p^mns^
i
Complementary Octads
Jp
2
2
522
mm w 322
£ H^ Pentad p^m^/?
^
p^ m^ n^ s^
i
I
J J
'^'i'^Ji^iiJ I
I
2
,2
s^
r
r^^«^iJ 21222
p2
$2
rr?
^^^B& O
r
"
•
Complermentary Heptad
J
£^ U?
"rrWftjjj 2 2
2
2
5-^ Connecting Hexods p m^n^'^dt
2
12
2 2
I
s^
it^'.'M^-fe
2222
Complementary Hexads
2
12
2
n' i
^ 8^ g V3
'
I
Is:
3
t
|iiM»:;iiB'f'
J
"r
„2 £^ ^2„l n rrr
2 2
)^mns^
I .X3L
J
"rr
2
3.fl^^n'i
4.^ t
Complementary Nonad
2
2 ^ le n.^ Tetrads
±1
46-4a
2
p^m'^nVdt^
319
I
COMPLEMENTARY SCALES Com p.
Inv. of
Heptad
p^m^nVd^t^
7-
2
2
1
2 2 2
3
Jp^m^s^nU
p6m5n5s6d*t2
8.^
2
12
I
1
J2^
2
t
2
I
2
2
p^m^s^
lbg"^°'"
t
p2 n't
n'
ii|iAlJlf 5
Comp.Nonad pSpn^nSs^dSfS
p^
2
n' t
I
I
^
2
2
3
2 2
of
Inv.
10
2
2
3
i jj^jjj^^^r 2
2
4^^^
Complementary Tetrads
2 2
p^m^n^s^dV
a,
Complementary Pentad
Comp.Octads
of
Inv.
}p2m2_s2
2
I
I
I
2
and Example
2
t
p
Complementary Triad
s nr
nn
2
5
I
I
t_2
gives the reverse relationship— the minor
46-4??
third plus the perfect fifth:
Example $
n^
Triad
y
n^
p't
Tetrads
t
rrii^A^i J iJJj
i"''
i
i
36
33
pmn^t
2I2IIII2
Compiementory Octads
if'''"'>j^^[^rVn-^'rrit^^WJJ 342
3.. t
n^
p'
IS
334
m324
Jn^F? -ti 4./(*-
2122
pmn^st
*
p^m^s^d^
Complementary Nonad
§^
^S
is 2
n^t
46-4Z7
j
tfB)|t S:
p^m^s^d
t^'::
I
Jii]
i
Pentad p^m^ n^s^t
2 12 112 212
p2 m^s^d'
"^^
2
Complementary Heptad
J
p^nn^s^
-ft
i ""l>jjJ^ 3 2
320
2
2
:*rr i^»^t^Jj i
2
1
2
2 2 2
;
tij^8
i 334
'
i*"h*
^
t
i
PROJECTION BY INVOLUTION Connecting Hexads
5^
4 2 3 4
pmrrsdt
^s 21222 i§
rr' Vii^J i
21222
i^jJ^'^r 3
2
2
^^^nMi^j
2
2
3
2 2 2 2
Comp. Heptad
Inv.of
p'^m^n'^s^d^t^
7^
Complennentary Hexads
^
iijjL N^f 12
1.811
i
2
2
p^m^s^
2 2
ii^
'fWiU2 3
Comp.Octads pSmSn^sSd^fS
^n^ p2
Complementary Pentad
2
2
Iny.of 8.>,
Jp^m^s^d't
J J iJi J
i J212. ji,j
|
2
II
pSm^nSsSd^tS
9.-
g |, ~
f
2'
*
i
1
I
342
HD
rl^
Inv.of '0/,
2
I
\
n
334
«2^2c2 p
p'
\m
[riip J||j] i
n^
i
n2
pi
t
-jt»
l^ gn^i^'rii^tJJ[ii^"rrT]rtig 3 3 4 3 2 4
2 2
Comp.Nonad p6m6n8s6d6t^
»iM, mrtf jjij J
^
i-
JJJr # jj ^jJt 12 2
Complementary Tetrads
$
p2m2s 2^2
Complementary Triad
3
2
6
^
^2
3 3
Example 46-5a gives the vertical projection of the perfect fifth and the minor second, and Example 46-5b the reverse relationship:
Example I
^^
P.
Tried p^s
46-5a
Complementary Nonod
tm^n^ _ _ _s^
j_
Complementary Octads
2„2 „2
„i
t HI
_ci
2.^$£
d^
t Tetrads
p ^ msdt
J]
p^
1 P
4
COMPLEMENTARY SCALES Complementary Heptad
J£^d2 Pentad p^m^s^d^t^
4
i
&
TiiKt
3'i¥
^ 14 ^4^
2
I
5.- Connecting
12
2 2 4
p4m2s2d^t3
i
iJ 4^llJ
j>
I
I
^.^ 4
I
Pui'l.'«B
f
I
I
nSs^d'^t^ II III a u M p'^m'^ |j 7/1
*f^2
I
(.
S
f,2
2 2
^^ ^P 4
I
uomp.i-iep Inv.of Comp.Heptod
^
Complementary Pentad
l>J-|g.W
114
2 2 Inv.of Comp.Octods .
2
J~
Hexads p^m'^ns'^d^t^ Complementary Hexads
2
8.
JjiJiiJ
2
^ ^^ g^
^
Ap2 ^2
4 2
Complementary Tetrads
pSm^n^s^d^t^
^-2
^jl
i
^JT^il^ l^_p'^
13 2 2 Comp. Nonad p8|n6n6s7(j6t3 I
I
I
Inv.of '<^/)
j ^i^
Complementary Triads
n2 52 p2
^^ rT,^&,^^^u-
^Jt^r'r ^^m2211nJ 121 i
d^
10
2.^
t
d^
i
d^ J '.ii
pi
±
^
Complementary Octads f^i ^l^liViiJiiJ 2
4
'
I
I
I
I
pmsd^t
I
I
i^CE
14 322
6
|
^2
^2 ^Z ^2
I
^s^ 6
3./1
I
p'tTetrads pmsd^t
=©«
46-5Z?
Complementary Nonad
sd^
Triad
p2
V' 5 2
Example J
j
16
4
I
I
I
2
r
I
»*
Im^n^
s^
d'
<''UPsi A_2 |m
„2 ^2 jKt n s d
PROJECTION BY INVOLUTION
^ 4.
I
d
p Pentad p^m
s
d
Complementary Heptad
W
t
^
14
2 4
5 - Connecting Hexads p^m^ns d^l^
5^?^ 2
I
4
I
2
m I
1
$
Inv of
8.^
t3 J
II I
I 1
I
I
!
m^
n^ s2
m2
_2 .2
I
Complementary Pentad
I
Inv.of
I
n
s
J d t Complementary Tetrads
(j2
d
2
t
J,*
I
m2 n^s2
d'l
J
d2
p't
2
I
I
4
'
I
I
I
^
I
J
14
'
'"" Comp. Nonad
'p'6;;'6';;6's?d'8t3
I
^
^«^ J i^X ^^^^^rl^^^^^^o^^i^tJjjl^vr^^
^
>.J
t
m^n^s^ d^
p2
^
p
^
g
I
* i^j^j I
^
-^^'•^
4
p5m5n4s5d6t3
10
^
j;^omD,Octads
p5m5n4s5do
122 9.
s
2 2
Inv.of
7
ir^
l"
s^ 14
4
Comp. Heptad p'^m'^n^s'^d^t^
n
¥iffm
i
2 4
p4m2s2d^t3
^
J|ja_[
2
m
Complementary Hexods
r
2
2
I
J
6'
Complementary Triad
^
Sfeo=
^2
14
Example 46-6a presents the relationship of the major third and minor third, and 46-61? presents the reverse relationship:
Example '
.,
^
m2
Triad
m^
46-6a
Complementary Nonod
I
2
I
I
2
I
t
£^m2d2n2
I
323
4
COMPLEMENTARY SCALES
^^
Complementary Octads
2^,jm^n't Tetrads pnrrnd
i
3
^-
4
I
2
I
I
i
2
I
I
I
p m^nd
J4
^A^^[^
jiJiiJ [r 4
3
1
*
JiJj] 14
p^m^n^d^t
Pentad
i4;^j^jiji|j
14
3
"
I
4
3
I
7T
t
;r"rr"^^it^tJ
1
I
I
Complementary Heptad
2
3
I
I
p^
rr\^
^
it''""i;jt
I
Complementary Hexads
'U^rhh
jju^j"^'!^ 2
2
I
m^d^ nU
p2
jt*>j1^^
r^r r^^ ^^^^JJ
i
^ ..Connecting Hexads p^m^n^sd^t
I
p^m^d^n't
p^^^^^^
^^rA
Im^nU i,un.
{
3
I
I
3
I
I
p^m^n^sd^t
^ 13
j^jjiJiiJ
Inv. of
7 i,
2
I
I
2
I
J
I
3
Complementary Pentad
I
p 2 m^cj 2
nU
J
I
14
p2m2d2n't
4
4
m
J
1
2
324
1
12
1
12
4 4
m^
J
m2
3
2 ^2^,2^2 Complementary Triad
jjiJiiJj^JY^rfei-::'^^
j
te j^ ^ 14
Inv. of
'0>,
rn^
n2
n'
4
i[3?:^^s'-":r^ri^^'i^[
2
Comp.Nonod p6m9n6s6d6t5
I
j
1
Complementary Tetrads
3'
I
jjjb^'r
2
^
3
9u£5m^nV^d^
^w
ETH!
11
3
1
J
Inv. of Comp. Octads 8-p5rT^7n5s4a5t2
*^r
12 14
Comp. Heptad
p^m^n^s^d'^t^
12
I
:r«rr»^iiJ»
11
3
^ j
ni2
n'T
3
projection by involution
Example '•
m^
n^
I
Triad
i m
^1 n2 n'^m't Tetrads
2.
^-
$
dt
i^it^ iiJiiJ)jj
2
I
I
TT r
3
;
JbJfcj J 4 2
J, 2 ^
m4 n 3 sd 3 t
^^
m^d^s'
#f:ttu)iB
itJiJ^j 1213111
^
ir"rr«^^iiJ«J
p2 ^2
,1
fc,
jj2
if^'tiB
3
I
Complementary Hexads
t^ i^
J|
J
it^
HI
I
I
Complementary Heptad
ji,j^jiJiiJ
Connecting Hexods p
P.
J p2
i
^2^2
fgi'^^mo-
i
2
pmn ^dt
n^ m^ Pentad p^m^n^d^t
i
2
r^rr
2
5.
I
J
5
iJ
(|,''iii');
p2 m^ d^ s^
Complementary Octads
I
J ^Jlr^J^j] J 351 315
ti I,.
I
I
pmn
n^m'^
J
#
3
I
J
rffrr'^«^>JiiJtJ^j]i"nt^tBtfe 2 2 II
^P
Ig
feEdfi
^>
Complementary Nonad
n^t
6
3
46-6??
i^r"^
i
3
1
I
^
3
p^m^n^sd^t
i
3
Jl
i
3
I
^
'
r'rr'i'^ 12 14
I
I
p^mSn^s^d^t^ O U p III 11
$
I
Complementary Pentad
p2 m^ d^
^^
§=
2 Inv.of ®-
-
3
I
I
I
,2
J
^
I
jt
I
I
I
pSmSn^s^d^tS " ~ L *"
I
r r 3
I
f4|J
i
m2 m "I? 2 n^
e
t
i'
Comp. Octads
p^mSnSs^dSt
jj 2 jjbJjiJ^iir 2 9-
r
Comp. Heptad
Inv.of 7-
''''^i
jbJuJ
j>
2
I
3
I
_2 d. m2 s* Complementary
pfm
Tetrads
(ui.!:teu: r^^r J 3
I
P^m^d^ il
* ii
IL'
1
2
t n*-
^ ^n2 —
5
"^
5
s't' 2.
3
„l m' ^
1
3
15 325
m'
t
COMPLEMENTARY SCALES of Comp. Nonad p^m^nSs^d^t^ p-m'"n'"s^a"'T^
Inv.
'0-A
j t
£2 ^2 £^ nn- ^2 a;-
Complementory Triad
s2 s;-
^
1111
12 12
Examples 46-7a and
j
6
3
46-7Z?
show the
n2
£
vertical projection of the
maior third and minor second:
Example '
^
m^
Complementary Nonad
m^
Triad
12
4 4 ^2^1 dt Jm
2.
3-
^-
pm^nd
Tetrads
I
3
I
I
I
3'
I
m^d^ Pentad p^nrrrTsd^
>J
13
fet
4 3
I
t^fJiy.! 2
4
I
I
p2
m^
%
^ ^ ^ ^^
%
m^
rf d^ p' t
^
^2
^2
2
I
Complementary Heptad
f
J"^
^^
j
2
pm^nd
\
5-
2
I
m^d'l
''^'
2'
I
1
Complementary Octads
4
J
f
46-7a
2
5^2
||llui »i|. l
Connecting Hexads p^m'^n^^d^ Complementary Hexads
jjg.ji^^r
'I
12 14
=
3
^^ 13
4
12
p'm^n^s^d^
13
12
Jm^n^d^
p^m^n^^s^d^t
I
12 14
3
Comp. Heptad
inv. of "^i.
4
2
326
I
4
I
2
Complementary Pentad |
13
4 3
^^
PROJECTION BY INVOLUTION Inv.of
8.
Comp.Octads
tm^n^d^p't Complementary Tetrads
p5m2nV*d5t2
^^ 12
j^
jj
i
i
i
2
2
I
I
2
I
jm^n^d^p^
I
d' t
jJ^ iS 13
4
Complementary Triad
.^2
4 4
2
I
[||
4 4 5
Comp.Nonad Lomp.Non p6m9n6s^d6t3
'°^
m^
J
2
I
Inv.of inv.oT
,-
tiJ
4
3
i.T#aj*rrtJ_J
i
3
I
I
r r
Jm^ n^ d^^'l
# ^j^ jtjJ J iJf I
^p
"
I
p5m7n5s*d5t2
9-
i;;^B"i«r
,ii l
112
3
1
JiiJp
^^Z^jl^
Example 46-7b I
^
I
^^
Complementary Nonad
Triad
sd^
iJ
r ir r
2
t
i
=o=
d^
^
II
10
I
^-
J
d^
»
4.^
df
I
4
I
M
13
p^^n^sd^
4
3
Connecting Hexods p m^n^s^d
g
jgi'^ r 12 14 3
iiJ
n^ d^ s^
I
f
^2
^2 ^z
I
I2I4II
'
^^'^^^
^^^^
^
^^
t
n?
