Steve Coleman Negative Harmony

7/27/2019 Steve Coleman Negative Harmony

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INTRODUCTION: BALANCE

We live in a world of immense beauty. There are a multitude of forms with countless variations onsimple themes. I want to speak here about balance and make some comments about how balancecan be achieved musically. There are countless ways that architectural balance can be musicallyachieved from the micro to macro level. Since attention to detail has always been an 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, using intuitive and logical methods. Symmetry is a fact of natureand one of the oldest fascinations of humanity. Some of the more obvious ways in which symmetricalmusical balance could be realized are through melody, rhythm, tonality, form, harmony andinstrumentation. As well as the structural considerations of symmetrical musical forms I will alsodiscuss 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 originally a melodic theory. I named it Symmetry because the motion of the melodiesinvolves an expansion and contraction of tones around an axis tone or axis tones (i.e. around a centerpoint). The expansion and contraction involved is almost always equal on both sides of the axis,hence the term Symmetry. This is basically a melodic system that obeys it's own laws of motion. Itmoves according to the gravity of this motion more than anything else but it can also be adapted todeal with the gravity of other types of tonality such as cells or the traditional dominant-tonic harmonicsystem. When I first started dealing with symmetry I only dealt with the laws of motion produced bythe system without any regard for other types of tonality. This is how I would suggest others to learnthe system to get a feel for thinking in these terms. Also a complete knowledge of intervals and theirrelationships (always thinking in terms of semi-tones) would be extremely helpful.

I began by writing symmetrical exercises for myself. Then I practiced these exercises to get myfingers and ears used to moving and hearing these ideas. It was only after doing this that I practiced

improvising within these structures playing at first in an open manner (not based on any outsidestructure such as a song). Later I adapted these improvisations to structures and forms. I did this byslowly integrating the ideas with the more traditional improvisational style I was already playing. Mygoal was not to play in a totally symmetrical style (as this would be as boring as playing all majorscales) but to integrate the style and give myself more options when I improvise.

The basic system involves what I call two spirals. They are tones that move out equally in half stepsfrom an axis (which is always at least two tones).

If the axis of the first spiral is the two tone C-C unison (one octave above middle C, it could be in anyoctave) then from that unison C, you move out (spiral out) each tone in a different direction in halfsteps, i.e. C-C, then B on the bottom and C sharp on the top; B flat on bottom and D on top; A on

bottom and D sharp on top; A flat on bottom and E on top; G on bottom and F on top; G flat on bottomand F sharp on top (at this point you are at the beginning of the spiral again, or the symmetrical mirrorimage of the spiral); F on bottom and G on top; E on bottom and G sharp on top; E flat on bottom, Aon top; D on bottom and A sharp on top; D flat on bottom and B on top; C on bottom and C on top (thisis your starting point one octave above and one octave below your original tones). You're thinking twotones at a time and they're spiraling out together. This I call spiral number one.

As you spiral out from C-C (the axis) and you think of the interval between the tones then C to C is aunison. The next tones in the spiral are B and C sharp, the interval between B and C sharp is a majorsecond. Next in the spiral is B flat on bottom and D on top, that's a major third. Then A on bottom andD sharp on top, that's an augmented fourth or a tritone. Continuing, A flat on bottom and E on top,that's an augmented fifth (could also be thought of as a minor sixth). Then G on bottom and F on topis a minor seventh. The next tones are G flat on bottom and F sharp on top, these tones are an

octave apart and are really the same as the beginning of the spiral. All symmetry has two axis and inthis system they are always a tritone apart from each other, more on this later. As you keep spiraling


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out until you reach the two C's two octaves apart the important thing about the spiral is not the tonesthemselves, but the intervals between each of the tones as you spiral out each half step. It is theseresulting intervals that are formed in spiral number one (Unison, Maj 2nd, Maj 3rd, Tritone, min 6th andmin 7th) that I call Symmetrical Intervals (see example 1). This is important to remember as it formsthe foundation for the laws of melodic motion. Note that beginning with the tones G flat on bottom andF 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 axis starting point instead of a unison.

