Axis: A symmetric approach to composition and improvisation

Introduction:

During the course of my life, like many others, I was lucky enough to have the immense pleasure of listening to the music of Mr. Steve Coleman. The very particular thing about this music especially, however, was that it sometimes resided in an uncanny valley, situated somewhere between a perfectly normal jazz composition and the most abstract of contemporary music. This fact interested me to such a high degree that, throughout my studies, I have invested a good amount of energy into its full understanding or, at the very least, into giving an account and reading of it. In the process of doing so, I realised that the complex musical language employed by Coleman's music is itself reflective of simple prescriptive rules, much like those of so called "Western Harmony", which contain in themselves the potential energy to explode into a complete and coherent system. The effect of this, is that it is inevitable to feel, at least to some extent, projected into a parallel universe: one in which the habitual rules of tonality, which usually play a decisive role in our listening experience (whether we know it or not) are no longer, in themselves, present. Despite this, however, it is not as if the music itself is utter chaos -- there is clearly a series of organizing principles, as the elements of tension and release, in and out, are clearly perceivable, and permeate the music to its most microscopic level.

It is this kind of opaque logic, clearly present but hardly discernible, that grants this music its extremely peculiar character. It is the hope of this post to provide an account of the organic development of a similar system, proceeding from similar (if not simplified) premisses (namely, the idea of symmetry in all its declinations), and wondering at their conclusion.

Above and Below - Tonal Symmetry:

Ernst Levy's A Theory of Harmony, in its second chapter, concerns itself with the dispute between polarity theory and Trübungstheorie (1), and in advocating for the first, he employs the concept of subharmonic series, of which he was one of the main proponents.

The minor triad, Levy maintains, is not derived from a pejorative alteration of the major, but rather from the inferring of a harmonic series, symmetric to the first: for while the first partials of the harmonic series design a major triad, the first six partials of this subharmonic series will design a minor one:

If we take the first seven partials of a sound, then, the natural seventh (2) is added to our previous pitch set, creating a dominant chord -- the other major building block of Western tonal harmony:

One might wonder, then, what the first subharmonic partials of C6 would be:

Interestingly, what we get is a minor 6 chord, where the tonic is F, or the fourth of C. One could say (as Mr. Coleman does) that we now have sufficient grounds for making a distinction: there is the "above", the harmonic series, where the generator of the series (i.e, the frequency the series stems from) is also the root of the chords that are enscribed in the series, and the "below", the subharmonic series, where the generator of the series is the fifth degree of the root of the chords.

What is also an interesting obesrvation to make, then, is that both the dominant seventh chord described by the harmonic series and the minor sisxth chord described by the subharmonic series offer very strong resolutions, from a voice-leading perspective, to chords with the generator of the opposite series as their root:

Perfect and minor plagal cadences based on the C harmonic and subharmonic series

This observation is, in many ways, one of the fundamental premises of Coleman's playing, as he is able to put it to use by weaving harmonic and melodic paths (as he calls them) to guide him through the musical field, making his way from one point to the other through oblique and unexpected routes.

The first obvious application of such a principle would be to conceptualise tonal space in an essentially polarised way, creating an axis with its poles in the root and its tritone, obtaining an above, a positive, a sharp side and a below, a negative, a flat side, where pitches stand in an equivalent and symmetrical relationship to one another:

What can be done, then, is to approach a certain target chord from the above as well as from the below, creating an harmonic path that moves in ways familiar to us, though applying radically different rules (3):

The technique employed in these two examples, above fully analysed, is a combination of the concatenation of positive and negative perfect cadences, mixed with triton substitutions, and a seemingly casual, but perhaps consciously planned, blurring of the lines, by using a ii-6 sound over positive V7 chords.

What we obtain from this type of technique is a series of substitution that create a sound somewhere between in and out, as, at the end of the day, we employ nothing more than tonal strategies and modal interchange, but we do so out of completely different principles, which redefine the realm of tonal relations themselves.

In observing the specific type of symmetry implied by the polarisation of the tonal space, one necessarily needs to wonder about other possible applications of such a concept. The next three paragraphs will be dedicated to exploring three different kinds of symmetries, namely modal, integral, and spatial, as well as their practical applications in composition and improvisation.

Bright and Dark - Modal Symmetry

Modal symmetry is the result of symmetrisation and gradation of musical modes. It shares its premise with tonal symmetry, in that it proceeds from the symmetrisation of predetermined series of intervals, which are, in this case, the modes of the major scale (4).

From performing this operation, we are able to group the modes in pairs of symmetric series of intervals. The only outlier from this grouping is the dorian mode, which is identical with its own symmetric. From these facts we can make the very important observation that this grouping of modes seems to reflect their value in the brightness scale, as Dorian, sitting at the centre, acts as a centre, while modes one shade darker are connected with ones one shade brighter, and so on (5):

Locrian	Phrygian	Aeolian	Dorian	Ionian	Mixolydian	Lydian
-3	-2	-1	0	+1	+2	+3

What we might then consider, then, is to approach these two polarities as the plates of a scale, on which we are called to place more or less equivalent weights as we traverse musical time, taking care to balance the darker with the brighter and granting them both equal weight:

As we can see from the transcription above, there seems to be a similar logic in Coleman's playing, as he employs pitches pertaining to different modes, assigning to each a duration in time such that there is an arguably perfect balance between the bright and the light.

