INTRODUCTION: INTRODUCTION: BALANCE
We live in a world of immense beauty. beauty. There are a multitude multitude of forms with countless countless variations variations on simple simple themes. I want to speak here about balance and make some comments about about how balance can be achieved musically. musically. There are countless countless ways that architectural architectural balance can be musically achieved from the micro to macro level. Since attention to detail has always been an important important factor for me, and these things are not usually discussed, I would like to initiate some dialog on this subject. The most obvious kinds of balance that come to mind are the various forms of symmetry (i.e. bilateral, etc. that can be applied musically, musically, using using intuitive intuitive and logical methods. methods. Symmetry Symmetry is a fact of nature and one of the oldest fascinations fascinations of humanity. Some of the more obvious ways in which symmetrical musi musica call bala balanc nce e coul could d be real reali! i!ed ed are throu through gh melo melody dy,, rhyth rhythm, m, tona tonalility ty,, form, form, harm harmony ony and and instrumentat instrumentation. ion. "s well as the structural structural consideration considerationss of symmetrical symmetrical musical musical forms I will also discuss these structures from a dynamic point of view, i.e. as they progress through time.
MELODIC MATERIAL GENERATED BY SYMMETRICALLY DERIVED LAWS OF MOTION
This theory was was originally a melodic theory. I named it Symmetry because the motion of the melodies involves an e#pansion and contraction of tones around an a#is tone or a#is tones (i.e. around a center point. point. The e#pansion e#pansion and contraction involved involved is almost always always e$ual on both sides of the a#is, hence the term Symmetry Symmetry.. This is basically basically a melodic melodic system that obeys obeys it%s own laws of motion. motion. It moves according to the gravity of this motion more than anything else but it can also be adapted to deal with the gravity of other types of tonality such as cells or the traditional dominant&tonic harmonic system. system. When I first started started dealing with symmetry symmetry I only dealt with the laws of motion motion produced by the system without without any regard for other types types of tonality. This is how I would suggest others others to learn the system to get a feel for for thinking in these terms. "lso a complete knowledge of intervals intervals and their relationships (always thinking in terms of semi&tones would be e#tremely helpful. I began by writing symmetrical symmetrical e#ercises e#ercises for myself. myself. Then I practiced these these e#ercises to get my fingers fingers and ears used to moving and hearing these ideas. It was only after doing this that I practiced practiced improvising within these structures playing at first in an open manner (not based on any outside structure such as a song. 'ater I adapted these improvisations to structures and forms. I did this by slowly slowly integrating integrating the ideas with the more traditional traditional improvisation improvisational al style I was already playing. playing. y goal was not to play in a totally symmetrical style (as this would be as boring as playing all major scales but to integrate the style and give myself more options when I improvise. The basic system system involves what what I call two spirals. They are tones that move out e$ually e$ually in half steps from an a#is (which is always at least two tones. If the a#is of the first spiral is the two tone )&) unison (one octave octave above middle ), it could be in any octave then from that unison ), you move out (spiral out each tone in a different direction in half steps, steps, i.e. )&), then then * on the bottom and ) sharp on the top+ * flat on bottom bottom and on top+ " on bottom and sharp on top+ " flat on bottom and - on top+ on bottom and / on top+ flat on bottom and / sharp on top (at this point you are at the beginning of the spiral again, or the symmetrical mirror image of the spiral+ spiral+ / on bottom and on top+ - on bottom and sharp on top+ - flat on bottom, " on top+ on bottom and " sharp on top+ flat on bottom and * on top+ ) on bottom and ) on top (this is your starting point one octave above and one octave octave below your original tones. 0ou%re thinking two tones at a time and they%re spiraling spiraling out together. This I call spiral number one. "s you spiral out from )&) (the a#is and you think of the interval between the tones then ) to ) is a unison. The ne#t tones in the spiral are * and ) sharp, the interval between * and ) sharp is a major second. 1e#t in the spiral is * flat on bottom and on top, that%s a major third. Then " on bottom and sharp on top, that%s that%s an augmented augmented fourth or a tritone. tritone. )ontinuing, )ontinuing, " flat on bottom bottom and - on top, that%s an augmented fifth (could also be thought of as a minor si#th. si#th. Then on bottom and / on top is a minor seventh seventh.. The ne#t ne#t tones are flat on bottom bottom and / sharp sharp on top, these these tones are an octave apart and are really the same as the beginning beginning of the spiral. "ll symmetry has two a#is and in this system they are always a tritone apart from each other, more on this later. "s you keep spiraling
out until you reach the two )%s two octaves apart the important important thing about the spiral is not the tones themselves, themselves, but the intervals intervals between between each of the tones as you spiral out each each half step. It is these resulting intervals that are formed in spiral number one (2nison, aj 3nd, aj 4rd, Tritone, min 5th and min 6th that I call Symmetrical Intervals Intervals (see e#ample e#ample 7. This is important to remember as it forms the foundation for the laws of melodic motion. 1ote that beginning with the tones flat on bottom and / sharp on top, the intervals of the spiral repeat themselves if you perform octave reduction on the intervals. EXAMPLE 1 [Spiral # 1]
