Compression of Harmony in Polymicrotonality: From Whole-Tone to Polymicrochromatic — Revisited, Part LV
Techniques of Polytempic Polymicrotonal Composition
Compression of Harmony in Polymicrotonality: From Whole-Tone to Polymicrochromatic — Revisited
Blog post, 2026
I want to return to an earlier blog (XIIb, June 2025) on harmonic compression in polytempic polymicrotonal composition. Having recently revisited Joseph Straus's post-tonal theory text and Bartók's String Quartet No. 4, I can now name what that blog was demonstrating more precisely than I did at the time.
The Existing Device
In post-tonal practice, the compression of the whole-tone tetrachord to the chromatic tetrachord functions as a directed harmonic device. The whole-tone tetrachord — four equally spaced pitches, each a whole tone apart — spans a tritone. The chromatic tetrachord compresses this to a minor third, with each interval a semitone.
Bartók's String Quartet No. 4 is the concentrated demonstration. His chromatic clusters — four consecutive semitone-adjacent pitches, for example [C, C♯, D, D♯], with interval pattern [1, 1, 1] in semitones — function as maximum-compression states within an equal-interval tetrachordal continuum. Release toward whole-tone adjacencies operates as harmonic narrative, not coloristic ornament. Straus's voice-leading geometry makes the structural adjacency of these tetrachord types legible, though the device itself — directed compression and expansion as a compositional principle — remains more assumed than theorized in the literature.
The chromatic tetrachord has been treated as the terminus of compression. I am breaking that assumption.
The Chromatic Is Not a Point
Before extending below the chromatic level, it is necessary to recognize that the chromatic itself is not a fixed value but a historical band. The Pythagorean limma — 256/243, approximately 90¢ — and the Pythagorean apotome — 2187/2048, approximately 114¢ — define the lower and upper boundaries of what has been understood as the chromatic semitone. 12-EDO's 100¢ semitone splits this range; it is a midpoint, not a definition.
This means each EDO has its own chromatic step that locates itself within or outside the limma-to-apotome band:
11-EDO at 109.1¢ falls near the apotome — upper chromatic boundary.
12-EDO at 100.0¢ occupies the center of the chromatic band.
13-EDO at 92.3¢ sits near the limma — lower chromatic boundary.
14-EDO at 85.7¢ falls below the limma: it is already microchromatic.
Every EDO from 14 upward into finer divisions produces a single step that is sub-limma — microchromatic by the boundary the historical chromatic zone itself defines. The chromatic is not a wall; it is a band with measurable edges, and those edges have always implied what lies below them.
The Compression Continuum
The full continuum operates across three zones: the whole-tone region, the chromatic band, and the microchromatic region. Each zone is EDO-dependent — the absolute cent values shift with every tuning system, while the structural relationships are preserved.
Table 1 shows whole-tone equivalents across EDOs 13–24 and selected others, sorted by interval size. Note that some smaller EDOs produce only approximate whole-tone equivalents:
Table 2 shows the single-step chromatic and microchromatic values across the same EDO range, color-coded by zone. The limma (~90¢) is the boundary between chromatic band and microchromatic:
The structural implication is significant: not only does each EDO produce its own whole-tone interval, it also produces its own chromatic step — which may fall within the historical chromatic band, near its limma boundary, or already below it into microchromatic territory. The continuum is a relational structure internal to each tuning system, not a fixed cent-value spectrum anchored to 12-EDO.
Polymicrochromatic
In polytempic polymicrotonal composition, multiple simultaneous independent tuning systems run at multiple simultaneous independent tempi. This means the compression continuum is not a single shared spectrum but several independent spectra operating simultaneously. Each voice inhabits its own EDO and therefore its own version of the continuum — its own whole-tone interval size, its own chromatic band position, its own microchromatic compression level.
I call this polymicrochromatic: simultaneous independent microchromatic compression states across independent tuning systems and independent tempi.
Whole-tone → Chromatic band → Microchromatic → Polymicrochromatic
Each term is a compression of the previous. The first two belong to the existing post-tonal literature. The third extends it below the chromatic threshold — more precisely, below the limma boundary — into sub-semitonal equal-interval tetrachords. The fourth is specific to polytempic polymicrotonal composition. It cannot exist in any system with a shared tuning reference.
Two voices could occupy structurally parallel whole-tone compression levels within their respective EDOs while producing entirely different absolute intervals — structurally parallel, acoustically divergent. One voice at its chromatic compression level might be wider in cents than another voice at its whole-tone level, because the EDOs differ sufficiently. The compression relationships are real but incommensurable across voices. And the chromatic band itself — the historical ground of the device — becomes a zone that different voices may occupy at different positions simultaneously.
SQ12 Revisited
The stretto figure in String Quartet No. 12, Big Bad Mother Fucker From Outer Space (2019), demonstrated in the earlier blog, compresses microintervallically across four simultaneous independent tunings: 20-EDO at 60¢, 22-EDO at 55¢, quartertones at 50¢, and 26-EDO at 46¢ — while simultaneously compressing rhythmically through independent tempi of 66, 72, 79, and 86 BPM. All four single-step values fall below the limma boundary: all four voices are operating in the microchromatic zone, each in its own tuning universe, at its own tempo.
At the time I framed this as Bartók's compression instinct extended to the micro level. Returning to Straus and the Fourth Quartet, and now with the chromatic band properly bounded by limma and apotome, I can say something more precise: SQ12 was operating below the entire historical chromatic zone — not merely below 12-EDO's 100¢ midpoint, but below the limma itself. The theoretical terminus the literature assumed was not even the terminus of the chromatic band. It was its center.
Bartók compressed to the chromatic tetrachord. In SQ12 I compressed past the chromatic band entirely, polymicrochromatic ally, with each voice doing so independently in its own tuning universe at its own tempo.
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