Lab iii · the drum kit
Your left hand taps three across a bar. Your right taps two. Both are even, neither is following the other — so when do they land together? The answer is the lowest common multiple, and here you can hear it.
Put your left hand on the table and tap three times, evenly, in the space of one bar. Now put your right hand down and tap two — same bar, also even. Both hands are right. Neither is following the other. So when, exactly, do they land together?
Left hand — hits in the bar
Right hand — hits in the bar
one bar, cut into 6 equal ticks — the finest grid both hands land on
the grid — lowest common multiple
LCM(3, 2) = 6
The left hand needs the bar cut into 3 equal pieces. The right hand needs 2. The smallest number of ticks that gives both of them a whole number of ticks between hits is 6 — left every 2, right every 3. That is the lowest common multiple, doing the job it was invented for.
the lock-ups — highest common factor
gcd(3, 2) = 1
3 and 2 share no factor, so the hands land together once in the bar — right at the start, and never again. That is why it feels like it's always about to fall over.
3 against 2 needs 6 ticks, and 3 × 2 = 6. So far so easy. But 6 against 4 needs 12 ticks, not 24 — and the reason is the same reason those two hands lock up twice a bar instead of once.
Two numbers only need the full A × B grid when they have nothing in common. The moment they share a factor, that shared part is counted twice — so you divide it back out:
LCM(a, b) × gcd(a, b) = a × b
| hands | a × b | shared | ticks needed | lock-ups |
|---|---|---|---|---|
| 3 : 2 | 6 | 1 | 6 | 1 |
| 4 : 3 | 12 | 1 | 12 | 1 |
| 5 : 4 | 20 | 1 | 20 | 1 |
| 6 : 4 | 24 | 2 | 12 | 2 |
| 8 : 6 | 48 | 2 | 24 | 2 |
| 6 : 3 | 18 | 3 | 6 | 3 |
Look at the last row. 6 against 3 isn't really a polyrhythm at all — 3 divides into 6, so the slower hand lands on top of the faster one every single time. Three lock-ups out of three hits. Musicians don't call that a polyrhythm; they call it playing along.
For the classroom