In spiral number two, you have two different tones as the a#is starting point instead of a unison. Instead Instead of )&) as the a#is, you have ) and flat together together as the a#is () is under flat. "nd then you spiral spiral out the same as you did in spiral spiral number one. So to begin with, with, you have the ) and the flat right above, above, then * on bottom and on top+ then * flat on bottom and - flat on top+ " on bottom and - on top+ " flat on bottom and / on top+ on bottom and / sharp on top+ flat on bottom and on top (the beginning of the spiral+ / on bottom and " flat on top, - on bottom and " on top+ - flat on bottom and * flat on top+ on bottom and * on top+ b on bottom and ) on top+ finally ) on bottom and b on top. So again you%re you%re spiral spiraling ing up in half steps but you%re getting getting complete completely ly differen differentt intervals intervals between between the tones in the spiral. spiral. To start off, you have have the minor second second between the ) and the flat. When you spiral out with the * on the bottom and the on top, you have the interval of the minor third+ with an * flat flat on the bottom and - flat on the top the interval is a perfect fourth. With an " on the bottom and and an - on top the interval is a perfect perfect fifth. fifth. " flat on the bottom bottom and / on the top produces produces a major si#th. on the bottom and / sharp on top, a major seventh+ seventh+ then the a#is again, again, etc. and you just keep spiraling spiraling out. "gain, "gain, its not the tones that are important in spiral number two but the intervals intervals between the tones which which are formed as you spiral out. These intervals intervals formed formed in spiral number two are a different set from the intervals formed in spiral number one, as a matter of fact they are all the intervals missing from spiral spiral number one. I call the intervals in spiral number two two (min 3nd, min 4rd, 8erfect 9th, 8erfect :th, aj 5th and aj 6th 1on&Symmetric 1on&Symmetrical al Intervals (see e#ample 3. EXAMPLE 2 [Spiral # 2]
The basis of the laws of movement are as follows. Thinking monophonically if you have an initial tone tone which you mentally consider to be the a#is, when you move in one direction from this a#is then general generally ly you must must move the same distance distance in the opposite opposite direction direction from that same a#is. /or e#ample, you play a ), then the ne#t tone you play is a above the ) , which is a major second away from ), then the following following tone you must play is the tone a major second second below ), which would be * flat. In other words, for the same distance that you moved above ) you must play the tone that is that same distance below ) (see e#ample 4 & measure 7 & beats 7 and 3, in these e#amples the a#is are circled. circled. "ctually "ctually it doesn%t matter matter whether the is above or below below the ) as long as you remember which direction you moved to get to the first tone, so that you move in the opposite direction (the same distance to reach the second second tone (see e#ample 4 & measure 7 & beats 4 and 9. ;bviously, you must know your intervals intervals very well to think like this $uickly $uickly in an improvisatio improvisational nal conte#t. /or another e#ample if you played and consider the as an a#is and then p lay * flat as your ne#t tone, then the
following following tone you play play must be -. With being being the a#is, * flat is a minor minor third above and - is a minor third below (see e#ample 4 & measure 3 & beats 7 and 3. 1ow you do not have to play the a minor third below , you could could play the - or the * flat flat in any octave. *ut you need to be thinking in in terms of being the a#is and the other tones %surrounding% the a#is (see e#ample 4 & measure 3 & beats 4 and 9. ;ne of the e#ceptions to this rule is when the interval that you play is one of the Symmetrical Intervals in spiral number one, those intervals being a major second, major third, tritone, minor si#th, minor seventh, octave etc. Then you don%t have to make make the e$ual movement in the opposite direction. direction. 0ou can chose to, but you don%t don%t have to. /or e#ample e#ample if you play the tone ) as the a#is, and the you play the tone , you don%t necessarily have to play * flat after that. 0ou could pick any tone at that point. *ut if you play one of the 1on&Symmetri 1on&Symmetrical cal Intervals Intervals in spiral number two, for e#ample, a#is ) to the tone / (an interval of a perfect perfect fourth, then you must play a after that, according according to these laws of movement (actually it would be more accurate to say that after playing the tone / you must then %complete% %complete% the symmetrical symmetrical motion, motion, more on this below. below. In this e#ample e#ample the tone can be in any octave, octave, but you need to play a because / is a perfect fourth above ) and is the perfect perfect fourth below ) (see e#ample 4 & measure 4. EXAMPLE 3
There are many variations to the above laws of movement that are still considered symmetrical movements movements according according to this theory. /or e#ample, e#ample, instead of moving away from an a#is you can do the reverse reverse and move towards an a#is. 0ou can play the tone /, then the tone , and then play the a#is tone ) (see e#ample 4 & measure measure 9 & beats 7 and 3. ;r you could think of the a#is as two tones further apart than a unison or a minor 3nd (actually, to be technically correct, even when it seems like there is only a one tone center all a#is are two tones as in the initial a#is of spirals spirals one and two, and they are all either either a unison or a minor second as in e#amples e#amples 7 and 3. So you could initiall initiallyy play ) and - flat. 1ow, logically, you might think think that the ne#t tone you have to play is ", ", because if ) is the a#is and if the - flat is a minor third above ) then you would think you have to play an " because " is a minor third third below the a#is tone tone ). *ut you can play play ) and - flat and then play * and - natural< natural< This is because you can think of the ) and - flat flat together as an a#is. Then you can e#pand out a half step on either side of this a#is and play * and - natural (the true a#is in this situation situation are the tones flat= or =" flat. This movement would still be within the rules because because mentally you%re using ) and - flat together as an a#is (see e#ample 4 & measure 9 & beats 4 and 9. So there are many variations depending on what you you mentally think of as the a#is. Some sample symmetrical movements are listed in e#ample 9, try and follow follow the logic of the movements. movements. -#ample -#ample : is a symmetrical symmetrical melody melody which connects different movement logics in one idea, demonstrating how the movement laws can flow together. together. The circles circles will give give you some hints on where where to look for the a#is. This is a more comple# comple# e#ample, e#ample, notice the %nested% %nested% a#ial movements (%nested% meaning meaning some tones of one a#is overlap and share tones tones with adjacent adjacent a#is. It is possible to generate generate shapes that contain contain an entire chain of nested nested a#ial a#ial movemen movements, ts, however however the result resulting ing melodi melodies es would would be e#treme e#tremely ly jagged jagged and not necessarily sound musical, unless of course that is the desired effect. EXAMPLE 4
EXAMPLE 5