2
I
I
4
I
d^
^
IIII4I2
Complementary Heptad
^
j
m^
n^ d^
2
Complementary Hexods
^ Sg^ I
3 4
p^m^n^s^d^
J.
I;
J ^ 3
iJ^J^
i*r"r^^r^
12
12 14
4
gi
rr]:^r*rtVitJiiJ«JtJ iite^Btt'
113
j''^ r ^ r j 173 113
jT^ Pentad
I
Complementary Octods
pmnsd^
m'l
""^°'"
I
I
m't Tetrads pmnsd^
137
m^
^^^^^m
mP€»-
i
|
3
327
s' *
|
COMPLEMENTARY SCALES Comp. Heptad
Inv.of
^
111
1^
II
a
I
{ J
jl^
Complementaryy reiiiuu uuiii^iBiiieinui Pentad
3
Comp.Octads
Inv.of
tn?Q? d^l'*
14
2
1
im^
p5nn5n5s5d6t2
t
3^11
7
II
Comp Nonod p6m6n6s7d8t3
'°.
|0«
n^ d^s't
Inv.of
,»
113
7
^
^plj
^jZ
-=
'»
V5
d^ m't
i'Hil;;^e^^;tf|Mlrjj|hM„j]^^
7*^ * jM^i^A^^r 14 12 I
f
4 3
Complementary Tetrads
13
I
m^
Q'd^
t |
^^^^^^^^^
p5TT,5n5s5d6t2
9.
qc d^2
rnc m2 nc
14 12
2 8-
u
| rn2 n2
d2 s^
3
r«^rvi«™«^r,+„..„ Triad Complementary t^:«^
^
^2
Ji>Jl|J^JljJ I
I
4
I
I
I I
I
10
I
Examples 46-8a and
46-8Z? give the vertical projection of the
minor third and the minor second:
Example I.
^
n_2
S
m 3
^-
$
s
^^ 2
I
Complementary Nonad
^m^s^ d^
Complementary Octads
t
mn^sdt
s
^"r'Trnt
6
II
12
3
^
1
m^
J n 2
m2
t
XiiiriiV f'""aijJrfri-^ijJ-rV II 12 13 2 12 6 i
3
^
i
t
^
^ I
328
2 6
2
Complementary Heptad
r^"rVrii^
112
4
2
p'
I
J
m2
ii
p' t
u^b,,^'!
i
iij
s2 d2
6 2
n2 d^ Pentad m^n^s^d^t
^^
s^ d^
«=
mn^sdt
tn^d'l
p^
^^^^i^p
6
n^d't Tetrads
fc^
3-
n^t
Triad
46-Sa
s2 d^
^p
:
PROJECTION BY INVOLUTION ^Connecting Hexods pm^n'sV't
Complementary Hexads
^F^
i^
5W.
116
I
2
I
6
I
2
pm4n2s4d2t2
6.
^^1'^^
jj 2J 2 J 4 ^ 2r
-
12
I
2 4
2
Comp.Heptad
Inv. of
7v,p2m^n^s5d4t2
m ^^ I
g
I
I
I
2 4
$
tj,i
2
of Comp. Octads p4^m5ri6s5d5t 3
m^ >
Complementary Pentadj n^ d^
d^
s^
^^g P
.^^bt!a
12
Inv.
2
4
2
12
I
^
12
3
9> p^mSnSsSdSfS
^
m^/
I
I
I
2
A
2
I
I
f^2 g2
d^p'4
t d*^
2
J
d2n't
^J
I
mns^d^
82
2
.k d t
f J J ^r^ ^fr^ ^p|.^[il.,[^fq_pg. i
12
6 2
6
j ^,2
46-8Z?
Complementary Nonod
10
Tetrads
A Jn
[
I
3
i
2
Triad sd
I
I
o
1^
^
6
p2 Complementary Tried
jj2
Example 1.^
2
I
^u^^ ^^_: ^Hi^^H/.^^rJ II 12 13 2 bo nv.of Comp.Nonad '°>)p6m6nSs6d6t4
^^ fc^
2
U Complementary Tetrads
j2
^^^^^^ I
6
128
I
I
' I
I
4
I
J
m2
s^
d2 n2
I
Complementary Octads
$
H)^!^
IIIII42 329
^2
n't
COMPLEMENTARY SCALES
i 5.^
m2n2c3H2f m^n^s^d'^t
PontnH Pentad
-|2n2
^ ^ci^pia
4.A ^
ii^jt'^ 2
Complementary Heptod
^
J
2
4
2
I
©=
2
Complementary Hexads
Connecting Hexads pm^n^s'^d^t
^i'^
:
^UbbJb
116
I
I
h2 d
c2 s^
^^
'r'lrVrii JtJ r
6
m^2
r^r^r'Ttpg
2
I
I
6 2
I
pm''n2s^d2t2
^^ 2
2
7y)
/I
4 2
p2m4n4'^d'^t^'^
i
IlL^
J"r i ^M^ 112 I,.
.
I
I
.
r^rrr^ riJ ^ 12 2 4 2 bfc>9
l
^
£
ni^
Complementary Pentad
;'l#H
:r^rrh.l.J ly J
Inv.
Q-*
l
Comp. Octads
of
*
1^2
p^^m^n^s^d^t^
^Jl^p J I
!>
•
^^ s
2
I
4
I
^
2
I
m2
^2
d2
£t bo p ^ '
:
i^ji^j^j^j'iJ^^r
K).
1
4 2
1
I
p^r|,|^3°'
111 The
I
I
4
J
m^
g
D
'
/*
.
^
td'^
n't
^
=&
2
^
^^
10
I
vertical projection of the perfect fifth
duphcates the perfect-fifth
and major second
the combination of the major
series;
second and the major third duphcates the major-second
and the
1
8
s2 d2_n2 Com^plementary Triad
I I
ii..o
^ 8
2
2
^
4
«fR^ t
6
Complementary Tetrads
nU
d^ tfe
I
9- p4m5n5s6d6t2
1
?3^
12
4 2
^ ^S
2
"i..
..
vertical projection of the
series;
minor second and major second
duplicates the minor-second series.
The
vertical projection of the
results in a curious
following chapter.
330
minor third and major second
phenomenon which
will
be discussed
in the
47
The
'Maverick' Sonority
The vertical projection
of the
minor third and major second
forms a sonority which can be described only as a "maverick,"
because
it is
the only sonority in
twelve-tone scale which
plementary
scale. It
complementary examine
it
is,
scale.
is
not
of the tonal material of the
all
itself
a part of
its
own comits own we should
instead, a part of the "twin" of
Because of
its
unique formation,
carefully.
In Example 47-1, line
1 gives
the tone
and major second above and below
it.
forms the descending complementary
C
with the minor third
The second
scale,
half of line 1
beginning on
G# and
containing the remaining seven tones which are not a part of the original pentad, arranged both as a melodic scale and as
two perfect fifths, two major seconds, and two minor secondsone above and one below the tone F#. In line la we follow the usual process of projecting upward from C the order of the complementary heptad, producing the scale CiC#iDiEbiEI::|3G2A— also arranged as two perfect fifths, two major seconds, and two minor seconds, one above and one below the tone D. We find, however, that the original pentad of line 1
is
not a part of
its
corresponding heptad (line la). There
can therefore be no connecting hexads. Line 2 gives the tetrad CsDiE^eA with
its
octad, while line 2a forms the octad projection.
give the tetrad CgEbeAiBb with
its
complementary Lines 3 and 3a
octad projection. Lines 4 and
4a form the projected octad of the tetrad CsDiE^^Bb, and lines 331
t
COMPLEMENTARY SCALES 5 and 5a form the projected octad of the tetrad CoDjAiBb. The tetrads in Hnes 2 and 3 will be seen to be involutions, one of the other. In the same way, the tetrads of lines 4 and 5 form involutions of each other.
Example
i
n5
^
Pentad
47-1
Complementary Heptad
p2nir,2s2d2|
rirB^j^jj^r^V'MriirViirr 16 2 3 2 1
Comp. Heptad
Inv. of
la.
^^^ If
I
.9 2/5
I
#
_s2
.ii-:^,;^^;
p2 s2
$
Complementary Octad
"rV^nir'ntrri iJii
"^2"^ t^
re
I
I
I
I
3
2
i
d^
m'
p2 j2
^2
ppl
ii
fj
I
p^m'^n^s^d^t^
$
^
iJbJ^J ^*^F7^ 13 ^^T 2
^ $2
p2
$
Comp.Octod
d2
^f^
2
3
d2
I
pn^sdt
I S
Inv.of
I
I
p'^m^n^s^d'^t^
Tetrad
1.
*,-^^
2o.
I
I
p2
j
^
XT «jt^-8-h
1
Tetrad
ln2s'| —
3
~
^C
"lib,
3 Inv.of
jbp
^j
(f 3o^
Complementary Octad * h i* ii
pn'^sdt P" sdt
6
I
4.^ t
*
I
$2 -
i ^^
I
I
2
Inv.
of
I
2
Comp Octad
II
I
d2
ii
mU
ii"'^ Ty
2
p2
s2
d2 m' t
^^ Complementary Octad $
17
I
p5m5n5s6d5t2
2
^s2
2
^^ 3
I
t
bri^rV«r'ir'iri
jftjj^j^J 332
I
p2 il.
'
^
n't Tetrad-^ pmnsHJ j—
2
^5,-
I
p^m'^n^s^d^t^
.^JbJ^JtfJ I
i^rT^fi^r'rriirr
1
Comp. Octad
t
r 2
I
I
I
i
3
p2
^
d 2 £> t
rrJ/::« e>ftB"" 2
2
t
p2
-nr
s2
d2
^^8-
nU
THE c
5-
i 5g
»
t
? _s^
Tetrad
.
jiU I
2
of
Comp. Octad
7
2"^
111112
I
p^m^n^s^d^t^
Example 47-2 shows the previous illustration to
s2
$ _p2
^^
bo- ^Q
SONORITY
Complementary Octad
pmns2d
^'
Inv.
MAVERICK
d_2 n' j
d^ n't
tP'
relationship of the pentad of the
its tw^in,
the pentad C-Cfl:-D-E-G, which
has the same intervallic analysis, p^mnh^dH. The the two pentads, each with
line gives
first
complementary heptad. Line 4
its
two complementary heptads but
gives the involution of the
with the order interchanged, the
first
heptad of
4 being the
line
involution of the second complementary heptad of line
vice versa.
The "maverick" pentad C-D-E-F-B
a part of the complementary scale of line 4.
The
both of
first
^or
^
will
1,
and
be seen to be
"twin"— second part of
its
pentad, C-CJ-D-E-G, will be seen to be a part
owTi related heptad and the related heptad of
its
its
maverick twin.
The connecting hexads the
first
also
show an
interesting relationship,
connecting hexad of line 2 being the "twin" of the
second connecting hexad of line
2;
and the
first
connecting hexad
of line 3 being the twin of the second connecting
Example I-
Pentad A p2mn?s2d^
Comp. Heptad
ujjJ Iff
I
23
2 3
(I)
I 3./^
r
I I
i' I
131 I
Hexad p'^m2n3s3d2t
^ Ifl
In^f
42
Comp. Heptad (2)
:^rMJjjjijjJr'^"'ih-^''^Mi II2I2 2I6 16 IIII32 12 12 2 2 3 I
I
I
I
I
Hexad twin
i2r P^m^n3s3d4i
l~
line 3.
47-2 pTmr^s2d2t
Hexad 2./)
hexad of
12
Comp. Hexad
I I
I I
O 2
p2m2n3s3d4t
c 6
I
I
Comp. Hexad
2 2
12
I
Hexad
2
I I
c 6
twin
p4m2n3s3d2t
i^4JJj 3 2
iTI
w
jJ J JJf 2 2
12
4
Comp. Hexad
^m I
I
I
I
3
Comp. Hexad
^^
iV-
12
3 2
333
COMPLEMENTARY SCALES
X^^ST""^ Comp Pentad I*
III 32
334
2216
'S°'3n^%^r'""c°"X>.ftntod
1*12124
1123
48
by Involution
Vertical Projection
and Complementary Relationship There
a type of relationship which occurs when
is
intervals are
by involution, as described in the previous two chapters, which explains the formation of the hexad "quartets" described in Chapter 39. If we compare in Example 48-1 the projection of two perfect fifths and two major thirds, one below and one above the tone C, together with its complementary heptad, with a similar projection of perfect fifths and minor thirds, together with its complementary heptad, we shall notice projected
a very interesting difference.
Example
J^
m2
Complementary Heptad p2 n2 m2
The complementary heptad
48-1
^
.
_
V^
It
Complementary Heptad $2 p2 m2
of
fG
E
C iF Ab that
is,
a perfect fifth
and major
third above
and below C, 335
is
)
COMPLEMENTARY SCALES
TCt A
A#
n which forms a perfect fifth, a minor third, and a major above and below FJf. The complementary heptad of
TG Eb C iF
.
,
A
a perfect fifth and a minor third above
E
jB which forms a perfect and below F#.
fifth,
and below C,
is
D
major second, and major third above
In other words, the projection of the projection of Xp^n^
third
is X'p'^m-s^.
ij^p^m^ is X'p'^'m^n^,
In the
first
whereas
pentad, the vertical
p and m is a part both of its own complementary heptad and of the complementary heptad of the vertical projecprojection of
p and
tion of
n.
In the case of the second pentad, however, the
and n is not a part of the vertical projection of its own complementary heptad, hut is a part of the vertical projection of the complementary heptad of the pentad vertical projection
Xp^rrt^,
that
This relation tion,
is,
of p
Xp^m^n^.
phenomenon makes
possible
a
fascinating
between pentads and heptads formed by
resulting
members
in
quartets
of
"diagonal"
vertical projec-
connecting hexads
all
of
the
which have the same intervallic analysis. In each consists of two hexads having differing formations but with the same intervallic analysis, each with its own involution. ( See Chapter 39. case
the
If
the
Chapter 336
of
"quartet"
student will re-examine the material contained in 46,
he
will observe that the
same phenomenon which
VERTICAL PROJECTION
we have
just
observed
in the vertical projection of the projection
p^n^ also occurs in the vertical projections of p^cP, mrn^, mh^, and n^(P.
We
have already discussed
in detail in
Chapter 47 the
peculiarities of the vertical projection of n^s^.
