Music Theory

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Music Theory

Postby Parodites » Sun Feb 14, 2021 11:39 pm

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.
buujin12.PNG (29.95 KiB) Viewed 370 times

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:

dyadictransformation.PNG (5.16 KiB) Viewed 370 times

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:


" 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.
Qui non intelligit, aut taceat, aut discat.

-- Hermaedion, in: the Liber Endumiaskia.

in formis perisseia mutilata in omnia perisarkos mutilatum;
omniformis protosseia immutilatum in protosarkos immutilata.

Measure the breaking of the Flesh in the flesh that is broken.
[ The Ecstasies of Zosimos, Tablet
the First.]
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Re: Music Theory

Postby Parodites » Thu Feb 25, 2021 4:30 am

Elaborations on what I said previously here. As one might gather from my previous posts, I intend to speak of music theory in such a way as to interest philosophers, mathematicians, etc.- or more generally, people not previously invested in the world of music theory, namely by connecting it to other disciplines you might not expect.

The overtones are a physical phenomena that results from one string resonating the others in an actual instrument, producing what we call 'notes' as composite acoustic entities. Theoretically, this is understood by dividing the lengths of actual strings in accordance to the mathematical ratios: "The one [the overtone series] derived by dividing the lengths of actual strings on an instrument according to the mathematical ratios, and the other [the undertone series] 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."

The undertone series does not exist in nature, it is not a physical phenomena. There is no physical reality behind them. It is a purely abstract structure that results from symmetrically reflecting the overtones over an imaginary axis, think of it like the musical equivalent of imaginary numbers. Hence why it is better to call them the meta-tonal series, as in a metaphysics of music, or a kind of philosophic dualism. The reason for doing that classically was just to note the undertone-overtone reciprocity as a useful teaching tool (Specifically, I am thinking of Reimannian theory, ie. Riemann's dualist system, in which minor tonality is just an upside down or reflected major tonality and vice versa.) for pointing out certain musical structures. However, I have come upon the realization of a far deeper reality behind the metatone spiral. My first inclination as to this deeper reality was Coleman's notion of musical symmetry:

"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 main terminology. So when I speak of improvising with regard to a ‘sum 11 tonal center’ I am speaking of 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 this means 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 the mind 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 analysis of the axial progression can be thought of in any number of different ways. In other words it is possible to analyze these same passages differently and still be well within the given laws. Notice that the above examples require thinking in small cells of ideas, at least initially. However there is a linear gravity involved in the thinking 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 backward and forward in time. This is a different skill than the normal way of thinking as retention of individual tones, 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 the preceding tones (Db and Ab)! Overall this produces a sort of accordion effect in time. Not that this will be heard by the average person, especially given the speed of execution, but it will be felt and it does have an effect."

If you draw out the entire overtone-undertone series as a symmetrically reflected spiral:

image_2021-02-24_220110.png (24.88 KiB) Viewed 229 times

(Image from Coleman's essay; click to expand it.)

then when you reflect a major chord over that line of symmetry, ie. into 'geometric space', the resulting chord suddenly becomes minor. A minor chord becomes major. This phenomena is "negative harmony", something very interesting in and of itself if you want to google/youtube it. One can flip an entire song in this way and generate automatically a new song that sounds entirely different: this 'new' song is a geometric projection of the original, an expansion of its harmonic structure through this kind of symmetry noted by Coleman.

So I went through every single scale, reflecting it in the same way, and then realized that when you reflect the Lydian scale and then add the notes of its reflection to the original notes of the Lydian scale, you get all 12 notes of the chromatic scale- no other scale does that. The eight notes of Lydian get four new notes added to their collection through this operation. Hence I conceived of a new "Lydian-Chromatic scale". The harmonic structure of this L-C scale was then used to create a system in which the 8 notes of Lydian act as tone-generators around which scales and chords built up from the remaining 4 notes of the chromatic as their roots "self-organize" in sequences of triads or hexatonic scales, with this self-organizing pattern being what I just call "orbits", that is, chromatic orbits around Lydian's generator-tones. These triads/hexatonic structures can in turn be superimposed to reconstruct the 12-tone chromatic scale.

