Music of moduli spaces

TL;DR

This paper introduces a novel musical instrument inspired by hyperbolic geometry and moduli spaces, translating geometric structures into sound to audibly explore mathematical concepts like mapping classes and Markoff triples.

Contribution

It presents a new way to sonify moduli spaces using a decorated Farey tesselation, linking hyperbolic geometry with musical expression and mathematical visualization.

Findings

Sound frequencies encode geometric data of the tesselation.

Audible paths represent elements of Riemann moduli spaces.

Chords correspond to generalized Markoff triples.

Abstract

A musical instrument, the plastic hormonica, is defined here as a birthday present for Dennis Sullivan, who pioneered and helped popularize the hyperbolic geometry underlying its construction. This plastic hormonica is based upon the Farey tesselation of the Poincare disk decorated by its standard osculating horocycles centered at the rationals. In effect, one taps or holds points of another tesselation tau with the same decorating horocycles to produce sounds depending on the fact that the lambda length of e in tau with this decoration is always an integer. Explicitly, tapping a decorated edge e in tau with lambda length lambda produces a tone of frequency 440 xi^{lambda-12N}, where xi^{12}=2 and N is some positive integer shift of octave. Another type of tap on edges of tau is employed to apply flips, which may be equivariant for a Fuchsian group preserving tau. Sounding the frequency for the edge after an equivariant flip, one can thereby audibly experience paths in Riemann moduli spaces and listen to mapping classes. The resulting chords, which arise from an ideal triangle complementary to tau by sounding the frequency of its frontier edges, correspond to a generalization of the classical Markoff triples, which are precisely the chords that arise from the once-punctured torus. In the other direction, one can query the genera of specified musical pieces.