Forte Pitch Cells for Microtonal Systems, Part LVI

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 the row — was historically urgent and compositionally productive for roughly three decades. It is now exhausted. Not because it was wrong, but because it solved a specific problem and that problem is no longer the one composers face.

The pitch cell is a vocabulary. It answers a different question: within a given pitch universe, what are the characteristic interval clusters — the two-, three-, and four-note groupings that have a recognizable sonic identity and can be transposed, inverted, and recombined without dictating order? Forte’s set classes are an enumeration of those clusters for mod 12. The interval vector of each set class tells you how many times each interval class appears within it — the sonority’s fingerprint. A composer using these tools is not following a row. They are choosing a palette.

This distinction matters because the vocabulary concept is entirely portable to other equal temperament systems. The procedure concept is not, or at least it offers nothing new in transport.

What Has Been Done

The mathematical generalization of pitch-class set theory to n-tone equal temperament is not new. Robert Morris’s work on set groups and mappings (1982) and David Lewin’s Generalized Musical Intervals and Transformations (1987) provide frameworks that operate on any cyclic group of pitch classes, not only Z12. More recently, category-theoretic approaches have formalized mappings between n-tone and m-tone systems. The theoretical machinery exists.

More practically, the Estonian composer and theorist Hans-Gunter Lock published in 2021 what is to my knowledge the most substantive practical realization of this idea. His chapter “Organizing Microtonal Pitches: Tables of Forte-Like Labelled Pitch-Class Sets for 13 to 24 Equal Divisions of the Octave as a Tool for Composers,” in Mikrotöne: Small Is Beautiful III (Mackinger Verlag, Salzburg), extends Forte’s and Solomon’s 12-EDO set-class methodology — prime forms, interval vectors, the A/B/* classification for narrow, wide, and symmetrical sets — to every equal temperament from 13 through 24 divisions of the octave. Lock’s naming convention mirrors Forte’s, adding a superscript modulo prefix (e.g., 224-164A, where 22 is the EDO, 4 is cardinality, 164 is the catalog number, and A indicates a narrow form). This is serious, systematic work and it deserves to be far better known than it currently is.

A 2025 thesis by Marc Lujan i Ballester at the Estonian Academy of Music and Theatre, supervised by Lock, demonstrates the analytical application of these tables to an original composition — Lullaby in 22 EDO for electric keyboard and viola. Lujan combines Lock’s prime-form classifications with a root-based chord taxonomy developed by microtonal composer Sean Archibald (Sevish) to produce a dual-lens analysis: one abstract and structural, the other perceptually grounded in 22-EDO practice. The result is the clearest demonstration yet of what Lock’s tables make possible analytically.

Robert Kelly has done related work adapting Forte’s set theory specifically for 22-EDO. Paul Erlich’s contributions to microtonal theory are foundational in a different but adjacent domain — his work on regular temperament theory and the harmonic structure of 22-EDO informs how composers understand that system — but his approach is not set-class based in Forte’s sense.

What Remains Open

Lock’s work is the foundation. But it has limits that define the remaining open territory, and they are substantial.

First, the tables cover set classes only up to pentachords. Hexachords and above are unaddressed — the combinatorial volume having made systematic enumeration impractical within the scope of a symposium chapter. For composers working with denser harmonic textures, this is a real constraint.

Second, Lock stops at 24-EDO. Systems beyond that boundary — 26, 31, 34, 41-TET and others that composers are actively using — have no comparable reference. My own String Quartet No. 17 uses 26-TET as one of its four simultaneous tuning layers; that system is entirely outside Lock’s coverage.

Third, and most consequentially: Lock’s tables and Lujan’s application both treat pitch-cell theory as an analytical tool applied to finished music. The question of how a composer uses these tables generatively — how to select cells before composing, how to understand which ones are structurally distinctive and why, how to build a working compositional practice from them rather than a retrospective analysis — remains unaddressed in the literature. Lujan is candid about this: the Sevish-DQ taxonomy “was of limited use” analytically, and “many harmonic shapes used here may not yet have established labels.” This points directly at what is still missing: not the tables, but the compositional commentary that would make those tables a living vocabulary rather than a catalog.

What the Vocabulary Actually Looks Like: Two Cases

With Lock’s framework as context, let me sketch what native pitch-cell thinking looks like in two systems within his range, as a way of gesturing toward the compositional dimension the tables alone do not provide.