"ll of the e#amples above are written in atonal space. The analysis of the a#ial progression can be thought thought of in any number number of differ different ent ways. ways. In other words words it is possibl possible e to analy!e analy!e these same passages passages differently differently and still be well within within the given laws. 1otice 1otice that the above e#amples e#amples re$uire thinking thinking in small cells of ideas, at least initially. initially. >owever >owever there is a linear gravity gravity involved involved in the thinking which re$uires that the improviser become fluent in thinking in two directions simultaneously. It should be clear from -#ample : that it helps to be able to think at least two or three tones backward and forward forward in time. This is a different different skill than the normal way of thinking thinking as retention retention of individual tones, as well as phrases, needs to be practiced. /or instance, in -#ample : at the end of measure 3, the tones / and - are the a#is, not only of the tones immediately following (" and ) but also of the preceding tones (b and "b< ;verall this produces a sort of accordion effect in time. 1ot that this will be heard by the average person, especially given the speed of e#ecution, but it will be felt and it does have an effect. "fter this what has to be obtained is fluency in progressing from one idea to the ne#t in seamless motion, building up to higher levels levels of comple#ity in the communication of ideas. This is similar to the progression progression from words, to sentences, sentences, to grammar, grammar, and finally to communicatio communication n of conceptual ideas in linguistic e#pressions. "lso, with some imagination, the same ideas could be merged with any other logic. It is not my goal here to write down ideas for others. I simply want to demonstrate that there are many possibiliti possibilities es to be e#plored, e#plored, definitely definitely more than have already been e#plored. e#plored. I have been working with the ideas above for at least 33 years now and I still have not found any end in sight<
HARMONIC MATERIAL GENERATED IN SYMMETRICAL SPACE
;ver the years I have been e#ploring several ideas which could be e#pansions of the symmetrical laws of motion mentioned mentioned above. above. ost of these ideas are based on the various concepts concepts of %gravity% %gravity% and what can be generally called %binding% and %unbinding% (i.e. different types of laws of attraction. The melodic concept discussed above and other related harmonic concepts all deal with tonal centers in terms of spatial geometry, as opposed to the standard tonality which deals in tonal key centers in terms of tonics. These different different approaches approaches can be looked at as different types of %gravity%. %gravity%. >ere we could borrow two terms coined by music theorist -rnst 'evy, calling the concept of gravity that results in the traditional traditional tonic&based tonic&based tonality Telluri Gr!"ity or Telluri Telluri A#!$t!ti A#!$t!ti%& %& and the concept of gravi gravity ty that that is at the the basi basiss of cent center erss of %geom %geomet etri ricc spac space% e% A'(%lute C%&e$ti%& Telluric C%&e$ti%& . In Telluric "daptation out perception of gravity is based on laws of attraction that are influenced by our sense of %up% and %down%. Thus we tend to look at the harmonic series series only from the %bottom&up% %bottom&up% perspective, perspective, with the %fundamental% on the bottom. This is a %terrestrial% mode of thinking influenced by the fact that we live on -arth and tend to locali!e locali!e our concept of space space according to our everyday everyday situation. situation. In "bsolute )onception what is important is the position of the tones tone s in space and a nd their distance. >ere the harmonic series is seen as %spiraling% out from a %generator% (as opposed to a tonic or fundamental so as to produced both an %;vertone% and %2ndertone% series (see e#ample 5&a< "bsolute )onception is based on a %universal% mode of thinking that results when you look at the -arth, other planets, satellites and stars from the point of view of how they relate to each other in space. So the difference is the way the gravity operates from a %terrestrial% or %telluric% perspective (on -arth we tend to think of
the gravitation pull in one direction, %down% and how gravity operates from a %universal% or %absolute% perspective (in space we tend to think of objects orbiting around a gravity source or being pulled towards towards the source source from a multid multidire irecti ctional onal perspecti perspective ve.. In the absolu absolute te concep conceptio tion n partial partialss are thought of as %orbiting% around a generator tone producing both overtone and undertone energy. These two different concepts of gravity, telluric and absolute, will be e#plored in more detail below. E)AMPLE *+!
"bsolute conception of the >armonic Series based on the generator tone ) O"ert%&e Serie(
U&#ert%&e Serie(
I just want to say a few things about the informatio information n series above. The tones shown shown are the closest e$ual temperament temperament e$uivalents e$uivalents to the actual actual tones that are in the series. In some books the 77th partial is listed as / natural instead of / sharp. >owever I believe that this is technically wrong as the ratio 77?@ is closer closer to /A by a very small amount (if ) is generator. generator. There are 7BB cents to an e$ual e$ual temperament half step (for e#ample between between / and /A there are 7BB cents. The ratio 77?@ (which is 77?7 sounded sounded in the same octave octave as the generator is ::7 cents above the generator. generator. "n e$ual temperament perfect 9th is :BB cents above the generator and an e$ual temperament augmented 9th is 5BB cents above the generator. Since ::B cents would would be an e#act $uarter tone between a perfect 9th and an augmented 9th then 77?@ is closer to /A as the distance between 77?@ and / natural is :7 cents and the distance distance between between 77?@ and /A is only 9C cents. The tone is actually actually closer to /A and books that list / natural as being the 77th partial in a harmonic series with ) as generator are not correct. ;f course this means that the corresponding tone, the 77th partial in the undertone, series is b. >owever >owever since most books do not deal with the undertone series series we don%t have to worry about that. "lso the 74th partial is listed as being " natura l instead of " flat. "gain this is technically wrong as the ratio 74?@ is definitely definitely closer closer to "b (again if ) is generator. generator. The ratio 74?@ is @97 cents above the generator generator (74?@ is 74?7 octave reduced. reduced. "n e$ual temperament temperament minor 5th is @BB cents above the generator and an e$ual temperament temperament major 5th is CBB cents cents above the generator. So the ratio 74?@ is closer closer to being "b because the distance between 74?@ and "b is 97 cents and the distance between 74?@ and " natural is :C cents. The tone is closer to to "b and books that list " natural as being the 74th part partia iall in a harm harmon onic ic seri series es with with ) as gener generato atorr are not corr correc ect. t. This This also also means means that the corresponding tone, the 74th partial, in the undertone series is - natural. In terms of the nomenclature that can be used to e#press the actual tones which act as a#is (melodic or generators (harmonic (harmonic in the "bsolute "bsolute )onception )onception I propose propose using Sum N%t!ti%& as the main terminology terminology.. So when I speak of improvising improvising with regard to a %sum 77 tonal center% I am speaking speaking of