Instead of C-C as the axis, you have C and D flat together as the axis (C is under D flat). And thenyou spiral out the same as you did in spiral number one. So to begin with, you have the C and the Dflat right above, then B on bottom and D on top; then B flat on bottom and E flat on top; A on bottomand E on top; A flat on bottom and F on top; G on bottom and F sharp on top; G flat on bottom and Gon top (the beginning of the spiral); F on bottom and A flat on top, E on bottom and A on top; E flat onbottom and B flat on top; D on bottom and B on top; Db on bottom and C on top; finally C on bottomand Db on top. So again you're spiraling up in half steps but you're getting completely differentintervals between the tones in the spiral. To start off, you have the minor second between the C andthe D flat. When you spiral out with the B on the bottom and the D on top, you have the interval of theminor third; with an B flat on the bottom and E flat on the top the interval is a perfect fourth. With an Aon the bottom and an E on top the interval is a perfect fifth. A flat on the bottom and F on the topproduces a major sixth. G on the bottom and F sharp on top, a major seventh; then the axis again,etc. and you just keep spiraling out. Again, its not the tones that are important in spiral number two

but the intervals between the tones which are formed as you spiral out. These intervals formed inspiral number two are a different set from the intervals formed in spiral number one, as a matter of factthey are all the intervals missing from spiral number one. I call the intervals in spiral number two (min2nd, min 3rd, Perfect 4th, Perfect 5th, Maj 6th and Maj 7th) Non-Symmetrical Intervals (see example2).

EXAMPLE 2 [Spiral # 2]

The basis of the laws of movement are as follows. Thinking monophonically if you have an initial tonewhich you mentally consider to be the axis, when you move in one direction from this axis thengenerally you must move the same distance in the opposite direction from that same axis. Forexample, you play a C, then the next tone you play is a D above the C , which is a major second awayfrom C, then the following tone you must play is the tone a major second below C, which would be Bflat. In other words, for the same distance that you moved above C you must play the tone that is thatsame distance below C (see example 3 - measure 1 - beats 1 and 2, in these examples the axis arecircled). Actually it doesn't matter whether the D is above or below the C as long as you rememberwhich direction you moved to get to the first tone, so that you move in the opposite direction (the same

distance) to reach the second tone (see example 3 - measure 1 - beats 3 and 4). Obviously, you mustknow your intervals very well to think like this quickly in an improvisational context. For anotherexample if you played G and consider the G as an axis and then play B flat as your next tone, then the


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following tone you play must be E. With G being the axis, B flat is a minor third above G and E is aminor third below G (see example 3 - measure 2 - beats 1 and 2). Now you do not have to play the Ea minor third below G, you could play the E or the B flat in any octave. But you need to be thinking interms of G being the axis and the other tones 'surrounding' the axis (see example 3 - measure 2 -beats 3 and 4).

One of the exceptions to this rule is when the interval that you play is one of the Symmetrical Intervalsin spiral number one, those intervals being a major second, major third, tritone, minor sixth, minorseventh, octave etc. Then you don't have to make the equal movement in the opposite direction. Youcan chose to, but you don't have to. For example if you play the tone C as the axis, and the you playthe tone D, you don't necessarily have to play B flat after that. You could pick any tone at that point.But if you play one of the Non-Symmetrical Intervals in spiral number two, for example, axis C to thetone F (an interval of a perfect fourth), then you must play a G after that, according to these laws ofmovement (actually it would be more accurate to say that after playing the tone F you must then'complete' the symmetrical motion, more on this below). In this example the tone G can be in anyoctave, but you need to play a G because F is a perfect fourth above C and G is the perfect fourthbelow C (see example 3 - measure 3).

EXAMPLE 3

There are many variations to the above laws of movement that are still considered symmetricalmovements according to this theory. For example, instead of moving away from an axis you can dothe reverse and move towards an axis. You can play the tone F, then the tone G, and then play the

axis tone C (see example 3 - measure 4 - beats 1 and 2). Or you could think of the axis as two tonesfurther apart than a unison or a minor 2nd (actually, to be technically correct, even when it seems likethere is only a one tone center all axis are two tones as in the initial axis of spirals one and two, andthey are all either a unison or a minor second as in examples 1 and 2). So you could initially play Cand E flat. Now, logically, you might think that the next tone you have to play is A, because if C is theaxis and if the E flat is a minor third above C then you would think you have to play an A because A isa minor third below the axis tone C. But you can play C and E flat and then play B and E natural!This is because you can think of the C and E flat together as an axis. Then you can expand out a halfstep on either side of this axis and play B and E natural (the true axis in this situation are the tones Dflat/D or G/A flat). This movement would still be within the rules because mentally you're using C andE flat together as an axis (see example 3 - measure 4 - beats 3 and 4). So there are many variationsdepending on what you mentally think of as the axis. Some sample symmetrical movements are listedin example 4, try and follow the logic of the movements. Example 5 is a symmetrical melody which

connects different movement logics in one idea, demonstrating how the movement laws can flowtogether. The circles will give you some hints on where to look for the axis. This is a more complexexample, notice the 'nested' axial movements ('nested' meaning some tones of one axis overlap andshare tones with adjacent axis). It is possible to generate shapes that contain an entire chain ofnested axial movements, however the resulting melodies would be extremely jagged and notnecessarily sound musical, unless of course that is the desired effect.