The reason for this phenomenon becomes clear if we examine Example 48-2. Here again we have the circle of perfect fifths "stretched out" with C at one extreme of the ellipse and F# at the other. The letters p, s, n, m, and d at the top of the figure represent the intervals which the tones G, D, A, E, and B, and the tones F, B^, E^,
A^ and
form above and below the tone
and p below the figure represent the tones E#, A#, D#, G# and C#, and the
C; while the letters d, m, n, the relationshhip of
D\), s,
tones G, D, A, E, and B, below and above the tone,
Example
Now
if
we
Ffl:.
48-2
project the intervals of the perfect fifth
and the
major third above and below the tone C, the remaining tones,
which constitute the complementary heptad, become the perfect major third, and minor third above and below Ffl:. However, if we project the perfect fifth and the minor third above and below C, the complementary projection above and below F# becomes the perfect fifth, major second, and major third. Hence it becomes obvious that the projection of the minor third above and below the axis, C, cannot be found in the complementary scale above and below the axis, F#, since the minor
fifth,
337
COMPLEMENTARY SCALES
C
and below below and above FJf. third above
are the
same tones
as the
minor third
There follows the list of pentads formed by the projection of two intervals above and below the axis C, with their complementary heptads arranged above and below the axis F#: X p Vn^
t fs^
It will
p-n
p-^s''m''
22 p^m^
p'^n-m-
p^(P
s^n^m^
2
2
9
s^n^
ph^d^
s^m^
p"n^(P
sH^
s^nH^
n^m^
p^m^d^
n^d^
s^m^d^
m^d^
n^m^d^
be noted that
in four of the ten possible projections,
the pentad contains the
plementary heptad. In
same
six of
vertical projection as
com-
its
the projections, the heptad does not
contain the vertical projection
of
the
same
intervals
as
its
pentad prototype.
Example 48-3 works out all of the relationships based on this which result in the formation of the hexad "quartests." Lines 1 and 2 give the two pentads formed by the vertical projections p^m~ and p^n^. The heptad of line 1 is the principle
projection of the pentad of line is
2,
while the heptad of line 2
the projection of the pentad of line
dotted
lines.
The
as indicated
1,
four connecting hexads,
by the
upon examination,
prove to have the same intervallic analysis, the second hexad of line 1 being the involution of the
first
hexad of
line 1;
the second hexad of line 2 being the involution of the
first
and
hexad
of line two; the four together constituting a quartet having the
same
intervallic analysis.
All of the other hexads in this illustration are
same principle and each quartet 338
of scales has the
formed on the
same
analysis.
VERTICAL PROJECTION
Example p2
—Q
-^
S
i
,^.
m^
s^
.09,
^p^mSgi
ITTW m2
p2
n2
/^3
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rr,2
^^
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m2 ^2 ^
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s^aa
p2 n^ m'
._h2
m2 ^::
^
2
e2
° bS
J iJ
f
^ ^ *=«
12
g ''^
r r
13
J3J ^jJ 3
i
d3t
J
K^ 12
1
1
„ „
^ii
^
J
i
I
iJ
p^m^n^s d^t
m^2plf ,
0^\
I
3
I
3
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I
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p4m2h3s3d2t
d^ b'^8
£2s2d'
t
t
il/JtlJ ^ ^^r P 3 2 3
p2s2d'i
rn'O. 3 2
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,5.2
i
£?
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p'*m2n3s3d2t
_n2 t
n2s2^'
t
i"lJ
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£2
l?
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t|
m2 b'H
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rtr J JlJ
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2
4 z
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;;'""'-
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fp2m2dlt
^
^m^
*
p2
2
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2
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n2 i
Jj
lp2n2mU'
J J
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P'
l
}
^^ t _5
Pl^p 2' 2
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^
3
n2
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ijl
2
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1
JlJ
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a 2 r^-
,p3m3n3s3d2t
I
J
I
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$p2
i P^
48-3
^Q
4
3
m2n2sit
Jb^
J 2
I
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2
p3m3n2s2d3t2
^^
^
r 4
'
I
I
3m2n2s3d3t2
p2
O
d2
s2
£2d2slT
L
#
ibj j,^ (n2
,j2
n2 '''11
i
13 ii
I
^ ^B
$
n2s2m'
I'
2 4
' I
3
2^
p3m2n2s3d3t2 t
t
n2s2mT
^
l^^;
*•? 2
r 1
15R 1^
1 1
2'
I
I
^ 339
t
'
COMPLEMENTARY SCALES
S&^— »
^k
f^
ta
m2
t
i
b
-
£
'
-
#w* 12
m^ d2 _^
.
n''^B
\
,n
|
^
m^dSg
iJ
J
p3m4n3sd3t
^ '
4
1
?2i
2i«
4
'
I
I
t
'
^ "" r 3
^
nrr^i I
'3
17
3
I
I
There
remains
only
one
I
other
5 3
I
I
group
of
of a projection
from a tetrad
to
its
may be
15
hexads
considered, the isometric twins discussed in Part
48-4 indicates that these sonorities
1'
1
III.
3
1
to
be
Example
considered as part
related octad. Line la gives
the tetrad formed by the projection of two minor thirds and a perfect fifth above C. Line lb gives the isometric twins, the first
formed by the simultaneous projection of three minor thirds
and three perfect of
two minor
fifths,
and the second formed by the
relation
thirds at the interval of the perfect fifth.
combination of these two hexads forms the octad of line
which
is
Ic,
the projection of the tetrad of line la.
Line 2a
is
perfect fifth
similar in construction to line la except that the is
projected below C. Line 2b
except that in the
340
The
first
is
similar to line lb
isometric twin the perfect fifths are
VERTICAL PROJECTION projected below C, and the second twin
formed of two minor
is
below C.
thirds at the interval of the perfect fifth
(It will
be
observed that the twins of line 2b are merely different versions of those of line lb since, is
begun on A,
first
it
will
the order of the
if
twin in line lb
first
be seen to contain the same intervals
twin of line 2b: AgCsDiEbgCbiGt]. In the same way,
order of the second twin of Hne lb
is
begun on G,
cate the intervals of the second twin of line 2b:
it
as the if
the
will dupli-
G3Bb2CiDb2Eb3
Gb.)
Line 2c of line
the octad formed
is
by the combination
of the hexads
2b and proves to be the projection of the tetrad of
line 2a.
In similar manner, lines 3a, 3b, and 3c show the projection of the tetrad formed of two minor thirds and a major third above
show the
C, while 4a, 4b, and 4c of
two minor
thirds
above
C and
projection of the tetrad formed a major third below C.
Lines 5a, 5b, and 5c explore the projection of two minor thirds
and a major second above C, while
two minor
projection of
thirds
lines 6a, 6b,
and 6c show the
above and a major second below C.
Lines 7a, lb, and 7c and lines 8a, Sb, and 8c are concerned
with the projection of two minor thirds and a minor second. Lines 9a,
9fc,
projection of
and 9c and
two perfect
lines 10a, IQb,
fifths
and a major
and 10c concern the third.
Lines 11a, lib, and lie and lines 12a, 12b, and 12c show the projection of
two minor seconds and a major
third.
The relation of two perfect fifths and a minor second, or of two minor seconds and a perfect fifth, does not follow the same pattern. It
is
interesting,
however, to observe in
that the combination of the hexads p^
seven-tone scale which
is
@
lines
d and
d^
13a and 13b
@
p form a
the involution of the basic perfect-
fifth-tritone heptad.
341
complementary scales
Example n^+ p
pmn
Tetrad
=
iWSr^^
3
dt
^
JbJ
3
48-4
I
Hexad
Hexad =
+ P^
p3m2nVdgt2
^s ^
>o"
lb.
^^
2
Combinotion Ic.
Hexods
of
^a
i I
I
£^+
p
2a.
3
I
^^0=^
n^
4
-f
3
3
£* I
^^
yi^ftJfaJ ]|tf§
I
^p
12313
|^5rj6s4(j5f 3 Comp.Tetrod
=
^ te ^?^
1
B jt^ 3 2
r^l
b^g.
p3m2n4 s2d2t2
pn^sdt
= Tetrad
I
:^
1312
^ 2
I
@
n2
I
Hexad
+ p^* 2b
^m2n^s£d2t2
=
^t^°""^»bo>jJ^J^^r 3 2
Combination of Hexads
^^
2c
2
3
2
I
i
=^^6s^3 Comp.Tetrad
rn
2
Hexod p2m3n4s2d2t2 _n2
=
=
1221
12
n2 + m'l
I
1
Hexad
~
p2m3n4s2d2t2
tletl^'^''
*
jbJljJt^^
^
'31213
n^+nn' 4
^^ w^^
II
3
2
I
pmn^st
= Tetrod
^gm
^
3
^
n2+ p'f
m
@
5^^6s5H5t3^®^P-^®*'"°**
=^^^ ^^ 3
3
^W
2
>JilJb>^^'^ l^bot^^ 3
3c.
4
12
^
Combination of Hexads
4o.
3 2
3
^°" ^oil^
^
^^m
1
I
I
p3m2n4s2d2t2
s£
^3^^^,^'
b^g-fte-
i
xx:
Hexad *
Tetrad mn^sdt
=
+ 3 b.
I
^ -
|@ rc^
bobi.hi^bol'"
n2+rn' 3o.
13
"."^
+
3
2
m2^
= p"
m^n4s2d2t2
m
@
n^
*
=
pll^gn4s2d2t2
4b 3
Combination of Hexads
4c
*
l
342
I
12
2
2
^21323
1
Comp.Tetrad -§ai§in6fe5crt5
>(bbo)bo ^"t^°'^"'^^^
2
12
12
n5+
p^lf
^^^ ^^
^
3
3
2
i
2
VERTICAL PROJECTION +
n
mrrsdt
Tetrad
=
s'
mS
5a
-^
.^k>
Hexad
s3
+
^
5b
Combination of Hexad s= 5c
n2 + s
6a.
^
'
3
p2m2nVd2t2
i
2
^r^
ji
2
1
a ^^^ 12 12
33C
CompTetrod
l
112
11
2
112
n4m5n6s5d5t3
t7ok-%
I
iJbJtlJfr
2 Octad
@
^
iE ltv>
Hexad
o
r
p2rAAVt2
I
i
J
il^
.Ka'^^^-'t
3
m i^^ 3
3
4
Hexad
+
'^
"bou,t,„ of
Hexads
2
I
n2 +
i
2
I
i
^^i,»"
"^""^
3
I
p2m2n4s3d2t2
».>jU ^J^'''r i
^12 12
i_n2 + s'
4
t
m
ii^J ^
1^
t>^a
I
i^b^fe",
1'
I
=
4
Tetrod pn^dt
=
d'
2
2
^
D§m5n6c5ri4t3 CompTetrod
=
^^^^^
6c.
7a
j^jt,jiJiiJbp
*3'3
Combination
Hexad
@
_p2rn2n4s3d2t^2 n2
=
_s3l
^o»
b.
^
l'
I = Tetrad pnr^n2st
n3 6
*"
I
2
3
Hexad
Hexad
p2m2n4s2d3t2 7b. I
Combinotion of Hexads
^^^^
=
3 3
6s5d5t3
7c.
F^^lJ
,ljv,l^e>k^botlotli I'
1
dU
n2 t
!%
^
n^
+
8a.
8b.
12 12
I
Tetrad
=
I
2
2
I
^
.li|tf§*^
2 3
m
Hexad
d3
•^-
=
I
3
p2m2n4s2d3t2
vs^vs^
>bo"t^°" 2
11^+^
*
I
i,^^r,r 335
i^ ^ *
I
2
pmrftit
Combination of Hexads = 8c.
I
^'^^''®^'^°*^
^
I
I
311
Octad -5j^5p6s4ci5t3
bo qo l
^^
3 3
^2
@
Hexad _d
^^ Comp.Tetrad
I
=
p2m2n4s2d3t
XT >^ijbJJ^^r 2 12 15 i
n^-k-6^
t
^g^ ^^P 7^ 343
U
j
COMPLEMENTARY SCALES ,2 f p'^
,r.l _-r ^ ^2^„^2 t sTetrod + m' p^mns
4.
_2
I
+
neXOa
-
m
^^
9b.
2 2 3
P2
I
m'
i Q^-^-
=
I
2 2 2
o
j^
£3
+
Combination
i
^ 2 I
d^ Ila
2
,
@
3
^
*^2
I
4
I
2
_p^-»-m't
^r'r^jj ^ i
bo be
I
I
m'
=
5
I
I
mns2d2
Tetrad
^^
8- -J* l«-
I
2
Hexad IIl^=p2m3n2s3d4t
Hexad
@
i
^°^° jj^M ^ i ^->" 4
° """
'
1
I
lie
I
:
r
I
I
I
2
J J
[>b|» I
I
2
^
=p2m3n2s3d4t
ID
Combination of Hexads =p4^°5n5s6d6t2 CompTetrad
344
^S
=p4m3n2s3d2t
b-O-*^
Hexods _=p^jJ|5°^4g5jj5^3Comp.Tetrad
1^
I
Hexad
,
i
k> ^
«-
2 2 3 11 Octad
2
3
I
«»
'
m
^
j^i
+
If
^g
^
^i
* «#<=" i
2
£2
L^obii i^bo :
+ lib.
of
+
r'r^r^rk^
HexQd p4m3n2s3d2t
=
o "bvn
lOb.
^
p2
I
,
_r]n2i
I
S
^2^5
,
lOc.
Comp. Tetrad
p^msdt
Tetrod
10a. b1
a
tt""UJjji^^r 14
=olP=t
ott"Qft^" II 12 2 2
2
Hexad
-
m r«
@
1
Octad Combination of H9)iods=p6m5n5s6d4t2 9c.
^
,^2
-p4m3n2s3d2t
ill
i
d2
# +
^obon
m'
^
VERTICAL PROJECTION d2
mU
+
I2a.
Tetrad pmsd^t
=
^m i«^i
I2b
:xx
I
Combination of Hexods 12
^
i
c.
l~
_p2
II 1
I
@
d
fe
:3o.
at
r
-
*
p^^T4s5d6t3
-
;|
14
11
^
S
Comp.Tetrod
Hexad p^m^ns^d^t ^
Combinotion of Hexods
13b
=
b L *| c ib < _
s
a
g
^ w^ 114
=
i
i^^3j5t3
^F'
1
p
I I
I I
+
m' t
pWns^d^2
Hexad
S
OBO $ iff
CompPentod
xsr
d^
=&
@
d2
|
15K I
I 1
I
^^f ^ s*
e^ is*
c^if^ 5^1'
I
Note: The tetrads of Example 48-4 have
all
been discussed
in
as projection by involution. For example, tetrad la Example 48-4, (n^ + p^), is the same chord as the tetrad of Example 46-6fo, lirie 2, {%n^m}^), and is itself the involution of tetrad 8a of Example 48-4, ( n^ + d\^), which appears in Chapter
Chapter 46 of
46-6Z?, line 3, as Xrem}\^.
