Musical Symmetry as a Compositional Strategy in Bartok’s “Subject and Reflection”
by LEUNG Tai-wai, David
Submitted to the Department of Music for the Course Advanced Music Analysis II (MUGS 5170)
The Chinese University of Hong Kong 2004/4/27
All rights reserved by David LEUNG Tai-wai on 2004/4/27 1
Preface:
“Subject and Reflection” is No. 141 from Book 6 of Bela Bartok’s Mikrokosmos, a set of short pedagogical pieces designed for piano learners.
In this paper, I shall
argue that this piece, though commonly be notable for teaching piano techniques, it actually reflects compositional ingenuity.
Not only does it train the independence of
hands and fingers in young pianists through two-part counterpoint, it also explores the musical symmetries as the main compositional strategy, which is not a common conventional compositional practice.
In particular, as the title suggests, Bartok
employs various compositional techniques, the small-scale of melodic gestures to the large-scale of structural construction, to corroborate the musical symmetry for the delineation of the “Reflection” of an image.
Hence, musical symmetry is the integral
part of organicism of this piece.
In the outset of my discussion, I will investigate Bartok’s underlying idea and the historical context of Mikrokosmos, and then, I will examine the selected piece “Subject and Reflection” to comprehend what and how the symmetry is used to serve as the compositional strategy constructing the organic structure of the piece.
Simple Historical Background and Underlying Ideas
Mikrokosmos is a cycle of 153 pieces for piano, written by Hungarian composer Bela Bartok (1881-1945) with a didactic purpose between 1926s and 1939s.
The
whole collection, giving piano pieces to be used from the very beginning and further on, is graded in six volumes according to different levels of technical difficulties. The pieces of the last two volumes are technically and musically different, and were 2
intended to serve as the material for professional pianists in concerts. included most of them in his concert programs and recordings.1
In fact, Bartok
His own
interpretation of the work is regarded as ‘a many-sided project’, not only for teaching purpose.2
He also stated that the Mikrokosmos appears as a synthesis of all musical
and technical problems, which were treated and in some cases only partially solved in the previous piano works, particularly with regard to his own new styles of piano writing.3
It is clear that Bartok regarded every piano piece of this collection, albeit
short and small, not merely pedagogical exercise, but is a kind of artistic creation. Furthermore, Peter Bartok, son of Bela Bartok, commented on his father’s cosmological idea about this set work, that, “Mikrokosmos may be interpreted as a series of pieces in many different styles, representing a small world. interpreted as a world of the little ones, the children.”4
Or it may be
Mikrokosmos, therefore, is
embedded with Bartok’s own profound, artistic and cosmological ideas of creation indicating that it is an inventive composition, not a mere finger exercise. “Subject and Reflection” is composed under such underlying ideas and historical context, and so, it is worth exploring how the symmetry, which is a conception of aesthetic, can be depicted throughout the music by means of various compositional devices.
Part I: Large-Scale Symmetrical Strategy
The Centric (Tonal) Plan of the Music
A structural-level symmetry can be experienced from the centric or tonal plan of 1
2 3 4
Benjamin Suchoff, “Synthesis of East and West: Mikrokosmos,” in The Bartok Companion, ed. by Malcolm Gillies (London: Faber and Faber Ltd. 1993), 203. Ibid., 190. Ibid., 190-91. Peter Bartok, “Preface,” in Mikrokosmos 6, Bela Bartok (New York: Boosey and Hawkes, 1987), 4. 3
the entire piece.
Although the work is only 82 bars long, it can be divided into seven
sections plus a coda according to the phrasing, dynamic, tempo and centric.
All
centric notes of each section combined to form a symmetrical scale of what Erno Lendvai called 1:3 model.
The 1:3 model scale simply means that the successive
scalic notes are 1 and then 3 semitonal apart alternately.5
I will discuss the scale’s
inherent symmetrical features to see how it governs the compositional processes of the music later.
Now, let us explore the structural plan of the entire piece first.