EXAMPLE 4


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EXAMPLE 5

All of the examples above are written in atonal space. The analysis of the axial progression can bethought of in any number of different ways. In other words it is possible to analyze these samepassages differently and still be well within the given laws. Notice that the above examples requirethinking in small cells of ideas, at least initially. However there is a linear gravity involved in thethinking which requires that the improviser become fluent in thinking in two directions simultaneously.It should be clear from Example 5 that it helps to be able to think at least two or three tones backwardand forward in time. This is a different skill than the normal way of thinking as retention of individualtones, as well as phrases, needs to be practiced. For instance, in Example 5 at the end of measure 2,the tones F and E are the axis, not only of the tones immediately following (A and C) but also of thepreceding tones (Db and Ab)! Overall this produces a sort of accordion effect in time. Not that this willbe heard by the average person, especially given the speed of execution, but it will be felt and it doeshave an effect.

After this what has to be obtained is fluency in progressing from one idea to the next in seamlessmotion, building up to higher levels of complexity in the communication of ideas. This is similar to theprogression from words, to sentences, to grammar, and finally to communication of conceptual ideas

in linguistic expressions. Also, with some imagination, the same ideas could be merged with any otherlogic. It is not my goal here to write down ideas for others. I simply want to demonstrate that there aremany possibilities to be explored, definitely more than have already been explored. I have beenworking with the ideas above for at least 22 years now and I still have not found any end in sight!

HARMONIC MATERIAL GENERATED IN SYMMETRICAL SPACE

Over the years I have been exploring several ideas which could be expansions of the symmetricallaws of motion mentioned above. Most of these ideas are based on the various concepts of '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 centersin terms of spatial geometry, as opposed to the standard tonality which deals in tonal key centers interms of tonics. These different approaches can be looked at as different types of 'gravity'. Here wecould borrow two terms coined by music theorist Ernst Levy, calling the concept of gravity that resultsin the traditional tonic-based tonality Telluric Gravity or Telluric Adaptation and the concept ofgravity that is at the basis of centers of 'geometric space' Absolute Conception. In TelluricAdaptation 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 only from the 'bottom-up' perspective,with the 'fundamental' on the bottom. This is a 'terrestrial' mode of thinking influenced by the fact thatwe live on Earth and tend to localize our concept of space according to our everyday situation. InAbsolute Conception what is important is the position of the tones in space and their distance. Herethe harmonic series is seen as 'spiraling' out from a 'generator' (as opposed to a tonic or fundamental)so as to produced both an 'Overtone' and 'Undertone' series (see example 6-a)! Absolute Conception

is based on a 'universal' mode of thinking that results when you look at the Earth, other planets,satellites and stars from the point of view of how they relate to each other in space. So the differenceis the way the gravity operates from a 'terrestrial' or 'telluric' perspective (on Earth we tend to think of


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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 pulledtowards the source from a multidirectional perspective). In the absolute conception partials arethought of as 'orbiting' around a generator tone producing both overtone and undertone energy.These two different concepts of gravity, telluric and absolute, will be explored in more detail below.

EXAMPLE 6-aAbsolute conception of the Harmonic Series based on the generator tone C

Overtone Series

Undertone Series

I just want to say a few things about the information series above. The tones shown are the closestequal temperament equivalents to the actual tones that are in the series. In some books the 11thpartial is listed as F natural instead of F sharp. However I believe that this is technically wrong as theratio 11:8 is closer to F# by a very small amount (if C is generator). There are 100 cents to an equaltemperament half step (for example between F and F# there are 100 cents). The ratio 11:8 (which is11:1 sounded in the same octave as the generator) is 551 cents above the generator. An equaltemperament perfect 4th is 500 cents above the generator and an equal temperament augmented 4this 600 cents above the generator. Since 550 cents would be an exact quarter tone between a perfect4th and an augmented 4th then 11:8 is closer to F# as the distance between 11:8 and F natural is 51cents and the distance between 11:8 and F# is only 49 cents. The tone is actually closer to F# andbooks that list F natural as being the 11th partial in a harmonic series with C as generator are notcorrect. Of course this means that the corresponding tone, the 11th partial in the undertone, series isGb. However since most books do not deal with the undertone series we don't have to worry about

that.

