Euclidean Rhythm — Jake Richter

Today I’m going to be talking about an incredibly profound rhythmic concept that I’ve been integrating into my drumming and composing over the past few years. Study and practice of this rhythmic concept has helped me get much closer to three of my long term drumming goals:

1. To achieve equal comfort no matter the meter,

2. To achieve more creative and fluid orchestration around the drum set while improvising, and

3. To integrate the sound and feeling of some of my favorite folkloric drumming ensembles into my improvisations.

The concept is called Euclidean Rhythm.

Euclidean Rhythms are made by distributing notes as evenly as possible throughout a fixed rhythmic cycle that doesn’t allow for the subdivision of its beats. This is a mouthful, but is easy to understand with examples. Imagine you have a rhythmic cycle that’s 12 beats long. The 12 beats repeat over and over again, and you can’t divide the beat into anything smaller than 12 equal units. You’re limited to assigning either “note” or “rest” to each of the 12 spaces in your cycle.

A 12 beat cycle visualized in western notation

There are many possible combinations of notes and rests across 12 beats, but to make a Euclidean Rhythm, you have to space the notes as far away from each other as possible without breaking up the 12 beats into smaller units. If you put two notes in the 12 beat cycle, you can evenly distribute those two notes by placing them 6 beats apart. This is a Euclidean Rhythm.

Two notes evenly distributed in a 12 beat cycle

But, if you have 5 notes to distribute throughout the 12 note space, then you can’t evenly distribute the notes without breaking the 12 beats apart. You have to approximate even distribution by spacing some of the notes 2 beats apart and some of the notes 3 beats apart. Furthermore, since we’re trying to distribute the notes as evenly as possible, we can’t put the threes next to each other and the twos next to each other, so we split those up as best we can. This process also gives us a Euclidean Rhythm that looks like this:

5 beats distributed as evenly as possible throughout a 12 beat cycle (with no further division of the 12).

The 5 notes above are spaced as evenly as possible throughout a 12 beat cycle (so long as you don’t allow the 12 notes to be broken up). You can then displace any of these rhythms, bumping them back by a certain number of beats for further rhythmic possibilities.

5 notes in the space of 12 beats displaced by one 8th note

They’re called Euclidean Rhythms because Euclid’s Algorithm can be used to show us the most even possible distribution of notes. This post won’t dive into the math of Euclid’s Algorithm, but regardless I think it’s very cool that the first ever integer algorithm, an algorithm that was created around 300 BC, has a wide range of mathematical applications today, and I think it’s even cooler that some of those applications are musical. But if we examine the 5 out of 12 rhythm above more closely, we can see something that absolutely blows my mind. If we start the cycle on the 3rd beat, we find that it gives us a rhythm that is fundamental to Afro-Cuban music: The 12/8 clave.

When I first learned about Euclidean Rhythms (from a Zach Lapidus video, link below), this was the point that really stuck out to me. It would be one thing for a mathematical process to provide some incidentally interesting rhythmic material, but it’s another thing entirely for a mathematical process to spit out a rhythm that is the backbone of one of the richest rhythmic traditions in the entire world.

And, going even further, you find that the 12/8 clave is just one of many culturally significant “clave” rhythms that can be classified as Euclidean. In fact, on further investigation, you’ll find that a huge amount of Euclidean Rhythms have found their places as important rhythms all around the world for hundreds or even thousands of years. Learning about the universality of these rhythms, that they are often fundamental fixtures in music all around the world, was what made me want to dig deeper.

The first published instance of someone using the Euclidean Algorithm to analyze rhythm was the paper The Euclidean Algorithm Generates Traditional Musical Rhythms by Godfried Toussaint. The computer scientist at McGill University published his paper in 2005, a staggering 2300 years after Euclid’s Algorithm was invented. Among other things, it goes to show that there’s still juice to be squeezed out of the classics.

Since it was published, Toussaint’s paper and Euclidean Rhythms have become fairly well known in electronic music/production circles, but as far as I can tell are mostly unknown to jazz musicians and contemporary classical composers (at least among my peers). I’m hoping this post does a small part in changing that.

Toussaint’s paper is fairly short, and it’s worth a read (link below), but it essentially goes through the math of Euclid’s Algorithm, talks about how the algorithm can be applied to music, and then lists a whole slew of examples of these rhythms as they’re found in cultures around the world.