345
49
Relationship of Tones
Equal Temperament
in
We
come finally
to the formidable but fascinating task of
attempting to show the relationship of these galaxies of tones within the system of equal temperament. The most complete presentation,
and
in
many ways
the most satisfactory, would
seem to be that involving the abstract symbolism which I have employed in the large diagram accompanying this text. Although this symbolism may at first glance seem foreign to the musician's habit of thinking tones only through the symbol-
ism of written notes, and may, therefore, seem "mathematical" rather than musical,
it
has the great advantage of presenting a
graphic,
all-embracing picture
from the
artificial
tone
of
relationship
and awkward complexity
For example, the symbol
divorced
of musical notation.
p^5^ indicates the simultaneous pro-
two perfect fifths and two major seconds on any tone, up or down, and in any position. This one symbol therefore represents the sonority C-D-E-G in any of its four positions: C-D-E-G, D-E-G-C, E-G-C-D, and G-C-D-E, together with its
jection of
iC-B^-Ab-F,
involution
in
its
four
positions:
C-Bb-A^-F,
Bb-Ab-F-C, Ab-F-C-Bb, and F-C-Bb-Ab, plus the transposition of these sonorities to the other eleven tones of the chromatic scale.
The one symbol presentation
assume a
of
size
therefore represents ninety-six sonorities.
beyond the realm
of the practical. It should
noted that the order of half-steps of the chart as
346
The
such a chart using musical notation would
this sonority,
223(5)-C-D-E-G-(C)— may
also
be
represented in
appear in the ver-
,
RELATIONSHIP OF TONES IN EQUAL TEMPERAMENT sions 235 ( 2 ) 352 ( 2 ) or 522 ( 3 ) 225(3), 253(2), or 532(2). ,
I
;
and
322 ( 5 )
in involution as
cannot overemphasize the statement which has reappeared
in different
forms throughout
this text that
my own
concern
is
symbohsm but with sound. The symbols are a means in clarity of thinking. They have value to the composer only if they are associated with sound. To me the symbol p~s^ represents a very beautiful sound having many not with to
an end, a device to aid
its position, doubling, and which precede and follow it. One other word of caution should be added before we take off into the vast realm of tonal space which the chart explores. The student who has worked his way slowly and perhaps painfully
diflFerent
connotations according to
relationship with other sounds
through the preceding chapters cannot only with the vast scale,
number
fail to
be impressed, not
of possibilities within the chromatic
but also with the subtleties involved in the change or the
He may
overwhelmed both by the him in the apparently simple chromatic scale, and wonder how any one addition of one tone.
amount and the complexity
feel
of the material available to
person can possibly arrive at a complete assimilation of
this
material in one lifetime.
The answer, were
to
of course,
is
that he cannot. For
if
a composer
have a complete aural comprehension of
all
of the
he would know more than all of the composers of occidental music from Bach to Bartok combined. This would be a formidable assignment for any young composer and should not be attempted in a one-year course! The young composer should use this study rather as a means of broadening his tonal understanding and gradually and slowly tonal relationships here presented,
increasing his tonal vocabulary. tionships
which appeals
to his
absorbing this material until will
then
naturally,
speak in
and
as
this
it
He may
find
one
esthetic tastes
becomes a part
"new" language
as
series of rela-
and
set
about
of himself.
He
confidently,
as
communicatively, as Palestrina wrote in his
idiom, providing, of course, that he has Palestrina's talent.
347
COMPLEMENTARY SCALES
One that
much contemporary music
of the greatest curses of
is
uses a wide and comphcated mass of undigested and
it
The end
unassimilated tonal material.
chaos not only to the listener but,
The complete
himself.
used with mastery
is
I fear,
becomes tonal often to the composer
result
assimilation of a small tonal vocabulary
be preferred
to
infinitely
to
a
large
vocabulary incompletely understood by the composer himself.
Let us
now
pocket of corner
turn to an examination of the large chart in the
this text.
we
Beginning
at the
find the letters p, d,
six basic intervals:
the perfect
s,
fifth
extreme right-hand lower
m, and
n,
t,
symbolizing the
or perfect fourth, the minor
second or major seventh, the major second or minor seventh, the minor third or major sixth, the major third or minor sixth,
and the augmented fourth or diminished fifth. Below each of the letters you will find a number of crosses, 5 under p, 5 under d, 6 under 5, 5 under n, 6 under m, and 3 under t. These crosses serve as abbreviations of the interval symbol, that is, every cross under the letter p represents that interval. A cross indicates that the interval, of which the symbol appears at the top of the vertical column, of which the symbol appears which the cross is located.
is
included in the triad,
to the left of the horizontal line in
Proceeding laterally to the
left
we
devoted to triad formations,
III.
Here, again, the crosses repre-
find the section of the chart
sent abbreviations of the triad symbols. In other words, each
on the
cross
laterally
sents
the triad
with the triad symbol p^s repre-
line
The same thing is true of the crosses triads pns, pmn, pmd, and so forth. divided by dotted lines into groups— the first
p^s.
marking the positions of These four
triads are
all
contain the perfect
minor second; ms^
is
basic minor-third triad; last
two
fifth;
m^
is
triads are those in
predominates.
all
contain the
The numbers
nH
is
the
the basic major-third triad; and the
which the
interval of the tritone
to the right of the triad
indicate the order of half-steps
348
the next three
the basic major-second triad;
which form
symbols
this triad in its basic
RELATIONSHIP OF TONES IN EQUAL TEMPERAMENT
C becomes C2D5G5(C),
position— p^s above the tone
having
the order of half-steps 25(5). Each cross in this section of the
whose symbol appears
chart indicates that the triad,
at the left
included in the tetrad identified by the
of the horizontal line,
is
symbol at the top of the
vertical
column in which the
Proceeding upward from the
triads,
we
above them the section of the chart devoted
cross occurs.
immediately
find
Here
to tetrads, IV.
again the crosses represent the tetrad symbol proceeding ver-
downward. The tetrad P^, for example, will be found below the symbol on the first, second, fourth, and fifth spaces of tically
the chart.
For the sake of space the interval analysis of the tetrad given as
and
t.
six
numbers, without the interval
The numbers
letters p,
should therefore be read: three perfect 301,200, three perfect
fifths,
fifths,
no major
The
half-steps
being
C2DgG2A(3) (C). Each cross in that the tetrad,
s,
is
d,
252(3),
tetrad P^
having the analysis
thirds,
one minor
two major seconds, no minor seconds, and no of
n,
to the right of the interval analysis represent
again the order of the sonority in half -steps.
order
m,
that
is,
third,
tritones;
the
above
C;
this section of the chart indicates
whose symbol appears
at the top of the vertical
included in the pentad identified by the symbol at the
column,
is
extreme
left of
the horizontal column in which the cross occurs.
Proceeding laterally and to the
left
we come
to the section of
pentads, V, which occupies the large lower left-hand section of
the chart. Here, again, the crosses indicate the pentad on the
same first,
lateral line.
The pentad
P*, for
example,
is
found on the
second, fourth, and sixth spaces of the lateral line following
the symbol P* This pentad has the analysis 412,300, and the order of half-steps 2232(3), which might be represented by the
Each cross in this section of the chart indicates that the pentad, whose symbol appears at the left of the horizontal line, is included in the hexad identified by the symbol at the top of the vertical column in which the cross occurs. The six-tone scales, or hexads, VI, will be found above the
tones C-D-E-G-A-(C).
349
.
COMPLEMENTARY SCALES pentads and forming a connection between the pentads below
and the heptads above. The
which
crosses, again, indicate of
hexads the individual pentads below are a part. The pentad P^ will
be seen to be a part of the hexads
^pV^\ and
F^, pns,
p^m^.
P° has the analysis 523,410, indicating the presence of five perfect
fifths,
two major
thirds,
three minor thirds, four major
seconds, one minor second, and no tritones.
order of half-steps 22322 ( 1
)
,
has the indicated
It
which would give the
scale,
above
CsDsEaGaAsBd, ( C ) The portion of the chart above the hexads gives the heptads, VII. These scales are the involutions of the complementary scales of the pentads below and are so indicated by the letter "C." The heptad VII p^ is, therefore, the corresponding scale of C, of the tones
the pentad
V
P^.
The
scale C, pns/s, corresponds to the
pns/s, the heptad C, p7?in/p, corresponds to the pentad
and so
pentad
pmn/p,
is used as an abbreviated form of the symHere each cross in this section of the chart indicates that the heptad, whose symbol appears at the left of the horizontal column, contains the hexad identified by the symbol below the vertical column in which the cross occurs.
bol,
forth.
@
pns
(Pns/s
s.)
Proceeding
now
laterally
the right
to
VIII, above the tetrads. These scales are scales of the tetrads below, so that
it is
we all
find
the octads,
the corresponding
not necessary to repeat
the symbol, but only to give the intervallic analysis and the order of half-steps. For example, the corresponding scale to the tetrad, P^, is
the octad opposite, with the analysis 745,642 and the order
1122122(1), giving the
scale,
above C, of CiCSiDoEoFJfiGsAs
Ba,(C). Proceeding vertically upward to the top of the chart are the nonads, IX, which are the counterparts of the triads at the
bottom
of the chart.
Proceeding horizontally to the
right,
we
find the relationship
between the nine-and ten-tone scales. It will be observed that the six ten-tone scales which are on the upper right hand of the chart are the counterparts of the six intervals which are repre350
RELATIONSHIP OF TONES IN EQUAL TEMPERAMENT sented at the lower right hand portion of the chart.
At
first
glance, this chart
mendous amount ship.
may seem
to
be merely an interesting
but careful study will indicate that
curiosity,
it
contains a tre-
of factual information regarding tone relation-
For example, the relation of two-tone, three-tone, four-tone,
and five-tone sonorities to their corresponding ten-, nine-, eight-, and seven-tone scales will be discovered to be exact. If we begin with the pentads on the left of the chart and, reading down, we add 2
to the
number
of intervals present in each sonority— except
where we add one-half two, or one— we automatically produce the intervallic compo-
in the case of the last figure, the tritone,
of
the sonority's corresponding heptad. For example, the
sition of first
pentad has the
this
the
found
number
to
intervallic analysis 412,300. If
we produce
222,221,
634,521,
we add
which
will
to
be
be the analysis of the corresponding heptad. The
second pentad has the analysis 312,310. Adding to intervals 222,221,
we produce
the
this
the analysis 534,531, which
is
the
analysis of the heptad C. pns/s. In like manner, the analyses of all
of the heptads
may be produced
from that of
directly
their
corresponding pentads.
we have already pointed out that the and octads have a corresponding relationship. This may
Proceeding further, tetrads
be expressed arithmetically by adding to the
intervallic analysis
of the tetrad four of each interval, except the tritone,
The
again add half of four, or two.
444,442,
we we produce 745,642, which
of the corresponding octad.
211,200.
655,642,
which proves is
be
intervals
tetrad, p^s^, has the
444,442,
we produce
the analysis of the corresponding
true, again, of all tetrad-octad relationships.
The triad-nonad
relationship
the triad analysis of the addition
to
proves to be the analysis
The second
Adding the
analysis
octad. This
analysis of the four-tone
observe to be 301,200. Adding to this
chord
perfect-fifth
where we
is
six of
is
expressed by the addition to
each interval except the
one-half of
bottom of the chart
is
six,
or three.
The
tritone,
where
first triad at the
p^s or, expressed arithmetically, 200,100.
351
COMPLEMENTARY SCALES Adding
to this 666,663,
we produce
866,763,
which
be found
will
to be the analysis of the corresponding nine-tone scale at the
top of the chart.
The
becomes
triad pns, 101,100,
relationship 101,100 plus 666,663, or 767,763,
in
and
The single interval may be projected to its by the addition of eight of each interval, and four tritones. The decad projection of
p,
m,
889,884,
and so
tice to
read
it
and
s,
of the
The
d,
fifth
projec-
minor
third,
forth.
Since this chart
fifth, p, at
n,
the perfect
therefore becomes 100,000 plus 888,884, or 988,884.
becomes 898,884;
nine-tone
ten-tone counter-
part
tion of the major third
its
so forth.
is
of necessity biaxial,
we
accurately. If
it
may
begin with the interval of the
the lower right hand of the chart
ing laterally to the
pmn, pmd, and
we
pdt. Conversely,
from the triads to the tetrads
we
find
by proceed-
contained in five triads
left that it is
triad, p^s, contains the intervals
take some prac-
p and
we
p^s, pns,
find that the perfect-fifth s.
Proceeding
now upward
find that the triad p^s
tained in the tetrads p^, ph^, p^m^|,
p^d^'l,
and
is
con-
p^d^. Conversely
the perfect-fifth tetrad p^ will be seen to contain the triads p^s
and
pns.
Proceeding laterally to the
we
left,
observe that the tetrad P^
is
from the tetrads found
in the
to the pentads,
pentads
P^, pns/s,
^p^n^l, and p^d^. Conversely, the pentad P* contains the tetrads P^, pV, and p/n.
Proceeding upwards, from the pentads to the hexads, that the pentad P^
is
contained in the hexads
and p^m^. Conversely, the hexad, pns/s,
P^,
we
find
P^, pns, p^s^d^l,
contains the pentads P^,
and pmn/ p.
Proceeding again upwards, from the hexads to the heptads, find that the
hexad P^
is
a part of the three heptads
P®,
we C.
and C. pmn/p. Conversely, the heptad P^ contains the hexads P^, pns, n^s^p'^X, and p^/m. Proceeding now laterally and to the right, from the heptads to the octads, we find that the heptad P^ is a part of the octads pns/s,
352
RELATIONSHIP OF TONES IN EQUAL TEMPERAMENT
P\
C.
pV, and
C. p/n. Conversely, the octad P^ contains the
heptads P^ C. pns/s C. jp^n^j, and C. p^d^.
we
Proceeding upward, from the octads to the nonads, that the octad P^
is
found
in the
nonads P^ (C. p^s) and C. pns.
pV,
Conversely, the nonad (P^) contains the octads P^, C.
p^m%
C.
nonad P^
find that the s.
C.
