The de Bruijn torus * and the related de Bruijn sequences (as well as the concept of Lyndon words, ie. a nonempty string that is strictly smaller in lexicographic order than all of its rotations.) have to do with computing, over an n-ary array of symbols usually formed by a binary alphabet, every possible sequence, in order,- meaning, every m-by-n matrix, exactly once. To obtain the binary alphabet I have (working from the exemplary studies by Hindemith into a vast recategorization of the harmonic series, beyond any diachronic formality, in terms of pure intervals instead of functionally harmonic ratios built up from triads, as practically demonstrated in his Ludus Tonalis) converted the 12 notes of the chromatic scale into a list of musical dyads, (one note plus one other note, describing a single fundamental interval, ie. the binary conformation of the dia-chromatic tonal system) and utilized the Bruijn torus to create a complex array of every possible musical structure that can be re-composed from these elementary tone-dyads, such that I can then represent any chord-structure as one of these m-by-n matrices over the torus, and even more importantly: I can describe any possible movement from one such matrix to another, that is,- any conceivable harmonic motion from one chord to another chord or, even more generally, any harmonic structure to another,- using a variety of algorithms and topological functions related to Bruijn tori, like spanning trees for example. (Entire systems of harmonic motion and by extension melody can be similarly described, as I have done; I use them instead of simple scales or the equally vapid serialist technique of the tone-row.) Such algorithms and functions offer analytic scope far beyond anything found in conventional functional harmony or even in more advanced post-jazz musical theory. Instead of scales, one uses matrix permutations,- permutations that can be further analyzed in terms of Euler/Hamiltonian cycles over a directed graph; instead of keys, one uses more complex mapping-functions to move between matrices,- this movement is the alternative to modulating keys, etc. etc.

*

We can go even deeper by extending the idea to directed graphs, eg, Eulerian and Hamiltonian cycles. We can use dyadic transformation * to formalize the binary directional Bruijin map as a Bernoulli map, (formally, the n-dimensional m-symbol De Bruijn graph is understood as a Bernoulli map) which is an ergodic dynamic system, ("The trajectories of this dynamical system correspond to walks in the De Bruijn graph, where the correspondence is given by mapping each real x in the interval [0,1) to the vertex corresponding to the first n digits in the base-m representation of x. Equivalently, walks in the De Bruijn graph correspond to trajectories in a one-sided subshift of finite type.") allowing a bridge between two of my favored subjects: the theory of dynamical systems, and music theory. Dynamical systems theory covers things like strange attractors, chaos, ergodicity, etc.- all things that would be very interesting to connect to music theory, as I have done.

* as an iterated function map of the PLF:

I have recently posted a few of my symphonies: 21,22,23, and 9. All of these works were enabled by this new theory of harmony,- a theory of harmony beyond major and minor tonality, yet a theory suffering none of the intellectual defects of the so-called 12-tone serialist composition.

I would include a few passages from one on my textbooks of music theory:

