How do two rhythmic cycles align?

Two musical lines can be individually regular yet create a more intricate pattern together. One line might begin three evenly spaced sounds while another begins four within exactly the same span. They share the beginning and end of the larger cycle but divide its interior differently. This relationship is three against four, a form of polyrhythm.

A sound's starting moment is its onset. Dividing the common span into twelve equal ticks makes the example precise. Line A begins at ticks 0, 4 and 8, so its onsets are four ticks apart. Line B begins at 0, 3, 6 and 9, so its onsets are three ticks apart. The markers specify starts, not how long each sound continues.

The lines coincide at tick 0. After that, one may begin while the other is between starts. Their next simultaneous onset is tick 12, the first positive tick that is a multiple of both three and four. Their distinct spacings produce a shared reunion without either line changing its own regular pace.

One cycle includes tick 0 but excludes tick 12, because tick 12 begins the next cycle. Counting that boundary in both would double-count the shared start. This is why the first cycle still contains three onsets in A and four in B rather than an extra one in each.

Tempo changes the elapsed time without changing that relationship. At one second per tick, the cycle lasts twelve seconds; at half a second per tick, it lasts six. The proportions and onset order stay the same. A complicated combined rhythm can therefore arise from two simple repeating rules, with their coordination understood through the time they share.