Elementary spectrum for the dissonance curve: from biophysics to number theory of musical harmony

Musical harmony, as the ancient problem of relating tunings and scales to the commensurability of sound waves, has been approached by Helmholtz in terms of a dissonance curve in the frequency domain. This model for musical intervals is here recast in an elementary number-theoretic and thermodynamic formalism, connecting the biophysical insight with Riemann's zeta function on the critical line and Minkowski's question mark measure. The former models rational relationships from a harmonic timbre in the Mellin-Fourier domain conjugate to log-frequencies, capturing the acoustic aspect. The latter minimally models the neurocognitive effort to assess commensurability in frequency pairs, emerging as the maximum entropy distribution on rational approximations generated by Euclid's algorithm. Motivated by just intonation, this ensemble extends the primon gas, a statistical mechanics model based on prime energy contributions. This framework gives Helmholtz's phenomenological dissonance curve a theoretical basis. The spectrum of the resulting fractal curve predicts the quasi-periods of widely used scales, from the pentatonic to microtonal divisions of the octave, providing a biophysical and mathemusical common ground for harmony across musical genres and cultures.