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" Western musical theory traces its beginnings to Platonic and Pythagorean philosophy, and to the metaphysical tradition more broadly conceived, such that a dualism has been transmitted through its generations, however tenuously, given the fact that philosophical dualism itself has been so grossly exaggerated, misunderstood, and repulsed with the tides of popular opinion concerning those subjects about which the great division of intellect and nature, man and God, spirit and matter has the most to say. The Overtone series is simple enough: when one plucks a string, this causes adjacent strings on the instrument to vibrate in response to it as well, such that every perceived note is actually an emerging resonance from a greater harmonic series, ie. a composite auditory phenomenon, whose mechanics have been tediously worked out by our integral mathematics. The Undertone series, however, is an entirely abstract structure conceived as a reciprocality between the Overtones of physis, (thus undertone is somewhat of a misnomer; we are speaking more of a metatone series) and thus,- existing solely within the luminant and inner universe accessible to man's rational faculty,- reflects the inward concentration and metaphysical intuition of that structure in accordance to which the Overtone series is generated as a phenomenon of mere Nature, and apparently sensible manifestation of the laws of physics, to which there belongs all enjoyment of harmony on the part of our anatomical investments, that is, our sense of hearing. The Undertone series, conversely, gives itself only to the silent apprehension of the intelligence, and the invisible sense by which that inner universe, as is properly the domain of man alone, is constructed and organized, or more; populated by its own materials,- just as the physical world is so populated by the resonant frequencies perceptible to us as notes and chords, propagated through whichever medium- namely by a corresponding intellective entity which Goethe called the tone-monad, serving as both a product and generator of tonal gravity along the metaphysical abstraction of the spiral of the Undertone-Overtone series, (depending on what orientation it occupies within the spiral) around which orthogonal and adjacent monads are energized, accumulated and fall into their own peculiar orbits between the dominant minor harmony of the major scale and the subdominant major tonality of the minor scale. While theory has thus far focused on the application of the overtone scheme and associated physics, using the undertone series merely as a teaching tool or to illustrate certain reciprocal harmonic relationships, the extrapolation of this spiral itself and the laws of these 'orbits' to a new theory of harmony has yet to be done. Every expansion of the tone monad, as Goethe stated, psychologically imparts major tonality, while every contraction imparts minor tonality, such that the 'orbits' within the metatonal spiral result from reciprocal introspecting and projecting harmonic movements. The spiral 'wants' to return to an equilibrium or grounding-state, a neutral entropic leveling of any excessive tension in one tonal direction or another, such that music effects and emotional significance are simply a result of the composer giving in or fighting with the 'will' of the metatones or ‘tone-monads’. The Undertone-Overtone series, (The one derived by dividing the lengths of actual strings on an instrument according to the mathematical ratios, and the other by infinitely extending corresponding 'imaginary strings' via a multiplication of these same ratios along the course of Apollo's more celestial lyre, and that as an activity of pure cognition intended to reveal a metaphysically derived series of harmonic symmetries between the two structures. Schenker inferred that the minor scale was a kind of modification of the 'chord of nature', extending its internal dynamism, through a variety of entirely artificial constructions, to the dynamism of human psychological intuition and the canons of literature, culture, and taste, and we may agree with that assessment,- though here the term 'artificial' would denote, not a degradation, but an ennoblement of the laws of harmony, in just the same way that the perfections of Euclidean geometry, though divorced from the physical world by the abstracting intelligence, reveal, precisely by their severance from it, the necessary perspective and truths of that world.) or the 'metatonal' series, as an abstract structure useful in automating the discovery of new harmonic materials, thus satisfies very beautifully our definition of the zairja as a cognitive machine for the generation of Thought from Non-thought.

The 15th chord (designated as a lost gem in the history of music by the bimodalist Enrique Aubieta) incorporates two tetrads: it is an eight-note chord. (These coincident tetrads may be conceived in relation to the diagrammatic representation of the ratios involved in the 15th-limit Otonality/Utonality diamond of Meyer's psychoacoustic theory, which likewise incorporates eight distinct tonal identities. Furthermore, the full 8-tone voicing can be abbreviated by eliminating the fifth scale-degree of each tetrad, yielding a six note chord,- that is, a synthetic hexascale like that demonstrated by several of the 19th and 20th centuries' greatest harmonic developments, eg. Scriabin's Prometheus, Stauss' Elektra, or Stravinsky's Petrushka. It is significant that the elimination of 5ths from the full voicing of the 15th, while not functionally altering it, does reorganize its negative polarities or symmetrical mirrors in the circle of fifths and thus alter the harmonic motion embedded in the chord, as mapped from the root to highest note. The possibility of a new form of harmonic motion is indeed one of the greater implications of this system of harmony, that is, the metatonal sequence. More precisely, such a harmonic motion weds itself inseparably to melodic motion, with the harmonic element of the resolution corresponding to the exchangeable tetrads, and the melodic element corresponding to directionality on either side of the overtone-undertone series projected as the spiral-like structure of the metatones. The perfect marriage of the laws of harmony and melody was notably the overwhelming dream of the late Scriabin, for which he invested the full store of his musical knowledge.) The four notes left out from the chromatic scale in the 15th's unabbreviated 8-note voicing are exchangeable, such that they can serve as tone-monads around which to energize the other constituent notes of the 12-tone scale, and therefor allow chromatic modulation between harmonic major-minor polarities by reorienting tetrads with corresponding roots drawn from their pitch collection and from their reciprocations within the circle of fifths, (ie. the metatonal spiral or negative harmonic mirrors of these four remaining notes) and that while not being bound by the conventional laws of functional harmonic resolution for which the inherent metatonal structure is not taken account of. It is of course hardly unsurprising, that this structure was not taken account of at the inception of the system of Western equal temperament, since that tuning was not derived procedurally after a theory of psychoacoustics or perception was first determined, ie. as a conformation to such a theory, but only after the fact, that is, as a kind of ad hoc solution. What the 15th affords conceptually is a polytonal chromaticism equivalent to the modal exchanges of polymodal chromaticism. Polytonality has been, ever since Mozart's ludical application of it in his divertimento for horns and string quartet, a kind of pseudomythological musical effect, and has been deprived of any working theory or guiding system for its compositional utility,- a deficiency the 15th chord provides a correction for.