Structural Plan Table:
Section
Measures
Centric
Tempo
Dynamic
Section 1. mm. 1 – 14 : centric B ; tempo: Allegro;
dynamic: f
Section 2. mm. 15 – 22 : centric B ; tempo: Piu Mosso;
dynamic: p
Section 3. mm. 23 – 29 : centric D ; tempo : Tempo I;
dynamic: mf
Section 4. mm. 30 – 39 : centric E ; tempo: Vivacissiomo;
dynamic: p
Section 5. mm. 40 – 46 : centric F ; tempo: Meno Mosso;
dynamic: f
Section 6. mm. 47 – 62 : centric G ; tempo: Vivacissimo;
dynamic: p to mf
Section 7. mm. 63 – 73 : centric B; tempo: Tempo I;
dynamic: f
Coda:
mm. 74 – 82 : centric B; tempo: Piu Mosso;
dynamic: f, ff, sf
From the above table, it is noted that the entire structure of the music is clearly
5
Erno Lendvai, Symmetries of Music: An Introduction to Semantics of Music (Budapest: Kodaly Institute Kecskemet, 1993), 30. 4
divided by locating the centric note in each section, that is B, B, D, E , F, G , B. Interestingly, if we consider meticulously to the tempo organization of every section, excluding Section 1, the overall structure of the piece is somewhat symmetrical designed to certain extent, that is, B – A – C – D – C – A – B.
The
tempo D (meno mosso)of Section 4 is the central axis and the other sections just relating to it.
We can refer to example 1 below for this symmetrical design of tempo
as following (see example 1): Example 1: Symmetrical pattern of the piece in relation to tempo setting
The Symmetrical Pattern of the Centric Plan
As mentioned before, the entire music is structurally constituted by different centric (tonic) in each section forming the 1:3 model scale (see example 2). Throughout the entire work, we can trace that this 1:3 model scale axes appears continuously in only one octave (see example 2). the scale is below middle C.
The beginning centric B note of
Each centric note of the scale then moves within a
range of an octave pertinent to the corresponding section.
The ending points of this
octave running scale are the pair of B notes (two octave apart between voices) with the same pitch level as the upper voice that opened the piece.
This 1:3 model scale 5
is a symmetrical scale in structure and it serves to provide more symmetrical features for the piece. Example 2: The centric plan of “Subject and Reflection”
In order to give more variety to the music, Bartok doubled the centric notes in an octave apart and distributed them to both hands (voices) in each section (see example 3).
This doubling of notes in the scale (excluding the final B centric,
which is regarded as an expansion to other axes in octaves) fall into a somewhat symmetrical pattern in design.6
This pattern may read as following:
Centric Doubling Plan: Section
Doublings
Section 1 and 2:
B and B centric notes doubled an octave higher
Section 3:
D centric note doubled an octave higher and lower
Section 4 and 5:
E and Fcentric notes doubled an octave lower
Section 6:
G centric notes basically doubled an octave higher and lower
6
Bernard W. Jonathan, “Space and Symmetry in Bartok,” Journal of Music Theory 30, no. 2 (fall 1986): 191. 6
Example 3: Symmetrical pattern from octave doubling of the centric note
Apart from the overall structural design of symmetry in this music, Bartok also exploited the symmetrical characteristics of the 1:3 model scale to construct the music of each section.
Augmented Triad Structure in the 1:3 Model Scale
According to Lendvai, 1:3 model scale was a favorite scale used in Romantic music and its “floating” quality was already recognized by Liszt and Wagner.7
This
“floating” quality is due to the character of “annihilating” the tonalities, which is originated from the use of scale’s augmented triad, which is a mirror-structured harmony.
Four augmented triads can be derived from the 1:3 model scale on B (see
example 4).
7
E. Lendvai, Symmetries of Music, 30. 7
Example 4:
Mirror harmony – the augmented triad
Unlike the music of Liszt and Wagner, Bartok’s compositional strategy to employ the 1:3 model scale in “Subject and Reflection” is not for its tonal annihilating character, but for its symmetrical attribute. triad is endowed with symmetry.
