Absolute, Relative, and Plural Independent, Part XXXVb

 Absolute, Relative, and Plural Independent

A new framework for understanding tuning philosophy in music

The conventional division of Western tuning thought into Pythagorean and Aristoxenian camps

— ratio mathematics versus perceptual judgment — is accurate as far as it goes. But it does

not go far enough. A more fundamental distinction underlies both positions, one that every

musician already understands from their own ear training: the difference between absolute and

relative pitch.

Overtone-series tunings are absolute systems. Aristoxenian tunings are relative systems.

Polytempic polymicrotonality is neither, and it is not simply the plural extension of either. It is

a plural independent system: multiple simultaneous systems, each internally consistent, each

derived by whatever means serves the compositional purpose, none subordinate to the others

and none sharing a common ground. Understanding this distinction clarifies not only what

separates these philosophical positions from each other but also why the third position constitutes a

genuinely new ontological category.

I. The Absolute System

A person with absolute pitch hears C as C. Not C in relation to something else — C as a fixed

point in acoustic space, identified independently of context. The note has an address that does

not change regardless of what surrounds it.

Overtone-series tuning systems are absolute in precisely this sense. The fundamental is the

fixed reference point. Every pitch in the system has an address — a ratio — that locates it

unambiguously relative to that fundamental. The Pythagorean perfect fifth is 3:2, always. The

just major third is 5:4, always. These are fixed positions in a hierarchical space that the

fundamental generates and governs. The hierarchy is total: every pitch traces back to the

fundamental through a chain of ratios. The system is closed, consistent, and absolute.

Pythagorean tuning, just intonation in all its extensions, Harry Partch's 43-tone system, Ben

Johnston's prime-limit lattices, Henry Cowell's overtone-series rhythms — all are absolute

systems. The fundamental may be implicit rather than sounded, the ratios may reach into the

11th- or 13th-prime limit, the system may be extraordinarily complex. But it remains one system,

one hierarchy, one absolute reference frame.

"The ear must submit to the mathematics, not because the mathematics is

imposed from outside, but because mathematics is nature's own hierarchy,

audible in the overtone series of any vibrating body."

II. The Relative System

A person with relative pitch hears intervals — the distance between notes — rather than fixed

positions in acoustic space. A minor third is a minor third because of how it sounds in relation to

what surrounds it, not because of its ratio to a fixed fundamental. Context is everything.

Aristoxenian tuning systems are relative in precisely this sense. Aristoxenus rejected the idea

that intervals have fixed mathematical addresses. We do not perceive sound as ratios — we

perceive the relationships between sounds as they unfold in musical time. The ear is the

authority; intervals are defined by function and perceptual experience, not by position in a ratio

hierarchy.

This produces a different kind of tuning space: continuous, infinitely divisible, defined by

perceptual function rather than mathematical position. Equal temperament is the logical

terminus of Aristoxenian thinking — a system in which every semitone is equal not because

nature demands it but because equal division serves the ear's need for functional consistency.

The system is open rather than closed: no master fundamental, no hierarchy of ratios, only the

ear's navigation of interval relationships in context.

Xenakis, Messiaen, Birtwistle, Ferneyhough, Ligeti, Carrillo, Haba — all operate within relative

systems. The ear navigates a single acoustic world defined by perceptual relationships. The

world may be extraordinarily complex. But it remains one world, one acoustic reality, one

relative reference frame.

"The ear does not perceive ratios. It perceives relationships. The tuning space

is not a lattice of fixed positions but a continuum of perceptual distances,

navigated in real time by a trained musical intelligence."

III. The Plural Independent System

Polytempic polymicrotonality is neither absolute nor relative — and it is not simply the plural

extension of either. The defining philosophical invariant is not the type of each individual

system but the independence between systems. Multiple tuning and tempo systems coexist

simultaneously, each internally consistent, none subordinate to the others, none sharing a

common ground.