Also the 13th partial is listed as being A natural instead of A flat. Again this is technically wrong as theratio 13:8 is definitely closer to Ab (again if C is generator). The ratio 13:8 is 841 cents above thegenerator (13:8 is 13:1 octave reduced). An equal temperament minor 6th is 800 cents above thegenerator and an equal temperament major 6th is 900 cents above the generator. So the ratio 13:8 iscloser to being Ab because the distance between 13:8 and Ab is 41 cents and the distance between13:8 and A natural is 59 cents. The tone is closer to Ab and books that list A natural as being the 13thpartial in a harmonic series with C as generator are not correct. This also means that thecorresponding tone, the 13th partial, in the undertone series is E natural.

In terms of the nomenclature that can be used to express the actual tones which act as axis (melodic)

or generators (harmonic) in the Absolute Conception I propose using Sum Notation as the mainterminology. So when I speak of improvising with regard to a 'sum 11 tonal center' I am speaking of


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an Absolute tonality that has an axis (or spatial center) of sum 11. Sum 11 means that the tones B-C(also F-G flat) are the spatial tonal centers of this section of the composition. For the improviser thismeans improvising with this spatial tonality in mind.

One necessary skill required for this mode of thinking would be to learn how to hear spatially with themind as well as with the ears (actually it is all 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 for using the term 'sum' comes from the concept of adding note numbers. If tones Cthrough B chromatically are represented by the numbers 0 through 11 respectively, then it is possibleto 'add' tones together to arrive at their sums. The sums represent the axis (or center point) betweenthe two tones being added together. For a 'sum 8' axis, any two tones that add up to the number 8would be considered a sum 8 interval. For example D sharp and F (3+5) would add up to 8. Thecenter (or axis) of D sharp and F is E and E (which is also sum 8 or 4+4, an axis always implies atleast four tones, in this case the E-E unison represents two tones but if thought of from anotherperspective B flat and B flat is also the axis, i.e. 10+10=20 minus 12 = 8). The same goes for C sharpand G (also sum 8). So the axis of a sum 11 interval would be B and C (i.e. 11 + 0 = 11. Since we aredealing with 12 tones the entire tonal system then , for the purposed of octave reduction, you cancontinually subtract the number 12 from any sum that is 12 or greater until the sum is below 12).

Notice that all of the even sums are the result of any interval in spiral number 1 above and all of theodd sums are the result of intervals from spiral number two. Example 6-b is a table that is a summaryof the relationships between spirals, axis and sums:

EXAMPLE 6-b

SPIRALS NUMBER ONE & TWO

AXIS

TONESSUM #

AXIS

TONESSUM #

C-C SUM 0 C-Db SUM 1

C#-C# SUM 2 C#-D SUM 3

D-D SUM 4 D-Eb SUM 5

Eb-Eb SUM 6 Eb-E SUM 7

E-E SUM 8 E-F SUM 9

F-FSUM

10F-F# SUM 11

F#-F# SUM 0 F#-G SUM 1

G-G SUM 2 G-Ab SUM 3

Ab-Ab SUM 4 Ab-A SUM 5

A-A SUM 6 A-Bb SUM 7

Bb-Bb SUM 8 Bb-B SUM 9

B-BSUM

10B-C SUM 11

This may initially be a little confusing but many things that are unfamiliar are confusing at first. With alittle work it can be as natural as any other internalized system.

A THEORY OF HARMONY


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The bookA Theory of Harmonyby Ernst Levy deals with a different approach to harmony and voiceleading coming from a standpoint of using perfect 5ths, major 3rds and their reciprocals (perfectfourths and minor 6ths) or Polarity Theory. The book talks a lot about upwards perfect 5ths anddownward perfect 5ths (or an upward perfect 5th) or 'dominants'. It then uses the same approach withmajor thirds (which Levy calls 'determinants'), using upward major thirds ( ) and downward majorthirds ( ). Levy then derives all of his harmonic and voice leading theory from these two concepts,

the only exception being his inclusion of the importance of the 'natural 7th' of the ratio 7:4. In Levy'sview the natural 7th is important for several reasons, "The seventh partial appears in the same octavewithin which the triad is completed by the introduction of the determinant." It is Levy's view that thenatural 7th "reveals the latent dynamism of the triad."