The intrinsic feature of an augmented
It is a chord with the 3rd note as the axis expanded
an interval of M3rd higher and lower in span.
This mirror inversion is a kind of
harmonic symmetry, arranged in vertical manner (see example 5).
In fact, not only
did Bartok exploit the symmetrical traits of an augmented triad, but also extended it to a quintal chord (chord by fifths), which is also a mirror-structured harmony, to constitute the music of Section 2 and 6 respectively.
The use of this strategy will be
discussed thoroughly in part II of the paper.
Example 5: The axis of the centric note in Section 2 and its expansion
Major-Minor Structure in the 1:3 Model Scale
Another symmetrical attribute built in the 1:3 model scale is the major – minor 8
In example 6, several major-minor triads can be derived from the B
triad feature. 1:3 model scale.
But how the presence of major-minor triad derived from the scale
contributing to the symmetrical design of the work?
Example 6:
Major-minor triad in 1:3 model symmetrical scale
Similar to the inherent symmetrical feature embedded in an augmented triad, we do realize that when a major triad disposes in mirror inversion, a minor triad is derived (see example 7), and thus, a minor triad can also be regarded as a symmetrical reflection of a major triad.
That is why a minor triad, [037], is classified to the same
set class of major triad, [047], under the concept of set theory.
Example 7:
Major-minor triad – mirror harmony
Therefore, the employment of major-minor triad (in vertical aspect) or major-minor scale (in linear aspect) materials simultaneously could attribute to the 9
symmetrical character of the music.
Bartok’s intention on using the major-minor
idea as the compositional strategy is, again, exhibited obviously. on the triads (vertical aspect).
But the focus is not
It is the materials derived from the scale (linear aspect)
that Bartok employed to construct the symmetrical melodies in both voices (see example 8).
Two-parts contrapuntal melody flowing in mirror-inversion is easily
found in many sections of “Subject and Reflection” (example 8).
Example 8: The major-minor scalic materials used simultaneously in Section 1
From the above example, the purpose Bartok employed major-minor scalic materials simultaneously in Section 1, is palpable not for “neutralizing” the tonalities in order that an atonal effect is created in the music, as what the Romantic music usually did.8 lower parts.
It is for constructing a symmetrical melodic line in both upper and The left hand melody, therefore, becomes a mirror inversion of the right
hand melody (see example 9).
This kind of symmetrical construction of melodies
can be found throughout the section 1 to 6 of the music.
The upper melody employs
materials from major scale while the P 4th higher apart natural minor scale is used for
8
Lendvai used a musical excerpt of Wagner ’s Parsifal to illustrate this tonal neutralized effect by employing major-minor triads in juxtaposition in his book Symmetries of Music. For further reading, please read E. Lendvai, Symmetries of Music, 30-31. 10
the lower part (see table below).
However, in Section 7 and Coda, some chromatic
tendency notes are completed (filled) in the gap of a diatonic gesture bringing a certain tension to the music. design.
But it cannot annihilate the musical symmetry in
Please refer to the table below for the major-minor materials employed in
each voice for constructing the mirror-inversion melody:
Table for Major-Minor Planning of the Two-Parts Melody in Each Section:
Section
Upper melodic part material Lower melodic part material forming mirror inversion
Section 1
B major scale
E natural minor scale
Section 2
B major scale
E natural minor scale
Section 3
D major scale
G natural minor scale
Section 4
E major scale
A natural minor scale
Section 5
F major scale
B natural minor scale
Section 6
G major scale
C natural minor scale
Section 7
B major scale
Chromatic completion in diatonic gesture
11
Coda Section
B major scale
Chromatic completion in diatonic gesture
Example 9: Major-minor scale materials used in the symmetrical melodic parts of Section 1
In short, the use of 1:3 model scale is extraordinarily important and indispensable to the symmetrical organicism of this piece.
The scale governs not only the
large-scale symmetry of the entire structure of the music, but also provides prodigious and valuable symmetrical resources for the composer to construct the smaller-scale symmetries in each section.