C. p^d't, and C. pH^.
Finally, proceeding laterally,
we
find
is
The arrows on
to the decads,
contained in the decads C.
p,
and
C
Conversely, the decad
(C.p^s), C. pns, C.
from the nonads
p (or P^) contains the nonads P^ C. pmd, and C; pdt.
pmn,
the chart which indicate the progression from
the intervals to the triads, from the triads to the tetrads, the tetrads to the pentads,
and
so forth,
may be
helpful in tracing
various "paths" of tonal relationship.
As the student examines the analyses of the various or
he
scales,
find
will
analysis of the triads
is
that
sonorities
The
they differ in complexity.
simple.
The
analysis of the tetrads
is
comparatively simple, but there are several forms that have at least
two possible
analyses.
The second
tetrad, p^
s^,
for example,
may be
analyzed as the simultaneous projection of two perfect and two major seconds (pV); or as the projection of a perfect fifth above and below an axis tone, together with the projection of a minor third above or below the same axis or, again, as the projection of a major second above and ( p^n^X ) fifths
;
below an
axis tone, together
above or below the same be analyzed
as
n
@
with the projection of a perfect
axis (s^p^).
The
p, since the result
tetrad p
is
tetrad of the tritone-perfect-fifth projection SLS
p @t, and so forth. The pentads have
some
There are
still
of
may
also
n
may
The
also
basic
be analyzed
members which have a double The hexads are more comthem having three or more valid analyses.
analysis, as indicated
plicated,
@
the same.
fifth
several
on the
chart.
other possible analyses which have not been
specifically indicated, since their inclusion
would add nothing
of vital importance.
353
COMPLEMENTARY SCALES
One
might be noted. In Chapter 48 the subject of a
curiosity
"diagonal" relationship was discussed in the case of the isomeric
among the hexads. The chart makes this The twins and quartets are indicated
"twins" and "quartets"
relationship visually clear.
by
Now
brackets.
we examine
if
the position of the crosses indi-
cating the doads, triads, tetrads, octads, nonads, and decads find that the
upper half of the chart
is
we
an exact mirror of the
lower part of the chart. In the case of the pentads and heptads, the upper half of the chart the connecting hexad lationship,
column
is
a
is
where the order
at the
extreme
a mirror of the lower except
member
left of
is
where
of the "twin" or "quartet" re-
exactly reversed. In the vertical
the chart, the three crosses indicat-
ing pentads one, two, and three are mirrored above by the heptads one, two, and three, in ascending order. In the second
column from the
left
the crosses marking pentads, one, two, four,
and twenty are mirrored by heptads in the same ascending order. The third and fourth columns, however, are fourteen, fifteen,
connected with their corresponding heptads by the isomeric hex-
ad "quartets." Here tads
is
it
will
be seen that the third column of pen-
"mirrored" in the fourth column of heptads, and, con-
column of pentads is mirrored in the third column of heptads. This same diagonal relationship will be observed wherever the twins and quartets occur, although there are four cases where there is a "double diagonal," that is, where one pentad and one heptad are related to both members of a
versely, the fourth
quartet family.
As
far as the order of presentation of the sonorities
cerned,
I
possible.
In the
have
tried
The hexads,
first
to
make
is
con-
the presentation as logical as
for example, are arranged in seven groups.
of these, the perfect fifth predominates or, as in the
case of the second hexad, has equal strength with
major second.
In
the
its
concomitant
second group the minor second pre-
dominates, except in the case of the second of the series where the minor second has equal strength with
its
concomitant major
second. In group three the major second predominates, or has
354
RELATIONSHIP OF TONES IN EQUAL TEMPERAMENT equal strength with the major third and tritone. In group four the minor third predominates with
group six
five the
its
concomitant
In
the tritone predominates, or has equal valency with the
perfect fifth and/or the minor second.
In the last group no
interval dominates the sonority, since in all of six intervals
solid
them four
of the
have equal representation.
This grouping the
tritone.
major third predominates throughout. In group
is
indicated by the dotted vertical lines and
"stair-steps"
which should make the chart more
easily readable.
355
50
Translation of
Symbolism
into
Sound
For those composers wHq have
difficulty
grasping com-
in
pletely the symbolism of the preceding chapter,
I
am
attempting
here to translate the chart of the relationship of sonorities and scales in equal
temperament back again
musical notation.
It
to the
symbolism of
should be stated again that this translation
cannot possibly be completely satisfactory.
A
example, will have nine different versions. involution, that involution will also
of these eighteen scales
nine-tone scale, for If
the scale has an
have nine
may be formed on
positions.
Each
.any of the twelve
tones of the chromatic scale. Therefore, in the cases of such
nine-tone scales, one symbol represents 216 different scales in
musical notation, although only one scale form.
The musical
translation of the chart can therefore give only
one translation of the many translations possible and must be so interpreted.
Example 50-1 begins with the twelve-tone and the eleven-tone scale, each of which is actually only one scale form, and then proceeds to the
six
ten-tone scales.
have seen, corresponds
of these scales, as
is
The
ten-tone scale C,
and
so forth.
it is
the projection.
d is presented with its corresponding interThe order of presentation will be seen \p
conform with the order of presentation of the scales are isometric,
356
we
two-tone interval. The ten-tone scale
presented with the interval p of which
C, p
val d,
to a
Each
in the chart. Since all
no involutions are given.
translation of symbolism into sound
Example
50-1
I
^'/i
I
(I)
Eleven -tone Scale
Ten -tone Scales
Conresponding Intervals
££m£n£s^dfl^
C.p
i ^ ^^ I"
P
o
o o
2
I
4n^
^
fl*^
jt
I
I
ff I
I
2
I
(I)
p8m8n8sQd9t^
C.d
A
o Iff Iff
r
1 I
k^
jt
o
k1 .
!
33:
iL^
-oI
I
I
I
I
? 2
1 I
9 2
II
I I
I
I
t| I
o bo
^* ^
bo
-
1
t;o
bo
*>>
^
2
I
I
!>") (I)
I
I
p8m8n9sQd6r4 bvs
Cm
5-0-
(3)
I
o ff
#
d
i^^
p8m8n8s9d8t4
C.s.
C.n
(2)
I
I
^i
(2)
I
p^m^n^sSd^t'*
*F7^
^^ 12 11
o to
v^ =#g= -o-
II
ivTr
(II
p^m^nQs^d^t^
C.t
ff I
I
l" l"
I I
? 2
I I
I
1
I
1
I
I
(?^ (2)
V
-«»-
357
COMPLEMENTARY SCALES Example 50-2 gives the nine-tone scales with their involutions, where they exist, and with the corresponding triads, of which they are projections, and the triad involution, if any.
The order
of presentation
again, the
is,
same
as that of the
chart for ready comparison,
Example Nine-tone Scales
50-2 Cor^^ponding
Involutions
involutions
IX C.
866,763
p^s
p2s
^EC^ts^
^^^ I'n 2 2
2
I
2
(1)
\^
2
I
2 2
I
-tnfcisx
^o(o)i olxibo,,b^(^ ^
^:tjO
"
(I)
I
I
I
2
I
=»^
safe 2 2
I
f^ f 12
ii?^
11
f
3
I
2(1)
I
2
I
i
I
2
I
4 3
(I)
I
I
(I)
I
I
3
I
2
I
ni >« 7 4(1) -»
(0
I
I
C.sd^ 666,783
sd
r
I
I
I
I
I
r
(4)
I
I
I
III I
I
I
I
I
2(3)
I
I
I
I
I
2
I
I
(3)
C .mnd 677,673
I
(10)
«3l^O
I
2
I
I
I
I
(I)
I
C.rns^ 676,863
2 2 2
I
I
1(1)
668,664
C.n£l
,i4 o(") '
r
358
I
I
2
i
(9) (
*'o
pr
I
2(91
3
I
2
I
I
I
I
I
^")'"ti ni l?
t^
(I)
3
I
(8)
2
2 8
n2t
bol^^il'otl* I
2
ms'
I
I
I
i
nnnd
i^^oit»°''-'^-^^°'^'^'""'^"^"oi>o„^ 3
/I 7 4 (H
nsd (")
I
/'?=-'
•,
:X«3fc:
667,773
II
(5)
2
rtEst
C.nsd
4 3
(5)
pnnd
12
I
B)^
/>")'^*^l>>
-^^I^v^bo^^t,,^^^
C.pnnd 776,673
r
7 2
pmn
ODOO' 2
7 2(3)
(I
C.pmn 777,663
i
5(5)
pns
767,763
C.pns
*
I
I
i^nt
FO
2
I
(2)
^^^» 3 3
(6)
i
3 1(8)
TRANSLATION OF SYMBOLISM INTO SOUND
Cm'
696,663
m^
^^^ 2
I
2
I
C.pdt
I
2
I
iW=
^ 4 4
(1)
I
766,674
^ (4)
pdt (,o)
I
I
I
2
I
I
I
(3)
I
I
I
2
I
I
I
I
I
6
(3)
^ C.mst 676,764
i
2 2
I
I
111
1«T
6
(5)
15)
1
mst
I
I
I
(2)
I
2 2
I
I
I
I
I
Example 50-3 gives the octads with same order as that o£ the chart.
4
2
(2)
4
2
(6)
(6)
their corresponding tetrads
in the
Example Eight-tone Scales ^"'
50-3
Involutions
'^p3 301,200
745,642
C.p"^
^^
^^ 12
2 2
5dS
ots^i :o:'=»
2 5 2(3)
2(1)
p^^ 211,200
Cp^s^ 655,642 tf..o^ov>^^-); 2 2 2
1
I
2
I
"^^-^^*^ 2 2 2
(I)
1
I
2
I
.boM
12
1
3
l
12 11
1
(2)
I
C.p/m 665,452
t
^ ^ 2 113
4
3
p^m'l ^ i
2 2
3
^^^R^ 3
°<>^obo
112
1
I
^
3(2)
211,110
,^"^i
"u |
4
1(2)
I
p/nn
re^^
FF
4 3
2(1)
IV
2(5)
4
I"
1
r
I I
II M
2(5)
221,010
^
4(1)
T,
^^
!D
II(5) I
w
=®=EE
Df 001,230
M=
r
(S)
joC")
L(2)
i ^o^ n O ^ ;uboN
(5)
nbo[^
212,100
:
C.p^m't 655,552 b
2 3
p/n
^^Qobetjo
12
2
(I
Cp/n 656,542
2
(i>):
(9)
359
COMPLEMENTARY SCALES C.d^s^ 455,662
I
I
I
I
I
d^s^ 011,220
2
I
Cd/norn/d
(4)
I
I
I
I
I
2
I
(4)
456,562
d/n
P^
#
r
I
I
I
C.d^m'
I
r 2
3(3)
I
I
d£ml
555,562
$
:tnti: (8)
111,120
<
^S
ii^
14
2
I
«^?« I
1
(I)
I
I
2
4
I
I
I
I
^^^«=
13
(f)'
7
'l^
121,020
M:
=
^ol obotio^S=
Ml
13
7(1)
d/m
C.d/m or m/d 565,462
(8)
012,120
^
jM^**#*
2
2 (8)
I
I
j
VIII
1(3)
3
I
IV
464,743
S7
i
3
I
I
I
^
S3 020,301
4.,o^o"ttv^^'-^i
2 2 2 11
2 2 2
(2)
I
I
555,652
C.s£n't
II
C s/n
i
I
I
3 2
I
111,210
bo("); "t?^
o 2
12)
546,652
n/s
or
II
2(2)
I
(6)
s£n"|
'^'°'"':--'°''°
4 ^^^^^^\>113
(7)
s/n
I
7(2)
or n/s
1
7 (2)
102,210
i^niJ
=^33 t54:
,bc» [|o" it«
2
'
2
i ^ob
2
2
(3)
. l> «jboln <
12
^^^
;
I
3
(21
556,543
C.n^p't
12
C n^d'
2
1
2
1
2
I
2
(I)
b.^ ^vto^^^) l
:
*
I
360
2 2
I
I
2
(if
'
3
3
(3)
112,101
4 2
n^d't
3
(3)
4 2
F
(3'
012,111
^ oboJ o/^)' .^ob ^ J.") 12 3 2 (T) 2 6(3)
:
^'o
;
t
r
I
^'obo^^
^bo job I
3
n^p'l
456,553
t
2 (7)
1
N^ 004,002 >bo("J
i
2
2
I
448,444
C.N^
I
II
11
2 3
I
2(1)
II
1
I
I
2
eBT^
TRANSLATION OF SYMBOLISM INTO SOUND C.n^s'
i Cn^m'
t
'
')
:
^''oboi.^k
3 2 1(2)
I
I
n^s'
^bc» (>
^^
II
i
546,553
J
I
I
I
2
(2)
I
^%M
^
"
3 2
I
102,111
t
556,453
^^\? i
;
2
6(3)
I
nfm'j
^^ 16
(3)
112,011
^ 2
I
2
I
C.n/m
I
I
(3)
I
2
I
2
I
I
I
3~r 5(3)
(3)
I
m/n 566,452
or
n/m
^^^^^^ 12 3
1
M^
C.
2(1)
I
T
I
2
I
2
(I)
I
2
I
I
I
^
3
(I)
^^^ 2
2
2
11
2
I
I
i
^=^
2
1
1
r
m/t >^'^^i
C.
3 2
2
I
I
(I)
I
3 2
I
I
Cp^d'
2
2
t
654,553
I
3
I
I
^
I
I
2
2
I
3
I
(h
^^^ r 3
2 2
I
I
3
I
I
2 2
I
3
I
(I)
I
(5)
^^^^ 2
4 2
13
(6)
2
(6)
210,11!
A"): I
I
111,111
p2d' ^
(I)
4 2
(5)
I
^„k.>.»>.°^°>"'"'""°"°^° I
^^ 111,11!
pmnsdt
(I)
(5)
020,202
2
(I)
I
^^^^^^^^ I
5
^.ii» °H^^^„hJ 4 2
pmnsdt 555,553
r
^
i^
pmnsdt
^4"°«°""°^"';" I
(4
4 2 4(2)
C pmnsdt 555,553
I
^p
i^c^
'
(2)
I
I 1
200,022
^^
(3)
H^"^
^^IPS 12 2
>
p/t
464,644
C.m/t
4 3
(4)
I
^g=
(I)
I
I
t) 131,010
^ 2 2 4(4)
^3CS
3
3
.e.og
Cp/td'^) 644,464
^^f ^^
3 (5)
m2s2 030,201
474,643
C.m2s2
"4 4
s'T^'P
^^
M^lm^p'
3
I
1
m/n 122,010
or
^
575,452
3
(5)
*
-
^o ^35= I
4 2
361
(B)
COMPLEMENTARY SCALES C.£^
d/p 654,463
or
p/d or d/p 210,021
i!^^Mv>^l^"°'"^''^ r
I
3
I
I
^
C.p2d2 644,563
i
^^
^
3(1)
I
6
(4)
I
p2d2 200,121
^°^
1^^ 33l^ r
4
I
d^p'
C.
