GAMUT — Geometric-Algebraic Music Theory

The shape of musical possibility.

A three-part essay series exploring why music wants a geometry — from cyclic pitch content and permutation fibers to the symplectic structure that binds them together.

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3-part public essay arc

From combinatorics to content, from content to order, from order to symplectic motion.

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Camera-ready volume, technical survey, revised AMS proof paper, figures, code, and CSVs.

Open research

Figures, datasets, code bundles, and formal papers — all freely available alongside the public essays.

Part 1

Part I. Why Music Wants a Geometry

Why the raw twelve-tone universe exceeds ordinary theory categories and why cyclic autocorrelation is the first serious fingerprint of content.

Part 2

Part II. Attaching Order to Content

How rooted cyclic order, gap words, and local permutation geometry turn a content map into a fibered object.

Part 3

Part III. Why the Space Wants a Symplectic Form

Why a Fourier lift of the layered construction naturally takes the argument from static geometry into a phase-space picture of musical motion.

The argument in miniature

From twelve tones to symplectic geometry.

Begin with the chromatic universe. Every pitch-class set $S \subset \mathbb{Z}_{12}$ is a pattern on a twelve-point circle. Its cyclic autocorrelation measures self-overlap under rotation:

A_S(\tau) \;=\; \sum_{n \in \mathbb{Z}_{12}} \chi_S(n)\,\chi_S(n + \tau)

By the finite Wiener–Khinchin correspondence, this is equivalent to the squared magnitudes of the discrete Fourier transform — the spectral fingerprint of content. That duality gives the first coordinate chart. But two genuinely different sets can cast the same intervallic shadow (the Z-related or homometric phenomenon), so content alone cannot be the whole map.

Order enters as a fiber above each content class. A set with cardinality $k$ has $k!$ orderings in total; choosing a root and quotienting by cyclic reindexing gives a rooted cyclic fiber of size $(k-1)!$ distinct traversals, connected by adjacent swaps. The total object is a layered bundle: content below, order above.

When both layers are lifted into Fourier coordinates — complex content modes $X_j$ and order modes $Y_l$ — the ambient space acquires a natural symplectic form:

\omega \;=\; \frac{i}{2}\sum_j dX_j \wedge d\bar{X}_j \;+\; \lambda\,\frac{i}{2}\sum_l dY_l \wedge d\bar{Y}_l

In action-angle variables this becomes $\omega = \sum_j dI_j \wedge d\theta_j + \lambda \sum_l dJ_l \wedge d\varphi_l$ , pairing intensity with phase in each spectral mode. Transposition acts as coordinated phase rotation — a Hamiltonian symmetry. Inversion acts as reflection — anti-symplectic.

The result is a cardinality-graded layered space: 223 twelve-tone content classes, rooted cyclic order fibers, gap-word periodicities, and a standard exact symplectic thickening in Fourier coordinates. The visible diagrams are controlled projections of this larger object, not substitutes for it.

Start from Part I Read the proof paper

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