Now, I begin my exploration on the use of small-scale
symmetrical strategy section by section.
Part II: Small-Scale Symmetrical Strategy
Generally, the texture of “Subject and Reflection” is polyphonic in two-parts 12
counterpoint.
In order to delineate the symmetrical character of the music in
relevant to its title “Reflection”, as I have mentioned before, the lower left hand part melody throughout each section of the piece is constructed in mirror inversion of the upper right hand part in the same rhythmic and melodic pattern.
But what are the
compositional devices used to develop the music within this symmetrical strategy?
Symmetrical Strategy in Section 1:
Section 1 of “Subject and Reflection” comprises two musical phrases. phrase is from mm. 1 – 7 and the second is mm. 8 – 14. centric-axis note, B.
The first
Both phrases end on the
From the example 9 above, melodies of both voices begin
from an octave apart from this axis note simultaneously and flow in a strict manner of mapping to each other.
The melodic span is an interval of P 5th, that is, B – F in the
upper part and B –Ein the lower part.
Besides the inner pedal B sustains
throughout the section, the contrapuntal melodies also move along the axis Bwithin the P 5th boundary portraying a melodic symmetry (see example 10).
Example 10:
Melodic span forms melodic symmetry
13
Symmetrical Strategy in Section 2:
The musical symmetry experienced in Section 2 much relates to the intervallic structure of an augmented triad derived from the 1:3 model scale. contains two phrases in eight bars long (mm. 15 – 22).
inner voice acts as a primary axis.
This section
The centric pedal note B in the
An interval of M 3rd is expanded above and below
this axis forming a mirror-structured chord of augmented triad (G B D) at the beginning of bar 15.
Both the upper and lower contrapuntal melodies are paired up in
strictly manner setting off their melodic journey from these two temporarily secondary axes, D and G.
They also function as the counter-melodies in mirror-inversion
symmetrically opposing to the sustaining pedal point B.
In addition, each melodic part
forms a turn-like gestural figure (intervallic span of 3rd) producing a
somewhat symmetrical melody in reflection to each other (see example 11) alone.
Example 11: Turn-like gestural figure in Section 2
Symmetrical Strategy in Section 3: Section 3 begins from mm. 23 – 29 and the centric note is D.
Both upper and 14
lower parts flow along D axis to provide melodic symmetry individually.
In order to
give greater variety and to portray a false canonic imitation, which becomes a real device in the Section 6 and 7, to the music, the left hand mirror-inversed melody in the first phrase (mm. 23-25) delays a quaver value (half beat) to enter but vice versa in the second phrase (mm. 26-29) forming an off-beat delay entry (see example 12). But after the first note entered, both melodies flow vertically mapping to each other strictly again.
Furthermore, D centric melody is transposed (or expanded) an octave
higher in the second phrase for developing the music.
It is an archetypal using of the
compositional device of off-beat entry to serve the musical symmetry.
Example 12: The off-beat delay entry of the melodic parts
Symmetrical Strategy in Section 4:
Section 4 embraces a continuous melodic phrase of 10 bars long (mm. 30 – 39).
The mirror-inversed left hand melody again moves against the right hand part in synchronized manner.
Its symmetrical trait and the varied melodic figure is
very similar to that of the opening in Section 1, except that the contrapuntal melodies 15
of both parts begin from the notes a M 2nd higher from the centric note, that is, the F and D notes, creating a dissonant clash with the centric note E(see example13) in the starting moment.
Example 13: Melodic Variation in Section 4
Symmetrical Strategy in Section 5:
Section 5 consists of two phrases from mm. 40 – 46 flowing around the centric-axis F note.
The left hand melody is transposed to an octave higher in the
upper melody of the second phrase for variety.
applied in both phrases.
Again, the off-beat delay entry is
This compositional strategy to reinforce the symmetry is
very similar to that of Section 3, not only forming a sectional repetition, but also
producing a sectional symmetry in structural level as well (see example14).
