Polyrhythm Calculator
Calculate polyrhythm beat timing relationships — find when two rhythms realign and the ms timing for each beat at a given BPM.
Understanding Polyrhythms
A polyrhythm occurs when two or more rhythmic patterns with different pulse divisions play simultaneously. The most common example is 3 against 2 (3:2) — three evenly spaced notes against two evenly spaced notes in the same time span.
How to Think About It
For a polyrhythm of M against N, both patterns fill the same span of time. One hand divides that span into M equal parts, the other into N. They start together on the downbeat and land together again at the end of the span, and nowhere in between unless M and N share a factor.
The least common multiple of M and N is not how long the cycle lasts. It is how finely you have to slice the span so that every hit from both hands lands exactly on a slot. For 3 against 2 that is LCM(3,2) = 6 slots, and the two hands between them strike 4 of the 6.
Common Polyrhythms
| Ratio | Description | Found In |
|---|---|---|
| 2:3 | Two against three | Classical, African, Latin |
| 3:4 | Three against four | Jazz, contemporary |
| 3:5 | Three against five | Progressive rock, Indian classical |
| 4:5 | Four against five | Jazz, 20th century classical |
| 2:3:4 | Cross-rhythm | African drumming |
Calculating Beat Spacing
At a given BPM (beats per minute), one beat lasts:
Beat duration = 60,000 ms / BPM
For a polyrhythm of M:N over one beat:
- Pattern A beat spacing = Beat duration / M
- Pattern B beat spacing = Beat duration / N
The Cycle Length
Cycle = the span itself. Take one beat as the span and the patterns realign every beat.
Worked through, 3:2 at 120 BPM:
- One beat = 60,000 ÷ 120 = 500 ms
- Pattern of 3: a hit every 500 ÷ 3 = 166.7 ms, so at 0, 166.7 and 333.3 ms
- Pattern of 2: a hit every 500 ÷ 2 = 250 ms, so at 0 and 250 ms
- Shared grid: LCM(3,2) = 6 slots of 83.3 ms
- They meet again at 500 ms, the top of the next beat
The composite, which is the thing your ear actually receives, is four attacks: 0, 166.7, 250 and 333.3 ms. Long, short, short, long.
The “Not Difficult” Mnemonic
That long-short-short-long shape is what makes 3:2 learnable. Say “not dif-fi-cult” with the natural stress of the phrase and you have played it: not on the downbeat, dif on the second triplet, fi on the offbeat of the duple, cult on the third triplet.
Four syllables, four attacks. The words carry the rhythm for you, which is why every teacher reaches for a phrase rather than a count. The 4:3 equivalent is “pass the gol-den but-ter”, and that lines up too: six syllables for the six attacks 4:3 produces.
Once the phrase is automatic, drop it and let each hand keep its own pulse. Trying to compute the offsets while playing is the thing that never works.
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This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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