an "bsolute tonality that has an a#is (or spatial center of sum 77. Sum 77 means that the tones *&) (also /& flat are the spatial tonal centers centers of this section of the composition composition.. /or the improviser improviser this means improvising with this spatial tonality in mind. ;ne necessary skill re$uired for this mode of thinking would be to learn how to hear spatially with the mind as well as with the ears (actually it is all in in the mind. In other words learning to construct mental images of the %geometric space% and to be able to %hear% inside of that space. The reason reason for using the term %sum% %sum% comes from the concept concept of adding note note numbers. If tones ) through * chromatically are represented by the numbers B through 77 respectively, then it is possible to %add% tones together to arrive at their sums. The sums represent the a#is (or center point between the two tones being being added together. together. /or a %sum @% a#is, any two tones that that add up to the number @ would be considered a sum @ interval. /or e#ample sharp and / (4D: would add up to @. The center center (or a#is of sharp and / is - and - (which is also sum @ or 9D9, an a#is always implies implies at least four tones, in this case the -&- unison represents two tones but if thought of from another perspective * flat and * flat is also the a#is, i.e. i.e. 7BD7BE3B minus 73 E @. The same goes for ) sharp and (also sum @. So the a#is of a sum 77 interval would be * and ) (i.e. 77 D B E 77. Since we are dealing with 73 tones the entire tonal system then , for the purposed of octave reduction, you can continually subtract the number 73 from any sum that is 73 or greater until the sum is below 73. 1otice that all of the even sums are the result of any interval in spiral number 7 above and all of the odd sums are the result of intervals from spiral number two. -#ample 5&b is a table that is a summary of the relationships between spirals, a#is and sums?
E)AMPLE *+' SPIRALS NUMBER ONE & TWO AXIS TONES
SUM #
AXIS TONES
SUM #
C-C
SUM 0 CC-Db
SUM 1
C#-C#
SUM 2 C#-D
SUM 3
D-D
SUM 4 DD-Eb
SUM 5
Eb-Eb
SUM 6 Eb-E
SUM 7
E- E
SUM 8 E-F
SUM 9
F-F
SUM 10
F-F#
SUM 11
F#-F#
SUM 0 F#-G
SUM 1
G-G
SUM 2 GG-Ab
SUM 3
Ab-Ab
SUM 4 Ab-A
SUM 5
A-A
SUM 6 AA-Bb
SUM 7
Bb-Bb
SUM 8 Bb-B
SUM 9
B-B
SUM 10
B- C
SUM 11
This may initially be a little confusing but many things that are unfamiliar are confusing at first. With a little work it can be as natural as any other internali!ed system.
A THEORY OF HARMONY
The book A Theory of Harmony by -rnst 'evy deals with a different approach to harmony and voice leading coming from a standpoint of using perfect :ths, major 4rds and their reciprocals (perfect fourths fourths and minor 5ths or 8olarity Theory. Theory. The book talks a lot about upwards perfect perfect :ths and downward perfect :ths (or an upward perfect :th or %dominants%. It then uses the same same approach with major major thirds thirds (which (which 'evy calls %deter %determin minants ants%, %, using using upward major major thirds ( and downward downward major thirds thirds ( . 'evy 'evy then deriv derives es all of his his harmoni harmonicc and voice voice leadin leading g theory theory from these these two two concepts, concepts, the only e#ception e#ception being his inclusion inclusion of the importance importance of the %natural 6th% of the ratio 6?9. In 'evy%s view the natural 6th is important for several reasons, FThe seventh partial appears in the same octave within within which the triad is completed completed by the introduction introduction of the determinant.F determinant.F It is 'evy%s view that the natural 6th Freveals the latent dynamism of the triad.F 'evy speaks of a %senarius%, i.e. the first si# ratios, as forming two mutually e#clusive triads, one major and the other other minor. minor. If unity is ) ( 'evy prefers prefers to use the term %generat %generator% or% which which has a broader broader meaning that unity, I agree with him then the upward triad is )&-& () representing the numbers 7 and 3, representing the number 4 and - representing the number :. The downward triad would be )&"b&/, (again ) representing the numbers 7 and 3, / representing the number 4 and "b representing the number :. The seneric intervals are the octave, perfect fifth and major third, corresponding to the numbers 3, 4 and :. In other words the octave is associated with the prime 3 since since it is a doubling, the perfec perfectt :th with the prime prime 4 and the major major third with with the prime prime :. "lso "lso note that this is the first first number numberss of the /ibona /ibonacci cci se$uence se$uence.. When When 'evy 'evy includ includes es the natural natural 6th then then this this senari senarius us is e#tended to an %octarium% or comprising the first eight ratios. 'evy also speaks a lot in psychological and sometimes almost mystical terms about music and music theory theory.. In this way you can see the influenc influence e on -rnest -rnest c)lain c)lain who is I believ believe e one of 'evy%s students students and also his colleague. It is a combination combination of 'evy%s >armonic >armonic 8olarity Theory and his philosophical and psychological point of view that I find useful. >is book of course has no mention of rhythm where the concepts of balance and form are even more important. What I find useful is the e#treme symmetry that 'evy is dealing with which reminds me of some of the work I%ve done as well as elements of *artok%s work, >enry Threadgill%s Threadgill%s work, W.". athieu, >oward *oatwright, Schwaller de 'ubic! and ancient -gypt, 8ythagoras, 8lato and the work of the ancient reeks, *abylonian ideas of reciprocity and the work of 2mayalpuram Sivaraman and other related Gedic symmetrical ideas. I am especially attracted attracted to the idea that 'evy has introduced introduced of the upward and downward %determinant% being of of e$ual importance as the upward and downward %dominant%. >e then links these concepts dynamically dynamically and show how they work in progressions progressions of triads, after which 'evy introduces his concept of consonance and dissonance, temperament, tonal function of intervals, triads, triads, non&triadic non&triadic and compound compound chords. 