16
Example 14:
Off beat delay entry and expansion of the centric an octave apart
Symmetrical Strategy in Section 6: Section 6 comprises three phrases (mm. 47 – 58) plus a small link of four bars long (mm. 59 – 62) before returning to the recapitulation section.
Since there is an
obvious increase of technical complexity of the compositional process, this passage is more exciting and dramatic comparing with the previous sections.
Not only does the
upper melodic line flow along the high tessitura around the high D6 note9, but also the symmetrical boundaries between the upper and lower voices firstly span more than two octaves apart contributing to the dramatic effect (see example 15).
Each melody
begins its movement along its own secondary axis, D6 and G4 notes and finally returns to it, engendering a melodic symmetry.
9
I regard that the middle C note is C 4 in this paper. 17
Example 15: High tessitura, symmetrical boundaries and canonic imitation of the two-parts melody in Section 6
Obviously, the primary axis is the centric G note.
The starting notes of the
two-parts melody, D6 and C4 notes are expanded in an interval of P 5th above and below from this centric forming a quintal chord (chords by fifths). mirror-symmetrical harmony.
This is also a
The strategy of intervallic expansion being used here,
undoubtedly, is a technical extension of Section 2 (see example 16).
Then, from mm.
59 , after the phrase 3 onward, A and F notes are expanded an interval of M 2nd above and below the centric G , forming a secondual chord dyad (chord by second) constituting another vertical symmetry (see example 16). Example 16: Vertical symmetrical strategy in Section 6
18
In the succeeding small link after phrase 3 (mm. 59 – 62), the left hand dyad G1 – F1 and the right hand dyad G6 – A6 disjunct across different octaves vigorously in contrary motion attaining the maximum boundaries of five octaves apart in bar 62 (see example 16).
It is the focal point consisting of the greatest span in boundaries of this
section, as well as the entire music.
This expansion of the symmetrical boundaries
over a certain octaves will occur again later in the section 7, the recapitulation part of the music.
Before moving forward, there is one more essential point deserved mentioning, that is, the contrapuntal melodies in both hands are imitated in two-parts strict canon with half beat delayed (quaver-value delay).
This interlocking of rhythmic and
melodic patterns producing a somewhat stretto effect, contributes much to the exciting and exuberant character of the section (see example 15).
This compositional
strategy, is explicitly a technical extension from the off-beat delay entry employed in Section 3 and 5 (see example 15).
The only difference is that the previous off-beat
delay entries produce only a “false” half-beat delay because except for the first note, the rest of the melodies are synchronized to each other in both voices, but the true canon here maintains the voices in half-beat delayed and never maps together strictly.
Symmetrical Strategy in Section 7 and Coda
The Section 7, also regarded as the recapitulation, consists of three phrases starting from mm. 63 – 73 while the Coda section is from mm. 74 – 82. of Section 1 recapitulates in an octave lower from mm. 63 onward.
The melody
Each phrase,
similar to the opening section, moves along the centric-axis B note producing the melodic symmetry.
The returning of the centric note, as well as the similar melodic 19
variation in right hand part, not only functions as the coherent unit, but also provides the large-scale structural symmetry for the music (see example 17).
Again, the
canonic imitation of the upper and lower voices with one beat delayed keeps the vigorous stretto effect as the previous section did.
The lower part melody, just like
in all other sections, maintains its mirror-inversion character, portraying the image of “Reflection” of the programmatic title.
Also, this left-hand melody continues to
maintain the mirror-inversion character throughout the rest parts of the music portraying the image of “Reflection” of the programmatic title.
The symmetrical
span expands to three octaves apart in the phrase 3 (mm. 70 – 73) and the melodic flowing in the high tessitura reappears again contributing to the dramatic and climatic effect of this section (see example 17).
In the Coda Section, the contrapuntal melodies also flow around the centric-axis B .
The off-beat delay entry technique (half-beat delay) is used again to produce
tension to the music.
As discussed above, the final centric Bnote returned in
the upper voice of this section in bar 82 retains the same pitch level as the opening section maintaining a large-scale structural symmetry for the entire work in a certain extent (see example 17).