While standard Western music notation would work to notate any of these rhythms, Toussaint chooses to notate the rhythms in a couple of alternate ways (one unique to Euclidean Rhythms and one that is sometimes used by ethnomusicologists). The Euclidean specific notation he uses is E(a,b). All this means is E for “it’s a euclidean rhythm,” “a” for the number of hits in the cycle, and “b” for the cycle’s total length. The notation sometimes favored by ethnomusicologists (for all rhythm, not just euclidean rhythm) uses x’s to imply the hits, and dots to imply the rests. For example, 5 beats spaced as evenly as possible within a 12 beat cycle would be called E(5,12) by Toussaint, and would look like this to ethnomusicologists:

[x . . x . x . . x . x . ]

There’s a slight flaw with Toussaint’s notation system in that it does not show any difference between displaced rhythms. For example, the following rhythm is also E(5,12), but starting on the 4th beat:

[x . x . . x . x . x . . ]

So, it needs to be supplemented with an additional letter: R. This way, E(x,y,r) also shows the rotation. It answers the question of “which 8th note of the original cycle are you going to begin with?” or “which 8th note in the cycle is the downbeat?”

That way, E(5,12,1) is [x . . x . x . . x . x . ] [fig 5]

and E(5,12,4) is [x . x . . x . x . x . . ] [fig 9]

This use of rotation is not my own invention; it has been adopted by many electronic musicians who use Euclidean Rhythms in their own music. I also favor the image of the clock I’ve been using throughout this paper because it gives a good sense of the cyclical nature of these rhythms.

Anyways, the choice of notation really doesn’t matter. Many (if not most) of the cultures who utilize Euclidean Rhythms don’t have a system of music notation at all, nor do they think of these rhythms as “Euclidean.” Most people seem to be playing them either because they’re a Traditional Rhythm (i.e. the 12/8 clave), or because they discovered it intuitively and just think that it sounds good. This is certainly true of the many Euclidean Rhythms I was already using in my playing and writing before I ever learned their definition.

This is an important point about Euclidean Rhythms: they have a tendency to lay really well on the drums, to be easy to play, and to sound great. People throughout history across the world haven’t been discovering these rhythms by crunching the numbers, they’ve been discovering them by learning music from their friends and mentors, playing music together, and creating something that sounds good to them. And it’s hard to overstate just how ubiquitous these rhythms are, which speaks to their universal and intuitive nature.

I’m not exactly sure why Euclidean Rhythms tend to feel natural. My guess is that their even distribution lends itself well to fitting the movements of the body, and that their cyclical nature makes them feel familiar and easy to remember. I would also be willing to bet that the well-documented human bias towards symmetry with regards to beauty is at play.

Regardless, perhaps more insight can be gained not from asking why humans seem to gravitate towards Euclidean Rhythms as their organizing rhythmic structures, but from recognizing that humans do gravitate towards Euclidean Rhythm, and that they do so in every culture. Therefore, the importance of Euclidean Rhythm is self-evident. Their importance is proven day in and day out at concerts, bars, and jam sessions all around the world.

In Toussaint’s paper, he lists many examples of Euclidean Rhythms from around the world. I’ve decided to include some musical examples of Euclidean Rhythms that I’ve discovered in my own listening too. I’m going to continue to expand this list even after I initially put this post out, so know that this is a drop in the bucket of what exists out there. If you find any Euclidean Rhythms in your own listening, please send them to me, and I’ll add them below.

Drum Set Rhythms or Jazz:

The uptempo jazz ride cymbal pattern: E(3,4,1) [x . x x ]

The shuffle ride cymbal pattern: E(8,12,1) [x . x x . x x . x x . x ]

The back beat: E(2,8,3) [. . x . . . x . ]

The walking bassline: E(4,4,1) [x x x x ] or E(4,12,1) if there’s a triplet pulse [x . . x . . x . . x . . ]

“Dotted Quarters” in 3/4 swing: E(2,9,1) [x . . . . x . . . ]

The modern jazz 7 “clave”: E(3,7,4) [x . x . x . . ]

The modern jazz 5 “clave”: E(3,10,5) [x . . x . . x . . . ]

The modern jazz fast five (each group of 5 is treated like a quarter note): E(3,5,3) [x . x x . ]

Fast five as a beat: E(12,20,3) (bass drum and snare indicated) [bs . x x . sn . x x . bs . x x . sn . x x . ]

Modern Jazz Examples:

Countdown (from Vijay Iyer’s album Break Stuff): E(4,17,13) [x . . . x . . . . x . . . x . . . ]

or E(9,17,11) [x . x . x . x x . x . x . x . x . ]

Tipico (from Miguel Zenon’s album Tipico): E(10,20,2) [. x . x . x . x . x . x . x . x . x . x]

Academia (from Miguel Zenon’s album Tipico): E(3,11,5) [x . . . x . . x . . . ]

Or E(6,11,5) [x . x . x . x x . x . ]

Inter-Are (from Mark Guiliana’s album Jersey): E(3,8,6) [x . x . . x . . ]

Folkloric/Traditional Rhythms:

Brazil:

This bossa guitar pattern: E(7,16,12) [. x . x . x . . x . x . x . . x ]

João Gilberto - Águas de Março

Partido Alto: E(7,16,15) [x . x . . x . x . x . . x . x . ]

Cuba:

The 12/8 clave: E(5,12,4) [x . x . . x . x . x . . ]

Bembe bell pattern: (afro-cuban 12/8, also found in West African music): E(7,12,4) [x . x . x x . x . x . x ]

I don’t know enough about the following musical cultures to know the names of the Euclidean Rhythms I’ve found in their recordings. They probably have names for the following rhythms, maybe they don’t, but I can still show the recording and the E name for each.