Levy speaks of a 'senarius', i.e. the first six ratios, as forming two mutually exclusive triads, one majorand the other minor. If unity is C ( Levy prefers to use the term 'generator' which has a broadermeaning that unity, I agree with him) then the upward triad is C-E-G (C representing the numbers 1and 2, G representing the number 3 and E representing the number 5). The downward triad would beC-Ab-F, (again C representing the numbers 1 and 2, F representing the number 3 and Ab representingthe number 5). The seneric intervals are the octave, perfect fifth and major third, corresponding to thenumbers 2, 3 and 5. In other words the octave is associated with the prime 2 since it is a doubling, theperfect 5th with the prime 3 and the major third with the prime 5. Also note that this is the first

numbers of the Fibonacci sequence. When Levy includes the natural 7th then this senarius isextended to an 'octarium' or comprising the first eight ratios.

Levy also speaks a lot in psychological and sometimes almost mystical terms about music and musictheory. In this way you can see the influence on Ernest McClain who is I believe one of Levy'sstudents and also his colleague. It is a combination of Levy's Harmonic Polarity Theory and hisphilosophical and psychological point of view that I find useful. His book of course has no mention ofrhythm where the concepts of balance and form are even more important.What I find useful is the extreme symmetry that Levy is dealing with which reminds me of some of thework I've done as well as elements of Bartok's work, Henry Threadgill's work, W.A. Mathieu, HowardBoatwright, Schwaller de Lubicz and ancient Egypt, Pythagoras, Plato and the work of the ancientGreeks, Babylonian ideas of reciprocity and the work of Umayalpuram Sivaraman and other related

Vedic symmetrical ideas. I am especially attracted to the idea that Levy has introduced of the upwardand downward 'determinant' being of equal importance as the upward and downward 'dominant'. Hethen links these concepts dynamically and show how they work in progressions of triads, after whichLevy introduces his concept of consonance and dissonance, temperament, tonal function of intervals,triads, non-triadic and compound chords. Levy summarizes the discussion in his book as follows:

1. Tone has a structure. Its validity can be tested on the physical-acoustical level(division of the string as well as on the musical-esthetic level (fertility and musical adequacy of

application.

2. Major and minor are manifestations of the general principle of polarity.

3. The triad being the norm of our tonal system, the third has a direct function within the tonality, equalin dignity to the fifth. Parallel to the term dominants for the upper and lower fifths, the termdeterminants will serve for the functions of the third.

4. A major triad tends to function as dominant, a minor triad as subdominant.

5. A chord is a conglomerate organized by one or several generators.

6. To distinguish natural from psychological consonance and dissonance, the concept pair of wordsontic-gignetic will designate the latter.


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each person, as opposed to what he refers to the natural concept of consonance and dissonancewhich is inherent in the phenomenon. He sums these ideas up as follows:>

The triad is consonant.All other chords are dissonant.The triad may be used as a dissonance.

Other chords - maybe all of them - may be used as consonances.


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In the sixth chord, C outranks E. Hence the primary tendency: C tends to become generator, t + t o.

In the four-six chord, G outranks E. Hence the primary tendency: G tends to become generator, d t."For clarification I would add the following. Even though there are six possible solutions for each of the

two inversions (sixth chord and four-sixth chord with each solution yielding two variations), this onlyresults in 20 different progressions (instead of 24) because four of the progressions are identical toothers (see below). Following what has been quoted by Levy above the order reflects the followinghierarchy (the direction of the tones are shown by the or symbols):

Since in the sixth chord in telluric adaptation (major triad), of the two exterior tones C outranks E (Cbeing the tonic and E being the determinant), then the progression with C becoming the absoluteconception tonic is first. This in effect determines the triads to be:

() E:G:C () C:Ab:F

The other exterior tone (i.e. E) becoming a tonic function is then listed next:

() E:G:C () E:G#:B

Next would be these same exterior tones becoming a dominant function, beginning with C:() E:G:C () F:A:C

