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The numbers behind the common ratios
For a ratio a:b, let L be the lowest common multiple of a and b. The counting grid has L slots: the a-note layer plays every L/a slots and the b-note layer every L/b slots. There are a + b − gcd(a,b) distinct attacks per cycle. When the numbers have no common factor, only the downbeat is shared and that becomes a + b − 1.
The cycle lasts b beats when the b-note layer sets the BPM. The table uses 60 BPM on that reference. At another tempo, multiply each duration by 60 divided by the new BPM. With 4:3 at 60 BPM, the cycle is three seconds, the four-note spacing is 750 ms and the three-note spacing is 1,000 ms.
| Ratio | Slots per cycle | Attacks per cycle | One cycle | Top-layer spacing | One slot |
|---|---|---|---|---|---|
| 3:2 | 6 | 4 | 2 s | 667 ms | 333 ms |
| 4:3 | 12 | 6 | 3 s | 750 ms | 250 ms |
| 5:4 | 20 | 8 | 4 s | 800 ms | 200 ms |
| 5:3 | 15 | 7 | 3 s | 600 ms | 200 ms |
| 7:4 | 28 | 10 | 4 s | 571 ms | 143 ms |
| 7:5 | 35 | 11 | 5 s | 714 ms | 143 ms |
| 7:3 | 21 | 9 | 3 s | 429 ms | 143 ms |
| 8:5 | 40 | 12 | 5 s | 625 ms | 125 ms |
| 9:8 | 72 | 16 | 8 s | 889 ms | 111 ms |
Ratios such as 4:2 and 6:3 reduce to an ordinary 2:1 subdivision. A reducible ratio can still be a cross-rhythm: 6:4 has the same relationship as 3:2, repeated twice across the full cycle. Use Grid to check shared attacks and the ratio lessons to practise the timing.