The Distance Geometry of Music

TL;DR

This paper explores the mathematical structure of Euclidean rhythms in music, showing they maximize evenness and are deeply connected to distance geometry, with implications for rhythms and scales.

Contribution

It establishes that Euclidean rhythms uniquely maximize evenness and characterizes deep rhythms as a subclass of generated rhythms with shelling properties.

Findings

Euclidean rhythms maximize pairwise distances between onsets.

Deep rhythms have unique multiplicities of distances, forming an interval.

Results apply to both musical rhythms and scales, linking to distance geometry.

Abstract

We demonstrate relationships between the classic Euclidean algorithm and many other fields of study, particularly in the context of music and distance geometry. Specifically, we show how the structure of the Euclidean algorithm defines a family of rhythms which encompass over forty timelines (\emph{ostinatos}) from traditional world music. We prove that these \emph{Euclidean rhythms} have the mathematical property that their onset patterns are distributed as evenly as possible: they maximize the sum of the Euclidean distances between all pairs of onsets, viewing onsets as points on a circle. Indeed, Euclidean rhythms are the unique rhythms that maximize this notion of \emph{evenness}. We also show that essentially all Euclidean rhythms are \emph{deep}: each distinct distance between onsets occurs with a unique multiplicity, and these multiplicies form an interval $1,2,...,k-1$. Finally, we characterize all deep rhythms, showing that they form a subclass of generated rhythms, which in turn proves a useful property called shelling. All of our results for musical rhythms apply equally well to musical scales. In addition, many of the problems we explore are interesting in their own right as distance geometry problems on the circle; some of the same problems were explored by Erd\H{o}s in the plane.