Then E:() E:G:C () B:G:E

Finally the middle tone of this triad (i.e. G, being the dominant) would become a determinant, firstbecoming the telluric adaptation determinant (i.e. upward major third):() E:G:C () Eb:G:Bb

Then becoming the absolute conception determinant (i.e. downward major third):() E:G:C () B:G:E(note that this progression is a repeat of one of the others above)

Also in the sixth chord in absolute conception (minor triad), of the two exterior tones C outranks Ab (Cbeing the absolute conception tonic and Ab being the absolute conception determinant), then theprogression with C becoming the telluric adaptation tonic (the opposite of the above case) is first. Thisin effect determines the triads to be:

() Ab:F:C () C:E:G

The other exterior tone (i.e. Ab) becoming a tonic function is then listed next:() Ab:F:C () Ab:Fb:Db

Next would be these same exterior tones becoming a dominant function, beginning with C:() Ab:F:C () G:Eb:C

Then Ab:() Ab:F:C () Db:F:Ab

Finally the middle tone of this triad (i.e. F, being the absolute conception dominant) would become adeterminant, first becoming the absolute conception determinant (i.e. downward major third):() Ab:F:C () A:F:D

Then becoming the telluric adaptation determinant (i.e. upward major third):() Ab:F:C () Db:F:Ab(note that this progression is a repeat of one of the others above)


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Since in the four-six chord in telluric adaptation (major triad), of the two exterior tones G outranks E (Gbeing the dominant and E being the determinant), then the progression with G becoming the telluricadaptation tonic is first. This in effect determines the triads to be:

() G:C:E () G:B:D

The other exterior tone (i.e. E) becoming a tonic function is then listed next:() G:C:E () E:C:A

Next would be these same exterior tones becoming a dominant function, beginning with G:() G:C:E () D:Bb:G

Then E:() G:C:E () A:C#:E

Finally the middle tone of this triad (i.e. C, being the tonic) would become a determinant, firstbecoming the telluric adaptation determinant (i.e. upward major third):() G:C:E () Ab:C:Eb

Then becoming the absolute conception determinant (i.e. downward major third):() G:C:E () E:C:A(note that this progression is a repeat of one of the others above)

Also in the four-six chord in absolute conception (minor triad), of the two exterior tones F outranks Ab(F being the absolute conception dominant and Ab being the absolute conception determinant), thenthe progression with F becoming the absolute conception tonic (the opposite of the above case) isfirst. This in effect determines the triads to be:

() F:C:Ab () F:Db:Bb

The other exterior tone (i.e. Ab) becoming a tonic function is then listed next:() F:C:Ab () Ab:C:Eb

Next would be these same exterior tones becoming a dominant function, beginning with F:() F:C:Ab () Bb:D:F

Then Ab:() F:C:Ab () Eb:Cb:Ab

Finally the middle tone of this triad (i.e. C, being the absolute conception tonic) would become adeterminant, first becoming the absolute conception determinant (i.e. downward major third):() F:C:Ab () E:C:A

Then becoming the telluric adaptation determinant (i.e. upward major third):() F:C:Ab () Ab:C:Eb(note that this progression is a repeat of one of the others above)

Keep in mind that what is normally called a minor triad is treated, in Levy's theory, as a major triadgenerated from the top down. In other words there are only unisons, perfect fifths and major thirds inthis theory. What would normally be called an F minor triad is a triad in absolute mode (designated bythe symbol o ) generated by C. This would be spelled C-Ab-F (thinking downward from the generatorC) and has the same interval structure as a C triad in telluric adaptation (i.e. C-E-G thinking up fromC), so symmetrical reasoning is necessary for thinking in absolute conception. So in absoluteconception C-Ab-F (thinking downward) is a triad in absolute conception 'generated' by C but thinkingin telluric adaptation this same harmonic cell is a minor triad with F as the 'tonic'. As I mentioned

before all major telluric adaptation cells produce the same result as upward absolute conception so the'generator' C would be identical with the 'tonic' C in this case (i.e. C-E-G thinking upward).


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In all of the symbol sets below the top line of each set of symbols shows the function of the commontone within each harmonic cell. These triadic harmonic cells are referenced from a generator (not atonic). If there are more than two lines of symbols, meaning there are two common tones, then the toptwo lines of symbols show the common tone functions and the top symbol line would be what isreferenced in the headings above the table row (i.e. Exterior tones to tonic, Exterior tones to dominantand Middle tones to determinant). In all the symbol sets the first symbol of the bottom line shows if the

first triadic cell is in telluric adaptation (major) or absolute conception (minor, or upside-down major),and the second symbol of the bottom line shows the relationship of the generator of the second triadiccell to the first triadic cell. In other words the second symbol of the bottom line shows which part of thefirst triadic cell (tonic, dominant, determinant or some other relation to these three functions) thesecond triadic cell is generated from (i.e. which part of the first triadic cell is the same tone as thegenerator of the second triadic cell).