The eight notes of the Lydian scale plus the four remaining chromatic notes form three tetrads: putting a Lydian tetrad plus a new tetrad with its root as one of those four chromatic notes creates a Lydian-chromatic 8-note chord, which I call the 15th chord, as it's been augmented at the 15th scale degree: such a chord embeds harmonic tension that can be released by following the laws inherent in the symmetrical construction of the metatones. If you remove the fifth-degree from each of the two tetrads, you get a six-note chord like Scriabin's Prometheus Scale.

It's highly related to the theory of Lydian-chromatic tonal gravity. Coleman talking about Levy's concept of tonal gravity:

"Levy also talks about two types of gravity, telluric gravity or telluric adaptation meaning the normal bottom to top gravity that we all know (the term telluric means terrestrial or earthly, which I take to mean from the ground) and absolute gravity or absolute conception which looks at things symmetrically being generated from a center. Generally when Ernst Levy discusses traditional concepts and names such as major and minor triads, etc. then he is speaking in telluric terms. Otherwise this entire conception is basically an absolute conception dealing with generators and polarity. For upward thinking (major tonality) there is no difference between telluric adaptation and absolute conception. Therefore, for all intents and purposes, when we are referring to 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 Levy sometimes uses the = symbol for telluric adaptation."

Anyone interested in this kind of deep-theory should read Coleman's entire essay on symmetrical movement and undertone-overtone/negative harmony:

Removing the fifth from the two tetrads out of which the 15th is composed (the 15th chord itself is: C-E-G-Bb-D-F#-A-C#) results in a six-note abbreviation of the 15th's 8-note voicing, with this abbreviation being Scriabin's Prometheus scale, (a hexachord or superimpostion of two triads) which is itself the Lydian-b7 scale with the 5th removed: C D E F# (G-removed) A Bb. Thus, removing the fifth from only one of the tetrads in the 15th results in a 7-note collection corresponding to the completed Lydian scale, ie. Lydian with the fifth included. An essential note is this: flipping the 15th chord/scale through negative harmony results in the same collection of pitches, save for one note: Bb is transformed into Ab. If you take that Ab and add it back to the 15th, you regenerate a new 9-note or "nonatonic/enneatonic scale," essentially the entire 12-tone chromatic scale missing a triad: [ C-E-G-Bb-D-F#-Ab-A-C# ] (Missing: D#, G#, E#)

This scale, when inverted through negative harmony, yields the same collection of notes. This structural symmetry (like that discovered by Messiaen, in his modes of limited transposition) offers a basis for super-tonal (but not serial/'atonal') harmonic mechanics, ie. harmonic theory beyond major-minor/diatonic tonality. This scale is, in other words, insusceptible to the "telluric influence" of the tonal gravity of the Lydian-chromatic and major-minor duality, and moreover: it preserves its emotive quality through negative harmonization, much as the enneagon, Hinaxian mizan, or the twin-gyre of Cusanus' metaphysical diagrams resists all disturbance of its mathematical harmony. I would note: a way of escaping the influence of telluric or earthly gravity, which brings all our music back down to earth, clips the wings of our muses, and defies all those who "weave the wind", etc.- this is a big deal: to cite both Joyce and Eliot:

"The void awaits surely all them that weave the wind ... " (Stephen's Telemachus.) As the Void awaits also, all our speculations like "vacant shuttles that weave the wind". (From TS Eliot, in his Gerontion. )

Thus I named the original 8-note scale used to construct Ubieta's 15th chord the Orpheus or Orphic scale, as Orpheus sung and moved stones, defying gravity; it is the 'music of the spheres', the heavenly or firmamental strings that bind the universe together.

The triad or remaining three notes can be used to replace any three notes in the enneatonic scale (this operation being equivalent to superimposing triads/hexatonic scales) to create a new enneatonic, 9-note scale, with this generalized nine-step structure being the fundamental insight in Tcherepnin's theoretical writings, as noted here:

Citation from: "The Nine-Step Scale of Alexander Tcherepnin: Its Conception, Its Properties, and Its Use"; by Veenstra, 2009.

" The nine-step scale is a symmetrical scale, like the whole tone, octatonic, and
chromatic scales, but it has been largely overlooked by music theorists. As is the case
with other symmetrical scales, one can arrive at the nine-step scale from a variety of
directions. It is a repeating stepwise pattern of two half steps and one whole step. It is the
combination of a major-minor hexachord and its inversion TnI such that n = any even
integer in mod-12 pitch space. It is the combination of any three augmented triads, and is
therefore also the complement of the augmented triad. The sum of all these attributes
results in the appearance of the nine-step scale in numerous late Romantic to modern
musical compositions.

