Rhythm: Maths That Makes You Dance

Around 300 BC, Euclid of Alexandria described an algorithm for finding the greatest common divisor of two numbers. One of the oldest algorithms in existence. In 2005, Godfried Toussaint at McGill University realised the same algorithm is found in all kinds of music.

That is not an exaggeration. It is a peer-reviewed observation.

The idea: take a number of steps (the pattern length) and a number of pulses (the hits). Distribute the pulses as evenly as possible across the steps. The Euclidean algorithm does this automatically. The results are rhythms humans independently discovered on every continent.

How the grid works

The step grid above is a 16-step sequencer with four voices. Click any cell to toggle a hit on or off. Press play and the playhead moves across the columns, triggering whatever is active at each step.

Euclidean patterns to try

Open the Euclidean mode panel and set pulses per row. The grid fills itself using the same algorithm Euclid described 2300 years ago. You can then manually edit the result.

4 pulses in 16 steps (kick) - four-on-the-floor. Foundation of house, techno, disco. Four equally spaced kicks at positions 0, 4, 8, 12. The most basic rhythm, and the one that makes the most people move.

3 pulses in 8 steps - the Cuban tresillo. One of the most widespread rhythmic cells in the world. Positions 0, 3, 6. You have heard it in thousands of songs without knowing its name.

5 pulses in 8 steps - the cinquillo. West African and Caribbean. The complement of the tresillo - hits become rests and vice versa.

7 pulses in 12 steps - a West African bell pattern. Found in jazz, samba, rock.

The rotation slider shifts the starting point of the pattern. Same hits, same spacing, different downbeat. This is how you get syncopation - the feeling that the rhythm pulls against the expected pulse. Rotate the tresillo by 1 step and it becomes the pattern behind reggaeton.

How the drums are made

Each drum voice is synthesised, not sampled. No recordings. Pure maths.

Kick - a sine wave that starts at a higher pitch and drops rapidly. The initial click is a short burst at 8x the base tone. The pitch drop is what gives a kick drum its "thump" - a sine wave doing a nosedive.

Snare - filtered noise (the "snap" of the snare wires) plus a triangle wave body (the resonance of the drum shell). Higher tone = brighter, thinner. Lower = fuller, more old-school.

Hi-hat - high-passed and bandpass-filtered noise. Short decay = closed hat (tight, clicky). Longer decay = open hat (shimmering, washy). The OH row uses the same synthesis with a longer decay.

Swing

Swing delays every other step. At 0% every step is perfectly even. At 100% the off-beats are pushed right up against the next on-beat, giving a heavy shuffle feel. Most classic drum machines sit around 50--60%.

The number of possible combinations across four voices, sixteen steps, every tempo and swing setting is larger than the number of atoms in the observable universe. That is not a metaphor. It is a combinatorial fact. Explore.