Maths & Music: Euclidian Rhythms

Posted by Esteban Miranda on November 3, 2022

A lot of music is surprisingly math-heavy.

When you study music theory, you notice that a lot of what governs the relationships between notes, scales, harmony and rhythm are mathematical concepts.

This shouldn't come as too much of a surprise, though. After all, we're all used to the fact that we count music in beats and bars, time signatures and Beats per Minute. In this sense, it's impossible to separate the idea of music and maths.

But there can be a lot more to this relationship than you'd think, and you can get some really cool results when you introduce some strange concepts into your music making.

So, today we are going to be talking about Euclidian Rhythms.

Euclidian Rhythms are a fun way to approach rhythm, and with modern sequencers and improved beat making technology available to music producers nowadays, these rhythms are increasingly being explored in ambient, dance, techno and other more generative forms of electronic music.

They're very popular with a lot of musicians who make generative modular music, as it allows the introduction of a certain 'structured randomness' into the music, which fits nicely with the styles of genres which incorporate generative sequences.

So, in this guide we will cover what Euclidian Rhythms are, how you can use them, and hopefully you'll get some cool tips and tricks along the way.

Let's get going.

What is a Euclidian Rhythm?

Euclidean Rhythms resulted from an ancient mathematical concept - Euclid's Algorithm- being applied to music making in 2004 by a Professor called Godfried Toussaint.

He discovered that Euclid's Algorithm could be used to create really interesting rhythmic patterns.

In fact, he discovered that it explains a lot of the interesting polyrhythms we find in many styles of world music.

The basic concept of the algorithm is to find the greatest common divisor between two numbers. The mathematics involved is very complex (and goes way over my head) but you can find out more about it here.

What is interesting, though, is that it can be used to create rhytmic patterns by spreading a specified number of beats across a specified length of time as evenly as possible.

We are all familiar with this concept, even if we don't realise it; 4/4 music separates 4 evenly spaced beats into a bar.

But then consider, if you added 3 beats, or pulses as they're sometimes referred to in Euclidian Rhythms, you'd have to space those 3 beats out as evenly as possible.

Where this becomes really interesting is when you start working with numbers that aren't perfectly divisible between the number of steps and number of events (we'll get to those terms shortly).

This then forces your software to to distribute the events as evenly as it can, but this is still going to create a decidedly groovy and interesting rhtyhm, due to the inherent qualities of the algorithm.

So, to go back to our example. When you play the 4 beats and 3 beats spread over the same measure of time, this creates some really interesting counter rhythms between the two patterns.

Let's take a bit of a closer look at it.

Steps, Events & Rotation

There are a bunch of terms used to describe the parameters of Euclidean Rhythms, and they can often be different depending on the device you use to create them.

In this case, I'm going to be using a Max for Live device called Euclidean Sequencer Pro, which uses a neat circular clock-like interface to display the rhythmic patterns. You can see this pictured below.

When we consider that the concept behind these rhythms is to space a set amount of beats equally in a specified time frame, a circular clock is a good way to do it. It's visual, intuitive and simple. Not to mention it looks pretty!

The circle is always a fixed size, so by adjusting the number of steps, you're changing the length of the sequence within that circle. You're always going to see your steps visualised within this specified space.

It's great for allowing you to visualise things clearly, and in this context, the controls have a very clear function.

Steps is the parameter which controls the length of a sequence. So, if we are using a 16 step sequence, that means there are 16 possible events in a sequence.

Events, then, are the beats in that sequence. If you check out the blue circle in the image above, I have 4 events in a 16 step sequence.

We are all familiar with this; this is a simple four to the floor kick pattern.

Rotation determines the start position of the events in the sequence. You can see on my green circle that I've rotated its value to 2, which changes when it plays in the sequence.

There are some other controls here which are quite interesting too.

Across the top bar we have a Triplets button, which changes the rhythm to triplets. And we also have two buttons which either double or half the speed of the pattern.

You'll notice also some different shaped arrows to the right of each of the four coloured channels. These dictate the direction of the playback within the cirlce. We can have simple forwards or backwards, but then we can also have more interesting options like forwards and then backwards, as well as random. This can really make for some interesting rhythms.

