Absolute, Relative, and Plural Absolute, Part XXXVa

 Absolute, Relative, and Plural Absolute

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 — it is something the history of music theory has not

previously named: a plural absolute system. Understanding this distinction clarifies not only

what separates these philosophical positions from each other, but why the third position

constitutes a genuinely new ontological category rather than a refinement of either

predecessor.

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 not approximations or perceptual conveniences —

They are fixed positions in a hierarchical space that the fundamental generates and governs.

This hierarchy is total. Every pitch traces back to the fundamental through a chain of ratios.

There is no pitch in the system that does not have a precise mathematical address, no interval

that is defined by anything other than its position in the hierarchy. The system is closed,

consistent, and absolute in the same way that absolute pitch is absolute: there is one fixed

reference frame, and everything is located within it.

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 the 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; the

same physical frequency can be heard as different intervals depending on what precedes it.

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, he

argued — we perceive the relationships between sounds as they unfold in musical time. The

ear is the authority; the tetrachord is the basic structural unit, not because of its ratio content but

because of its functional role in musical experience.

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 transposability and

functional consistency. The system is open rather than closed: there is no master fundamental,

no hierarchy of ratios, only the ear's navigation of interval relationships in context.

Aristoxenus, Xenakis, Messiaen, Birtwistle, Ferneyhough, Ligeti — all operate within relative

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

than ratio mathematics. The world may be extraordinarily complex — microtonally dense,

rhythmically stratified, parametrically independent. But it remains one world, one acoustic

reality, one relative reference frame in which everything is defined by its relationship to

everything else.

"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 Absolute System

Polytempic polymicrotonality is neither absolute nor relative in the senses defined above. It

does not operate within a single absolute reference frame — there is no master fundamental to

which all pitches trace back. But it does not operate within a single relative reference frame

either — the systems are not defined by their perceptual relationships to each other.

What it proposes instead is multiple simultaneous absolute reference frames, each internally

consistent and hierarchically complete, none subordinate to the others. Each tuning system has

its own fundamental, its own internal logic, its own absolute pitch space. Each tempo system

has its own pulse, its own internal consistency, its own time. They coexist without a common

ground — without a master fundamental that would make one of them primary and the others

derivative.

The closest analogy in physics is Einstein's special relativity: multiple simultaneous reference

frames, each internally consistent, none privileged. There is no absolute space, no absolute

time — only the consistent internal logic of each frame and the relationships between frames

when they interact. The Pythagorean composer assumes absolute space — one master

fundamental that all other pitches orbit. The Aristoxenian composer assumes relative space —

no fixed positions, only perceived distances. The polytempic polymicrotonal composer

assumes something closer to relativistic space: multiple frames, each locally absolute, none

globally privileged.

This is why the system cannot be heard as complex Pythagorean harmony — there is no

master fundamental from which to derive the relationships. And it cannot be heard as complex

Aristoxenian texture — the systems are not defined by their perceptual relationships to each

other but by their independent internal consistencies. The listener inhabits multiple

simultaneous acoustic realities, each complete in itself.

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 related to each other by ratio, not

streams perceived as relative distances from a common pulse, but streams each with their own

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

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

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

"Every musician understands the absolute/relative distinction from their own

ear training. What polytempic polymicrotonality proposes is that both positions

have assumed there can only be one reference frame at a time — absolute or

relative, but singular. The plural absolute is the discovery that this assumption

was never necessary."

IV. Implications

This reframing has several consequences for how the three positions relate to each other

historically and philosophically.

First, it explains why the Greek genera modulation documented by Ptolemy — metabolic

modulation between diatonic, chromatic, and enharmonic tunings — is the deepest historical

precursor to polytempic polymicrotonality. The Greek composers were modulating between

absolute systems sequentially, one at a time. The plural absolute makes those transitions

simultaneous.

Second, it explains why Schnittke's polystylism — the coexistence of historically incompatible

worlds within a single work — gave Thoegersen philosophical permission without being the

same thing. Schnittke's worlds are stylistic and sequential; the plural absolute makes them

structural and concurrent.

Third, it explains the precise limitation of every precursor. Ives's Universe Symphony

approaches the plural absolute from the polymicrotonal direction but keeps the systems

sequential. Nancarrow approaches it from the polytempic direction but keeps the tuning

singular. Wyschnegradsky comes nearest — multiple microtonal systems with rhythmic

independence — but does not achieve the full concurrent bonding of tuning to tempo that

constitutes the plural absolute in its complete form.

Fourth, it provides the clearest non-technical explanation of what polytempic polymicrotonality

actually is: not more complex harmony, not more complex texture, not more parameters under

independent control — but a different number of simultaneous acoustic realities. The question it

answers is not 'how complex can one musical world be?' but 'how many simultaneous musical

worlds can coexist in one composition?'

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