Entropy, Graining, and Polytempic Polymicrotonality, Part LIV

 

Entropy, Graining, and the Music Nobody Can Hear All At Once

A mathematician friend recently sent me a chapter he is writing on mathematical entropy and the arts. His intent was generous: here, he was saying, is a formal framework that might describe what I do. He was right that entropy theory is the right tool, rather than the simplistic calculus integration/differentiation metaphor I had previously considered. What I want to explore here is what happens when that tool is applied rigorously — where it fits my music, polytempic polymicrotonality, where my music strains it, and what new theoretical apparatus my music turns out to require.

The Framework

The chapter's central argument is elegant: entropy is never simply a property of a signal. It is always a property of a signal relative to a receiver — relative to what the author calls a graining strategy, the set of distinctions a listener is equipped and disposed to make. Two listeners hear the same piece of music; one has spent years immersed in the relevant tradition and can distinguish harmonic substitutions, rhythmic displacement, and the moment where a soloist falls back on a stock phrase. The other hears an undifferentiated texture. The piece has not changed. The entropy each listener experiences has.

The chapter extends this through a second tool: mutual information. If two simultaneous musical layers are sounding, mutual information measures how much knowing what one layer is doing reduces your uncertainty about what the other is doing. In Bach counterpoint, mutual information between voices is moderate — enough correlation to cohere harmonically, enough independence that each voice carries its own information. In Ives's Fourth Symphony, where a marching band, a parlour song, and a hymn tune proceed simultaneously in their own tempos and keys, mutual information approaches zero: the layers are nearly statistically independent, and shifting attention to a new layer delivers a completely fresh world.

Ives is the chapter's ceiling case for simultaneous independence. And this is precisely where my system begins.

The Hidden Assumption

The entropy framework, as the chapter presents it, inherits a quiet assumption from Claude Shannon: entropy is always relative to a probability model defined over a shared alphabet. Even when the listener is inexperienced, the framework presumes that a richer listener could, in principle, refine that alphabet until the structure emerges. Difficulty, in this model, is epistemic — a problem of training and attention, not a problem of kind.

My system shifts that from epistemology to ontology. The issue is not that the listener hasn't yet learned the right distinctions. It is that there is no single set of distinctions that preserves the identity of the object. Each layer is a different alphabet, and those alphabets are not refinements of one another — they are mutually incompatible coordinate systems.

In polytempic polymicrotonality, I work simultaneously with four independent tuning systems, each bonded to its own independent meter and tempo as a single irreducible identity. In String Quartet No. 17, the four bonded voices are precisely specified: 15-TET bonded to 5/4 at 60 BPM; 14-TET bonded to 7/4 at 72 BPM; 22-TET bonded to 11/4 at 75.4 BPM; and 26-TET bonded to 13/4 at 66.85 BPM. These are not independent parameters that happen to coexist. The tuning system and the tempo-meter pair are fused: the 22-TET layer does not merely accompany the 11/4 meter at 75.4 BPM — it is that meter at that tempo, constituting a single acoustic identity that cannot be disaggregated without destroying both.

The tempi are not arbitrary. The ratios among 60, 72, 75.4, and 66.85 BPM produce a polytempic field in which the four voices move at genuinely independent rates. The meters — 5/4, 7/4, 11/4, and 13/4 — are drawn from prime numbers, ensuring that their cycles resist periodic realignment. The independence is not approximate. It is engineered.

Four Consequences

Applying the entropy framework rigorously to this system produces four consequences that clarify exactly where the framework gives way.

First: mutual information collapses into triviality across layers. In Ives, near-zero mutual information is contingent — he chose not to make the layers interact. In my system, it approaches zero as something closer to a theorem. Given incommensurable pitch partitions and prime-related temporal cycles, the joint distribution factorizes almost completely. But mutual information is only defined once you have already chosen a joint state space. My system effectively says there is no natural joint space that does not distort at least three of the four layers. The near-zero figure is not just low correlation; it is the byproduct of a forced projection.