Finally, the boxes to the right of the arrows control the MIDI output notes and velocities of each channel.

It's worth pointing out before we continue that the plugin I'm using is paid, but there are other options out there.

Polyrhthmus is a free Max for Live device which lets you generate and play with Euclidean Rhythms, but it doesn't have the circular interface seen in Euclidean Sequencer Pro.

So, let's go through some basics, and then we can explore some more complex ideas.

Basic Euclidean Rhythms

As we mentioned earlier, the concepts behind Euclidean Rhythms can account for many of the interesting polyrhythms present in almost all of world music, and this is our goal when using this type of concept to make some music.

If you're unfamiliar with the term polyrhythms, it simply means a groove which consists of two or more contrasting rhythms playing at the same time.

If we begin with the basics, we will focus on having one event playing, but experiment with the effects that setting different amounts of steps in a sequence can have on your rhythms.

So, in whatever DAW you're using, let's get some rhythmic interplay going.

As you can see below, I have two patterns in play. The first is a 4 step sequence, while the second is a 7 step sequence.

We see that even though they are lined up on the top of our clock, the interplay between them is complex. They aren't always hitting at the same time, but this does occasionally happen.

What we're hearing is the algorithm trying to fit both 7 steps and 4 steps into the same time as evenly as it can, which creates some really interesting polyrhythms.

This can be a really good method for cementing your understanding of how Euclidean Rhythms work; setting the number of events to 1 and adjusting the steps value can really help demonstrate how these rhythms relate to each other, and how changing the values can create different patterns and rhythmic relationships.

But you can take this a step further by increasing the number of events, and adjusting the both the step and the rotation values to get some really cool grooves.

You can see this in practice in the first image of Euclidean Sequencer Pro I included, there are several complex rhythms playing simultaneously.

How you can use Euclidean Rhythms.

I had a lot of fun messing around with this device, and I didn't only use it for drums.

While the focus is on rhythm, it's really good for creating a wide range of interesting rhythmic patterns which can be applied to not just percussion or drum sounds, but to synths and samples too.

I created a track with two instances of my Euclidean Sequencer running. One of them was doing a drum pattern, while the other was triggering some nice melodic samples.

It's worth noting here that while I usually take some time to rifle through my percussion samples to create a nice kit, in this instance I just used a default 808 Drum Rack in Ableton. The interesting rhythms from the sequencer made it sound really cool, and I didn't feel the need to dig for different drum sounds.

Across the drums and samples I was launching, there was enough interest there that with a bit of arrangement, I could turn it into a fully fleshed out track.

If you're in a position where you need to inject a bit of unpredictability into your music, using Euclidean Rhythms could be a great way to do that.

It's also a great tool if you struggle coming up with interesting sounding drum patterns; let the algorithm do some of the work for you and you'll likely get some fun and exciting results.

What's really cool about these rhythms is, as I've mentioned earlier, that many rhythms found in world music are naturally Euclidean. The Brazillian bossa nova is 5 events distributed as evenly as possible across 16 steps, Cuba’s tresillo is 3 events distributed as evenly as possible across 8 steps, and Turkey’s Asak rhythm is 4 events distributed as evenly as possible across 9 steps.

This speaks to a deeper connection between mathematics and music that isn't just a theoretical one. Human being must instrinsically feel the patterns created by this algorithm, since so much of our music across different cultures naturally falls into it.

To me, I think it shows that there's something to this music / math relationship that we really connect with, and I think that's really cool.

So, using Euclidean Rhythms in your music is a great way to inject some calculated randomness; to have some unpredictability which follows a loose set of rules, and to create some really interesting rhythmic interplay that's got the potential to sound really really good.

So, I hope you learned something from this guide, because I certainly had fun writing it!

There's so much you can do with this concept, so thanks for checking in with us here at Top Music Arts, and keep making good music! While you're here, check out our deals on Ableton Live Project Templates, which are a great way to learn the intricate details of how a professional level track is put together.

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What is a Euclidian Rhythm?

Steps, Events & Rotation

Basic Euclidean Rhythms

How you can use Euclidean Rhythms.

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