Afro-Carribean (multiple musical cultures):

By combining two Euclidean Rhythms you can get the classic calypso beat: E(2,8,1) for the bass and E(3,8,1) for the snare (but skipping the first snare note so it doesn’t blur the bass note).

Haiti:

(This super prevalent stick-on-wood pattern: E(2,3,1) [x . x ] (the rhythm begins appearing after time comes in)

Ghana:

This stick on wood pattern: E(5,8,1) [x . x x . x x .]

This stick on wood pattern, same as the 12/8 clave but this time with more of a 16th note emphasis than a triplet emphasis: E(5,12,4) [x . x . . x . x . x . . ] This example features one of my musical heroes, Bernard Woma, on Gyil:

(The clave enters around 4:30)

Finally, I’ve included some incredibly important rhythms that are just barely not Euclidean. You can decide whether or not they’re relevant to this discussion:

Cascara: E(9,15,3) + E(1,1,1) [x . x x . x . x x . x . x x . ] + [x ] = [x . x x . x . x x . x . x x . x ]

Clave: In 4/4, the Clave is one 8th note away from being a Euclidean Rhythm (it’s perfectly euclidean if you bump the last x back 1 space): almostE(5,16,11) [x . . x . . x . . . x . x . . . ]

I can’t emphasize enough that many of the above rhythms are more than just arbitrary patterns that happen to sound good. These rhythms are foundational to their genres. They’re so important that they get named. In other words, they fall into the broad category of “claves.”

I’m describing the idea of “clave” as a broad organizing principle of rhythm that exists in many types of music as opposed to the specific Cuban rhythm (the Clave). I’m talking about clave as a way of organizing the tendencies of accents and syncopations within a rhythmic cycle.

While many folkloric claves happen to be euclidean rhythms, it’s also true that many euclidean rhythms feel like claves, even if they aren’t. In other words, even the Euclidean Rhythms that don’t actually exist (to my knowledge) as “claves” in a specific musical culture can often be used as such in your own compositions. Even if a Euclidean Rhythm hasn’t been adopted by a folkloric musical tradition, it still has a tendency to feel natural and traditional.

It’s also my experience that these rhythms tend to feel natural no matter their meter. When I’m playing or listening to something based on a Euclidean Rhythm with a 17 beat cycle, ironically, it doesn’t feel like a bunch of math homework to me. It feels like a groove, it feels like a beat, it swings. There are many cultures who groove in 7, 11, or 13, who have traditional dances in these meters, and don’t consider rhythms in these meters to be fundamentally difficult or overtly complex by their nature. I believe Euclidean Rhythm has something to do with this fact.

This is super exciting because I know a lot of modern jazz musicians (especially drummers) who are hungry to include this type of rooted, drum-centric, cyclical feeling in their music, and who want to incorporate more odd meters, but who then take the route of learning and using traditional rhythms from other cultures verbatim. This can have some problems, namely that incorporating other people’s traditional rhythms can feel forced and really awkward. How many times have I had to ask myself “Why am I listening to the Bembe Bell pattern in this person’s modern or straight-ahead jazz tune?”

This is all to say that I think Euclidean Rhythms provide an amazing tool for composers and improvisers who want to create a churning complex wash of rhythm without being beholden to any particular culture or tradition. You can use Euclidean Rhythm to compose your own claves or your own multi-layered grooves. You can change anything about them any time you want, and you’ll have nobody to answer to. You can flip the clave whenever you see fit. You can use a Euclidean Rhythm as the basis for an improvisation and depart from it based on your intuition and feelings. You can use the Euclidean Rhythmic concept with complete freedom because it's not part of a culture, it’s as universal as math.

This has been the bridge I’ve needed. Traditional drumming has been a huge part of my musical life for the past several years, but I’ve had no way to get into that world of sound in a way that tells my own story, in a way that is authentic to myself rather than authentic to a culture that I don’t belong to. It’s been the tool that I’ve needed to achieve the goals I mentioned at the start: achieving comfort in odd meters, achieving greater creativity in orchestration, and achieving some of the spirit of traditional drumming while still being authentic to my own voice and heritage as an artist.

