From Ratio to Scaling Law: The Mutual Properties of Music and Mathematics, and the Fenophone as a Measured Instrument

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Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-15
DOI
https://doi.org/10.5281/zenodo.22760052
Primary Topic
Advanced Mathematical Theories and Applications
Type
preprint
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preprint

From Ratio to Scaling Law: The Mutual Properties of Music and Mathematics, and the Fenophone as a Measured Instrument

Evan Tabak Atlas
Zenodo (CERN European Organization for Nuclear Research)
Advanced Mathematical Theories and Applications
preprint

From Ratio to Scaling Law: The Mutual Properties of Music and Mathematics, and the Fenophone as a Measured Instrument

Evan Tabak Atlas
preprint en

Abstract

For twenty-five centuries the relationship between music and mathematics has run between two poles. At one pole music is mathematics — generated from a model fixed in advance, whether Pythagorean ratio, Rameau's corps sonore, the twelve-tone row, or a Xenakian probability distribution. At the other, mathematics is what one measures in the sounding phenomenon — Aristoxenus's ear, Vincenzo Galilei's weights, Helmholtz's resonators, Voss and Clarke's 1/f spectra. This essay reads that history as a single convergent story. It argues, first, that the two poles were never reconciled for long: the generative tradition repeatedly attached a mathematical claim to music that it never verified on the output (the blacksmith legend, Kepler's fudged planetary intervals, Rameau's non-existent subharmonics, Xenakis conceding that a formal structure "cannot be confirmed or invalidated," serial music's post-hoc analysis), while the descriptive tradition measured real music rigorously but built no instruments. Second, that the mutual property binding the two traditions has itself narrowed and deepened across the centuries — from a static ratio between two magnitudes, through single-scale periodicity and the harmonic spectrum, through finite-group symmetry, through the probability distribution, to scale-invariant, multifractal burst-and-lull: a power-law "symmetry" holding across all time-scales, and one genuinely shared by turbulence, financial volatility, physiology, and music. We then position the Fenophone — a generative instrument whose intensity core is, by construction, the Markov-switching multifractal of Calvet and Fisher — as the point where the two poles fuse under engineering discipline. Its controls are the model's own parameters (the generative pole taken to its limit); the multifractal and 1/f statistics of its emitted note stream are measured on every build with detrended fluctuation analysis, surrogate-tested, and enforced as continuous-integration invariants (the descriptive pole hardened into a release gate). Design target and verification estimator are the same object — Ptolemy's "reason proposes the ratios, perception confirms them on a calibrated monochord," now automated, quantitative, and applied to a scaling law. The instrument extends the tradition by importing a mature, estimable, invertible generative model from a sister domain that shares the mutual property, rather than borrowing a metaphor or a noise source — the disciplined modern successor to musica mundana's "same proportion at every scale," recast from a metaphysical assertion into a testable invariant. We close by locating what the Fenophone does not settle: whether the structure is heard remains the open Aristoxenian question, and the ear keeps the final vote. Note on the section 5 coda The labyrinth coda reads every number from a sealed record of the wider Feno program (cited as Atlas 2026e), including the exact enumerable null over all valid designs at depth 12, the L1–L6 pre-registered register and both honesty flags. Two of its claims were corrected on 2026-08-25 and the corrections stand in the text: the depth-8/depth-12 self-dual census, and L5 restated as the resolution-invariant part only (the normalized compression comparison is a sampling-density artifact and is not quoted). Part of the Feno program. The source repository is private; the reproduction materials for this record are contained in the record itself. Version 2 (2026-09-01): reference-list publication-status notes corrected (the program's reserved DOIs were published 2026-08-31); content otherwise unchanged. Version 3 (2026-09-15). Carries the audited readings of two external results this paper cites: Levitin, Chordia & Menon (2012) on 1/f rhythm spectra (Feno-program audit card 07, Fenocosm record v59, 10.5281/zenodo.22759705) and Roeske, Kelty-Stephen & Wallot (2018) on multifractal structure in songbird rhythm (audit card 08, Fenocosm record v60, 10.5281/zenodo.22759707). A dated correction note near the head of the paper states what each reconstruction reproduces and what it does not; no result of this paper's own measurements changes. The paper now also carries its concept DOI in its header.

Zenodo (CERN European Organization for Nuclear Research)
Peace, Justice and strong institutions
Advanced Mathematical Theories and Applications
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