Stage 1 · A rhythm with no single beat
Two against three
Listen: one sound hits three times in a bar, another twice. Both are steady. Together they make a lilting pattern that seems to lean forward.
The amber ring holds three evenly spaced hits; the blue ring holds two. One hand sweeps both, once per bar, and each hit flashes as you hear it.
Which is the beat? Tap along with the three and the two sounds like it is dragging. Tap with the two and the three sounds like it is rushing. Both are right.
Only the first hit of each bar lands together. Everything in between interlocks: that is three against two, the simplest polyrhythm.
Stage 2 · Pulse, bar and tempo
A cycle is a circle
Draw time going round: one turn of the hand is one bar, each spoke a beat. Then change the speed and listen for where it stops feeling like a beat.
Four clicks per turn, about 600 ms apart: close to the tempo people pick when asked to tap at a comfortable rate.
Faster. Below about a tenth of a second between clicks, you can no longer count them or tap along; they become a buzz.
Slower. Beyond about two seconds, each click arrives on its own; you stop feeling the gaps as a steady beat and start waiting.
Between those edges, the shaded band on the dial, a repeating pattern feels like a beat. Every rhythm in this story lives inside it, until the stage where we leave on purpose.
Stage 3 · Accents make the meter
Meter is grouping
Identical clicks have no meter. Yet people hear them in groups anyway. Add an accent every third click, or every fourth, and the same pulse becomes a waltz or a march.
Every click here is the same sound at the same loudness. Many people still hear them in pairs or fours, as if some were louder.
Now one click in three is accented. The pulse has not changed, but it is grouped: ONE two three, ONE two three. That grouping is the meter.
One in four: ONE two three four. Meter is not in the pulse itself; it is the pattern of strong and weak beats laid over it.
Stage 4 · Polyrhythm: one bar, two divisions
Two rings, one axle
In a polyrhythm the layers share the bar and divide it differently. Draw each layer as a ring on the same axle: three spokes on one, two on the other, one hand for both.
Three and two share one bar. The rings turn together and their first spokes always line up: polyrhythm layers start every bar together.
To write it down, find a grid both fit: six steps, the least common multiple. The three lands on steps 0, 2, 4; the two on 0 and 3. The faint shaker plays the grid.
Together they make one composite rhythm with gaps of 2, 1, 1, 2: the lopsided figure you hear. Drummers learn it as “nice cup of tea”.
Four against three: a grid of twelve, and a longer, rolling composite.
Five against four: twenty steps. The more the numbers share no factor, the more intricate the interlock, and the harder to play.
Stage 5 · A ratio drawn on a torus
The knot
Wrap the bar around a doughnut. One layer turns the point around the ring, the other around the tube. Three against two draws a trefoil knot that closes every bar.
A point travels on the surface of a torus. Between two hits of the three-layer it goes once round the ring; between two hits of the two-layer, once round the tube. In one bar: three times round the ring, twice round the tube.
The amber hoop marks the ring’s start, the blue line the tube’s. The point crosses the amber hoop three times a bar and the blue line twice: every crossing is a hit you hear.
Five against two winds five times round the ring and twice round the tube. Untangled, it is the cinquefoil, the trefoil’s five-crossing cousin.
Four against six closes after half a bar, then draws the same knot again. Two numbers with a common factor are a smaller ratio, played twice: 4:6 is 2:3 twice per bar, a trefoil wound the other way round the torus.
Stage 6 · Rhythm turns into pitch
Faster becomes harmony
Keep three against two and speed it up. Somewhere past twenty hits a second the clicks stop being clicks: they fuse into two tones, and the two tones form a perfect fifth.
Three against two, one bar every two seconds. Scroll to speed it up; the dial shows the hits per second for each layer.
Ten hits a second: still rhythm, but a blur of it. You are past the fastest beat anyone can tap.
Somewhere around 20–30 hits a second, each layer becomes a buzz with a pitch. The flashes give way to a steady figure.
Hundreds of hits a second: two clear notes. The pitch you hear is the hit rate, and the two rates are still exactly 3 : 2.
600 and 400 hits a second: a perfect fifth. The figure is a Lissajous curve, the knot seen edge-on.
Four against three, sped up: a perfect fourth.
Five against four: a major third. The simple ratios of rhythm are the consonant intervals of harmony.
Stage 7 · The overtone series as rhythm
The Rhythmicon
In 1931 Leon Theremin built Henry Cowell an instrument that played rhythm and harmony from one rule: pulse k beats k times per bar, at k times the pitch.
The fundamental: one hole in a disc, one hit per bar, at a low pitch. A light behind the disc turns once a bar; the hole flashes as the light passes it.
