Music of moduli spaces

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 tessellation of the Poincaré disk decorated by its standard osculating horocycles centered at the rationals. In effect, one taps or holds points of another tessellation $\tau$ 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 fixed 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 frequencies 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.