Integral Symmetry:

Integral symmetry is, concisely put, the mapping of the plane described by the points of the twelve pitch classes through its subdivision in the three subsets defined as diminished quadriads.

Let's reconstruct the reasoning by assuming we are playing in respect to a certain root, namely C, and assigning polar values to notes based on their distance from the root, just as we did for the modes:

...Bb	B	C	C#	D...
-2	-1	0	+1	+2

Through this procedure, we once again have a system of gradated and polarysed pitches, that we could potentially use to creat self-sustaining musical architecture. It does, however, seem like a fairly limited system, as well as one that requires continuous computation. C is the center, and in moving to the right and to the left of it of the same amount of places, we are able to derive from one pitch its symmetrical with respect to C as the axis. If we try and compute the whole series of twelve sounds, we will soon find that the tritone holds a peculiar quality, in that it divides the octave squarely in two, and is thus its own symmetric. Therefore, we can view this tritone (in this case, F#) as itself a symmetric axis, so that we get:

...E	F	F#	G	Ab...
-2	-1	0	+1	+2

In doing this, however, we are also admitting a kind of equivalence between the sounds relating to one axis and the other, as their coexistence necessarily means that the gradation we assign to pitch classes is enough to identify said pitch classes, and therefore all pitch classes with the same gradation and polarity are between themselves identical, and all pitch classes with same gradation but different polarity are opposites.

Through this same process, we learn that there are two more axises, namely in correspondance of the last two pitches that constitute the diminished seventh chord with our root as the tonic. The insight we gain from this "squaring of the circle" is a total mapping of musical space into three subsets of four sounds each, all a minor third apart from each other.

C	C#	D	Eb	E	F	F#	G	Ab	A	Bb	B
0	+1	-1	0	+1	-1	0	+1	-1	0	+1	-1

Composition might benefit from the most thorough usage of this knowledge, in the sense that the composer should be mindful and resourcefull in their balancing of these polar values, while improvisation might be more greatly helped by the same mental image of the scale from a moment ago, only with three plates -- one for each diminished seventh chord.

Spatial Symmetry:

Another interesting approach to the use of symmetry is the translation of pitch classes and their harmonic relationships into a sort of relation, making use of its graphical representation as the object of pseudo-geometrical speculation. An example might be the very famous Coltrane Changes, an harmonic device invented by St. John Coltrane, relying upon the subdivision of the octave in three equal parts:

It's worth pointing out that, while yielding somewhat similar results (i.e, the division of the octave in equal parts) to the system of Integral Symmetry above described, this example seems to better fit in the category of spatial symmetry, as the geometrical and graphical componeent where more essential to it than the deeper implications of any of the symmetric axioms earlier proposed.

While appearing somewhat less grounded than the roads previously explored, this method provides us with very interesting ideas, like, for example, Tonnetz:

Tonnetz are a tool for musical analysis appearing originally in Hostinsky's, Die Lehre von den musikalischen Klängen (1879), and later in Riemann's Ideen zu einer ‘Lehre von den Tonvorstellungen. Essentially, they are used to visually represent the functional relationships between chords present in a certain key. By weaving together the intervals of perfect fifth (horizontal) and major thirds (diagonal), Hostinsky and, later, Riemann, effectively constructed a spacial mapping of harmonic 'space', highlighting the geometrical element of harmonic practice. This class of diagrams, visually representing a set of relations between pitch classes, can be said to have two intens: on the one hand to facilitate musical reasoning through spacial analogy, bringing the purely theoretical and imaginary field of musical pitches (as conceptual objects, not as vibrational physical events) into the realm of bodily experience, effectively (and, perhaps, subconsciously) acting upon the bodily nature of what we may call our musical cognition. On the other hand, they appeal to our extended nature, meaning that, in working through space, which is a medium we know extremely well, since we are immersed in it at all times, they enable us to employ the categories of reason that depend on this spatial conception.

Conclusion:

We have experimented in the devisement of self-coherent non-tonal musical systems, choosing the idea of symmetry as our guide, precisely because of its twofold nature: symmetry is linked both to the form of space, insofar as it is defined through geometry, and to

that of the intellect, insofar as geometry itself is defined through reason, and through the use of this geometric idea of symmetry (as displayed in relation, for example, to the subharmonic series) we are able to create a play between body and intellect. The most important lesson to be learned is not some strategy for composition or improvisation, but the realisation of the possibility to find a unifying organising archè of musical phenomena and devising around it a coherent system, that feels to the uninformed listener reasonable and absurd at once.

Notes:

Two theories essentially describing the relationship intercurring between the major and the minor triads, advocating respectively for their essential identity as mirror images of each other (as maintained by Ernst Levy himself) and for the turbid (from the German Trübung) transformation leading to the creation of the minor in respect to the major (as argued by Heinrich Schenker). ↩
Only a few commas off from what"WesternHarmony"considers to be the dominant seventh. ↩
Both of the following are excerpts of one of Steve Coleman's masterclasses ↩
Although an analogous process is applicable to modes of any scale.↩
In a way reminiscent of George Russel's ingoing and outgoing modes described in his Lydian Chromatic Concept of Tonal Organisation ↩

Axis: A symmetric approach to composition and improvisation

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