Now I’d like to discuss methods for practicing them. I’ve broken my practice recommendations up by instrument type: monophonic (instruments that play one note at a time like trumpet and voice), polyphonic (instruments who can play multiple notes at a time like piano and drum set), and ensemble. It’s far from an exhaustive list, but I hope it serves you well as a starting point. Any of these exercises can work in either improvised or composed settings.

But first, there’s an incredible resource at the following link that can build Euclidean Rhythms for you. https://dbkaplun.github.io/euclidean-rhythm/ This is so you don’t have to do the math on how to evenly distribute 13 notes across a 23 beat cycle, and can get right to practicing.

Monophonic Instruments:

1. Take any scale, melody, interval exercise, or set of chord changes, and just jam them into a euclidean rhythm. As you work with the rhythm, feel the natural cadence of accents that arises from the varied note lengths, and lean in to it. Ask which notes in the cycle feel like downbeats and which feel like upbeats, and experiment with accenting each.

2. Improvise but strictly using a euclidean rhythm. This can take many forms, but if you’re looking for a place to start, try to use the musical devices at your disposal like tempo, melodic contour, dynamics, and articulations to spin the repeating rhythmic cycle into a longer piece of music with distinct sections. For example you might play a three minute improvisation, the first minute of which is staccato, quiet, and mid register, the second minute is legato, loud, and low register, and the third is a mix of staccato and legato, moves across the range of the instrument, and has wide dynamic variation, all while staying strictly within the Euclidean Rhythm. This one works great in a linear context on the drums, playing the Euclidean Rhythm over and over again while finding seemingly infinite possibilites for orchestration.

3. Use a sparse euclidean rhythm and play it as the highest or lowest note of a phrase, while improvising in-between. (bass pedal ostinato or soprano pedal ostinato).

A low F played in E(3,22,9) with improvisation in-between

Polyphonic Instruments:

1. Play a euclidean rhythm in one voice, improvise in another (ostinato).

a. Side note: I was talking about folkloric rhythms in a lesson with Jeff Ballard (I took some zoom lessons in 2020) and he brought up a really great point: the coolest traditional rhythms (in his opinion and mine) are the ones that don’t put the low/grounding/resolving/bass sound on the downbeat, but rather save them for upbeats. It gives the groove a really nice forward momentum when the cycle’s resolution and the sound’s resolution don’t line up.

Bass note (resolving sound) arrives on the downbeat.

Bass note (resolving sound) arrives on an off beat.

2. Layer several euclidean rhythms on top of each other and compose your own etude/coordination exercise. They can either have the same length of cycle (like the example I composed below) or have different lengths of cycle. Depending on the numbers you choose, you can make the rhythmic cycles line up on a lot of hits, or make it so their cycles don’t line up for a very long time, creating a long complex wash of rhythm.

The soprano, alto, tenor, and bass voices have 12, 5, 9, and 11 notes respectively out of the 17 beat cycle, and they switch values every measure.

3. For drums, choose 1-3 limbs to play specific euclidean rhythms and then improvise in the remaining limbs.

Ensembles:

1. Use Euclidean Rhythms to compose a groove for the whole band, give everybody composed Euclidean parts (and maybe put a non Euclidean melody over the top).

2. Take a Euclidean Rhythm loop, record it, and then improvise over the top of it. Pay attention to the rhythmic rub you hear when you line up with its rhythms vs when you don’t.

3. If you’re really diving in to this concept, you can also ask me to compose some etudes for you. I’ve been making a lot of programs in Max/MSP which create Euclidean Rhythm etudes for a variety of instruments at the press of a button, and I’m planning on publishing these etudes at some point in the future.

4. Finally, I’m working on creating a program that algorithmically generates a play-along track that exclusively uses Euclidean Rhythms. I’m hoping to eventually make this resource available on my website for people to practice with, but my coding abilities aren’t quite at the level needed to make this happen yet, so that is more of a long term goal for now.

I hope this technique gives you a new lens to analyze rhythm, and I hope it sparks some creativity in your own musical life. More than anything, I hope it adds a new tool to your composer/improviser tool belt that helps you compose grooves that are authentic forms of self expression. As I’ve said already, working on Euclidean Rhythm has been a real game changer for my playing and composing, and the rabbit hole only gets deeper the more I dive in. If you’ve made it this far I really seriously appreciate you taking the time to read my blog. If you know someone who you think would be interested in learning about Euclidean Rhythm, please send this post their way. I’m planning on editing this post further, this is the first draft, so all feedback is highly appreciated.

Links:

Link to Toussaint’s Paper: http://cgm.cs.mcgill.ca/~godfried/publications/banff.pdf

Zach’s video on Euclidean Rhythm: https://www.facebook.com/zach.lapidus/videos/10115944123770019