One algorithm, most of the world's rhythms
Take n beats and spread k strikes across them as evenly as whole numbers will let you. That one instruction, with nothing else added, gives you a surprising number of the rhythms people actually play.
The problem itself is ancient. It's the same greatest-common-divisor argument Euclid wrote down around 300 BC, which is where the name comes from. What's recent is Godfried Toussaint noticing, in 2004, that the output keeps landing on traditional patterns. Not close to them. The same ones, in places that never had contact with each other.
That's less mysterious than it sounds. A rhythm has to be learnable, and holdable, by a group of people playing together. There aren't many ways to fit five hits into eight beats without leaving an awkward gap somewhere, so people keep finding the same handful.
| k | n | pattern | where it turns up | |
|---|---|---|---|---|
| 2 | 5 | x.x.. | Korean / Greek folk | |
| 3 | 4 | xxx. | Colombian cumbia, Trinidadian calypso | |
| 3 | 8 | x..x..x. | Cuban tresillo, and half the world | |
| 5 | 8 | x.xx.xx. | Cuban cinquillo | |
| 4 | 9 | x.x.x.x.. | Turkish aksak | |
| 5 | 12 | x..x.x..x.x. | South African Venda | |
| 7 | 16 | x..x.x.x..x.x.x. | Brazilian necklace | |
| 9 | 16 | x.xx.x.x.xx.x.x. | Central African Aka |
x is a strike, . a rest. All of them come out of the same short piece of arithmetic, given two numbers and nothing else.