Similar to the conception in Mathieu's book "Harmonic Experience" the only intervals that are used inthis thinking are powers of 2 (unison and octaves here called tonics or generators), powers of 3(perfect 5ths, here called dominants), powers of 5 (major thirds, here called determinants), and laterpowers of 7 (dominant 7ths, here called natural 7ths). All other tones are derived from somecombination of these four functions.

Using the second symbol group in the first line group below as an example (Four-Six triads -Exterior tones to tonic - Minor side - second group) the translation would be as follows:

[ Keep in mind that all of the symbols in the first three line groups have a four-six triad, either inabsolute conception or telluric adaptation, as their first cell. This means that, in the key of C, thetelluric adaptation for this cell would be identical with a standard triad in four-six inversion spelled G-C-E (upward thinking). Again in the key of C, the absolute conception of the four-six inversion is F-C-Ab(thinking downward from F). Note that this is the structure described by the first symbol in the last lineof the second group (in the first line group below), however the generator of this cell is the tone C!The absolute conception may be slightly confusing at first. In terms of the four-six interval structurethe cellular structure is designed thinking downward from F (i.e. F-C-Ab), however the root positiontriad is C-Ab-F (again thinking downward). Note that if you think of this same triad (i.e. Ab-C-F) in

telluric adaptation you will find that it is a sixth chord (thinking upward from Ab), i.e. an F minor triad insecond position. So this cell is a minor triad in telluric adaptation and that is why this symbol groupfalls on the 'Minor side' ]

In the first line o t describes a common tone that changes in function. The meaning is that thesame tone that is the absolute tonic of the first triad (i.e. the generator of a downward triad) becomesthe upward determinant (major third) in the second triad.

In the second line t describes a second common tone that changes in function. The meaninghere is that the same tone that is the absolute downward determinant in the first triad (i.e. a major thirddown 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 it is a triad in absoluteconception. The second symbol tells us that the second triad's generator is the same tone as theabsolute downward determinant of the first triad, however since there is no absolute conceptionsymbol 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 or symbols):

() F:C:Ab () Ab:C:Eb

Using the fourth symbol group in the first line group below as an example (Four-Six triads -Exterior tones to tonic - Major side - second group) the translation would be as follows:


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In the first line o t describes a common tone that changes in function, the tone that is the upwarddeterminant of the first triad (i.e. the upward major third) becomes the absolute tonic (i.e. thegenerator) of the second absolute conception triad (i.e. the generator of a triad constructeddownward).

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. amajor third down from the generator) in the second triad, which is an triad in absolute conception (i.e.constructed downward).

The first symbol of the third line (i.e. T ) describes the first triad, in this case it is a triad in telluricadaptation (i.e. a normal upward triad). The second symbol o tells us that the second triad'sgenerator is the same tone as the upward determinant of the first triad. However since there is anabsolute conception symbol, in front of the upward determinant symbol, this alerts us that the secondtriad is constructed in absolute conception (i.e. constructed downward). This describes the followingprogression (the direction of the tones are shown by the or symbols):

() G:C:E () E:C:A

Using the first symbol group in the sixth line group below as an example(Sixth triads - Middle tones to determinant - Minor side - first group) the translation would be asfollows:

In the first line o d describes a common tone that changes in function, the tone that is theabsolute downward dominant of the first triad (i.e. the subdominant) becomes the absolute downwarddeterminant (i.e. downward major third) of the second triad.

The first symbol of the second line (i.e. o T ) describes the first triad, in this case it is a triad inabsolute conception. The second symbol o s tells us that the second triad's generator is the sametone 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 thereis an absolute conception symbol in front of the upward determinant symbol, this alerts us that thesecond triad is constructed in absolute conception. This describes the following progression (thedirection of the tones are shown by the or symbols):

() Ab:F:C () A:F:D

In Example 7 the table is divided into two progressions each beginning with a minor triad (absoluteconception) on the left side and two progressions each beginning with a major triad (telluricadaptation) on the right side.


