Euclidean Rhythms

What are Euclidean Rhythms?

Maximally Even Rhythms are also called Euclidean Rhythms (named for the Greek mathematician Euclid). The term was coined by Godfried Toussaint in his influential paper "The Euclidean Algorithm Generates Traditional Musical Rhythms".

In this paper, Toussaint showed that an ancient algorithm for finding the greatest common divisor of two numbers can be used to generate maximally even rhythms. He also showed that these rhythms exist in cultures around the world—from West African bell patterns to Cuban claves to Bulgarian folk dances.

We write Euclidean rhythms as E(k, n), where k is the number of onsets and n is the total number of pulses. For example, E(3, 8) means 3 onsets distributed as evenly as possible across 8 pulses.

How Can We Generate Euclidean Rhythms?

Toussaint proposes several different ways to do this. The most intuitive is Bjorklund's algorithm, originally developed for nuclear physics applications but perfectly suited for rhythm generation. Let's see how it works.

Example 1: E(3, 8) — The Cuban Tresillo

We'll start with a simple example: E(3, 8) — 3 onsets in 8 pulses.

Step-by-Step Explanation

1. Step 1: Write onsets as 1's and rest pulses as 0's in the least even way: [1] [1] [1] [0] [0] [0] [0] [0]
2. Step 2: Pair each 1 with a 0 to create cells: [1,0] [1,0] [1,0] with remainders [0] [0]
3. Step 3: Distribute remainders to cells: [1,0,0] [1,0,0] [1,0]
4. Step 4: Recombine to form the final pattern: [1,0,0,1,0,0,1,0]

This produces the Cuban Tresillo rhythm, one of the most common rhythms in Latin music!

Example 2: E(5, 12) — The Fume-Fume

Now let's try E(5, 12) — 5 onsets in 12 pulses. This generates the Ghanaian Fume-Fume rhythm.

The Algorithm in Action

1. Initial: [1] [1] [1] [1] [1] [0] [0] [0] [0] [0] [0] [0]
2. Step 1: Pair 1s with 0s: [1,0] [1,0] [1,0] [1,0] [1,0] + remainders [0] [0]
3. Step 2: Distribute remainders: [1,0,0] [1,0,0] [1,0] [1,0] [1,0]
4. Step 3: Continue redistributing (3 cells of [1,0] as "remainder" to the 2 cells of [1,0,0]):
   [1,0,0,1,0] [1,0,0,1,0] [1,0]
5. Step 4: Final distribution: [1,0,0,1,0,1,0] [1,0,0,1,0]
6. Result: [1,0,1,0,1,0,0,1,0,1,0,0] → onsets at 0, 2, 4, 7, 9

Example 3: E(7, 16) — The Bembé Variant

Let's look at a more complex example: E(7, 16) — 7 onsets in 16 pulses.

Interactive Euclidean Rhythm Generator

Try generating your own Euclidean rhythms! Enter any values for onsets and pulses.

Common Euclidean Rhythms Around the World

Toussaint's research revealed that many traditional rhythms from around the world are Euclidean rhythms:

Euclidean Rhythms in World Music: A Catalog

Toussaint's research catalogs a remarkable number of traditional rhythms from around the world that are Euclidean rhythms. The following table summarizes these findings, organized by the number of onsets. In some cases the rhythm as generated by the algorithm is a rotation of the traditional version—both belong to the same necklace (the same pattern, just started at a different point in the cycle).

Source: Toussaint, G. "The Euclidean Algorithm Generates Traditional Musical Rhythms," Proceedings of BRIDGES: Mathematical Connections in Art, Music and Science, 2005.

Further Reading

• Toussaint, G. "The Euclidean Algorithm Generates Traditional Musical Rhythms" — The original paper
• Euclidean Rhythms (Medium article) — Detailed algorithm explanation with animations
• Pulse Lab — Generate and hear Euclidean rhythms and other rhythm types interactively
• Maximally Even Rhythms — Related lesson on the mathematics of even distributions