vuza

A rhythmic tiling canon of period n is an inner rhythm R (a set of onsets, counted in slots) and an outer rhythm S (the moments each copy of R enters) such that every translate R + s lands on fresh slots and together they cover all n exactly once. Written additively: R ⊕ S = Z_n. Each ring here is one entry, the outer ring is the sum, and the sum is always a flat pulse when the tiling holds.

Most tilings are cheap: either R or S is periodic, a smaller pattern repeated on a sub-cycle (the 12 and 16 presets; in the 24, 36 and 60 ones S repeats only every half cycle, and at 48 every quarter). Hajós guessed that one side must always repeat. For every period below 72 he was right, but not beyond: Sands worked out exactly which periods force a repeat, and the first that does not is 72, then 108, 120, 144, 168. Dan Tudor Vuza, coming from the music side in the early 1990s, constructed canons where neither side repeats. The one loaded here comes from Vuza's own construction with n = 2·3·3·2·2.

Set hear to sum and the twelve voices collapse to one click on every slot, a metronome. Set it to parts and each voice gets its own pitch: the same slots, now heard as twelve staggered, irregular figures. Nothing changed in the timing. Drag any ring to rotate its entry and the tiling breaks: doubled slots flare red, holes go dark. Click a colored dot above to solo a voice; that also grabs its ring, so on a phone you can then drag anywhere on the circle.