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EXAMPLE 7

The standard notation for the symbolic expressions above are as shown in Example 8 below in thesame format, i.e. the first chord (four-six or sixth) of the first and second measures are in absoluteconception (minor) and the first chord (four-six or sixth) of the third and fourth measures are in telluricadaptation (major):

EXAMPLE 8 - Standard musical notation for Harmonic Voice-Leading Progressions usingtelluric adaptation and absolute conception.


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On The Natural Seventh

On page 45 of "A Theory of Harmony" Levy states "for nearly three hundred years the interval of theminor seventh has been recognized as a dissonance different from all other dissonances. Whereasdissonances in general are produced by a tone or tones disturbing a chord, and may therefore beresolved within that chord, the seventh is an integral part of a chord to be resolved as a whole intoanother chord. A dissonant tone is understood as a function of a chord; a dissonant chord, as afunction of another chord. The seventh confers a definite function to the chord of which it is a part".

Here Levy makes two statements that set up the rest of his discussion on natural sevenths:

a) the minor seventh added to a major triad characterizes it as a dominant;b) the minor seventh added to a minor triad in absolute conception characterizes it as

a subdominant:As an example I used saxophonist Charlie Parker as my model when first learning how to improvise.Being basically self-taught I remember the initial steps that I took in learning how to distinguish tonalfunctions. I generally recognized chords as having one of two functions, stationary and changeable(or fixed and mutable). Chords that had a dominant function I considered changeable, they soundedlike they were going somewhere. For example I looked at the fourth measure of a blues as "going toIV" (the IV 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 Charlie Parker play that I associatedwith "going to IV." I also remember other sounds that I called "minor iv to I", "minor vi to I" and so on.These sounds were based on, respectively, a minor seventh chord built on the subdominant degree of


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the destination tonality and a minor seventh chord built on the flatted sixth degree of the destinationtonality.

I instinctively knew that all of these sounds could be played in place of a normal dominant to tonicchord progression. I also knew that there was something different in the feel of "minor iv to I" and"minor vi to I". Even though I could see that it was a substitute for a dominant function I also knew that

the normal dominant sound was 'bright' and the "minor iv to I" and "minor vi to I" sounds were darkerfunctions. Not because there was minor tonality involved, the progression itself was 'dark' in relationto the tonic tonality. I realize now that what I was hearing was the difference (implied by the equaltemperament tuning system) between 'overtones' and 'undertones', the latter being 'darker' in sound inrelation to the 'generator' tonality or fundamental tonality. The minor seventh chord in 'absoluteconception' definitely has an 'undertone' quality to it, despite equal temperament tuning as our earstend to compensate for this anyway (more on this later). We will see that it is the dominant seventhchord in 'absolute conception' with the tonic of the key as generator which can be substituted for thedominant 7th chord in 'telluric adaptation" built on the fifth degree of this same tonic. In other words, inthe key of G, G:Eb:C:A (in absolute conception , i.e. thinking downward) can be substituted forD:F#:A:C (in telluric adaptation, i.e. thinking upward). In the case of the G7 chord in absoluteconception the 'undertones' represent the fundamental (or generator), 5th partial, 3rd partial and 7thpartial respectively. In the case of the D7 chord in telluric adaptation the 'overtones' represent the 3rd

partial, 15th partial (i.e. 3 times 5), 9th partial (two fifths up) and the 21st partial (i.e. 3 times 7). Soone chord is all 'undertone' energy and the other is all 'overtone' energy. This may be apparent in thiscase but things go much farther, as will be seen later. As will be discussed later the 'overtone' energycan generally be associated with the Sun and with brightness, the 'undertone' energy with the Moonand with darkness.

Ernst Levy then goes on to prove how the minor seventh tone that he is referring to represents theseventh partial of the harmonic series (in other words representing the ratio 7:4 as opposed to theratios 9:5 or 16:9, which are also minor 7ths. Conversely the 'undertone' minor 7th is represented bythe ratio 4:7, or 8:7 octave reduced). To keep things moving I will skip this discussion of the tuning ofthe natural seventh, interested readers will find this discussion on pages 45 and 46 of Levy's "ATheory of Harmony". What I am going to discuss is what is happening from the standpoint ofprogressions using these symmetrical ideas.

On page 48 of "A Theory of Harmony" Levy shows the following progression (Examples 9 through 13):


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Steve Coleman Negative Harmony

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