'evy summari!es summari!es the discussion discussion in his book as follows? follows? 7. Tone Tone has has a struc tructu ture re.. Its Its val validit idityy can be test tested ed on the the phy physica sical& l&a acous cousti tica call lev level (division of the string as well as on the musical&esthetic level (fertility and musical ade$uacy of application. 3. ajor and minor are manifestations of the general principle of polarity. 4. The triad being the norm of our tonal system, the third has a direct function within the tonality, e$ual in dignity dignity to the fifth. fifth. 8arall 8arallel el to the term dominant dominantss for the upper and lower lower fifths, fifths, the term determinants will serve for the functions of the third. 9. " major triad tends to function as dominant, a minor triad as subdominant. :. " chord is a conglomerate organi!ed by one or several generators. 5. To distinguish natural from psychological consonance and dissonance, the concept pair of words ontic&gignetic will designate the latter. Hby the term ontic 'evy means the state of %being% or what I call stationary and by the term gignetic he means the state of %becoming%. *y psychological consonance and dissonance 'evy is referring to our active mental participation in the concepts of consonance and dissonance, which is a little different for
each person, as opposed to what he refers to the natural concept of consonance and dissonance which is inherent in the phenomenon. >e sums these ideas up as follows? The triad is consonant. "ll other chords are dissonant. The triad may be used as a dissonance. ;ther chords & maybe all of them & may be used as consonances. HThe first two statements of the above summation reflect the natural concept of consonance and disson dissonanc ance. e. The last two statemen statements ts of the summation summationss reflec reflectt the concept concept of consonan consonance ce and dissonance from a psychological point of view. /or the psychological conception conception of consonance and dissonance 'evy calls this ontic and gignetic, in order to differentiate from the terms consonance and dissonance dissonance (which (which refer only to the natural conception. conception. 'evy notes that if a consonant consonant character could be described as Fan impression of restfulnessF and Fin musical terms, we can say that a consonant chord is apt to be used as a closing chordF, the triad represents that. >owever, Fsometimes we can observe that chords which are not perfect triads and therefore do not posses that naturally cons consona onant nt char charac acte terr are are yet yet bein being g used used as clos closin ing g chor chords ds and and do give give us an impr impres essi sion on of restfulnessF. F;n the other hand, triads may be used in a way as to produce an impression of tension henc hence e of impe imperf rfec ectition on,, as in a half&c half&cade adence nceF. F. -lse -lsewh where ere 'evy 'evy stat states es that that Fthe Fthe prog progres ressi sing ng psychologi!ation of music had reached a culmination point in the period after the first world war, when the e#istence of consonance and dissonance was largely disregarded or even denied, and when solely the ontic&gignetic concept concept pair was relied upon for producing the desired effects of binding and unbinding.F unbinding.F That leads to the summation summation above above about consonance consonance and dissonanc dissonance. e. I intuitively intuitively am attracted to this way of looking at things as it makes room for both the idea that there are universal $ualities that are inherent in a phenomenon and that these $ualities can change according to the perspe perspecti ctive ve of the observer observer and the observe observer%s r%s active active mental mental particip participatio ation. n. "lso "lso I think think that that the introduction of a concept of Fbinding and unbindingF that is not necessarily connected to traditional concepts of Fconsonance and dissonanceF is very important as this makes room for radical e#tensions of previous ideas 6. The seventh in the dominant seventh chord is the natural seventh (i.e. the 6th partial @. In analogy to calling the fourth the complement of the fifth in the octave, the minor third is recogni!ed as the complement of the determinant in the fifth. C. Temperament arises from the necessity to represent the infinite within the definite. 7B. Traditional and newly introduced nomenclature is indicated by shorthand symbols. Hin my opinion some of these shorthand symbols go too fa r 'evy also talks talks about two types of gravity, gravity, telluric telluric gravity gravity or telluric telluric adaptation adaptation meaning meaning the normal bottom to top gravity that we all know (the term telluric means terrestrial or earthly, which I take to mean mean from the ground ground and absolu absolute te gravity gravity or absolute absolute concepti conception on which which looks at things things symmetr symmetrica ically lly being being generat generated ed from a center center.. enera enerally lly when when -rnst -rnst 'evy 'evy discuss discusses es tradit tradition ional al concep concepts ts and names names such as major and minor minor triads, triads, etc. etc. then he is speaking speaking in tellur telluric ic terms. terms. ;therwise ;therwise this entire conception conception is basically basically an absolute conception conception dealing dealing with generators and polarity. polarity. /or upward thinking thinking (major tonality there is no difference difference between telluric telluric adaptation adaptation and absolute absolute conception. conception. Therefore, Therefore, for all intents and purposes, purposes, when we are referring to absolute absolute conception we are talking about %downward% symmetrical thinking and this will be designated by the symbol o (there is no need to designate telluric symbols but in his book 'evy sometimes uses the E symbol fo r telluric adaptation. The table below (-#ample 6 is in shorthand symbolism and shows the progressions of triads and the change of function of the common tones. "ccording to 'evy? FThe general tendency tendency of both triad inversions inversions is that of the si#th to become a fifth as part of a new triad in fundamental position. The transformation hinges on one or two tones of the chord. There are si# possible solutions for each inversion. " certain order of precedence e#ists, e#ists, based in the first place on the greater %magneti!ing %magneti!ing power% of the e#terior tones, in the second place on the normal hierarchy hierarchy of the functions tonic, dominant, and determinant.