20
Example 17:
Expansion of the symmetrical boundaries and the Coda section
Symmetrical Design of the Musical Form:
After the scrupulous analysis of the small-scale symmetrical strategies used in each section, it is noticeable that the compositional devices used to reinforce those symmetries also assist in shaping the entire musical form.
Instead of using only the
conventional way, the design of motivic materials, to determine the structure of a piece of music, we now use the compositional strategy as the analytical model to discern the character of each section.
Interestingly, it is observed that the musical
form of “Subject and Reflection” is also symmetrical in nature.
This symmetry of
structure and form is very similar to the symmetry of tempo setting that we have discussed in the previous part of the paper.
Let us refer to the symmetrical design of
the musical form pertinent to various compositional strategies from the following table (see example 18) before further exploration:
21
Example 18:
Symmetries designed in organic structure
From the above analysis, we attempt to classify each section in parallel with the compositional devices employed in that section.
Of course, some of the techniques
are mere variations, and hence they are categorized into the same group. example, the compositional device B of Section 2 is the centric expansion.
For It
appears in different forms of variation in B’ of Section 3, the B’ of Section 5 and 6, the B” of the link passage.
They all show similar attribute of expanding (transposing)
the centric-note melody to a certain intervallic span, for instance, the third, fifth, octave or even across several octaves. figure 1 (A’) in Section 4.
Another example is the display of melodic
It can be regarded as the variation of the opening melodic
figure (A) in Section 1, because they possess similar melodic and rhythmic gestures. This Section 4, thus, becomes the symmetrical axis of the large-scale structure and the other sections just relate to it constituting the symmetrical rondo-like form, A – B – C – A’ – C – B’ – A” – B’ – C (see example 18).
Apart from this large-scale 22
symmetry, some small-scale symmetries are also embedded within section and between sections (please refer to example 18).
The compositional devices,
therefore, are used to serve for the construction of the musical symmetry enhancing the structural organicism of the entire music.
Conclusion:
In short, Bartok wrote “Subject and Reflection” is not merely for the pedagogical purpose to discipline the virtuosity of playing contrapuntal piano music. This piece, in fact, reflects Bartok’s life-aimed compositional idea searching for a new musical language to express his artistic creation.
Instead of using conventional
compositional practices, musical symmetry is the main strategy used to govern the entire piece.
In term of large-scale strategy, he deliberately employs the symmetrical
1:3 model scale on Bto design the centric plan of the piece.
He further utilizes the
symmetrical traits of this scale by using the derived augmented triad and its extension, quintal chord, as well as the major-minor materials, to construct different kinds of vertical and linear symmetries.
As a result, the musical symmetries contribute
greatly to the organic structure of the entire piece.
Apart from the using of 1:3 model scale strategy, Bartok also uses multifarious compositional techniques, such as mirror inversion, canonic imitation, stretto effect, octaves displacement, intervallic expansion of the symmetrical boundaries, development of the secondary axes, off-beat delayed entry and turn-like gestural figure, to enhance the musical symmetries in the small sections, as well as for the 23
design of the entire form.
Therefore, “Subject and Reflection”, albeit short and
small, just as the title suggested, successfully delineates the visualized image of reflection by the virtue of various musical symmetries that makes it one of the most important and virtuoso pieces in the 20th Century piano music.
24
Bibliography:
Bartok, Peter. “Preface.” In Mikrokosmos 6, Bela Bartok. New York: Boosey and Hawkes, 1987.
Jonathan, W. Bernard. “Space and Symmetry in Bartok.” Journal of Music Theory 30, no. 2 (fall 1986): 185-202.
Lendvai, Erno. Symmetries of Music: An Introduction to Semantics of Music. Budapest: Kodaly Institute Kecskemet, 1993.
Suchoff, Benjamin. “Synthesis of East and West: Mikrokosmos.” In The Bartok Companion, ed. by Malcolm, Gillies. London: Faber and Faber Ltd. 1993.
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