Pulse 2 plays twice per bar, an octave up. Pulse 3, three times, a fifth above that. Each new ring of holes is the next harmonic, in rhythm and in pitch.
Sixteen rings: sixteen interlocking polyrhythms, 1 against 2 against 3 and so on up to 16. Every pair is a ratio from the stages before.
Speed the whole machine up and the rhythms fuse into the chord they were all along: the harmonic series, the overtones of a single note.
Stage 8 · Polymeter: one pulse, two loops
Gears that share a beat
Polymeter is the other way to layer: the layers share the pulse, not the bar. A loop of four and a loop of three step together, drift apart, and meet again every twelve pulses.
One gear, four teeth, one tooth per pulse: a loop of four. The tooth under the pointer is the beat you hear; when the marked tooth comes up, the loop starts again.
A second gear with three teeth meshes with it. Meshing gears share their tooth rate: both step on the same pulse. But the small one comes round sooner.
The marked teeth come up together again after 12 pulses: three turns of the big gear, four of the small one. The hammer rises through the cycle and strikes the bell at each meeting.
Compare the axle. On the rings, four against three share one bar and divide it, so the three falls between the pulses. The gears share the pulse and loop it. Same numbers, different sharing.
Stage 9 · When loops take minutes to meet
Long cycles
Loops of four, five and seven on one fast pulse. Any two meet often; all three only every 140 pulses, nearly half a minute here.
A steady four on the drum: the bar everyone can follow.
A loop of five over it. Four and five meet every 20 pulses.
Add seven. Now all three line up only every 140 pulses: 28 seconds at this speed. Round the dial each loop ticks off its first beats; pairs meet at the white marks, all three only at the top.
Bands use this deliberately: a riff of odd length cycles against a steady four, so it starts in a different place every bar while the drums keep the listener anchored.
Stage 10 · Hemiola: one pulse, two groupings
The same six
Six pulses can be two groups of three or three groups of two. Shift the accents and the music seems to change meter while nothing else changes.
Six even clicks per cycle, no accents. Do you hear them in threes or in twos? Either is possible.
Accent every third click: ONE two three FOUR five six. Two groups of three.
Accent every second click: ONE two THREE four FIVE six. Three groups of two. The pulse is identical.
Play both accent layers at once and you are back where you started: three against two, now heard as a tug of war over the same six pulses.
Stage 11 · Euclidean rhythms
As even as possible
Ask an algorithm to spread three hits as evenly as possible over eight steps and it plays the tresillo. Five over eight gives the cinquillo; seven over twelve, the bell pattern of West Africa.
Three hits, eight steps. Eight does not divide by three, so the gaps cannot all be equal: the most even you can do is 3, 3, 2. That is the tresillo, the heartbeat of Cuban son and much of popular music.
The recipe is the same one Euclid used to find common divisors: share the remainder out, again and again. Godfried Toussaint noticed in 2005 that it reproduces rhythms from all over the world.
Five hits over eight steps: the cinquillo, gaps 2, 1, 2, 1, 2.
Seven over twelve: gaps 2, 1, 2, 2, 1, 2, 2. Start it from its third hit and it is the bell pattern that anchors many West African drum ensembles: 2, 2, 1, 2, 2, 2, 1.
Stage 12 · Many layers, one cycle
Cross rhythm
A simplified West African ensemble on a twelve-pulse cycle: the bell, a four-beat dance pulse, a three-beat cross pulse and a shaker, entering one by one.
The bell plays the seven-over-twelve pattern from the last stage. In many ensembles everyone keeps their place by it.
A low drum plays four even beats per cycle, where dancers’ feet fall.
A second voice plays three even beats per cycle against it. Four against three, held together by the bell: this is cross-rhythm, the source of the name.
A shaker marks all twelve pulses, the grid the others share. Every layer here is a polyrhythm of the twelve.
Listen for the four and then for the three: the same music changes character with where you put your attention. Real ensembles add lead drums that play against all of it.
Stage 13 · Build one
Your rhythm
Stack your own layers: divide the bar or loop the pulse, spread hits evenly, speed it up into a chord, and tap along to see how close you are.
Each layer either divides the bar (÷ bar: polyrhythm) or loops over the shared pulse (loop: polymeter). Change the views to see the same rhythm as rings, shapes, a knot, gears or a score.
Press a step to set or clear a hit, or let the hits spread themselves as evenly as they can. Changes to hits, sounds and levels are heard at once without losing the beat; a new number of steps starts on the next bar line.
Tap along with any layer (the pad, or Space): each tap is drawn on its ring, early or late against the moment the hit reached you. Mute the layer and play it yourself.