In the si#th chord, ) outranks -. >ence the primary tendency? ) tends to become generator, t D .
to
In the four&si# chord, outranks -. >ence the primary tendency? tends to become generator, d t .F /or clarification I would add the following. -ven though there are si# possible solutions for each of the two inversions (si#th chord and four&si#th chord with each solution yielding two variations, this only results in 3B different progressions (instead of 39 because four of the progressions are identical to others (see (see below. /ollowing /ollowing what has been $uoted by 'evy above the order reflects reflects the following hierarchy (the direction of the tones are shown by theJ or K symbols? Since in the si#th chord in telluric adaptation (major triad, of the two e#terior tones ) outranks - () being the tonic and - being the determinant, then the progression with ) becoming the absolute conception tonic is first. This in effect determines the triads to be? (K -??)
(J )?"b?/
The other e#terior tone (K -??) (K -?A?*
(i.e.
-
becoming
a
tonic
function
is
then
listed
ne#t?
1e#t 1e#t woul would d be thes these e same same e#ter e#terio iorr tone toness becom becomin ing g a domi dominan nantt func functition on,, begi beginni nning ng with with )? (K -??) (K /?"?) Then (K -??)
-? (J *??-
/inally the middle tone of this triad (i.e. , being the dominant would become a determinant, first becoming the telluric adaptation determinant (i.e. upward major third? (K -??) (K -b??*b Then hen beco ecoming the absolute conception determinant (i.e. .e. (K -??) (J (note that this progression is a repeat of one of the others above
downwar nward d *??-
major ajor
thir hird?
"lso in the si#th chord in absolute conception (minor triad, of the two e#terior tones ) outranks "b () being the absolute conception tonic and "b being the absolute conception determinant, then the progression with ) becoming becoming the telluric adaptation tonic (the opposite of the above case is first. This in effect determines the triads to be? (J "b?/?)
(K )?-?
The other e#terior tone (i.e. "b becoming a tonic function is then listed ne#t? (J "b?/?) (J "b?/b?b 1e#t 1e#t woul would d be thes these e same same e#ter e#terio iorr tone toness becom becomin ing g a domi dominan nantt func functition on,, begi beginni nning ng with with )? (J "b?/?) (J ?-b?) Then (J "b?/?)
"b? (K b?/?"b
/inally /inally the middle tone of this triad (i.e. /, being the absolute conception conception dominant would become a determ determinan inant, t, first first becomi becoming ng the absolu absolute te concep conceptio tion n determi determinant nant (i.e. (i.e. downwar downward d major major third? third? (J "b?/?) (J "?/? Then becoming the telluric adaptation determinant (i.e. (J "b?/?) (K (note that this progression is a repeat of one of the others above
upward major b?/?"b
third?
Since in the four&si# chord in telluric adaptation (major triad, of the two e#terior tones outranks - ( being the dominant and - being the determinant, then the progression with becoming the telluric adaptation tonic is first. This in effect determines the triads to be? (K ?)?-
(K ?*?
The other e#terior tone (K ?)?(J -?)?"
(i.e.
-
becoming
a
tonic
function
is
then
listed
ne#t?
1e#t 1e#t woul would d be thes these e same same e#te e#teri rior or tone toness beco becomi ming ng a domi domina nant nt func functition on,, begin beginni ning ng with with ? (K ?)?(J ?*b? Then (K ?)?-
-? (K "?)A?-
/inally the middle tone of this triad (i.e. ), being the tonic would become a determinant, first becoming the telluric adaptation determinant (i.e. upward major third? (K ?)?(K "b?)?-b Then hen beco ecoming the absolute conception determinant (i.e. .e. (K ?)?(J (note that this progression is a repeat of one of the others above
downwar nward d -?)?"
major ajor
thir hird?
"lso in the four&si# chord in absolute conception (minor triad, of the two t wo e#terior tones / outranks "b (/ being the absolute absolute conception dominant dominant and "b being the absolute absolute conception determinant, determinant, then the progression with / becoming the absolute conception tonic (the opposite of the above case is first. This in effect determines the triads to be? (J /?)?"b
(J /?b?*b
The other e#terior tone (i.e. "b becoming a tonic function is then listed ne#t? (J /?)?"b (K "b?)?-b 1e#t 1e#t woul would d be thes these e same same e#te e#teri rior or tone toness beco becomi ming ng a domi domina nant nt func functition on,, begi beginn nnin ing g with with /? (J /?)?"b (K *b??/ Then (J /?)?"b
"b? (J -b?)b?"b
/inally the middle tone of this triad (i.e. ), being the absolute conception tonic would become a determ determinan inant, t, first first becomi becoming ng the absolu absolute te concep conceptio tion n determi determinant nant (i.e. (i.e. downwar downward d major major third? third? (J /?)?"b (J -?)?" Then becoming the telluric adaptation determinant (i.e. (J /?)?"b (K (note that this progression is a repeat of one of the others above
upward major "b?)?-b
third?
Leep in mind that what is normally called a minor triad is treated, in 'evy%s theory, as a major triad generated from the top down. down. In other words there are only unisons, perfect fifths and major thirds in in this theory. What would normally be called an / minor triad is a triad in absolute absolute mode (designated by the symbol o generated by ). This would be spelled )&"b&/ (thinking downward from the generator ) and has the same interval structure as a ) triad in telluric adaptation (i.e. )&-& thinking up from ), so symmetr symmetrica icall reasoni reasoning ng is necessa necessary ry for thinking thinking in absolu absolute te concep conceptio tion. n. So in absolut absolute e conception )&"b&/ (thinking downward is a triad in absolute conception %generated% by ) but thinking in telluric telluric adaptation adaptation this same harmonic harmonic cell is a minor triad with / as the %tonic%. %tonic%. "s I mentioned before all major telluric adaptation cells produce the same result as upward absolute conception so the %generator% ) would be identical with the %tonic% ) in this case (i.e. )&-& thinking upward.