13-TET

Thirteen equal divisions of the octave gives a step size of approximately 92.3 cents. Because 13 is prime, the group Z13 has no non-trivial subgroups — its interval-class structure is maximally uniform, with six interval classes:

IC1 = 92¢ — slightly narrower than a 12-TET semitone (100¢); less than 8 cents difference. The smallest native step of 13-TET is semitone-adjacent, not exotically microtonal.

IC2 = 185¢ — recognizable as a major second (200¢), about 15 cents flat. Still within the gravitational pull of a familiar interval.

IC3 = 277¢ — alien territory begins here. 23 cents below a minor third, 77 cents above a major second. Falls between 12-TET categories rather than approximating either. No convenient label applies.

IC4 = 369¢ — the alienness is fully established. 31 cents below a major third, 69 cents above a minor third. Neither family claims it. The ear has no existing template.

IC5 = 462¢ — a compressed perfect fourth, 36 cents narrow of 498¢. Reads less as a flat fourth than as its own interval.

IC6 = 554¢ — between a perfect fourth and a tritone, claiming neither.

The system’s strangeness does not live in its smallest step. The drift from 12-TET accumulates linearly — each interval class adds roughly 7.7 cents of deviation — but the perceptual effect is nonlinear. IC1 and IC2 are still recognizable; IC3 and IC4 are where the system crosses into territory the ear has no map for. That is where the compositional interest lives.

A characteristic trichord: {0, 2, 5} — intervals of 2 and 3 steps, outer span of 5. Contains IC2, IC3, and IC5, three distinct classes, none repeated. Interval vector [1, 1, 0, 0, 1, 0]. Open, widely-spaced. Compare {0, 3, 6}: two stacked IC3s, outer IC6. Interval vector [0, 0, 2, 0, 0, 1]. Denser, self-similar — the functional analog of what a diminished triad does in 12-TET. These are not translations. They are native structures that need to be heard in their own terms.

14-TET

Fourteen equal divisions gives a step size of approximately 85.7 cents. Here 14 = 2 x 7, so Z14 has subgroups of order 2 and 7 — a richer algebraic structure than the prime-order 13-TET. Seven interval classes:

IC1 = 86¢ — semitone-adjacent.

IC2 = 171¢ — narrow major second; recognizable but flat.

IC3 = 257¢ — between major second and minor third; alien territory.

IC4 = 343¢ — near a neutral third; unmoored from both minor and major.

IC5 = 429¢ — between major third and perfect fourth; claims neither.

IC6 = 514¢ — slightly wide perfect fourth; the closest 14-TET comes to a just fourth (498¢).

IC7 = 600¢ — exact tritone. Because 14/2 = 7, IC7 is self-complementary — the structural axis of the system.

A structurally significant consequence: no interval class approximates the perfect fifth (702¢) well. IC9 = 771¢, too wide; IC8 = 686¢, too narrow. Major and minor triads have no native home in 14-TET. The system is genuinely non-triadic. A composer who approaches 14-TET by approximating 12-TET triads is fighting the system rather than working with it.

Native cells: {0, 2, 6} — IC2, IC4, IC6, a symmetrically-spaced trichord with wide, cool sonority. {0, 4, 9} — IC4 and IC5, a denser quartal structure. Neither translates from 12-TET. Both need to be heard as themselves.

Generative Methods: Thinking Past the Triad

The central obstacle to generative pitch-cell work in microtonal equal temperament is not mathematical. It is perceptual and habitual. Even a composer who has Lock’s tables in hand will instinctively scan them for near-triads — cells that approximate the major third plus perfect fifth, or the minor third plus perfect fifth, that Western ears have been conditioned to treat as the basic unit of harmonic stability. In systems where those approximations are poor, this instinct leads directly away from the system’s own character.

The alternative is to begin generative work from the interval class itself rather than from the named chord type. This means identifying which interval classes in a given system are structurally central — not because they approximate a just ratio, but because of their behavior within the system’s own internal logic. In 14-TET, IC6 (514¢) is the nearest native fourth; it becomes a structural anchor not because it approximates 4:3 (it does, poorly) but because it is the widest stable dyad before the system’s self-complementary axis at IC7. In 13-TET, IC5 (462¢) plays a similar role. These are not perfect fourths. They are the fourths of their respective systems, and they need to be treated as such.

From these dyadic anchors, trichords and tetrachords can be built by asking which additional interval classes complement rather than duplicate the anchor’s character. A generative principle: prefer cells whose interval vectors contain no repeated interval classes — maximum internal diversity. This is the microtonal analog of the all-interval tetrachord in 12-TET, a cell with no redundancy in its intervallic content, each class appearing exactly once. In 13-TET, {0, 2, 5} exemplifies this at the trichord level. In 14-TET, {0, 2, 6} does the same. These are cells that use the system’s resources efficiently, without the gravitational pull of a single dominant interval class.

