Why Polymicrotonal Music, Part XXXI

Why Polymicrotonal Music: A Theoretical Restatement

(Notes Toward a Third Edition of Book I)

by Peter Thoegersen

Some theoretical insights are only fully articulable after the fact. The system precedes its own justification. You build the architecture first, inhabit it for years, and only later find the precise language for why it had to be built exactly the way it was. What follows is that language — a restatement of the foundational argument for polytempic polymicrotonal music that has become clearer to me in the years since Book I was written. It is intended as the basis for a future third edition preface, but I set it down here while the thinking is sharp. Since I discussed Ives and Johnston more in Book I, I will not address them here. 

I. The Primary Justification: Microintervallic Voice Leading

The reason for polytempic polymicrotonal music is not only for expanded pitch resources. It is not complexity for its own sake. It is not even the elimination of harmonic hierarchy as an end in itself, though that elimination is a necessary precondition.

The reason is this: only by running genuinely independent simultaneous tuning systems can one generate microintervallic voice leading that emphasizes the specific differences between those systems. The intervals I am most interested in are the ones that exist between systems rather than within any single one — intervals that cannot be reduced to or explained by any single tuning framework, because they are produced by the friction of incommensurable systems in proximity. And due to polytempic behavior of the individual systems, microtonal simultaneities will result, which were not planned, but happen by accident. Many of these "chords" will result in many new types of microtonal chords. 

The music first requires the theory. The voice leading is the point resulting from a new type of 21st-century polyphonic counterpoint. Everything else — the independent tempi, the equal temperament choices, the absence of a master fundamental — these are the architectural conditions that make this particular kind of voice leading possible. The systems, therefore, are not the subject of the music. They are the precondition for it.

II. Equal Temperament as Architectural Elimination of Hierarchy

Most microtonal practice, even the most sophisticated, evaluates equal temperaments by how accurately they approximate the intervals of the overtone series. The question the field keeps asking is: how close does this temperament get to the acoustically pure intervals generated by a fundamental?

I am asking the opposite question. I choose equal-tempered microtonal scales precisely because they do not generally answer to the overtone series, except for 19, 31, and 53 tone equal temperament, which are meantone adjacent tunings. Nevertheless, a perfect fifth implies a fundamental. A fundamental implies an overtone series. An overtone series implies hierarchy — a vertical arrangement of tonal authority in which some pitches are structurally primary, and others are structurally subordinate. I prefer no hierarchy in the tuning system.

Equal-tempered microtonal scales — particularly those that do not contain a native perfect fifth — sever the connection to acoustic physics entirely. The intervals are not derived from natural resonance phenomena. They carry no embedded resolution tendencies, no gravitational center, no implied root. This is not a limitation. It is the point. The elimination of hierarchy is architectural rather than compositional: it is built into the choice of tuning system itself, not imposed afterward through avoidance strategies.

III. Architectural Elimination vs. Compositional Discipline

A clarification is necessary here. I have worked in equal temperaments both with and without a native perfect fifth. Temperaments like 13-TET, 11-TET, and 23-TET do not contain a usable fifth — the hierarchy is architecturally absent. But temperaments like 12-TET, 31-TET, 53-TET, and 72-TET do contain a fifth, some of them very accurate ones.

In every case, regardless of whether the temperament contains a fifth, my compositional practice is identical: I avoid the fifth entirely and concentrate on the smallest available interval in the tuning. The perfect fifth is present in some of the systems I use, but I simply do not go there. If I do use perfect fifths, they are not functional and lead to no tonic.

This means the anti-hierarchical principle is primary and the architectural condition is secondary. Choosing a temperament without a fifth is the ideal condition — the hierarchy is eliminated before a note is written. But even when working in temperaments that contain one, I maintain the same discipline compositionally. The goal is not the absence of the fifth as a structural feature of the scale. The goal is the absence of hierarchy as a compositional value. The two can coincide, but the value is prior to the architecture.

This also means my approach cannot be reduced to a simple temperament selection argument. Someone could arrive at 13-TET by accident or aesthetic preference without sharing any of the underlying theoretical commitments. What defines my practice is the consistent anti-hierarchical voice leading discipline maintained across the entire catalog — in temperaments that support it architecturally and in those that require compositional will, or intent, to sustain it. This is why compositional intention is important to me. 