Of the monists, who might be more properly described as natural theorists in their insistence that the laws of music are grounded, not upon any Platonizing dualism, but solely upon the material physis of acoustical propagation in atmosphere, on the chord of nature and the like, Hindemith stands as the most contemporary contributor. The theory of harmony, be it expounded from the depths of Nature or from the Olympus of abstraction and form, exists to guide the artist in his appropriation of sounds and intervals, such that, as is often the case when relying solely on passion or instinct, he does not become entangled in the kind of babel-like repetitions we observe in those stricken with glossolalia, and which are difficult to avoid otherwise, given the unmanageable complexity of the raw table of musical intervals themselves, absent any higher-level, abstracted language for dealing with them. Theory exists to guide and to temper, not to abnegate or revile: instinct. However, to return to Hindemith as naturalist, we find a man who, in the need to pursue the naturalist philosophy, went so far as to classify all the musical intervals individually- a monstrous task; a feat even by itself, which no other composer before had found the courage or masochism to undertake. Of course, if one refuses the basic monistic premise and accepts the existence of the undertone series, as well as the idea of the laws of harmony being established on their basis, and on that of a mathematical-metaphysical abstraction, then the fundamental modality by which Hindemith organized the intervals into more or less consonant and dissonant families becomes untenable, in that this distinction itself becomes, as I have before discussed, arbitrary- that is, 'un-natural,' while the concept of symmetry between intervals and chordal structures on the circle of fifths, reflected in their mirror-forms between the overtones and undertones, remains a truly objective criteria for a new theoretical conformation of the laws of harmony, though one for which the adopting of a musical Platonism is necessary, as unpalatable as the fact might be for the still predominant naturalists. *

* Toward a less arbitrary classification of the subfunctional intervallic sequences, we have of course the Limited Modes of Messiaen's system, for whose advancement the composer maintained a common interest: the discovery of inherent musical symmetries, namely by dividing and transposing the bare harmonic series, that can be used in the reworking of melodic and harmonic materials in accordance to a higher sphere of musical laws than that of the functional diatonic harmony which dominated the classical era,- though a sphere at once less dehumanized than Hindemith’s system and less arbitrary than the algorithmically generated serial-amalgamations of the five remaining notes of the chromatic scale left over at the conclusion of musical research conducted in the previous three centuries,- conglomerate ‘tone-rows’ derived by an entirely route mathematical procedure in Schoenberg’s system at the incept of the 21st. The five notes left out of the 13th chord- a chord which stood as the implacable limit, and the impermeable limitation, of Romantic and Impressionist-era theory, so tortured modern theorists that composers took to Schoenberg almost out of necessity, given the fact that no other musical system emerged that could deal with the impossible pentachord. This mistaken leap of faith into serialism directly moved us (skipping the potential of a 15th chord,- a blindsight that could have provided the necessary system, that is, a true alternative to serial composition) from the inspired discovery of the Romantic-era's thirteenth chord (while passing through, both hastily and uncomprehendingly, the final developments of the same from out of the Impressionist-dominated 20th century and the most recent applications of that final Romantic genius in the Lydian-chromatic theory of tonal organization) into an uncertain musical posterity dominated as much by the converts to serial composition as the educational system is dominated by an overtly left-leaning professorship.

While various abbreviations of the 15th, as well as the conceptual system underlying its existence presented here, has been utilized in nearly all my works. However, my 23rd symphony opens up with rare full voicings of the 15th, and the work in toto is meant to demonstrate various contextualizations of this chord and the four-fold modal root exchanges of its constituent tetrads- specifically as the basis for a new form of harmonic motion and resolution. The first movement of my 23rd uses 15th chords to cycle through the spiral of the over-undertone series in such a way as to simultaneously build and release tension continuously, always reversing one movement in the spiral and thus sustaining a kind of perfect symmetry and tonal neutrality, as a backdrop against which other harmonic inventions are gradually placed in later movements, fulfilling what was truly the great dream of Impressionism- to discover a perfect musically neutral atmosphere, that is, a musical atmosphere that could, for that very reason, be populated with any note from the total chromatic without inciting any contradictory motion.