I
I
2
I
r
(1)
554,563
J
d^p'^
^ i^^^M^^ 12 III
*-" '»'' OPO c^ 'Oj^^bfiob
12
4(1)
1
Example
5
I
1
I
14
I
f So I
(5)
110,121
o(o)
;
it -
6 4(1)
^—IKJ 6 4
I
r
(f
manner, shows the relation of the hep-
50-4, in like
tads to their corresponding pentads and involutions.
Example
50-4 Conresponding
Seven -tone Scales
Involutions
412,300
634.521
p6
T^m
oo t^>°
Q
{
2 2 2
2 2
I
3 2 2
C.
I
1
:
(I)
»»^o=
'boo 2
3
2
2
1
^
2 2 3
C
t
I
2(1)
2
2 3
1
(I
H^ 12
2
I
2
2
i I
^4
362
(3)
I
2 2
r
•^^toi 5 2 2 (
2
(I)
^
2 2(1)
o(,o)!"bo|^
I
2
2
I
2
3) (3)
^
2 2 3 4
2 2 3 4(1)
2
4
I
2
2
(3)
IT
4
2
:^^ ^^ 12 ^^,
C") *^
14
2
I
(3)
311,221 ;
obo
=^
5
2(3
^5~
I
1^
312,211
«-»' p'd^
TT^ 1
2 5
tp^n^l '(©)
I
C.p^d^ 533,442
r
(1)
p^rf i 534,432
^P r
>U>):Ob<
*'
pmn/p 322,210
a^o(");"b ei^ I
1
o
^
l>
pmn/p 544,431
m
4
2 3 2(3)
pns/s 312,310
=^=si botlolvy 2
^^
O
l-l
2
(I)
pns/s 534,531
C,
Involutions
Pentads
VII
5
2(3)
I
r
— TRANSLATION OF SYMBOLISM INTO SOUND pmd/p
C.pmd/p 543,342
i 2 3
3
I
1
2 3
(I)
I
f^
13
1
1
J„o^^"^'"H. 4
2
VII
D^
234,561
^^
I
I
"^^f
^^
o )ot>o'
(,ki);not> «
t
I
2 5
I
I
(I)
I
I
2 5
I
I
III
(f)
I
Cmnd/d 344,451 >'>*J
^^^ I
I
I
4
I
I
7
2
I
tipbc^ I
I
I
I
I
2
I
(5)
I
^ '
C.
12
l^'^l
VII
c -6
I
I
2
I
I
I
=
13
4
I
(I)
I
I
4
2 2
1
C.t£n2|
2 2 2
I
(or
I
2
3
I
I
I
I
2 (2)
(
^
W.6
2
I
I
^
(7)
I
(6)
I
I
I
3
(6)
t)
(fc .
4
I
(5)
I
I
I
r
4
(I
221,131
m
^Rf= oC^) I
tl olyc^
:
3
(I)
I
:
m ^33|
6
13
I
r
(H
040,402
2
(4)
tsfr^lor ££n2<) 222,30!
2 2 2 3(3)
(2)
(3)
3
2 2 2
p^n^) 444,522
I
(7)
I
211,231
S^
I
I
I
I
O P Ui
'
C.s3p2 443,532
2 2 2
ri i*n
(5)
262,623
2
r
Il2,23i
pmd/d
V):"ot>OL TStP i^^^^
ii
2
IT^ ^
443,352
pmd/d
i
(5)
I
1
r
dV
C.d5p2 433,452
4 II
2{
I
I
I
|
'
d^n^
(5)
(7)
>')i"o^. ris^:
III 1144(3)
(3)
I
2
*'o
122,230
mnd/d :
C.d^n^ 334,452
l"l
(8)
I
nsd/s 112,330
bot|o'»'J ,b
4 Ifp
1
:^
C.nsd/s 334,551
i
4
2
(I)
cr«-
(6)
I
4
I
012,340
D."*
^^P^
321,121
s3p2
2
2
2
I
I
I
221,31!
2 2 2
1(5)
2
2 2
363
I
(^)
COMPLEMENTARY SCALES C.s^d^ 343,542
s^d^
121,321
i r
I
2 2
I
I
(4)
I
2 2
I
I
I
r
14)
i
^^
jbo^
r
II
I
^" _n6
2
336,333
^«sboC"): 2
C. pmn/n
445,332
=^33
k>^^<
3
3
> ,
12
(2)
I
12 i^^^
^
2
C.
3
I
I
(2)
2
I
3
I
2^^
I
435,432
pns/n
2
2
I
2
I
2(3)
I
C.mnd/n
C«
i^ 2
1
^
^^^ 3
3
(3)
2
1
(3)
223,111
W*-*):* V,bc .
^^l?olyoll< I
3
pmn/n
>,l7o("):^^i^
2
022,321
114,112
|" l"k i
^
J
2
I
I
)
2(6)
I
Vm4
^obo^^' 12 3
(6)
^^^ ^
2 4(2)
1
2 2
I
s^n^ lor n£d ^
C.s^n^ior n£d5t) 244,542
^^^ "3
3 3(2)
I
3
3 3
I
(2)
pns/n 213,211
2
I
2
I
2
I
13)
4 2
4
2
12 (3)
mnd/n
345,342
2(3)
I
123,12!
^r
^»oit"°^"'l"''^^^'^"«be^^;U"°""°'^ 12
3
1
I
I
III 122 Ill I
m6
12 II
3
1(3)
3(31 3 (3)
I
2 3,1^'
I
I
I
ofloo o^o^^"^ I
3
3
I
(I)
:"boo b t 2
M»?
113
(I)
I
364
I
2
3
(I)
2 2 2
3
I
I
(I)
I
(5)
2(61
I
2
(4)
n£d2
I
I
2
I
I
2
(4)
1
3
2
1
I
(g)
I
2
1
2
(6)
r
3
(I)
^^^
4 3
I
i^
23!,2!!
2 2 3
^^^^^^^^^^^ I
I
^^ ok"): "k 4 3
p2m2
353,442
C.m2d2
1*^
I
2
I
=«a:o^
C.p^m^ 453,432
2 2 2
2
1
Vm4 M'^ 242.020
464,24!
f
3
nsd/n 113,22!
335,442
C.nsd/2
V!!
(3)
I
(4)
2 2 3
I
[4)
!3!,22l
^^^^^ W
;^x» I
2 4
(4)
I
I
2 4
(4)
TRANSLATION OF SYMBOLISM INTO SOUND
Cm^n^
3
I
mV
354,432
2
2
I
I
3
(2)
I
2
2 {^)
I
I
*- '»
1
I
C.m^d^l 454,341
m£d_^
iV «^2^^ ^^^^ r
C.m^n^
4
Vtl
2
I
2
2
2
I
(4)
1
232,120 booC^')
3 4 3
(I)
(I)
,2„2
mfrf J 232,021
^^
j?o|:io I
I
14
3
(3)
VjS
532,353
1
(3)
310,132
=
j)»o^s")|"c>bot, I" I"
C.
4 /I
I
I
I
I
I
I
I
1
11/1111 4
/'Z\
(3)
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{31
I
I
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2
^
434,343
[f\
I
3 2
i
3
I
I
(5)
4
I
(5 (5)
I
4 2 4
(I)
I
(I)
r
3 2
I
13
(5)
2
1
(5)
pmn/t(l-3) 122,212
3 2
C.t^2|
4
imn/t (l-5) 212,122
344,433
C.pmn/t(l-3)
I
5^n)=
4(1)
pmn/t(l-5)
r
p2d2j 220,222
p^d ^l 442,443
^^^^^^ 12 11 C
(4)
^^
$
3CSt »iasi}&i
3
,-3
3
eJ^oO
454,242
$
I ^uMo' I
I
(4)
232,120
1
12
4
(3)
(«0
2 2
I
^S^^
2
I
3
p£m2
Cp^m^t 454,341
^otlo
132,211
2
I
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(2)
444^441
4 2 tp2
3
1
(2)
222,220 botjo ^Vl *^te= '
12
2 4
1
I
(I)
2 4
I
2
I
I
3
I
3
(2)
5 3
2
p2n2
i ^^^ ^^^^^^^ I
(rT
444,342
C.p2n2
r
I
I
I
I
3
I
I
(^) 21
2
2 5 3
' I
(f)
222,121
^s 3
(I)
I
3
Ul);tl|yo^,
p^
(5)
2
I
3
365
I
^ (5)
COMPLEMENTARY SCALES 444,342
C.tdfn^l
td^n^l
r
2
I
I
4
I
>
2 4(1)
1
C.nVj
I
3(3)
4 3
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1
(3)
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2
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>>) :"otyo t;^
2 4
I
(^''
2 3
i^f^i
n^^j
(5)
I
2 3
I
(5)
212,221
3 2(3)
I
Example 50-5 presents the
Finally
4
I
^=^
434,442
I
I
r
(2)
I
p2s2d2 212,221
o(« ^):^>obog^
l<^ 2
2
I
434,442
p^s^d^
C.
(2)
I
^«w
RX
^
4
I
^iob<^
^^lys)
^.,o»t^^^">;"'=>t>ot>o.4 j
222,121
most
involutions. In
hexad
tion of the
complementary
is
scale,
complementary
Where
third part of the line.
already seen, the involu-
complementary
also its
is
"quartet," the scale
we have
cases, as
of the isomeric "twins," the
six-tone scales with their
In the cases
scale.
scale
is
given in the
the original scale
is
a part of a
given with
its
involution, followed
followed in turn by
its
by the
involution.
Example 50-5 Six-tone Scales VI
p5
523,410
i ^"Q
(also
pVn'
X
Involutions
Six-tone "Twin"
Involutions )
T^JF^
tl
2 2 3 2 2
PNS
(I)
423,411 (also
(p^s^ )
|..o»^"'^i"'^l j
22212 Jr£s££l
(3)
nsd/p
366
I
(2)
m )
2
p^s^m')
423,321
tp^s^d'
>>c^^) ;^^bo
4 2
I
t
(3)
423,321 .
2
Jpfn^s";
^^
22212
i^^ iaiso
pmn@s;
^^
14
2
1
.bo'»^) (2)
^ I
I
3 2 3 :
(2)
:
"ote= I
I
3 2 3
(2)
)
)
TRANSLATION OF SYMBOLISM INTO SOUND
pym
432,321
p^m3
oM= ^torO^ 2 ^!-
2 2
D^
2 2 3
123,450
NSD
(also
(s^d^) 123,441
r
I
2
I
I
1
I
2
I
s; t
I
6
I
:»o|yo
^
(2)
I
I
I
n^dS
6
{^)
I
(3)
;
t s'^d^m'
223,341
^bo |o(") ll
' I
5 3
I
I
:
(1)
"ot?o 1
I
5 3
I
^&ar
dVm
W^ d^m^
232,341
232, 341 it p
VI
r
(6)
I
060,603
_Sf
;
$
-otta **'
I
I
Vt
^=^
:tg3=i:
?cr«v 2
^^«^ r
m^s^
(also t
^^^
IT
I
4
(4)
)
?cy^
2 2 2
2
ts^2|
2
(2)
242,412
^^D
;^^)
2 2 2
sV
2
I
1
(also I
2
(3)
241,422
m^s^p';
" '^^"t^^
($
^.^^
l
2 2 2
m^s^n';
I
p^n^d')
t
l
(3)
t P^d^n'
i 2 2 2
I
I
I
I
2
2
(4)
^
2 2 2
(4)
142,422
sln2
2
[I
pns/d)
(also
I
)
sx^?fe
I
s^d^p'
^«
I
mVt _2^2.
(6)
^
,Jto
^o^ot>o
1
.A ;|
)
mnd@
(olso
(6)
(^ i)
I
s^d^n'
J
223,341
tn^s^fl'
O ^^
S-C3
411)
1
2.2* pVt
432,321 (t
(J
I
m^s^d'
2
I
I
2
I
;
(4)
J
n^dV
2
(4)
)
367
s^d^t
)
;
COMPLEMENTARY SCALES pmd/s 322,431 -crg» (ct)
i
T
*^
5 2 2 U)
I
I
I
5 2 2
(I)
323,430 (also tp^d^i)
s^/n s-/r
fi
^55^
ll i?i\
\
5B^
P^ ^2 I
2
I
I
VI
(5)
225,222
iP
boHo' ,bo
t
'3
3
I
2
i^)
i 't
:
,
5R^ 12 1*^^
3 3
(2)
I
324,222
n^/p
W»^)
n^+p^
,^\>^^[ 2
I
3
I
>oljo^°fa'
r
2
I
nVs
2
^oN«"^'^^:
3(2)
2
^
224,232
n^/d
2
2
I
3
3
I
1
3
^
2 2
(I)
^
,||^o^*>):"bokv.
PMD
3
1
(I)
13
2 2 3
342,231
,.„>.»> ''i'H... 2 4
I
MNP
3
12
368
2 4
I
^
^r
3
I
(I
243,231
i ^^^ 3
1(1)
^fe°^
t^^^^°^ ^izsx I
I
(4)
t n^d^t
2 3(3)
j?ot|ot^i»
3 (2)
343,221
2 2 3
I
i
it°^
PMN
1
n^-i-m S 234,222(4
363.030
^'iM6
pVt
jb^W^ 2
234,222
1
(^
^(?^
^otlot'^H 3
(3)
224, 322
n^ + s^
I
bo(^^)
2
I
224,232
3(3)
.b^ obo^-'-"-'i 2 2 (4)
n2/fn
3
^ ^^^^ 1113
224,322
I
I
n^ + d^
(5)
I
324,222
;
3
12
1
(4)
1(3)
mVti^n^s^t)
)
)
TRANSLATION OF SYMBOLISM INTO SOUND pmd/n
343,230
(also t m^d^n'; t p^m^n')
boljo^^*)
* :
-^^
^^^ 2
4 3
1
p^m^d
i
i *^
2
[I)
I
^
3
l7f)
JmVp'
343,131
'
b..(.")i^^ok.
