First pulse
2 3 4 5 6 7 8 9
against
Second pulse
2 3 4 5 6 7 8 9
r(3:2)
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r(3:2) = 2+1+1+2 (6 units, 4 attacks)
Start from next attack
Play it backwards
Copy as text
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Try it. Leave it on 3 against 2 and press Play. The orange ring clicks every 3 units, the green ring every 2. The purple ring is what you'd clap if you clapped both: long, short, short, long . That's a bossa nova bell, half a clave, the bass line of a hundred songs.
Count it. The two pulses meet again after 3 × 2 = 6 units, so the rhythm is 6 units long. Go to 4 against 3: 12 units, 3-1-2-2-1-3 , a rhythm that reads the same backwards. 5 against 4 gives 20 units and starts to sound like a drummer thinking.
Play it on the guitar. Pick a chord. Strum only on the purple attacks, muting the strings in between. Then strum on the orange ring only, then green only, then both. Your right hand is now playing two things at once.
Change where it starts. "Start from next attack" rotates the same durations. Rotations of one rhythm are what a band does when the drummer and the guitarist each hear "one" in a different place.
Fractioned. Tick the box and the short pulse restarts every time the long one lands, so the rhythm grows to 3 × 3 = 9 units: 2-1-1-1-1-1-2 . It's the same two numbers with a different relationship between them, and it's where fills and turnarounds come from.
Joseph Schillinger called these resultants of interference and filled the first book of his System with them. Everything here is from his Theory of Rhythm, chapters 2 to 4, with the arithmetic done for you.
Practice task
Set the metronome to your tempo and play r(3:2) on one chord for two minutes, then r(4:3) for two minutes.
Write both formulas in your practice journal with the chord you used.
Tomorrow, do the same with the rotations. Practice it in the polyrhythm trainer Open the metronome Open the journal