The relationship between tuning and tempo in a given work is compositionally chosen rather

than systematically fixed. Sometimes tempo generates tuning: a chosen tempo produces equal

divisions of the octave that define the pitch system for that stratum. Sometimes tuning

influences tempo: the intervallic logic of a pitch system suggests corresponding rhythmic

values. Sometimes, tuning and tempo systems are derived entirely independently and simply

coexist. What remains invariant across all these possibilities is the separateness — the

systems never collapse into each other, never share a common fundamental, never reduce to a

single governing hierarchy.

This means the absolute/relative question applies separately to each system in each piece,

depending on how it was derived. A tempo-generated equal division of the octave is locally

relative — an Aristoxenian move, dividing a continuous space by a chosen unit rather than

deriving pitches from a natural harmonic series. A ratio-derived tuning system applied

independently is locally absolute. The piece as a whole is plural independent because multiple

separate systems coexist, but the individual systems may be absolute, relative, or mixed

depending on their compositional origin.

The philosophical claim is therefore not about what kind of systems are used, but about how

many simultaneous ones exist and what relationship — or non-relationship — obtains between

them. Competition implies a shared arena. Independence means no shared arena exists. The

systems in polytempic polymicrotonality do not compete, do not refer to each other, do not

generate each other unless the composer chooses that relationship for a specific purpose.

They simply coexist as separate acoustic realities.

The genesis of this insight was physical rather than theoretical: four-limb drumming

independence at age eighteen. The discovery that four limbs could maintain genuinely

independent simultaneous rhythmic streams — not streams competing for primacy, not

streams defined by their relationship to a common pulse, but streams each with their own

internal consistency — was the bodily precondition for the compositional system. The

polyrhythmic structure 3:4:5:7 arrived at through this practice is itself plural independent: four

simultaneous systems, each internally consistent, none the master of the others.

"Neither absolute nor relative alone captures the position. The question is not

how complex one musical world can be, nor how many reference frames a

single world can contain — but how many simultaneous independent worlds

can coexist in one composition, each complete in itself, none requiring the

others to define it."

IV. Implications and Precursors

The plural independent framework clarifies with precision why every historical precursor falls

short of the threshold — and where exactly the shortfall occurs.

Charles Ives's Universe Symphony is the most instructive case. Ives partially used competing

tunings and tempos — but they were still not independent systems. The word competing is

exact: the tunings and tempos in the Universe Symphony are in tension with each other, aware

of each other, vying for primacy within a shared acoustic arena. Competition presupposes a

shared arena. Independence means no shared arena exists. Ives imagined multiplicity; he did

not achieve the structural separation that makes each system genuinely self-contained.

Conlon Nancarrow crossed the polytempic threshold — genuinely independent simultaneous

tempi, mechanically realized. But his tuning is fixed throughout: equal temperament, one

system, one acoustic world. The tempo systems are independent of each other; there is only

one pitch system for all of them to share. The plural independent dimension is absent.

Ivan Wyschnegradsky came nearest: polymicrotonal systems with rhythmic independence,

closer to the threshold than any other composer in the literature. The full concurrent bonding of

independently derived tuning and tempo systems — plural, independent, simultaneous — does

not occur.

Alfred Schnittke's polystylism gave philosophical permission: if incompatible worlds can coexist

within a single work, why not structurally independent ones? But Schnittke's worlds are

sequential and stylistic — they take turns within a single acoustic arena. The plural independent

makes them concurrent and structural.

The Greek genera modulation documented by Ptolemy — metabolic modulation between

diatonic, chromatic, and enharmonic tunings — is the deepest historical root. The Greek

composers modulated between systems sequentially. The plural independent makes those

transitions simultaneous and removes the requirement that they share the tetrachordal frame

that made Greek modulation possible in the first place.

In each case, the precursor achieves one dimension of independence without the other, or

achieves competition without true independence, or touches the polymicrotonal territory without

the polytempic, or the polytempic without the polymicrotonal. Polytempic polymicrotonality is

the first compositional system in which multiple tuning and tempo systems — however derived,

whether absolute or relative in their individual characters — coexist as genuinely independent

simultaneous realities with no shared ground between them.

Peter Thoegersen · Polytempic Polymicrotonality · Jenny Stanford Publishing 2022, 2024