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

14 thoughts on “Understanding Euclidean Rhythms”

1. John

what nonsense.

1. Space Captain

Are the numbers too hard for you?

2. stub

I think one of the things that makes the concept of “euclidean rhythms” interesting is the idea that you have superimposed “event rates” that seem to interact. I.e., you have the regular pulse (the beat) and you have some other fairly regular (but not quite) event rate that overlaps and interplays with the beats.

Calling them Euclidean seems to be something of a catch-all for rhythms that are for the most part regular but not quite.

1. goat monster

I wonder why the video triggered John. I thought it was well presented.

1. randmcnasty

I don’t think John bothered to watch the video, seems like he is just being dismissive.

3. alacazam

Euclidean Sequencer iOS app baby! Math can be fun.
Great video. Cool stuff.

2. Kristi Brud

Great content! More stuff like this please.

3. Werner+van+Nuffel

Interesting to look in this way to euclidean set ups…

4. Thanks so much for posting about my video! This one was a bit more of a digression to a more general topic rather than just being about a specific module, and it’s great to see that so many people are interested in the topic. Now I need to figure out what to do next! 🙂

As to John’s comment, if it was about my video specifically then I’m really sorry that it didn’t work for you, and I’m always open to constructive criticism. But if it’s about the basic idea of Euclidean Rhythms, then I can’t agree that it’s nonsense. When you go look at the original paper where it is described, you can see that the technique really does recreate patterns found in music from all over the world.

Thanks again for the post!

1. Oskar

Great video, very well explained and laid out with graphics. Thanks for taking the time 🙂

5. Frank Ullrich

More comprehensive and detailed information on the subject can be found in the German-language book by Peter Giger, Kunst des Rhythmus: Professionelles Know How in Theorie und Praxis, Schott book publisher.

6. Cool thoughtful approach, I like it!!!

7. stub

Just to throw out a look at the same thing from a slightly different angle:

Let’s call the basic beat division the “Native Step Grouping” i.e., if we’re in 4/4 with 16ths, then the native step grouping is 4 steps per beat; which is expressed as the regular beat (an event or accent every 4 steps). Let’s say we take those same 16ths, but super-impose a secondary grouping onto them, like groups of 3 (which in this case would equal dotted 8th notes, or just accents on the first of every group of three 16ths); now we end up with a left-over 16th. Rather than 3+3+3+3+3+1, we can take the last 3+1 and make it a 4 or 2+2 (3+3+3+3+2+2=16). And yes, shifting that pattern around makes a bunch of useful drum beats, fills, or strum patterns.

OR, we can just continue to let the groupings of 3 continue across the bar as long as we want.

This kind of secondary grouping is just another way to think of what is being expressed by this Euclidean concept. The fact that examples of these rhythms can be found around the world has less to do with the Euclidean math, and perhaps more to do with the fact that these superimposed secondary groupings give us a sense of complexity and depth that is fun and interesting.