On The Euclidean Algorithm: Rhythm Without Recursion

TL;DR

This paper introduces a non-recursive, matrix-based method for calculating Euclidean rhythms, making it easier to compute by hand and offering a combinatorial perspective, inspired by Bresenham's line algorithm.

Contribution

It presents a novel, non-recursive matrix construction for Euclidean rhythms, simplifying hand calculations and providing combinatorial insights.

Findings

The method is easy to perform by hand.

It offers a combinatorial interpretation of Euclidean rhythms.

It does not outperform traditional algorithms but enhances understanding.

Abstract

A modified form of Euclid's algorithm has gained popularity among musical composers following Toussaint's 2005 survey of so-called Euclidean rhythms in world music. We offer a method to easily calculate Euclid's algorithm by hand as a modification of Bresenham's line-drawing algorithm. Notably, this modified algorithm is a non-recursive matrix construction, using only modular arithmetic and combinatorics. This construction does not outperform the traditional divide-with-remainder method; it is presented for combinatorial interest and ease of hand computation.