On Straus Through a Synthesis of Webern and the Greek Tetrachord, Part LXII

 

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. What changes is the content of the interval-class space. In 31-TET, there are 15 interval classes; in 53-TET, 26. The set-class count grows ferociously with n. A catalog approach — already demanding in the 12-note case — becomes computationally absurd for larger microtonal systems. The compression of intervals as n increases means that what functions as a minor third in 12-TET has no direct structural analog in 19 or 53. You are not translating. You are in new territory.

Why Webern

Among the foundational post-tonal composers, Webern scales where the others do not — and the reason is architectural.

Schoenberg treats the row as primary. The aggregate is a surface phenomenon: the row is a sequence, and subset relations are often incidental rather than structurally generative. Berg is more pragmatic still, sometimes frankly arbitrary. Stravinsky tends toward centricity; Bartók toward bimodality. Neither offers a model free of tonal gravity.

Webern inverts the priority. He builds his rows out of subset structures — the trichord or tetrachord cell is primary, and the row is a consequence of cellular combination and transformation. The cell comes first. The aggregate is downstream of the cell. And the cell he favors — 014 being the most characteristic — is chosen for its interval-class properties: high internal symmetry, invariance under inversion, the capacity to generate its complement under transposition.

This inversion of priority is exactly right for microtonal work at scale. When the aggregate is large enough that cataloguing all set classes is impractical, you need a generative vocabulary — a small number of cells whose interval-class properties do the compositional work — rather than an exhaustive accounting of the full pitch field. Webern was already working this way within 12-TET. The methodology ports; the specific cells do not.

There is, however, one Webernian inheritance to set aside: aggregate completion. Even when Webern’s cellular logic is the structural engine, the serial obligation remains — the aggregate must saturate before pitches repeat. That creates an accounting relationship between composer and pitch field that is foreign to polytempic polymicrotonality, where each stream has its own independent pitch space and no obligation to complete anything. Webern as methodological ancestor, then. Not as a rule.

The Greek Genera

The other tributary is older and, for this purpose, more liberating.

The ancient Greek system of tetrachords organizes melodic space not as a scale to be completed but as a bounded interval — the perfect fourth — filled by two interior movable pitches. The genus is determined by the characteristic positioning of those interior pitches: diatonic, chromatic, and enharmonic. But Aristoxenus, the most theoretically sophisticated of the ancient theorists, understood the genera not as three fixed categories but as a continuous spectrum. The shades — the chroai — describe the sliding of the inner notes across a range of positions: soft chromatic, hemiolic chromatic, tonic chromatic. Qualitative character as the operative category; continuous variation as the underlying reality.

This is already, conceptually, microtonal territory. Aristoxenus was describing positions that resist integer rationalization, an interior space defined by characteristic sound rather than by ratio or arithmetic proportion. The specific interior positions a given n-TET system makes available within a fourth are not translations of Greek intervals. They are native to that tuning system. But the conceptual structure — bounded fourth, characteristic interior positioning, qualitative melodic identity — transfers completely.

Equally important is what the Greek system does not require. It does not require completing a mode. It does not require conjunct or disjunct tetrachord linkage to fill a larger system — that logic belongs to the Greater and Lesser Perfect Systems, which are the Greeks’ own superstructure, their own aggregate-completion drive. What I am taking from the genera is narrower and more useful: the bounded fourth as a characteristic interval space, the interior pitches as definers of that space’s sonic identity, and the use of the resulting cell as melodic and rhetorical rather than syntactic and obligated.

This is not an appropriation of Greek music, nor of the Arabic maqam tradition that developed its own elaborate conjunction and modulation logic on similar foundations. It is the extraction of one conceptual layer — interval-space character — and the deliberate discarding of the superstructure built on top of it in both traditions.

The Synthesis

The cell is the tetrachord. The container is the perfect fourth, which every n-TET system possesses in near-equivalent form, close enough to the acoustic 4:3 that the structural logic of bounded interval space transfers even when the exact cent values differ. The interior pitches are positioned according to the continuous chroai space — not selecting from three fixed genera but placing pitches within the fourth at positions native to a given tuning system, producing a characteristic interval-space quality.

