Euclidean Rhythms

**Euclidean rhythms** are a species of rhythm named and elucidated by Godfried Toussaint in [some 2005 paper] after he found out that many irregular but vital rhythms in music throughout the world could be generated algorithmically. The algorithm, most closely related to Euclidean long division, distributes a certain number of pulses or notes (*n*) as equally as possible over an arbitarily long sequence of equal rhythmic "steps" (*m*), where *n* ≤ *m*. These are notated as E(*n,m*) in the paper, but Kaiveran Lugheidh proposes a different and more intuitive nomenclature, derived from the numbers as follows:

*m* subdivided by *n*

**5** **sub**divided by **2**

**E**uclidean **R**hythm **5s**ub**2**

**ER 5s2**

This notation allows rhythms to be sorted first by the number of steps they cover, and then by their density.

The trivial case, where *n* = 1, will be ignored in this article and related pages; there will also be little attention paid to *isochronous* Euclidean rhythms which occur when *n* divides *m* without remainder forming a regular pulse; these rhythms build interest and groove when they form uneven steps.

## Example of the process

Let's look at the building of ER 8s3, that is, 3 notes distributed over 8 steps.

First, lets have 1 represent a note or "hit", and 0 represent "downtime" between notes in steps (known academically as an "interonset interval"). Since there are 8 steps in this pattern, and 3 notes to be played, we have 5 steps of downtime.

* 111|00000

When attempting division of 8 by 3, we get a quotient of 2, and remainder of 2, meaning each rhythmic group (hit + rest time) covers at least 2 steps, and there will be two steps left over. With this, we set the remainder of 2 aside and distribute 0s to our ones to form 3 cells of 2 steps.

Alternatively, we could say that we attempt to divide the 5 rest steps by 3, giving a quotient of 1 and telling us that there's at least one rest step after every hit, and the same remainder of 2; the result is the same.)

* 10-10-10|00

Now, we distribute our remaining 2 rest steps as evenly as possible. 3 divided by two leaves a remainder of 1, so one cell has to do without and remain two steps long. Our finished rhythm is

* 100-100-10

with two groups of 3 followed by one group of 2. This is the most even distribution for this set, and it also happens to be an ubiquitious rhythm pattern in many kinds of music, perhaps best known as the Cuban *tresillo*.

A more complex example is 13s8:

* 11111111|00000

13 divided by 8 = 1R5. 8/5 = 1R3, so 3 groups will consist solely of one step. We distribute the remainder of 5:

* 10-10-10-10-10|111

Then we distribute the remaining 3 hits among the groups of two:

* 1011011010110

## Relationship to MOS (Well-Formed) Rhythms

Euclidean rhythms are a special case of MOS/well-formed rhythms, in which the generator is an integer division of the period, and the relationship between the large and small step sizes is a superparticular number.

Thus, Euclidean rhythms are the direct analogy of maximally even MOS scales in equal temperaments.

## Relationship to *aksak*

*Aksak*, a Turkic/Balkan rhythmic concept in which a meter (usually uneven) is divided into cells of two and three steps, is represented in any metric length, given a certain density. So not only are the majority of traditional aksak rhythms represented, Euclidean rhythms offer a way to extend the concept even further.

Euclidean rhythms consisting of only groups of two and three will be labeled *aksak-compatible*.

## Modified Euclidean Rhythms (modER)

Euclidean rhythms may be modified by simple permutation of their rhythmic cells, or by altering an interval by one step (analogous to the way MODMOS scales are formed).

There is also the possibility of creating a rhythmic structure analogous to omnitetrachordal scales, as of yet unexplored. One could construct ER or modER rhythms in which every rotation contains a certain rhythm of smaller cardinality as a subset.