The nine-step scale offers a tonal system that is theoretically sound, but few are
aware of it. The most extensive theorization of the scale until now has been provided by
the Russian composer, Alexander Tcherepnin. He explained the construction of the scale
as it developed from his instinctive use of two harmonic structures—the major-minor
tetra/chord and the major-minor hexachord. He discussed a tonal system that involves
tonic pitches, modal uniqueness, arpeggiations in a mod-9 pitch-class universe (although
he did not have the language with which to fully explain it), and harmonic structures with
tonal tendencies. Tcherepnin’s Basic Elements of My Musical Language provides
evidence of his brilliance as a modern music theorist."


I will also include the bulk of Ubieta's discussion of the 15th chord, because it's an obscure text you can't readily find without the Internet Archive:

IF WE WERE TO SUPERIMPOSE a major third over the top of the thirteenth chord, we would realize that such a third would correspond to the 17th overtone (C#) of the natural harmonic series.

But, if in keeping with the established norm, we were to classify this major third in a tertian chordal order, we would then denominate it as the augmented-fifteenth degree of the chord root, an interval consisting of a high double octave (fifteen notes) plus one chromatic semitone.

On this basis, I have appropriately coined the term, “augmented-fifteenth chord,” to refer to this chord. A chordal composition of eight sounds looks like this:


You can represent this chord either as +15 or 15+.

As you can see, once the seventh and eleventh degrees of a thirteenth chord are altered—relative to their approximate pitch within the natural series—the eight component sounds of the augmented-fifteenth chord could be divided in two superimposed tetrads: the lower one, a dominant seventh; the upper, a major seventh. Take for instance the following:



[lower tetrad]

—[upper tetrad]

This order of sounds is entirely unalterable, both in the inversion of the two tetrads that make it up, as well as in the pitch of each of their sounds.

Although the augmented-fifteenth chord should not be subject to any sort of alteration, it could, however, be reduced to seven or six of its eight notes by eliminating the fifth degree of one or both tetrads of the chord, respectively:

C-E-[ ]-Bb—D-F#-A-C#
(7 sounds)

C-E-G-Bb—D-F#-[ ]-C#
(7 sounds)

C-E-[ ]-Bb—D-F#-[ ]-C#
(6 sounds)

Moreover, the lower tetrad can be in either close or open harmony, while keeping its seventh degree on top in either case; whereas the upper tetrad should always remain in close harmony. For instance:

[close position]—[close position]

or also

[open position]—[close position]

Although the upper tetrad must remain in close position, this, however, does not hinder it in the higher pitch range from being doubled to its high octave, or placed a major-tenth chord apart from the lower (close or open) tetrad.

This latter option is more suitable for piano and string instrument scoring because these instruments are sufficiently rich in harmonics and homogenous in sound to bridge such a large harmonic interval. The union of these two tetrads that form the augmented-fifteenth chord now comes to us as the live embodiment of a harmonic alliance between the secular dominant-seventh chord (lower tetrad) plus the major-seventh chord (upper tetrad), a chord whose repeated use, goes back to the early days of musical impressionism, and which becomes mundane later on in western pop music.

This chord is in fact the most emblematic icon of impressionist harmony (cf., Claire de Lune). Therefore, this union consisting of both seventh chords could have been of great historical importance to music, considering it represented a genealogical relationship between an ancestor (the dominant seventh chord) and its analogous descendant (the secondary major-seventh chord), which has ever since habitually practiced its harmonic function over the first and fourth degrees of tonality.

(Precedent: J.S. Bach began writing his 24 preludes and fugues for The Well-Tempered Clavier in 1722. Take note that on the 12th measure of his first prelude, you can already see the broken chord consisting of C major seventh in its third inversion, as well as secondary seventh chords over the 2nd, 4th, and 6th degrees of the tonality across the entire prelude. A century and a half later, these chords would become the first flashes of musical impressionism at the dawn of its stylistic history.)

Given its well-balanced sound, the complete or incomplete use of this hitherto unnoticed chord could have harmonically emulated the novel effect caused by the free motion of parallel ninths, which constituted the first wonders of musical impressionism.

As for its harmonic lineage, the augmented-fifteenth chord could have been perhaps the last fruit borne of the genealogical tree of musical impressionism—a bright, crystalline, compact chord, both in the lower and upper registers.