That quality is then used with full melodic and rhetorical freedom. Repetition is encouraged. Return is encouraged. Emphasis and avoidance are compositional choices, not violations of an accounting rule. Nothing obligates the completion of anything beyond the fourth itself.

From Webern: the cell as a structural primitive, the interval-space character of the cell as the compositional engine, the methodological priority of the small generative unit over the large aggregate.

From Greece: the specific shape of the cell — the fourth as container, the continuously variable interior — and the freedom from any obligation to complete or extend toward a larger system.

The two are not in tension. Webern’s cellular primacy needed a cell. Greece provides one with a natural acoustic authority that predates common practice tonality, survives it, and appears across cultures that developed entirely independently of European functional harmony. The microtonal filling of the fourth does not evoke tonality because the interval-class relationships that make tonal hearing possible — the leading tone, the tritone’s instability within a diatonic context, the fifth’s structural dominance — depend on very specific configurations in 12-TET that do not exist in the same form in 19, 31, or 53. The genera framework, especially the chromatic and enharmonic shades, foregrounds precisely the interval regions that common practice systematically suppressed.

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 and acoustic realization is the defining feature of NIIC.

This produces a pitch organizational principle that is neither tonal, atonal, nor centric. Tonal music organizes pitch by function. Atonal music negates that function. Centric music orbits a gravitational pitch object. NIIC does none of these. The organizing principle is the abstract numerical relationship itself — invariant at the structural level, irreducibly multiple at the acoustic level simultaneously.

Bartók compressed equally spaced chords intervallically as a structural principle. I am extending that vision down to the microlevel, across four tuning systems that share interval class integers but not cent realizations. The compression/expansion technique is NIIC in motion.

A further consequence follows. Perfect pitch is predicated on a finite, learnable set of absolute pitch classes. 12-TET makes this possible — 12 is fixed and culturally reinforced over centuries. As n increases toward infinity, absolute pitch recognition becomes statistically improbable, then perceptually impossible, then a category error. No one can reasonably be expected to have perfect pitch for n equals infinity divisions of the octave. Therefore, absolute pitch class loses its theoretical primacy as n grows.

What takes its place is interval content — the relational structure between pitches rather than their fixed identities. This is not merely a perceptual observation but a theoretical necessity. NIIC is its formalization.

The implications extend outward. Tonal, atonal, and centric systems are each organized around pitch class as primary. As n grows, all three categories erode simultaneously — their organizing principle becomes increasingly irrelevant. NIIC steps into that vacancy. It is not a modification of the existing three categories but the principle that supersedes them when pitch class can no longer bear the organizational weight.

This also retroactively justifies the notation philosophy behind my scores. If absolute pitch class is not the primary organizational fact in microtonal music, then crowding the horizontal line with absolute pitch symbols is not merely aesthetically undesirable — because the symbols get in the way of the polyrhythms, which are dense in my musical narrative — it is theoretically misguided. The interval content lives in the system and is always explained by the tuning legend; the score carries the temporal argument.

Society of Pitches

In the third to last paragraph of “Society and Solitude,” Emerson writes of what happens when people gather with freedom for conversation: a rapid self-distribution takes place into sets and pairs — not through love or hatred, not through interference, but through intrinsic affinity. Each seeks his like. Any interference with those affinities would produce constraint and suffocation. “All conversation is a magnetic experiment.”

I am transposing this observation wholesale into pitch space. Given a company of pitches from any microtonal system and freedom to choose their own pairs and sets, a rapid self-distribution takes place based on the magnetic principles between their interval identities. The pitches seek their own mates. This is not a metaphor — it is a structural claim about how pitches behave when released from hierarchical control.

Emerson’s magnetic experiment is the NIIC relationship made audible. The interval ratios are the personalities; the numerical invariant interval class identities are the affinities that govern attraction and repulsion; the self-distribution into sets and pairs is the compositional methodology. The composer does not assign relationships — the relationships emerge from the intrinsic interval identities of the pitches themselves.