In all of the symbol sets below below the top line of each set of symbols shows the function function of the common tone within within each harmonic cell. These triadic triadic harmonic cells are referenced referenced from a generator (not a tonic. If there are more than two lines of symbols, meaning there are two common tones, tones, then the top two lines of symbols show the common tone functions and the top symbol line would be what is referenced in the headings above the table row (i.e. -#terior tones to tonic, -#terior tones to dominant and iddle tones to determinant. In all the symbol sets the first symbol of the bottom line shows shows if the first triadic cell is in telluric adaptation adaptation (major or absolute conception conception (minor, (minor, or upside&down upside&down major, major, and the second symbol of the bottom line shows the relationship of the generator of the second triadic cell to the first triadic cell. In other words the second symbol of the bottom line shows which part of the first triadic cell (tonic, dominant, determinant or some other relation to these three functions the second triadic cell is generated from (i.e. which part of the first triadic cell is the same tone as the generator of the second triadic cell. Similar to the conception in athieu%s book F>armonic -#perienceF the only intervals that are used in this thinking are powers of 3 (unison and octaves here called tonics or generators, powers of 4 (perfect :ths, here called dominants, powers of : (major thirds, here called determinants, and later powers powers of 6 (domin (dominant ant 6ths, here called called natural natural 6ths. 6ths. "ll other other tones tones are derived derived from some combination of these four functions. 2sing the (e%&# (ym'%l ,r%u$ i& t-e .ir(t li&e ,r%u$ 'el%/ as an e#ample (/our&Si# triads & -#terior tones to tonic & inor side & second group the translation would be as follows? M Leep in mind that all of the symbols in the first three line groups have a four&si# triad, either in absolute absolute conception conception or telluric adaptation adaptation,, as their first cell. This means that, that, in the key of ), the telluric adaptation for this cell would be identical with a standard triad in four&si# inversion spelled &)& - (upward thinking. "gain in the key of of ), the absolute conception of the four&si# inversion is /&)&"b (thinking downward from /. 1ote that this is the structure described by the first symbol in the last line of the second group (in the first line group below, however the generator of this cell is the tone )< The absolute absolute conception may be slightly confusing confusing at first. In terms of the four&si# interval interval structure structure the cellular structure is designed thinking downward from / (i.e. /&)&"b, however the root position triad is )&"b&/ (again (again thinking downward. downward. 1ote that if you think of this same triad (i.e. "b&)&/ in telluric adaptation you will find that it is a si#th chord (thinking upward from "b, i.e. an / minor triad in second second position. So this cell is a minor triad in telluric adaptation adaptation and that is why this symbol group falls on the %inor side% N In the first line o t describes describes a common tone that changes changes in in functio function. n. The meaning meaning is that the same tone that is the absolute tonic of the first triad (i.e. the generator of a downward triad becomes the upward determinant (major third in the second triad. In the second line t describes describes a second common tone that changes changes in function. function. The meaning meaning here is that the same tone that is the absolute downward determinant in the first triad (i.e. a major third down from the generator becomes the tonic (i.e. the generator of the second telluric adaptation triad (which is an upward constructed triad. The first symbol of the third line (i.e. o T describes the first triad, in this case case it is a triad in absolute conception. The second symbol tells us that the second triad%s generator is the same tone as the absolute downward determinant of the first triad, however since there is no absolute conception symbol present here this alerts us that the second triad is built %upward% (i.e. in telluric adaptation. This describes the following progression (the direction of the tones are shown by the J or K symbols? (J /?)?"b
(K "b?)?-b
2sin 2sing g the the .%urt- (ym'%l ,r%u$ i& t-e .ir(t li&e ,r%u$ 'el%/ as an e#ample (/our&Si# triads & -#terior tones to tonic & ajor side & second group the translation would be as follows?
In the first line o t describes a common tone that changes in function, the tone that is the upward determinant of the first triad (i.e. the upward major third becomes the absolute tonic (i.e. the gene genera rato tor r of the the seco second nd abso absolut lute e conc concep eptition on triad triad (i.e. (i.e. the the gener generat ator or of a tria triad d cons constru truct cted ed downward. In the second line t describes a second common tone that changes in function. The same tone that is the tonic (i.e. the generator in the first triad becomes the absolute downward determinant (i.e. a major third down from the generator in the second triad, which is an triad in absolute conception (i.e. constructed downward. The first symbol symbol of the third line line (i.e. T describes describes the first triad, triad, in this case case it is a triad in telluric telluric adaptat adaptation ion (i.e. (i.e. a normal normal upward upward tria triad. d. The seco second nd symbo symboll o tells tells us that that the seco second nd triad triad%s %s generator generator is the same tone as the upward determinant determinant of the first triad. >owever >owever since there is an absolute conception symbol, in front of the upward determinant symbol, this alerts us that the second triad is constructed in absolute conception (i.e. constructed downward. downward. This describes the following following progression (the direction of the tones are shown by the J or K symbols? (K ?)?-
(J -?)?"
2sing the .ir( ir(t (ym' ym'%l ,r%u$ %u$ i& t-e (i0t0t- li& li&e ,r%u$ %u$ 'el%/ l%/ as an e#ample (Si#th triads & iddle tones to determinant & inor side & first group the translation would be as follows? In the first line o d describes describes a common common tone that changes changes in function, function, the tone that is the absolute downward dominant of the first triad (i.e. the subdominant becomes the absolute downward determinant (i.e. downward major third of the second triad. The first first symbol symbol of the second second line (i.e. (i.e. o T describes describes the first first triad, triad, in this case it is a triad triad in absolute absolute conception. conception. The second second symbol symbol o s tells us that the second triad%s generator generator is the same tone as the upward determinant of the subdominant of the first triad (i.e. the absolute downward dominant. So the second triads generator is minor third below the first triad%s generator. Since there is an absolute conception symbol in front of the upward determinant symbol, this alerts us that the second second triad is constructed constructed in absolute conception. conception. This describes describes the following following progression progression (the direction of the tones are shown by the J or K symbols? (J
"b?/?)
