Understanding Euclidean Rhythms


The latest episode of Sound + Voltage takes a look at Euclidean Rhythms – a way of using math to describe beats that are distributed as evenly as possible, and how this is a feature of many common musical rhythms from around the world.

The concept of Euclidean Rhythms was put forth in the late Godfried Toussaint’s paper, The Euclidean Algorithm Generates Traditional Musical Rhythms (pdf):

“The Euclidean algorithm (which comes down to us from Euclid’s Elements) computes the greatest common divisor of two given integers. It is shown here that the structure of the Euclidean algorithm may be used to generate, very efficiently, a large family of rhythms used as timelines (ostinatos), in sub-Saharan African music in particular, and world music in general. These rhythms, here dubbed Euclidean rhythms, have the property that their onset patterns are distributed as evenly as possible.”

The evenness of note distribution is important to drummers, because drummers traditionally need to be heard over all other instruments and they need to establish the tempo. And, whatever pattern they play, a drummer needs to be able to play the pattern for a long period of time, at the fastest tempo that’s needed.

Euclidean patterns meet these criteria, because they represent the steadiest distribution of notes across a rhythmic cycle.

When musicians play percussion instruments that require more physical energy to play, like a bass drum, they tend to play patterns with a lower number of evenly distributed notes per rhythmic cycle – like the ubiquitous 4 notes across 16 pulses. And when playing percussion instruments that require less physical energy to play, like a bell, they tend to play patterns with higher numbers of evenly distributed notes per rhythmic cycle – like African bell patterns. Put these physically sensible patterns together and you get polyrhythms that are easy to hear, make the rhythm clear, and are playable at the fastest tempo needed.

Topics covered:

00:00 – Intro
01:11 – History & Context
01:49 – Divisors & Relative primeness
04:50 – Euclidean basics
09:02 – Demos
12:18 – Odd time signatures
14:50 – Rhythm zoo


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