Second: Kolmogorov complexity stops being scalar. The usual move — short description, complex surface — still assumes a single descriptive language. My system has something closer to a vector of descriptions, each minimal in its own system, with no lossless translation between them. Standard Kolmogorov complexity measures compression relative to one universal machine. My system implicitly requires multiple non-equivalent machines running in parallel. The complexity cannot be reduced to a single number without destroying what makes it what it is.

Third: the implied receiver becomes plural rather than idealized. The entropy framework imagines an ideal listener at the limit of training — someone who has refined their graining strategy until the music's structure is fully accessible. My music replaces that with something structurally different: a bundle of partial listeners, each adequate to one layer, none adequate to the whole. Not a super-listener, but a distributed one. This is a qualitatively different model of what a musical audience is.

Fourth: attention becomes the real medium. The object is not the instantaneous acoustic field; it is the trajectory of attention through incompatible fields. In information-theoretic terms, this shifts from the entropy of a signal to something like the entropy of a path through multiple signal spaces. The listener's movement between, say, the 15-TET voice in 5/4 at 60 BPM and the 26-TET voice in 13/4 at 66.85 BPM is not incidental to the experience. It is structurally required by it.

Where Predecessors End

The genealogy of simultaneous independence in music runs from Ives through Nancarrow through Ligeti. Each pushes toward independence in different ways: Ives by layering culturally distinct materials in separate tempos; Nancarrow by driving temporal independence toward perceptual overload through mechanical means; Ligeti through micropolyphony, where pairwise mutual information between individual voices is low while ensemble-level mutual information is high.

What each of these systems preserves, in different ways, is at least one shared scaffold. Ives shares a common pitch space. Nancarrow works within a single equal temperament. Ligeti's voices coexist in a shared pitch field and a shared temporal grid. A listener can, if not hear everything at once, at least collapse everything into a single compromised grid without destroying the fundamental identities involved.

My system removes the last shared scaffold — common pitch space — and then binds that removal to tempo so that it cannot be perceptually reassembled. Without the bonding, a listener could cheat: hear all four tuning systems as a single mistuned blur, lose the precise microtonal identities, but retain the temporal relationships. With the bonding, any such collapse destroys the identity of each voice. The tuning and the tempo are a single thing. You cannot approximate one without losing both.

What the Framework Needs

The entropy framework, applied to my music, is not broken — it is incomplete. What it reveals, by straining, is that its current unit of analysis is too small. The unit cannot be 'the signal relative to a receiver.' It must become 'a set of signals relative to a set of incompatible receivers, plus the rules for moving between them.'

More precisely, the framework needs three additions. First, entropy must be defined over families of grainings rather than a single graining. Second, it must account for the costs of translation between grainings — in my case, these costs are effectively infinite if the identity of the layers must be preserved. Third, attention dynamics must be treated as part of the channel itself, not as something external to the signal. The listener's trajectory through incommensurable spaces is not incidental data about how the music is experienced. It is the structure of the musical object.

That is not a small patch. It is a different theoretical object.

What This Means for Listening

The obvious question is whether music that resists a unified graining strategy is listenable. I want to argue that this frames the question incorrectly.

A listener can enter any one of the four simultaneous worlds and follow it — can track the 22-TET layer in 11/4 at 75.4 BPM as a coherent voice, can shift attention to the 14-TET layer in 7/4 at 72 BPM, and find an entirely different coherent world running alongside the first. What the listener cannot do is perceive all four layers as a unified field from a single graining strategy. But this impossibility is not a failure of the music. It is its structural meaning.

The Sistine Chapel ceiling cannot be taken in as a single retinal act. You move through it. You shift attention. The totality exists in the architecture, not in any single moment of perception. My music asks for something analogous — not unified comprehension, but a mobile attention that can enter and exit incommensurable worlds and recognize that all of them are simultaneously real.

This music does not make something hard to hear. It formalizes the impossibility of hearing everything at once, and turns that impossibility into the content itself.

When the map runs out, that is where the territory begins.