IV. Webern and Bartók: Unfinished Projects

Anton Webern understood the problem before anyone had the tools to solve it. His use of pitch cells — particularly trichords like 014, built from a semitone and a minor third — was a deliberate avoidance of the intervals that imply tonal hierarchy. The perfect fifth implies a fundamental. The major third implies a fundamental. Webern's preferred cells are specifically those that do not specify these intervals, which is why his music feels so weightless, so suspended in a kind of permanent present tense, as both minor and major thirds are contained; there is a bi-modal element, just as in Bartok.

But Webern was working within twelve-tone equal temperament, which still contains the perfect fifth. The hierarchy he was avoiding was always structurally present in the system around him. He was swimming upstream — using pitch cell invariance under transposition and inversion to prevent the music from collapsing into tonality, while the gravitational field of the fifth remained active. His anti-hierarchical impulse was real, but the system he inhabited could not fully honor it, both from the limitations of 12tet, and by rote of the 12-tone row, which actually lends its own brand of hierarchy, even if Schoenberg tried to avoid it.

Equal-tempered microtonal scales complete what Webern's pitch cell thinking was reaching toward. When you work in a temperament that does not contain a native perfect fifth, you are not swimming upstream. You have changed the river. The hierarchy is not avoided; it is absent. Webern identified the problem. The solution required a different tuning system. I am quite sure that had Webern remained alive, microtonality would have been right around the corner for him.

Béla Bartók was pursuing a parallel anti-hierarchical project through different means and from a different starting point. His equal-interval chord structures are specifically anti-hierarchical in the way that matters here. A diminished seventh chord divides the octave into four equal minor thirds — it is perfectly symmetrical and therefore points to no fundamental. A whole tone chord divides it into equal major seconds — same symmetry, same absence of gravitational center. His systematic chromatic compression of intervals toward cluster harmony pushes further in the same direction. These are not chords that imply a root. They are constructed objects with no acoustic derivation, answering to geometry rather than physics. Nowhere is this more evident than in his Music for Strings, Percussion, and Celesta.

Bartók arrived at this partly through his folk music research. The modal and pentatonic scales of Eastern European folk music do not organize themselves around the overtone series hierarchy of Western common practice. In fact, many Slavic folk musics are oriented around a tetrachord featuring the minor third. It is very similar to the Webernian 014. His ear was trained on music that did not obey those rules, was bi-modal, and his compositional language absorbed that freedom at a deep level. His axis system extended this logic further — treating tonal areas as symmetrically equivalent rather than hierarchically ordered, with the tritone as structurally equivalent to the tonic rather than its most destabilizing opposite. That is a systematic dismantling of tonal hierarchy using equal division logic.

Like Webern, Bartók was working within 12-TET — the gravitational field of the perfect fifth was always structurally present even when he was working against it. Both composers were swimming upstream in a system that still contained the current they were resisting. The equal-interval structures, the axis symmetry, the pitch cell invariance — these are all compositional disciplines imposed on a system whose underlying physics undermines them. They are strategies of avoidance rather than architectural solutions.

V. The Distinction from Maneri

Joe Maneri and his circle — Julia Werntz, Mat Maneri, and others working in and around 72-tone equal temperament — represent a close historical precedent to my approach. Maneri rejected Western diatonic norms and had atonal inclinations that avoided reference to just intonation. This is genuinely adjacent to my concerns.

But Maneri was working with 72-TET as a single expanded pitch space. The question he was answering was: what happens when you give a composer seventy-two pitches per octave instead of twelve? That is a question about resources — about the enlargement of a single system.

My question is different. I am not enlarging a single system. I am running multiple independent systems simultaneously, systems that share no common fundamental and no common grid, and I am composing in the microintervallic space between them. The Maneri circle gets me to the door, but the door opens onto a different territory.

VI. Xenakis: Another Partial Predecessor

Iannis Xenakis presents a genuinely interesting case in relation to this framework — close in some respects, distant in others, and ultimately another partial predecessor who identifies part of the problem without arriving at the solution.

The closest point of contact is sieve theory. Xenakis developed a formal system using modular arithmetic to construct arbitrary scales — including microtonal ones — that are not derived from the overtone series and carry no inherent hierarchical structure. A sieve-generated scale is a mathematical object, not an acoustic one. In that fundamental sense, he shares the anti-derivation instinct: his pitch materials do not answer to natural resonance phenomena.

But Xenakis was primarily interested in the density and texture of pitch space rather than voice leading. His microtones in works like Metastasis emerge from orchestral glissandi — continuous sweeps through pitch space where the individual microtonal intervals are not compositionally designated but are byproducts of physical gesture. The microintervallic relationships are not chosen; they pass through. The between-system microintervallic voice leading that is central to my work has no equivalent in Xenakis because he was not composing at the interval level in the way I am.