*^
3
2
I
(4)
I
3
I
^
^^ r
I
4
' I
^^ 3
(4)
I
(also
)
343,131
M"^"^:"" !^ " 3
I
I
(3)
3
.
mnd@p)(al:so
t
13
obo/ I
I
r
(3)
m^d^'
^Pm 420,243
)il^{p^)
2
1
m^n^d'
lalso pnnn/d)(alsoi J
4
14
I
(also i p^d^t
3tnt (4)
I
224,223
pmn/t
||..oll'^'":"°l'"||.. |..„
:
l
"3
2
ri
3
I
^ 4
I
i *^
I
I
p^d^s'
(3)
I
3
:
4
I
" obo
(5)
I
I
I
I
2
(3)
i
3
I
I
(5)
4
I
r
(also
^ta^ >
r
(4)
322,332
3 2 4 pdt /s )
421,242
d^/p
t
;tnt "^1
^^ r fe
2
I
^ t
3(2)
421,242
p2/d
r
I
K\oa.
(la)
C") r
2
322,242
mst/d
I
13
(2)
422,232
mst/p
[I)
>W-
113 (also
I
I
2 4"(\r
p^d ^)
1
4 2
I
5
j|ovi^"): I
322,332
n^s^m'
tboCfct)
^-" M« 2
1
I
1(3)
5
1(2)
:
^bo^ ^bc '
2
I
I
5
I
i^d3
(3)
369
"^r (2)
COMPLEMENTARY SCALES ,2w2„l
t/
-^^o '?^'> 332,232
£ldfm'
t
-« k»o" 3
I
o(")i"o(,. 2 4
p2n2m'
^
OflOJ
3
1
I
2 4 (^) IFp
1
2
I
I
4
2
t3)
I
I
4
I
IT
I
2
3
2(3)
333,321
X p2rT?s'
^0= ^")!*VJ^V I
I
l
2
u-<* »!
"l?o
(4)
2
; |
i
^^^^
po-o^
f«^ 2
2
'
(3)
233,331
2
I
2
I
2
I
2
(^)
I
^^5
5 2
12 15
21
1
^
233,331
m2d2s'
t
-rr^to* obot] <
:"ot>ok^
k^o(»>)
^oi]«
112
2
4
3(1)
I
2
I
^F
4
3
"T (f)
nsd/m)
(also
,2 p*^
323,331
p^s^d^H- pi
mO r
^^e^
333,321
n^d^m'
, i>
^^^^^^S^
^
pns/m)
(also
%
13
(IJ
332,232
m^n^s'
pdt/m)
(also
*
1
X
(.")
" ^"bo ^"^^
2
I
2
(5)
I
I
2
I
2
+d;
323,531
^{yO
'
!
:
-o^e^*^
t«^) I
e2 s^ A^ d *^
(5)
I
I
2 3 4
(1)
iibiit^i,.
112
rr
3 4 0)
These relationships of tone
will repay endless study and them lies all of the tonal material of occidental music, classic and modern, serious and popular. Within them lie infinite and subtle variations, from the most sensuously luxuriant sounds to those which are grimly ascetic;
absorption, for within
from the mildest of gentle sounds
Each
to the
scale or sonority encloses
In parting,
let
is
its
own
character.
as
an example, the tetrad
pV
and
its
we
octad
a sweet and gentle sound used thousands of
times by thousands of composers.
370
and enfolds
us look at one combination of sounds which
have used before projection. It
most savagely dissonant.
It has, for
me, a strong per-
TRANSLATION OF SYMBOLISM INTO SOUND sonal association as the opening sonority of the "Interlochen
theme" from
my
"Romantic" symphony. You
will find
it
and
its
octad projection on the second line of Example 50-3. Note that the tetrad has the sound of C-D-E-G. Notice that
its
octad
is
saturated with this pleasant sound, for the octad contains not
only the tetrad
C-D-E-G but
tive composer,
it
on D, D-E-Fjf-A; on E, E-FJ-GJf-B, and on G, G-A-B-D. In the hands of an insensialso similar tetrads
could become completely sentimental. In the
hands of a genius,
it
could be transformed into a scale of
surpassing beauty and tenderness.
In conclusion, play for yourself gently and sensitively the
opening four measures of Grieg's exquisitely beautiful song,
"En Svane." Note the dissonance of the second chord as contrasted with the first. Then note again the return of the consonant triad followed by the increasinglv dissonant sound, where the Dt>
is
mark
substituted for the D. Listen to of genius. It took only the
it
carefully, for this
boding. But
of
the
its gentle pastoral quality to one of vague forehad to be the right note! If this text is of any help assisting the young composer to find the right note, the labor writing it will not have been in vain.
the sound from
in
is
change of one tone to transform
it
371
tl
Appendix
Symmetrical Twelve -Tone Forms
For the composer who is interested in the type of "tone row" which uses all of the twelve tones of the chromatic scale without nineteen of the six-tone scales with their comple-
repetition,
mentary involutions arrangement.
by
followed
If its
we
offer interesting possibilities for
symmetrical
present these scales, as in Example
each
1,
complementary involution, we produce the
following symmetrical twelve-tone scales:
Example
1
pns ->
i '^^2'
Jp
J
<-
fj^Ji
i
I
2
2^
#iijjtfj I
J ^
I
2:
->
tJ ^I'^'i^t^r UJ^Jt'-' i
I
I
I
I
I
I
I
I
I
i
'^^-'
I
i
ir*r"r
21222
I
nsd
<-
->
<-
Jur^r'^nijj j||jjJ ^JttJr ^2^2 2
^'22322 i
'^
I
-^
2
<-
iiJt^''^"rt''r 2
I
I
I
I
6 -etc.
1222
22222 «
^>'
^
^J^ r r^r'rV '
-0^lij, 2 2 2 i
ii
i
i
2
12
2
2?22
s4p2
2
2
2
»
n ijj.i
2
2
11
22222
m 2
373
2
.
2
—
1
APPENDIX s
2 4 n*^
pmd@
1152
9 2
P 2
9 2
1129 I
"^tt
I
s
> 9
«S
I
<-
P
P
s^@Il
iJt.JhJJJ-iiJtJ^t^rTiibJ^^fa^^^r 2
I
I
2
I
12
3
3
P
«S
I
I
^m 2
13
3
,6
^
<
<|>iJjJ«Jr 3 13
itJ^i ^'irV'^r
wf
i
i
13 13
3
13 13 ^ujJ^ir
"r^r^^
^
13 13
3
pmn
14 11
I
pmn@t
f|.i,i"jiiJJ
i
i^
13213
iiJ^tJ^r Tj^ji 13213 31231
nnst@ p
mst@d
<
-^.
t= b!
'
55^£fJNH
* ir —f^^'^r^ g a
1
I
yTt^ 31231 <
-
1
1
1
jiJ
-J -*
T
—— — —L_ ^Wt^^Ff^r^P 1--J
^^
In any of the above scales, any series of consecutive tones
from two to
five will
be found
to
be projected to
its
correspond-
ing ten, nine, eight, or seven-tone scale. For example, in the first scale, p^,
not only are the twelve tones the logical projection
of the original of the
first
projection
374
two of
hexad but the tones; the
the
first
first
three;
first
ten tones are the projection
nine tones will be seen to be the the
first
eight
tones
are
the
SYMMETRICAL TWELVE-TONE FORMS projection of the
first
projection of the
and the
four,
first
seven tones are the
first five.
In other words, the seven-tone scale C-D-E-FJj:G-A-B projection
is
the
C-D-E-G-A, the eight-tone scale C-D-E-F#-Gthe projection of C-D-E-G, and so forth, as illustrated of
G#-A-B is Example
in
2:
Example 2
h It
j'jNN
g
«^
r
iiJ
hip
|
"
*r
should be clear that the above relationship remains true
regardless of the order of tones in the original hexad as long as
the series
is
in the
form of a six-tone scale— or sonority— with
complementary involution. For example, the might be rearranged as in Example 3:
scale of
its
Example 2
Example 3 efc. etc.
-^
j_i-j
JjJ
The method
i
iJnrt.i^riiJ'ir
i
^. .irJj«riJ||^ i
of determining the "converting
tone"— that
is,
the
tone on which we begin the descending complementary scalewas discussed in Chapter 40, pages 266 to 269. A quicker, although less systematic, method is by the "trial and error" process, that is, by testing all of the possibilities until the tone is found which, used as a starting point, will reproduce the same order of intervals
downward without
tones. Referring, again, to
E#, or F,
is
duplicating any of the original
Example
the only tone from which
1,
p^,
it
we can
will
be clear that
project
downward
the intervals 22322 without duplicating any of the tones of the original hexad.
The hexad "twins" and "quartets" cannot be arranged in this manner for reasons previously explained. This is also true of the hexad
pmd
@
n which follows the general design of the 375
APPENDIX
unhke them,
"quartets" although, to
be
own
its
The nineteen hexads all
its
complementary
scale proves
transposition at the interval of the tritone. of
of the triads, tetrads
Example
contain in their formation
1
and pentads
of the twelve-tone scale
except the five pentads, p^m^t, m^d^t, m^n^^, p^s^d^, and
nV^,
the last of which will be recognized as the "maverick" sonority of
Chapter
ten-tone
47.
row
The
as in
four
first
Example
may be
projected to a symmetrical
4:
Example 4 p^m^
rn^d^
missing
t
i|,jjji.J 12 4
_
m2n2 —1
111
J *
^
>^
^
3
J
.ii
ii
ii
^i
14
1
112
uj.J':^r 4 3
>
tones
14 13
3
I
missing
<
3
missing tones
3
I
>.
15
p2s2d2
376
,'i;
|
1
2
11
missing tones
t
~zzz
Torres
14
1
<-
^W^' 3
4
3
<
14 13
missing tones
Index
Decads,
A
perfect-fifth,
Accent,
rhythmic, Analysis of intervals, by omission, Axis of involution,
p^m^n^sHH*,
minor-second, p^m^n^s^dH*, major-second, p^m^n^s^dH"^, minor-third, p^m^n^s^dH'^, major-third, p^m^n^s^dH'^,
58 58
agogic,
7
270
315 277 91, 278 119, 280 134, 281
31, 276,
66,
perfect-fifth-tritone.
20 - 21
149, 282
p^m^n^s^dH^, "Diagonal" relationship
B
of
Bartok,
From
the Diary of a Fly, Sixth Quartet, Fourth Quartet,
74 74, 127, 145 75, 145, 192
Beethoven,
35 35, 297 36
Leonore No. 3, Symphony No. 5, Symphony No. 8,
Nacht,
Les Illuminations,
9
of the perfect fifth series,
minor-second of the major-second
31, 276,
315
pl2^12„12jl2cil2i6^
66,
277
major-second. pl2ml2„125l2dl2t6^
92,
278
series, series,
of the minor-third series,
of the major-third series, of the perfect-fifth— tritone series,
Consonant symbols, pmn, Converting tone, Copland, A Lincoln Portrait,
275 276-277 278 279 280-281 282 11
266-269 214, 217
minor-third. pl2ml2„12sl2cil2i6^
119, 280
major-third. pl2OTl2„125l2dl2f6^
134, 281
perfect-fifth-tritone.
60 249
tones,
Complementary hexad, Complementary sonorities, of the
tritone.
minor-second,
38 96 83, 115, 156
progression,
4 57 139-140 49
perfect-fifth,
Clockwise and counterclockwise
Common
11 11
Dominant seventh, Dorian mode. Double valency of the Doubling, Duodecads,
pl2ml2nl2sl2(fl2i6^
Berg, Alban, Lyrische Suite, Britten,
336
hexad quartets,
Dissonant symbols, sdt. Dissonant triad, sd^.
pl2^12„12sl2cil2^6_
149, 282
E Enharmonic equivalent. Enharmonic isometric hexad, Enharmonic table. Equal temperament,
1
78 12 1
Expansion of complementary-scale theory. Exponents,
263 19
D Debussy, Voiles, La Mer, Pelleas and Melisande,
88 82 84, 95, 103, 115 202-203, 209 186, Les fees sent d'exquises danseuses, 116 81,
Fusion of harmony and melody,
3,
Gregorian modes, Grieg,
16
47 371
En Schwan,
377
INDEX pn5@d,Jn2s2
(
53
Harmonic rhythm.
{d~@m,
Hanson, Sinfonia Sacra,
128
Cherubic Hymn,
206 293
Elegy,
296, 371
"Romantic" Symphony, Harmonic-melodic material,
40- 47 67- 72
perfect-fifth hexad,
minor-second hexad, major-second hexad.
79- 81
98-103 125-126 141-144 153-154 270-271
minor-third hexad, major-third hexad, perfect-fifth-tritone hexad.
pmn-tritone hexad, Harris,
Symphony No.