(J
"?/?
In -#ample 6 the table is divided into two progressions each beginning with a minor triad (absolute conc concept eptio ion n on the the left left side side and and two two prog progres ressi sion onss each each begi beginni nning ng with with a majo majorr tria triad d (tel (tellu luri ricc adaptation on the right side.
E)AMPLE 1
The standard notation for the symbolic e#pressions above are as shown in -#ample @ below in the same format, i.e. the first chord (four&si# or si#th of the first and second measures are in absolute conception (minor and the first chord (four&si# or si#th of the third and fourth measures are in telluric adaptation (major? E)AMPLE 2 & Standard musical notation for >armonic Goice&'eading 8rogressions using
telluric adaptation and absolute conception.
O& T-e N!tur!l Se"e&t-
;n page 9: of F" Theory of >armonyF 'evy states Ffor nearly three hundred years the interval of the minor seventh seventh has been recogni!ed as a dissonance dissonance different different from all other dissonances. dissonances. Whereas Whereas dissonances in general are produced by a tone or tones disturbing a chord, and may therefore be resolved within that chord, the seventh is an integral part of a chord to be resolved as a whole into another another chord. chord. " dissonant dissonant tone is understoo understood d as a functi function on of a chord+ a disson dissonant ant chord, chord, as a function of another chord. The seventh confers a definite function to to the chord of which it is a partF. >ere 'evy makes two statements that set up the rest of his discussion on natural sevenths? a the minor seventh added to a major triad characteri!es it as a dominant+ b the the mino minorr seve sevent nth h adde added d to a mino minorr tria triad d in abso absolu lute te conc concep eptition on char charac acte teri ri!e !ess it as a subdominant? "s an e#ample I used sa#ophonist )harlie 8arker as my model when first learning how to improvise. *eing *eing basically basically self&taught self&taught I remember remember the initial initial steps that I took in learning how to distinguish distinguish tonal functions. functions. I generally generally recogni!ed chords chords as having having one of two functions, functions, stationary stationary and changeable (or fi#ed and mutable. mutable. )hords that had a dominant function function I considered considered changeable, changeable, they sounded like they were going somewhere. /or e#ample I looked at the fourth measure of a blues as Fgoing to IGF (the IG referring to the dominant seventh chord based on the fourth degree of the key of the blues. Then there were certain melodic sounds that I would hear )harlie 8arker play that I associated with Fgoing to IG.F I also remember other sounds that I called Fminor iv to IF, Fminor Fminor vi to IF and so on. These sounds were based on, respectively, a minor seventh chord built on the subdominant degree of
the destination tonality and a minor seventh chord built on the flatted si#th degree of the destination tonality. I instinctively knew that all of these sounds could be played in place of a normal dominant to tonic chord progression progression.. I also knew that there was something something different in the feel of Fminor iv to IF and Fminor vi to IF. -ven though I could see that that it was a substitute for a dominant function I also knew that the normal dominant dominant sound was %bright% and the Fminor iv to IF and Fminor vi to IF sounds were darker darker functions. 1ot because there was minor tonality involved, the progression itself was %dark% in relation to the tonic tonality. tonality. I reali!e now that what I was hearing was the difference difference (implied (implied by the e$ual temperament tuning system between %overtones% and %undertones%, the latter being %darker% in sound in relation relation to the %generator% %generator% tonality or fundamental fundamental tonality. tonality. The minor seventh seventh chord in %absolute %absolute conception% definitely has an %undertone% $uality to it, despite e$ual temperament tuning as our ears tend to compensate compensate for this anyway anyway (more on this later. later. We will see that it is the dominant dominant seventh seventh chord in %absolute conception% with the tonic of the key as generator which can be substituted for the dominant 6th chord in %telluric adaptationF built on the fifth fifth degree of this same tonic. In other words, in the key of , ?-b?)?" (in absolute conception , i.e. thinking downward can be substituted for ?/A?" ?/A?"?) ?) (in tellur telluric ic adaptatio adaptation, n, i.e. i.e. thinki thinking ng upward. upward. In the case case of the 6 chord in absolut absolute e conception the %undertones% represent the fundamental (or generator, :th partial, 4rd partial and 6th partial respectively. In the case case of the 6 6 chord in telluric telluric adaptation the %overtones% represent the 4rd partial, partial, 7:th partial (i.e. 4 times :, Cth partial (two (two fifths up and the 37st partial (i.e. 4 times 6. So one chord is all %undertone% energy and the other is all %overtone% energy. This may be apparent in in this case but things go much farther, as will be seen later. later. "s will be discussed discussed later the %overtone% energy can generally be associated with the Sun and with brightness, the %undertone% energy with the oon and with darkness. -rnst 'evy then goes on to prove how the minor seventh tone that he is referring to represents the seventh partial of the harmonic series (in other words representing the ratio 6?9 as opposed to the ratios C?: or 75?C, which are also also minor 6ths. )onversely the %undertone% minor 6th is represented by the ratio 9?6, or @?6 octave reduced. To keep things moving I will skip this discussion discussion of the tuning of the natural seventh, interested readers will find this discussion on pages 9: and 95 of 'evy%s F" Theory Theory of >armon >armonyF. yF. What What I am going to discus discusss is what what is happening happening from the standpoint standpoint of progressions using these symmetrical ideas. ;n page 9@ of F" Theory of >armonyF 'evy shows the following progression (-#amples C through 74?