Now, as we all know, the microtonal gamut of the natural series departs exactly from the 17th overtone (C#). Had this chord, however, stood out with this crowning overtone at the start of the 20th century, we would probably be pointing it out today as the first tertian chord of superimposed thirds to build onto its structure a representation of the microtonal gamut, without losing its impressionist harmonic features.


The big question would be: how do all the different enneatonic scales produced by swapping out the missing triad I noted interact or connect to one another? How do you modulate from one to another? Well that requires a new intervallic reclassification (like Hindemith's) of the total chromatic, as I drew out using the Brujin torus, as noted in my previous post here.

I put together a little sampling of pieces featuring extensive displays of everything I talked about here (as well as a bunch of stuff I didn't talk about here) in action; 8 of my concertos, at 30 minutes each:

Concertos no. 30, 31, 15, 17, 14, 32, 37, 40: [Op. 34, no. 1 & 5; Op. 37, no. 4; Op. 51, no. 1; Op. 52, no. 2 & 3; Op. 56, No. 1; Op.57, no. 2.] ... erto-no-30 ... erto-no-31 ... -15-rev-23 ... erto-no-14 ... erto-no-17 ... erto-no-32 ... erto-no-37 ... erto-no-40

And a few etudes from my book on theory and technique, they're pretty short, 7-15 minutes each:

Op. 21, No. 52; Study on Staggered Chords and Angular Melodic Motion: ... dic-motion
Op. 21, No. 64; Grande-Fantaisie on Symph. No. 6: ... o-6-rev-18
Op. 21, No. 60; Study on Angular Writing Based on Concerto No. 19: ... erto-no-19
Op. 21, No. 75; Studies Based on Concerto No. 16: ... oncerto-16

Finally, a Grande-Fantasy based on themes in my first Symphony, Orpheus; 15-16 minutes long.
Op. 22, No. 3; Grande-Fantaise sur Thema de Orpheus: ... de-orpheus

So there's several big ideas I have brought together from disparate unconnected sources here to arrive at my own musical system: 1) Ubieta's bimodal/bitonal analysis of the 15th chord, (composing something in two keys at the same time, that is, bitonality, is only possible in the context of the 15th's superimposed dominant-major tetrachords) 2) Coleman's symmetrical movement concept, 3) Levy's tonal gravity/Lydian-chromatic system, 4) a resurrected form of negative harmony from musical dualists and Riemannian theory, and 5) Tcherepnin's nine-note scale. I'd recommend reading into all five of those subjects with what I've said in here as a guide, if one is interested in exploring this new musical universe.
Last edited by Parodites on Thu Feb 25, 2021 3:53 pm, edited 2 times in total.
Qui non intelligit, aut taceat, aut discat.

-- Hermaedion, in: the Liber Endumiaskia.

in formis perisseia mutilata in omnia perisarkos mutilatum;
omniformis protosseia immutilatum in protosarkos immutilata.

Measure the breaking of the Flesh in the flesh that is broken.
[ The Ecstasies of Zosimos, Tablet
the First.]
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Re: Music Theory

Postby promethean75 » Thu Feb 25, 2021 3:17 pm

Bro. Imagine what you could do on the keys if you had a sixth finger on each hand.
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Re: Music Theory

Postby Parodites » Sun Feb 28, 2021 11:38 pm

I have a sixth finger, because I studied Busoni's legendary "sixth-finger technique", but that is beside the point. I hate soloists. Every one of my compositions is symphonic in scale, all of these ones were for piano duet and trio for example. If you want to see the actual scale of the music across the keys, check these out; I synchronized the audio of the recordings with synthesia's visual output. This is Concerto No. 30, which starts out deceptively simple with a motif in the bass, and then explodes orchestrally: (the blue and orange each represent a single piano, not one of the two hands; in the score, each color is broken down into its own two hands. Check out the 8-voice fugue that starts at like 8:49 in the first part) ...
Qui non intelligit, aut taceat, aut discat.

-- Hermaedion, in: the Liber Endumiaskia.

in formis perisseia mutilata in omnia perisarkos mutilatum;
omniformis protosseia immutilatum in protosarkos immutilata.

Measure the breaking of the Flesh in the flesh that is broken.
[ The Ecstasies of Zosimos, Tablet
the First.]
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Posts: 332
Joined: Wed Jan 08, 2020 12:03 pm

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