Techniques of Polytempic Polymicrotonal Composition

  Tetrachords, Genera, and the Interval-Space Character: A Webern-Greece Synthesis Peter Thoegersen A recent excursion into Joseph Straus’s Introduction to Post-Tonal Theory sent me down a productive path — not because Straus’s framework applies directly to polytempic polymicrotonality, but because examining where it breaks down clarifies what a more appropriate methodology might look like. Straus is writing about the 12-note chromatic aggregate. His set-class catalog, his interval vectors, his normal and prime form algorithms — all of it is predicated on a closed 12-element universe. Within that universe, it is rigorous and useful. The question I found myself asking was: what survives transplantation to an n-note microtonal aggregate, and what collapses under its own weight? The answer is that the relational machinery survives. Normal form, prime form, the Tn and TnI operations — these work identically modulo any n. The interval vector as a similarity metric still functions. W...

 Has anyone ever noticed how structurally and energetically similar both Ravel's Bolero is to Christopher Smart's Song to David? Both are sustained and rhapsodic as they swell into fiery cauldrons of spiritual heat as they build into frenetic ecstasy, bit by bit; part by part, not vertically, but horizontally. This poem could be considered a Horatian Ode, or Pindaric-Sapphic, as well, since it's not David Smart is lauding, but Jesus. I suppose since Jesus was always thought to have been within the bloodline of King David, that the thrust of the poem is really a genesis story of the ancestral beginnings of the Nazarene bloodline.   Ravel, on the other hand, is doing something very similar to Smart, not that it's Pindaric, but it is essentially an argument for snare drum with repetitious motifs in 9/8, just like how Smart uses words like Adoration, and builds his orchestration around the drum; and as for Smart, he then begins a series of words that begin the subsequent st...

 

  Beyond Book III: Naming What Was Already There Two posts from June of last year form the empirical foundation of what I am about to describe. The first, from June 9, 2025, introduced compression/expansion — the simultaneous compression of interval size and duration across four independent tuning systems and four independent tempos, realized as a polytempic polymicrotonal stretto. The second, from June 11, 2025, fused prosodic feet with microtonal voice leading, arriving at a key statement almost in passing: these interval structures are relative, not absolute, because they apply to all microtonal systems at play in any one given poly composition. Both posts are now incorporated in Book III, recently submitted to Jenny Stanford Publishing. The book crystallizes the theory at a specific moment. But theory does not stop at submission. Looking back at those two posts together, I can now name what they had already discovered. The governing principle is Numerical Invariant Interval...

Numerical Invariant Interval Class In a post from June 9, 2025, I described a compositional technique I called compression/expansion — the simultaneous compression of interval size and duration across four independent tuning systems and four independent tempos. The stretto in that work moves a motive from 20-TET in the first violin down to 26-TET in the cello, compressing the interval microtonally at each successive entrance. At the same time, the tempo difference simultaneously shortens the figure’s duration. Two parameters, one structural logic. I want to now name the governing principle behind that technique: Numerical Invariant Interval Class . The interval class — the abstract integer identifying a relational distance — remains invariant across all four tuning systems. What varies is its acoustic realization in cents. The same interval class 3 is 180 cents in one system, 212 in another, 166 in another. The number is fixed; the sound is plural. This split between numerical identity...

Polytextural Density: A New Property of Polytempic Polymicrotonal Music Peter Thoegersen / zipcaustic.blogspot.com May 2026 While composing String Quartet No. 17, I observed something already living in the music. Each of the four independent streams — each bonded to its own tuning system and its own tempo — was capable of developing its own internal textural character simultaneously with the others. One stream might be monophonic: a single unbroken line. Another polyphonic: multiple independent voices moving within the stream. Another homophonic: parallel chords moving as a block. Another might split a monophonic line across registers such that the upper and lower become functionally two separate voices. String instruments can play multiple stops, sounding different pitches simultaneously, which opens further internal textural possibilities — and if one voice is a piano, the range of internal texture expands again. All of this is happening at the same time, across four structurally ind...

Two Clarifications on Polytempic Polymicrotonality Peter Thoegersen / zipcaustic.blogspot.com Theoretical systems attract misreadings. Sometimes the misreadings are careless, sometimes motivated. Either way, precision is the best defense. Two points in my compositional system invite confusion and deserve explicit treatment. I . C Zero Is Not a Master Fundamental Each of the four independent tuning systems in my polytempic polymicrotonal work is described beginning from the note C. A reader encountering this for the first time might assume that C functions as a tonal center — a master fundamental around which the four systems are hierarchically organized, a hidden key of C governing the whole. That assumption is wrong, and the error lies in confusing a measurement reference with a structural authority. C in my system is what I call C zero: the synchronization pitch, the common origin from which four independent tuning systems launch. It is where I begin to describe the tunings — to expr...

I. If Everything in the Universe Is One Electron, What If Music Is Just One Note The Reductive Reading The one-electron universe is the right pressure to apply here, but the metaphor forks immediately depending on which direction you push it. The reductive reading — everything is one note, one vibration, the harmonic series is just elaboration of a fundamental — leads straight to just intonation ideology, Pythagorean mysticism, the universe humming at A=432. That is the boring version. It reinstates the master fundamental already dismantled. The Wheeler Reading Done Correctly The one electron is not simple — it is everywhere at once, tracing every possible path through spacetime simultaneously. If music is one note, it is a note that occupies four independent tuning systems at the same moment, each with its own tempo, none of them reporting to a center. The singularity produces the irreducible multiplicity rather than collapsing it. Which means polytempic polymicrotonality may be the ...

The Missing Grammar: Pitch-Cell Theory for Microtonal Equal Temperament Peter Thoegersen | zipcaustic.blogspot.com Let me begin with a distinction that the field has largely failed to make clearly enough. Allen Forte’s pitch-class set theory, as developed in The Structure of Atonal Music (1973), gave composers and analysts working in 12-tone equal temperament an enumerated harmonic vocabulary. Not a procedure — a vocabulary. The confusion of Forte’s contribution with Schoenberg’s twelve-tone row has caused no end of theoretical muddle, and the muddle has consequences: it has caused composers to dismiss the cell concept along with the row when they abandoned serial procedure, when in fact the two are categorically different things. The twelve-tone row is a procedure. It answers the question: in the absence of tonal hierarchy, what governs the order of pitch events? Schoenberg’s answer — exhaust all twelve before repeating, and derive structure from the four canonical transformations of ...