Polyrhythm Reference
Every ratio below comes straight out of the same grid math the polyrhythm generatoruses — the shared pulse grid, where the two sides coincide, and the millisecond timing of every hit at a fixed 90 BPM reference tempo.
Ratio, shared grid, and feel
| Ratio | Shared grid | Coincides at | Beat length | Tightest gap |
|---|---|---|---|---|
| 3:2 | 6 steps | 0 | 666.67 ms | 111.11 ms |
| 4:3 | 12 steps | 0 | 666.67 ms | 55.55 ms |
| 5:4 | 20 steps | 0 | 666.67 ms | 33.33 ms |
| 6:5 | 30 steps | 0 | 666.67 ms | 22.22 ms |
| 5:2 | 10 steps | 0 | 666.67 ms | 66.66 ms |
| 5:3 | 15 steps | 0 | 666.67 ms | 44.44 ms |
| 7:3 | 21 steps | 0 | 666.67 ms | 31.74 ms |
| 7:4 | 28 steps | 0 | 666.67 ms | 23.81 ms |
| 7:5 | 35 steps | 0 | 666.67 ms | 19.04 ms |
Grid and timing figures come from the same polyrhythmGrid() and polyrhythmTimingsMs()functions behind the generator above — nothing on this table is a separately hand-typed number.
Reading the table
The shared gridis the smallest number of even subdivisions where both sides of the ratio land on whole-number positions — the least common multiple of the two numbers. A denser grid (7:5's 35 steps, versus 3:2's 6) doesn’t necessarily mean a harder polyrhythm on its own; what matters more for actually playing it is the tightest gapcolumn, which shrinks fast as the ratios get more complex — from 111ms for 3:2 down to under 20ms for 7:5 at this page’s reference tempo.
Every ratio here is coprime (no shared factor other than 1), which is why every row shows exactly one coincidence point, at grid position 0. That single shared instant per cycle is what your ear locks onto as confirmation the two sides are still lined up correctly — everywhere else in the cycle, the two sides interlock without ever landing on the same beat.
Frequently Asked Questions
Why do all these ratios only coincide at one point?
Every ratio on this table pairs two numbers with no common factor other than 1 (3:2, 4:3, 5:4, and so on). Whenever the two sides of a polyrhythm are coprime like this, the shared grid — sized to LCM(a, b) steps — only lines both sides up at the very start and end of the cycle. A ratio that shares a common factor (6:4, for instance) would show extra coincidence points partway through, because it's really a smaller coprime ratio repeated more than once per cycle.
What does "tightest gap" actually tell me?
It's the smallest span of time between any two consecutive hits once you combine both sides of the ratio onto one timeline — the moment in the cycle where you have the least room to place a hit accurately before the next one arrives. A smaller tightest gap is a reasonable proxy for how demanding a ratio feels to play cleanly: 7:5's 19ms gap at this page's reference tempo asks for noticeably more precision than 3:2's 111ms gap.
Why is everything shown at 90 BPM?
90 BPM is just this table's fixed reference point, so every ratio's timings are directly comparable to each other. The underlying relationship — how the grid divides, and which ratios pack hits more tightly than others — holds at any tempo; only the actual millisecond values scale up or down. Use the polyrhythm generator directly to get timings at whatever tempo you're actually practicing.
Why isn't 4:2 or 6:3 on this table?
Those ratios reduce to 2:1 and 2:1 again — they share a common factor, so they're not genuinely new polyrhythms, just a simpler ratio played twice as fast on one side. This table sticks to ratios in lowest terms so every row represents a distinct interlocking pattern rather than a repeated one.
Educational reference only — use the generator directly for timings at your own practice tempo, and always confirm feel by ear.
Use it with
- Polyrhythm Generator— get the same grid and timing math for any ratio and tempo you’re actually practicing, not just the curated set above.
- What Is a Polyrhythm? 3-Against-4 Explained — the concept explained from first principles, if you’re new to polyrhythms.
- Playing Polyrhythms: Actually Feeling 3-Over-2 and 4-Over-3 — a practice progression for getting these ratios into your hands, not just your head.