The biggest difference is structural. Xenakis was working with a single continuous or constructed pitch space, however complex. His pitch fields are unified — stochastically distributed or sieve-generated, but occupying one sonic world at a time. He was not running genuinely independent simultaneous tuning systems, other than quartertones with third tones, which were sieve-derived, and predicated on his Byzantine modal idea. The 'between-system friction' that generates my primary expressive material is simply not present in his work.

On centricity, Xenakis largely abandoned it. His stochastic works do not employ centric tones even in the arbitrary and provisional sense I use them. This is actually a point where my approach is more compositionally controlled than his — I maintain local orientation within each system while refusing the acoustic derivation that would make that orientation hierarchical.

Xenakis, therefore, clears the overtone series — his pitch materials are not derived from acoustic physics — but does not build what I build in the space that opens up. Like Webern, he identifies part of the problem without arriving at the solution. He belongs in the lineage of partial predecessors: composers whose work points toward my territory without fully inhabiting it.

VII. Carrillo and Wyschnegradsky: The Continuum Trap

Julián Carrillo and Ivan Wyschnegradsky represent the two most significant early microtonal pioneers outside the German tradition, and both are instructive precisely because of where they fall short of the position developed here.

Carrillo developed his Sonido 13 system by subdividing the octave into increasingly fine divisions — quarter tones, eighth tones, sixteenth tones — and built special instruments to realize them. His motivation was essentially acoustic: he was pursuing the natural small intervals available in the upper reaches of the overtone series, the partials that standard Western tuning ignores. His microtones are physically derived. They answer to nature. This places him squarely on the opposite side of the anti-derivation argument articulated in this document, where I move away from the overtone series; Carrillo moved further into it. His pitch space is a single unified continuum, not independent simultaneous systems, and his theoretical frame remains tethered to acoustic physics throughout.

Wyschnegradsky is more interesting and more genuinely adjacent. He worked primarily in quarter tones but explored other divisions as well, and crucially, he wrote for multiple pianos tuned differently from one another — an instrumental apparatus that is structurally closer to my approach than almost anyone else in the early microtonal tradition. His concept of ultrachromaticism and what he called pansonority was about dissolving the boundaries between pitches into a unified continuum of sound, which echoes more of a Veresian sound world.

And that last word is the decisive distinction. Wyschnegradsky's goal was unity — the fusion of all pitch space into one continuous field. My goal is the opposite: the maintenance of genuine independence between simultaneous systems. Where he wanted the boundaries between tuning areas dissolved, I want them audible, felt, compositionally emphasized at the voice-leading level. The micro level. Where he was moving toward pansonority — one sound, one field — I am insisting on irreducible difference between systems that remain incommensurable.

The multiple pianos tuned differently are a genuine precedent for the instrumental apparatus. The philosophical direction is reversed. Wyschnegradsky arrives at the door from the wrong side.

The pattern across all seven predecessors is now complete. Webern avoids hierarchy within a system that still contains the fifth, using pitch cell invariance to resist a gravitational field he cannot remove. Bartók pursues the same anti-hierarchical project through equal-interval chord structures and axis symmetry, arriving at geometry rather than physics — but also within 12-TET, also swimming upstream. Maneri expands a single system without multiplying independent ones. Xenakis removes acoustic derivation without generating between-system voice leading. Carrillo goes deeper into the overtone series rather than away from it. Wyschnegradsky deploys multiple differently-tuned instruments but seeks their dissolution into unity rather than the preservation of their independence. Jean-Étienne Marie comes closest of all — actually running multiple simultaneous microtonal systems — but remains tethered to acoustic derivation and serial organization rather than anti-hierarchical between-system voice leading. Each clears one obstacle or arrives at one adjacent territory. None completes the full architecture.

VIII. Jean-Étienne Marie: The Closest Predecessor

Jean-Étienne Marie deserves special attention in this lineage because he comes closest — closer than Webern, Maneri, Xenakis, Carrillo, or Wyschnegradsky — to the structural position I occupy. In works like Tombeau de Carrillo, he simultaneously deployed multiple microtonal systems, including chromatic half, third, fifth, and sixth tone scales, and in Ecce Ancilla Domini, he wrote serial and polytempered music using rows in quarter, fifth, and sixth tones simultaneously. He is the one figure in the historical record who was actually running multiple different microtonal systems at the same time.

