The Five, or Six, Camps of Tuning Ontologies, Part XLI

 

Beyond Two Camps: A Taxonomy of Tuning Foundations

Peter Thoegersen

The taxonomy of microtonal and alternative tuning systems is commonly framed as a conflict between two positions: those who derive pitch from the ratios of the harmonic series, and those who divide the octave by formula, indifferent to ratio loyalty. This two-camp model is useful but insufficient. A closer examination reveals at least four structurally distinct foundations — and a fifth position that dissolves the question entirely.

Camp One: Harmonic Series Obedience

The oldest and most philosophically unified position holds that tuning is not a human construction but a discovery — that the overtone series, generated by any vibrating body with harmonic boundary conditions, constitutes the natural ground of all pitch relationships. Pythagorean tuning stacks 3:2 fifths. Just intonation extends this to higher partials. Harry Partch builds an entire 43-tone system directly from the series, with a sacred fundamental from which all else radiates. The hierarchy is explicit and total.

Camp Two: Equal Division of the Octave

Aristoxenus argued in the fourth century BCE that the ear, not arithmetic, is the final arbiter of tuning. The tetrachord could be divided into equal or near-equal steps without ratio obligation. This intuition reaches its modern form in 12-tone equal temperament and its microtonal extensions: 31-TET, 53-TET, 72-TET. All divide the 1200-cent octave into equal logarithmic units. The octave remains sovereign; the harmonic series is approximated or ignored. The slight mistune of the major third in 12-TET — 400 cents versus the just 386 — is not an error but an architectural choice, severing ratio hierarchy deliberately.

Camp Three: Inharmonic Spectral Tuning

A third position emerges from percussion acoustics. Most struck instruments — bells, gongs, gamelan metallophones — produce inharmonic spectra. Their overtones do not follow the 1:2:3:4:5 series but fall in irrational, instrument-specific relationships. Gamelan tuning stretches octaves not by formula but to match the instrument’s own physical reality. William Sethares formalized this approach: consonance is not absolute, but relative to a given spectrum, and tuning systems should be derived from the instrument’s actual partials rather than from a universal template. This camp obeys the harmonic series idea in principle — tune to your overtones — but finds those overtones are not harmonic, dismantling Camp One’s claim to universality. If even acoustic physics produces inharmonic spectra, then the harmonic series is not the ground of all tuning. It is a special case, applicable only to instruments with specific harmonic boundary conditions.

Camp Four: Non-Octave Tuning

A further disruption attacks an assumption shared by all three preceding camps: that the 2:1 octave is the primary container. The Bohlen-Pierce scale divides the 3:1 tritave into thirteen equal steps, eliminating octave equivalence entirely and aligning more naturally with the odd-partial series of cylindrical instruments. Wendy Carlos’s Alpha, Beta, and Gamma scales derive their step sizes from minimizing mistuning of specific just intervals, with octave equivalence falling only approximately where it happens to land. Even the conventionally tuned piano uses stretched octaves in practice, because inharmonicity in wound strings pushes the 2:1 slightly sharp. The octave, like the fundamental, is revealed as a convenience rather than a law.

Camp Five: Plural Independent Systems

Polytempic polymicrotonality does not choose among the preceding positions — it makes the choice itself structurally irrelevant. In this system, multiple independent tuning systems sound simultaneously, each bonded to its own independent tempo, with no master fundamental organizing the whole. Any of the preceding camps could contribute a layer: one voice in just intonation, another in 31-TET, a third tuned to inharmonic partials. But the moment any such system becomes one independent layer among equals, its claim to universality is dissolved by plurality. There is no container, harmonic or otherwise, from which the whole radiates. The architectural principle is not division, generation, or spectral matching — it is independence itself.

Why Does This Matter, Anyway?

Q: Why does this matter, anyway?

A: Because tuning is not just acoustics — it is ontology.

Every tuning system embeds a claim about what music is. Camp One says music is the discovery of a natural order that precedes human choice. The harmonic series exists whether or not anyone listens. Tuning in just intonation is submission to that order. This is not a neutral technical decision — it carries the full weight of a cosmology. Nature is hierarchical. The fundamental is sovereign. Everything else is derivative.

Camp Two quietly rejects this while pretending not to. Equal temperament is a political act disguised as pragmatism — it democratizes the chromatic scale, makes all keys equivalent, enables modulation, and in doing so dismantles the tonal hierarchy that Camp One enshrines. The wolf interval is not fixed; it is abolished by distributing it equally across all fifths. Nobody rules.

Camp Three says the question itself was provincial. You were arguing about strings and winds. Go hit something.

Camp Four says the octave — which neither side thought to question — is also a convention, not a law.

And Camp Five says: none of these systems has the right to govern the whole.

This matters because the history of Western music theory is largely the history of one system at a time, claiming to be the natural, universal, or logically necessary foundation. The harmonic series as cosmic truth. Equal temperament as rational progress. Each camp, when it wins, tends to legislate. Music that doesn’t conform becomes judged as primitive, irrelevant, error.

A taxonomy that clearly names five structurally distinct positions — and shows that the fifth dissolves the terms of the debate entirely — is an argument for permanent plurality. Not relativism, not anything-goes, but a principled architectural claim: independence is itself a structural category, not an absence of structure.

That is why it matters.

What About Mathematicians Who Design Tunings?

Q: What about mathematicians who design tunings?

A: This is a genuine gap in the taxonomy. It might require a sixth camp — or it fractures across several existing ones in a revealing way.

The Mathematicians

Erlich, Milne, Carey, Clampitt — working in regular temperament theory. They map tuning systems by how well they approximate just intervals, using wedge products, vals, monzos, and lattice geometry. The math is beautiful and completely abstract. A tuning is a point in a high-dimensional space. The question “which tuning is best” becomes an optimization problem.

Clarence Barlow quantified harmonicity with formulas, assigning numerical values to intervals based on prime factorization. Music becomes computable.

Xenakis applied stochastic processes, set theory, and group theory directly to composition — not deriving tuning from the harmonic series but from mathematical structures that have no acoustic origin at all.

Which Camp Are They?

They are not Camp One — they don’t start from the harmonic series as sacred. They measure approximation to it, which is different from obeying it.

They are not quite Camp Two — they’re not simply dividing the octave. They’re asking what divisions do, algebraically.

They might be Camp Zero — prior to the acoustic question entirely. Mathematics as the foundation, with acoustics as one possible application domain among others.

Why This Matters for Polytempic Polymicrotonality

Polytempic polymicrotonality is also mathematically grounded — the independence of simultaneous tuning and tempo layers is an architectural, structural claim, not an acoustic one. Arrived at through drumming and compositional instinct rather than abstract algebra, the claim itself is nonetheless formal.

The difference is that the regular temperament mathematicians still tend toward finding the best single system — optimizing toward an ideal. Polytempic polymicrotonality rejects the optimization frame entirely. There is no best. There is only plurality.

Camp Zero for the mathematicians: foundation in abstract structure. Camp Five for polytempic polymicrotonality: foundation in irreducible independence. Related postures, opposite conclusions.