How to Make Euclidean Rhythms on iPhone

A Euclidean rhythm comes from one simple rule: spread a number of hits as evenly as possible over a number of steps. Three hits over eight steps, five over eight, seven over sixteen. It sounds like a math exercise, and it is one, but the patterns it gives you are some of the oldest grooves in music.

The method traces back to an algorithm Bjorklund came up with to space out timing pulses in a neutron accelerator. In 2005 the computer scientist Godfried Toussaint showed that the same even spreading produces traditional rhythms from all over the world, and he drew them as shapes inside a circle: step one at the top, time running clockwise.

• 3 over 8 gives x..x..x., the tresillo, the backbone of a huge amount of Latin and pop music.
• 5 over 8 gives the cinquillo, x.xx.xx., once you start it on a different step.
• 7 over 16 turns into a samba pattern, x.x..x.x.x..x.x., if you start it on its last hit.

The second and third examples show the part people miss: the same hits can feel completely different depending on where the loop starts. That's what the rotate control is for.

Orbeat is a drum machine that draws every lane as a ring, step one at twelve o'clock and the playhead going clockwise, which is the same picture Toussaint used.

1. Open the Generator tab and pick a lane, for example the hi-hat.
2. Switch the lane from Manual to Euclid.
3. Set the pulses and the steps. E(3, 8) is the tresillo.
4. Turn the rotate stepper and listen to where the accent lands.
5. Like what you hear? Tap the ring to write it in. The generated pattern becomes steps you can edit by hand.

Stack a few lanes with different counts, say 3 over 8 on the kick and 5 over 16 on a percussion lane, and the patterns weave around each other. Add swing and probability to loosen it up.

• The Orbeat manual covers Euclid, Probability and the other generator modes.
• Lanes can have different lengths, so a 7-step ring against a 16-step ring only lines up again much later. That's a whole other way to make a pattern that never quite repeats.
• Euclid, swing and the six-lane sequencer are free. More lanes, polymeter and patterns B to H are part of Orbeat Pro.

Toussaint's paper, The Euclidean Algorithm Generates Traditional Musical Rhythms, lists dozens more. Most of them are two numbers away.