Marie is discussed in Book I of my trilogy. Kyle Gann noted in his blurb that I had rescued this almost totally forgotten microtonalist from obscurity — a characterization that is itself telling. That the closest historical predecessor to my approach was so thoroughly forgotten that his rescue required mention suggests something about how thoroughly the field has failed to follow the implications of the most radical work done within it. Both Marie and Xenakis were students of Messiaen, which is another interesting tidbit, considering the nature of materials that map onto themselves, as in Messiaen's modes of limited transposition, reflecting Webern and Bartok's symmetry.

But the critical distinctions hold even here. Marie arrived at polymicrotonality through Carrillo — the concept was suggested by his predecessor's acoustic framework. This means Marie's simultaneous multiple systems are still rooted in overtone series thinking, even when deployed together. The anti-hierarchical argument that drives my choice of multiple independent equal-tempered systems is not present in Marie. He was extending acoustic exploration into a more complex domain, not escaping acoustic derivation entirely.

His organizational principle was serialism — rows governing pitch relationships across the multiple systems. This is a fundamentally different approach from anti-hierarchical microintervallic voice leading that emphasizes the specific differences between systems. Serialism is a pitch ordering discipline; what I am doing is composing in the intervals that exist between incommensurable systems, using the friction of their independence as the primary expressive material. Marie organized the systems. I compose between them.

Marie, therefore, stands as the closest predecessor and the most instructive limit case. He got to the door and opened it slightly. The acoustic derivation and serial organization kept him from walking through it, into the territory where the between-system microintervals become the compositional subject rather than the byproduct of simultaneous deployment.

IX. Centricity Without Hierarchy

A further clarification is necessary, one that might appear to contradict the anti-hierarchical position but does not. Within each independent tuning system, I also employ centricity — an arbitrarily chosen note that acts as a temporary tonal center within that system. This requires a precise definition.

Centricity is not hierarchy. The distinction is fundamental. Hierarchy is generated — it flows from the overtone series, from the fundamental, from acoustic physics. A hierarchical tonal center owes its authority to natural resonance phenomena and is confirmed by common practice chord progressions that lead toward it, prepare it, and resolve into it. It is not chosen. It is demanded by the system.

Centricity, as I employ it, is chosen. The centric note within each system is designated by compositional will, not derived from acoustic physics. It carries no structural authority over the pitches around it. It is not confirmed by common practice progressions — there are none. It is temporary, revocable, and can be shifted or abandoned without the system demanding resolution. It is an organizational focus, not a structural authority. Yet, the harmony is not static; there is a feeling of forward motion, rather than stasis. 

This is close to what Stravinsky achieved through pitch centricity — returning to certain pitches by sheer compositional assertion and repetition rather than functional harmonic logic. But Stravinsky was working in 12-TET, where the gravitational field of the perfect fifth was always present in the background, always threatening to pull the music toward functional tonality. My arbitrary centricity operates within systems where that gravitational field has already been neutralized architecturally and compositionally. The centric tone floats free of the physics that would otherwise give it unearned authority.

The complete architecture is therefore three-level: no master fundamental across systems nor within systems, no perfect fifth privileged within systems, and an arbitrarily designated temporary center within each system for local organizational purposes. Each level of this architecture is a distinct anti-hierarchical decision. Together they produce a music that has orientation without gravity — a sense of local focus without the structural subordination that tonal hierarchy requires.

X. A Summary Statement

Polytempic polymicrotonal music is justified by the voice leading it makes possible. It requires genuinely independent simultaneous tuning systems because only genuine independence generates the between-system microintervals that are its primary expressive resource. It prefers equal temperament because equal temperament can architecturally eliminate the overtone-series hierarchy that would otherwise subordinate one system to another, but even in temperaments that contain a perfect fifth, the anti-hierarchical discipline is maintained compositionally by concentrating exclusively on the smallest available intervals. The absence of hierarchy is a compositional value prior to and independent of any particular scale choice. It extends the anti-hierarchical logic of Webern's pitch cell practice and Bartók's equal-interval chord structures into a domain where that logic can be fully realized rather than merely approximated. It is categorically distinct from the expanded-resource microtonality of Maneri and his circle, because it is not about more pitches within a single system but about the irreducible difference between independent systems. And within each independent system, it employs an arbitrarily chosen centricity — orientation without gravity, local focus without structural subordination — that is fundamentally distinct from the generated hierarchy of common practice tonality.

These formulations were not fully available to me when Book I (originally my doctoral thesis) was written. They are available now. They belong in a future third edition — stated directly, early, and without apology.