This is not aleatory. Chance is indifferent to interval content. The Society of Pitches is motivated by autonomy — the freedom is real, but the choices are not random. They are determined by the mathematics already inside the pitches. Nor is it serialism, which imposes external order. Nor spectralism, which derives relationships from acoustics but retains compositional control over outcome. The Society of Pitches gives pitches agency grounded in intrinsic interval identity and follows where that agency leads.

In a polytempic polymicrotonal system with four independent tuning layers, this produces a “society of societies” — each tuning population forming its own internal affinities while also reaching across tuning boundaries to form cross-system relationships. The aggregate is neither tonal, atonal, nor centric. It is self-organized from the numerical interval identities that NIIC names.

Call and Response as Generative Mechanism

The serialist tradition establishes the set before the line. The row, the derived set, and the rotational array are pre-compositional decisions; the music unfolds from them. The set generates the composition.

In my practice, the relationship is inverted. I write a line first, then answer it with each of the remaining three voices in succession. The second voice responds to the first, the third to both, the fourth to all three. The set is not determined before the music — it is precipitated by the unfolding of the lines. The composition generates the set structure.

This resolves the question of how a pitch determines its desire. A pitch in 13-TET does not choose from all available relationships simultaneously. It chooses from what the call has already made meaningful to answer. The preceding line narrows the field of significant responses. The desire is situationally generated by the call itself — not arbitrary, not predetermined, but intrinsically determined by the interval relationships the first voice has established.

Bach said a voice ends when it feels like it. The responding voice ends when the call has been sufficiently answered — when the interval relationships between voices reach a state of completion that the contrapuntal situation itself determines.

The subsets that emerge are therefore not abstract pitch collections. They are fossils of the voice leading — records of how the lines moved, encoding specific contrapuntal events. The same pitches in a different call and response sequence produce different subsets, even within the same microtonal system. The subset is a property not of the tuning system alone but of the tuning system as activated by a specific compositional sequence.

The interval class is the set; the set is its interval content. The invariance is not a property of isolated intervals that then aggregates into sets — it is a single structural fact that manifests at every level of pitch organization simultaneously. The same numerical interval class content appears across all four microtonal systems at once, each realization sounding entirely different due to its distinct cent values. A trichord defined by interval classes 2 and 3 exists in 13-TET, 17-TET, 20-TET, and 26-TET simultaneously — numerically identical across all four systems, acoustically four distinct objects. NIIC is fully realized: the conversation produces a common social grouping present in four different acoustic worlds at once.

Society of Pitches now has a complete mechanism. The call establishes the field. Each responding voice follows its intrinsic interval desire within that field across its own independent tuning system and tempo. The subsets are the social groupings that the conversation produced. NIIC governs the answer: numerically invariant across all four systems, acoustically plural in every realization.

Polytempic Polymicrotonality and the Polymelodic Dimension

The tetrachord-as-cell gives polymelody a precise compositional primitive. Each of the four independent streams — each bonded to its own tuning system and its own tempo — can carry a distinct interval-space character derived from a different position in the continuous chroai spectrum. The streams are not only tuning-independent and tempo-independent; they are melodically characterized differently at the cellular level.

This integrates with what I have termed polymelody — the third structural dimension of polytempic polymicrotonality, the simultaneous independence of melodic identity across streams. Each stream’s melodic character is defined by its characteristic interval-space, its genera-position within the fourth, usable freely and without obligation to the other streams or to any aggregate.

NIIC and the Society of Pitches operate within and across these streams simultaneously. The numerical interval class identities are invariant; the acoustic realizations are plural; the call and response mechanism is what sets the social field in motion. The four voices do not negotiate a shared pitch world. They inhabit four distinct acoustic worlds while conversing through the one thing they hold in common: the abstract numerical relationships between intervals.

The genera framework supplies the local character. NIIC supplies the cross-system coherence. Call and response supplies the generative mechanism. The Society of Pitches names what results.

The container holds. What is inside defines the character. Nothing beyond the fourth requires completion. And the pitches, released from hierarchy, find their own.