3,
Heptads, p^m^n'^s^d^, 29, 275, 315 minor-second, p^m^n*s^d^t, 66, 277 major-second, p-m^n~s^d^t^, 90, 232, 278 perfect-fifth,
119, 279
minor-third, p^m^n^s^dH^, major-third, p*m^n*s^dH,
133, 281
perfect-fifth-tritone,
148, 282
p^m^n-s^dH^, Heptads, complementary. of
pmn
projection.
of pus projection, of of
pmd projection, mnd projection,
of nsd projection, of prnn-tritone projection of pentads p^ + s^, p- + s^ oi pentads d3 + s2,d2 + s3. o{ pentad p3 + d2,p2 + d3 of pentad tp2-|-d24,_ of pentad p^ + m^, of pentad d^ + m^,
pentad p^ + n^. of pentad d^ + n^. of pentad s^ + n^. of pentad m~ + rfi. of pentad p^ + s^ + d^, of pentad Ip-m^, of pentad \p^n^. of pentad ^p^d^. of pentad fm'^n^, of pentad m^d"^. of pentad '^n'^d^. Hexads, perfect-fifth. of
;;
286 288 290 291 292 295 304 305 306-307 308 309 309 310 310 311 311-312 312 333-334 317-318, 335 320-321, 335 322-323 324-325 326-327 328-330 1,
\
\Xp2s2d\
p^@m,
p'^m^n'^s^d^t,
p3 + m2,lp2sH,lm2d%
259 212 211, 229, 231
Hexads, minor-second. pm2n^s*d,ls2d2n^, nsd, s3
-1-
d3,
pm^n^s^dH,
378
65
mnd@s, I n^d^s^ ,Js2d2mi,
p^m^n^s^dH,
259 216 215, 230, 231 78,
m^s^t^,Xm^s2t,
ts%24,, jm2s2pl, Ip2n2d^, p2m*n2s*dt2, s4 4- p2, 1 m2s2ni, l p^d^n^,
233, 234 233, 234
237 237
s2@n,fp^d^ ],,p^m2n^s*d^, Hexads, minor- third,
98 197 195
p^m^n^s^d^t^,
(n2@p, p^m^n'^s^d^t^, \n^ + p^,
)n2@d,
230
232, 234
p2m*ns*d2t2, Si + n2,lm2s2d'^,ln2d2pi, pm^n^s^d^t^, pmd@s, p^m^n^s'^dH,
p^m^n'^s^dH^,
n^ + d^,
)n2@s, p^nfin^s^d^t^, n3+s3,i;p2n2f,:I;"2d2i,
in2@m, p2m^n*s2d2t2, n^ +m2, J n2s2f J m^n^t, ,
208 207 205 204, 230, 231 201 200, 230, 231
Hexads, major-third, p^m^n^d^,
13,
pmn, p^m'^n^s^d^t, pmd, p^m^n^s^dH, mnd, p2m*n^s2dH,
124 168 178 183
pmd@n,
X rrfid^n^, % p^m^n^, p^m^n^s^d^, 237, 240 / 1 p^m^d^, pmn@d, I rrfin^d^, I p^m^nHdH, 239, 240, 255 Ilm2n2pi, mnd@,p,1vrfid2p'^,
239, 240, 255
Hexads, tritone, t^, p2@t,lp2d2t, p^m^s^dH^,
pmn@t,
140,
p^m^n'^s^d^fi,
mst@p, p^m^n^s^dH^^ mst@d, p^rrfin^s^dH^,
{p2@d,
p'^m^ns^dH^,
d2@p,
230 152 237 238 219 220
/|p2d2ji, pdt@s, p^ + d^,
p^m2n2sUH2^
< vX
219, 239, 240, 256
259
n^s^m^,
np^d^m'^, pdt@m, J p^m?rfis2dH2^ (Jm2n2si, Hexads, neutral, )
239, 240, 256
259
pns@m,Jp2n2mi, p^m^n^s^d^t,
239, 240, 256
259
\lp2m2s\ ifn^d^m''^,
nsd@m,
p^m^n^s^dH, Ilm2d2si,
<
239, 240, 257
259 258 258 254, 339- 40 340-345
j
p2-\-s2-\-d2-\-p\^,p^m2n^s^dH,
\
p2+s2+d2+d|
Hexad Hexad
quartets,
"twins,"
Hoist,
The 188
239, 240, 255
d^ + m2,lp2m%ls2d% Hexads, major-second,
/
p5m'^nHid,lp^s^n^, 29, 315 pns, pmn@s, p^ + s^,'lp^n^s'^,tP'is2mi. ptm^nh'^dt. 173, 236 (nsd@p,l n^s^p^ p^m^n^s^d-t, 239, 240, 257
p^m^n^sHH,
{ Kls^d^p^,
Planets,
Hymn
of Jesus,
171 199
ESTOEX major-third, p^m^n^s'^dH^, 133, 281, 324, perfect-fifth— tritone p'^m^n^s^d''t*,
55
Influence of overtones, Intervals,
symbol
9-10 10 10 10 10
p,
m. ^, s.
d,
number present
Nonads, complementary, of
in a sonority,
of of
8,
simple. isometric.
enharmonic.
pmd mnd
projection, projection,
of mst projection. of
40 17 18 18 19
Involution, theory of,
286 289 290 291-292 292-293 293 317 319
projection.
of nsd projection.
14-15
table of, Inversion,
pmn
of pns projection.
11 11
t,
of
tp2. lm2.
O Octads,
of the six-tone minor-third projection,
perfect-fifth, p'^m'^n^sHH'^,
of the pmn-tritone projection,
minor-second, p'^m'^n^s^d'^t'^. major-second, p'^m^n'^s'^dH^,
of the
pmn
hexad
of the pns hexad. of the
pmd hexad. mnd hexad,
of the of the nsd hexad,
Isomeric pentad, pmnsdt, Isomeric sonorities. Isomeric twins.
110 158 170 174 179 184 189-190 23 22-23 196
minor-third, p^m'^n^s^dH'^, major-third, p^ni^n^s^dH-,
315 277 91, 278 119, 279 133, 281
30, 275,
66,
perfect-fifth— tritone.
p6m*n4s4d6^4^ Octads, complementary, of pmn-tritone projection, of tetrad
p@m,
of tetrad
n@p, m@t,
of tetrad J
of tetrad
Just intonation,
1
of tetrad
M
148, 282
296 299 299 299 300 300 301 301 301 304 305 306-307 308 317 318 320 321 322 323 324 325 327 328 329 330 332 332-333 342 342 343 343 344 344-345
n@m, m@d.
of tetrad
n@s.
of tetrad
n@d, p@d.
of tetrad
Major-second hexads with foreign tone, 232 331 "Maverick" sonority, 333 "Maverick" twins, Messiaen, L'Ascension, 122, 135 135 La Nativite du Seigneur, 17 "Mirror," Modulation, 60 key, 56 modal, 63 concurrent modal and key of the perfect-fifth pentad, 61 of the minor-second pentad, 76 of the minor-third hexad, 109 of the major-third hexad, 131 147 of the perfect-fifth-tritone hexad, 157 of the pmn-tritone hexad, 155 Moussorgsky, Boris Godounov, 6 Multiple analysis, 5,
of tetrad p'^+s^.
of tetrad d^+s^. of tetrad p'^+d^. of tetrad s'^+nfi.
of tetrad Ip^m^, of tetrad Jm^pi, of tetrad Xp^'n-^i of tetrad jn^pi,
of tetrad tp^d^, of tetrad Id^p^, of tetrad Im^n^, of tetrad | n^m'^, of tetrad I m'^d^, of tetrad Jd^^i, of tetrad % n^d^, of tetrad Id^n^, of tetrad In-s'^, of tetrad ts^n^. of tetrad
n2+pi.
of tetrad n^-\-m'^.
N Nonads, perfect-fifth, p^m^n^s'^dH'^,
30, 276, 315, 320, minor-second, p^m^n^s'^d^t^,
minor-third, p^m'^n^s^d^t*, 119, 280, 310, 311, 312, 321, 326, 329
of tetrad
n^-\-s'^,
of tetrad
n^+d^,
of tetrad
p^+m^, d^+m^,
of tetrad
322
66, 277, 323, 328, 330 major-second, p^m'^n^s^d^t^, 91, 278
327
149, 282
p Pentads, perfect-fifth,
pns@p,
"Ip-s^, p*mri^s^,
29, 172, 226,
379
315
INDEX pns@,s, p^mn-s^d. pmn@p, p^m^rfis^d,
"[p+^n^i, p^mn^s^dt. p^+d^, p^mns^d^t,
pmd®p,
172 47, 167 174, 196
of the tritone. of the perfect-fifth-tritone
212, 221
of the pmn-tritone series.
47,
177
p^m-nsd^t,
Pentads, minor-second,
mn^s^di, nsd®d,ts-d2, nsd@s, pmn~s^d^. mnd@d, pm^n^s^d^,
65, 187, 228, 72, 71,
d2+n2, pmrfisUH,
188,
p'^mns^dH, pmd@d, p'^m^nsdH, Pentads, major-second, |m2s2, m'^sH^,
216,
d^-\-p~,
277 188 182 208 220 177
beyond the
81, 227
173, 226
s3+d2, ts2+d2 4,^ pnfins^d%
188,
Projection by involution, Projection at foreign intervals. Projection by involution with
complementary
174,
.,
[]p2m2, p2n2, [
",
lp^d2.
213 217
lTn2d2, tn2d2. Projection of the triad pmn.
pmn®p,
98
pmn@n,
102, 168
p^m^n^sdt. pns@n, p^mn^s^dt. mnd@n, pm^n^sd^t. nsd@n, pmn^s^d^t, Pentads, major-third,
pmn@n. pmn hexad,
228-229
Pentads, minor-third, pmn'^sdfi.
102, 172 103, 182 103,
187
Projection Projection Projection Projection Projection their
124,
pm^n^s^dt,
169,
^p^m^, p^m^n^sd^. '•m^d^, p2m3n2sd2.
215, 211, 200,
216 201 226 229 228
144,
220
178,
;:m2n2, p2m3n2d2t.
Pentads, tritone. p^TnsdH2, pdt@p, Ip2d2, p2m2s2d2t2. pmn@tC^ 5), p2mn2sd2t2. P7nn@f(i3), pm2n2s2dt2. \p2+d2i, p2m2n2s2d2, p2-\-n2, p2m2n2sd2t, 'td2+n2l, p2m2n2sd2t, p2+s2+d2, p2mn2s2d2t, In2s2, p2mn2s2d2t,
of the triad nsd, of the triad forms with
complementary
sonorities.
Perfect-fifth— tritone projection.
Phrygian mode, 1,
154 154 179 169, 196 183, 207 205, 257 200, 227 338 140 57 272-273
its
sonorities,
Projection of the perfect fifth. of the minor second. of the major second, of the major second beyond the six-tone series, of the minor third. of the minor third beyond the six-tone series. of the major third. of the major third beyond the six-tone series,
294-296 27 65 77
285-288 288-289 289-290 291-292 292-293 293
nsd. mst. Projection of two similar intervals at a foreign interval.
p@m. p®n, m@t.
144, 227
Pentad projection by involution,
380
pmd, mnd,
pmd. mnd,
169
p2-\-m2, p^rrfins-dt.
m2+d2, pmHs2d%
pmn-tritone projection with
of the triad
pns,
168, 177, 182
Symphony No.
of the triad, pns, of the triad
pmn,
pmn@m, pmd@m, mnd@m,
complementary
314 315 316-319 319-321 321-323 323-326 326-328 328-330 167 167 168 168 168 172 177 182 187
sonorities.
pmn@m.
189, 205
Piston, Walter,
236
Perfect-fifth series.
s2+n2 or InH^, rrfirfisH%
fji2-\-n~,
148 151 225
six-tone series,
im2n2, 7fi1,
i*2„2 or Ip2„2^ p^m^n^sH, s3-)-p2^ p^m^ns^dt.
p2min2d2,
139
n@m. m@d.
298 298-299 299 299 300
300 300 301 301
n@s.
n@d. p@d, Prokofieif,
Symphony No.
38 128
6,
Peter and the Wolf,
R Ravel, Daphnis and Recapitulation of the Recapitulation of the Recapitulation of the Relationship of tones
35 136 161 241
Chloe, triad forms,
tetrad forms. pentad forms,
in equal 346-355 temperament. Relative consonance and dissonance, 106-108 Respighi, Pines of Rome, 171 Rogers, Bernard, Portrait, 283 ,
90 97 118 123 132
S
"Saturation" of intervals. Scale "versions,"
140 34
INDEX Schonberg, Five Orchestral Pieces, No. 1, 150, 203, 218
Tetrads, minor-third,
nH2,
Scriabine,
I
Poeme de
I'Extase,
Prometheus, Fourth Symphony, Simultaneous projection, Sibelius,
235 235 302, 313
296
and perfect fifth, and major third, third and major second, third and minor second, of the perfect fifth and major third, of the major third and minor second, of the perfect fifth and minor second. third
195
of the
third
200 204 207 211 215 219
Simultaneous projection of intervals with their complementary sonorities, 303-304 p2+s2, 304-305 d^+s^, 306-307 p^-\-d^, 307-308 p3+d3,
s2+m2, p^-\-m^,
308 309 310 310 311 311 312 25 274
d^+m^
p^+n^, d2-|-n2, s2-)-n2,
m2-fn2,
p2^s2+d2, Six basic tonal series,
with their complementary sonorities Six -tone scales
Xn'^s^, pn^sdt,
tn^mi, pmn^dt, pm^n^d,
Tetrads, major-third,
pm^nd, m^-\-s^,
Tetrads, tritone,
pmnsdt,
p@d-d®p, p^md^t, p2^d2, p2sd% td^p'^, pmsd^t,
Theory of complementary scales, Theory of complementary sonorities, Tonal center, Translation of symbolism into sound,
150,
315 42-43 43-44
28,
p2mns^, p@n-n@p, p^mn^s, Ip^m''-, p^mnsd,
p^m^nd,
d2+s2, mns^d^, d@n-n®d, mn^sd^, td2mi, pmnsd^,
d®m-m@d,
222 181
p2-j-s2^
p@m-m@p,
pm^nd^,
11,
12-13, 79, 123
(major-minor),
11,
pmd, pdt, mst,
mnd,
42
nsd,
Twelve-tone Twelve-tone
41 41 41 100 79 67 3 337
circle,
"ellipse,"
U Undecads, perfect-fifth,
piOmWniOsWdiots^ 31, 276, 315 minor-second, piOmiOniOsio^io^s^ 66, 277 major-second, piOmiOnio^io^^io^o^ 91^ 278 minor-third, piOmiOnio^iOc^io^o^ major-third, piOmiOniOsiOt^iOfS^
ng^ 280 134^ 281
perfect-fifth— tritone,
piOmWnWsiodwt5^
149, 282
46 18,
44-45
65 68 69, 102 70 69-70
Tetrads, major-second,
m^sH, ts2ni, pmns^d, s®n-n@s, pn^s^d,
28,
121
106
Tetrads, minor-second, ns^d^,
315 65 77 42, 98
perfect-fifth, p^s,
minor-second, sd^, major-second, ms^, minor-third, n^t (diminished), major-third, m^ (augmented),
193
"Tension," Tetrads, perfect-fifth, p3ns2,
143 143 143 261 247 56 356
Triads,
pmn
Three Movements,
Concertino, Sacre du Print emps,
101 144
IpH^, p^msdt,
pns,
intervals,
in
80 142 80
p^d^t^,
m®t-t@m; s®t-t®s; m^sH^,
formed by the
Shostakovitch, Symphony No. 5, 176, 269 Sonority, 3 Strauss, Richard, Death and Transfiguration, Stravinsky, Petrouchka, 37, 128, 155, 198 210, 218 Symphony in C, 37, 214 Symphony of Psalms, 49, 120, 171, 234
Symphony
123-124 m^s^t,
p@t-t@p; d®t-t®d;
simultaneous projection of
two
101 101 101 101 101
mn^sdt,
n@m-m@n,
of the
minor minor of the minor of the minor
97
n^p^, pmn^st,
I n2
81,
77 46 46
Vaughn-Williams, The Shepherds of the Delectable Mountains, Vertical projection
complementary
176
by involution and 335
relationship,
W Wagner, Ring des Nibelungen, Tristan and Isolde,
185 283
381
Ha„so„,lw?002
00339 1492
Harmonic materials of modern music; reso
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