A second generative principle: build cells that exploit the interval classes unique to the system — the ones with no close 12-TET analog. In 13-TET, IC3 (277¢) and IC4 (369¢) are where the system is most itself. A cell built primarily from these intervals cannot be heard as a 12-TET approximation. It forces the ear into genuine microtonal territory. The same logic applies in 14-TET to IC3 (257¢) and IC4 (343¢). A generative practice that privileges these intervals over the system’s more 12-TET-adjacent ones is working with the tuning rather than against it.

A third principle: respect what the system cannot do. 14-TET has no good fifth. 13-TET has no good minor third. These are not deficiencies to be worked around — they are defining features that shape what kind of music the system can generate on its own terms. A compositional practice built on acknowledging these absences, rather than lamenting them, will produce music that could only exist in that system.

Ear Training as the Foundation

None of the above is available to a composer who has not trained their ear in the specific tuning. This is the point that the table-based approach most risks obscuring: the tables give you the mathematics, but mathematics does not compose music. The interval vector of a cell describes its structure; it does not tell you what the cell sounds like, how it behaves melodically, whether it has tension or repose within the system, or how it relates to the cells adjacent to it. Only sustained listening does that.

Ear training in a microtonal equal temperament is categorically different from ear training in 12-TET. The task is not to recognize whether an interval is in tune relative to a known standard. It is to learn the interval as a new object — to hear IC4 in 13-TET (369¢) not as a sharp minor third or a flat major third but as itself, with its own character and its own name in the perceptual vocabulary. This requires extended, repeated, focused listening: singing the intervals, playing them in isolation and in context, composing short studies that use a single interval class exhaustively until it becomes familiar rather than foreign.

The phenomenological approach to pitch-cell derivation begins here. Rather than selecting cells from a table on the basis of their interval vectors, the composer derives cells from direct sonic experience — noticing which combinations of native intervals produce a sense of coherence or tension, which dyads want to resolve and which seem self-contained, which trichords have a distinctive profile that the ear can recognize across transpositions. These observations are then cross-referenced against the tables to confirm their theoretical identity, but the primary source is the ear, not the catalog.

Messiaen’s modes of limited transposition offer a partial precedent: he arrived at his characteristic harmonic language through a combination of systematic symmetry logic and direct sonic habituation, hearing his modes repeatedly until they became native rather than exotic. The microtonal composer faces a steeper version of the same task, because the intervals are genuinely unfamiliar rather than merely unconventional. The steepness is not an argument against the task. It is an argument for taking ear training seriously as a compositional prerequisite — not an optional supplement to theoretical knowledge but its necessary foundation.

A practical consequence: a composer cannot meaningfully use a microtonal system they have not spent time living with aurally. The tables alone produce music that is theoretically organized but sensorially inert — correct on paper, empty in the ear. The cells that will sustain a compositional practice are the ones the composer has internalized, can hear internally, can sing, can recognize in context. These are phenomenologically derived cells: arrived at through the body and the ear before they are confirmed by theory.

This is the work that no table can do. Lock’s contribution is indispensable — the tables provide the map. But a map is not the territory. The territory is the sound itself, and it must be entered directly.

The Remaining Work

Lock’s tables are the foundation. What remains is the superstructure: compositional commentary on what the cells in each system actually sound like and why certain ones are structurally distinctive; coverage of systems beyond 24-EDO; enumeration of hexachords and larger collections; generative principles for building a compositional practice without triadic default; and systematic attention to ear training as the prerequisite for meaningful cell selection.

The difference between analytical and generative use matters. An analytical tool is applied after the fact to understand what happened. A generative vocabulary is what a composer reaches for before a note is written — the equivalent of knowing which triads and seventh chords are available before composing in a tonal system, and having internalized their sound well enough to hear them in the imagination. Lock and Lujan provide the former. The latter remains open territory.

The twelve-tone row is finished. The pitch cell is not. And for microtonal equal temperament systems, the work of turning Lock’s foundational tables into a living compositional language — one grounded in the ear as much as in the interval vector — has barely begun.

Peter Thoegersen is a composer, music theorist, and memoirist based in Urbana, Illinois. His system of polytempic polymicrotonal composition bonds independent microtonal equal temperament systems each to its own independent tempo and meter. His work is published by Jenny Stanford Publishing.