Artificial Wasteland · rhythmic canons

Which goes with which

Twelve drummers, one rhythm, a cycle of 144 beats. Choose the entries so that every beat is struck exactly once and neither the rhythm nor the entry pattern ever repeats itself, and you have what the literature calls a Vuza canon. The complete classification was computed in 2009, and the paper that did it says in one sentence, for the one case where it is not obvious, which rhythm goes with which partner. Every table that has summarised it since keeps the two counts and drops the sentence. It matters: one block of that pairing is empty, and it is the largest block there is.

The case is the one Kolountzakis and Matolcsi call the most interesting. It has 162 rhythms on one side and 60 partners on the other. If every rhythm took every partner there would be 9,720 canons here and the arithmetic would be a multiplication. Below is what actually happens, one cell per pair, filled where the two tile. Every cell was decided by enumerating, from scratch, the complete list of aperiodic partners each rhythm has.

The canon itself

The selected pair, drawn out. Each row is one drummer playing the same rhythm, entering at its own beat. Read down any column: in a canon exactly one row is lit, which is what "every beat struck exactly once" looks like. In a pair that fails, some column is lit twice and some column is dark, and the drawing marks the first of each.

The count that follows

Three arithmetics are available on the same four families, and they give three different answers. Only one of them is a fact about canons; the other two are facts about how a table is laid out.

The other three cases

There are four cyclotomic cases in all at 144 beats, and the other three are solid: every rhythm takes every partner, so there the counts do multiply. That is why the fourth is easy to read past. It is also why the same shape at 72 beats, where the single case is 3 rhythms by 6 partners and completely filled, gives no warning that 144 will behave differently.

Is 144 unusual?

The obvious next question, and the layer would be weaker for leaving it open. At 72 beats the single case is 3 rhythms by 6 partners and completely filled, so the counts multiply to 18. At 108, Fripertinger published the compendium as a drawn grid rather than a pair of counts, 252 rows by 3 columns with a midi file and a score in every cell where a canon exists, and a drawn grid cannot drop the pairing. All 756 of its cells are filled. We did not take that on trust either: enumerating every aperiodic complement of each of his three rhythms returns exactly his 252, set for set, and all 756 canons his table claims re-test as genuine factorisations. So 108 multiplies too, to 756.

Which puts it this way. At both orders where somebody drew the grid, the two counts were enough. At 144, where the summaries only counted, they were not. That is not quite the same as saying 144 is the first place this happens: 120 is also a Vuza order, it sits between 108 and 144, and nothing here touches it. Somebody should check 120. The lab has the command.

What is checked, and by whom

The tile lists are Kolountzakis and Matolcsi's, from the data files they published alongside their paper. Everything done to them here is ours, and it is the part you can rerun:

• Every published set was re-tested for aperiodicity and for having a size that divides 144. All pass.
• Every set's cyclotomic type, the set of divisors d with Φ_d dividing it, was recomputed by exact integer polynomial arithmetic and matched against the ten types their paper prints. All ten match.
• Each family was checked to be closed under the 48 multipliers coprime to 144, which map canons to canons. A missing tile would break an orbit. 443,376 images tested, none escaped.
• For each of the 177 tiles on the cheaper side of each case, the complete list of aperiodic partners was re-derived by exact cover. No tile has a partner the published lists lack, and no listed partner is missing. That is what fixes the incidence, and it is a stronger statement than testing the published pairs.
• The n = 108 comparison above is the same machinery pointed one order down: node n108.mjs re-parses Fripertinger's compendium, re-derives each rhythm's partner list by exact cover, and re-tests every canon his grid claims.
• In the two cases whose rhythms have six onsets, the 36 and the 8,640 partners therefore come out of the search rather than out of the file, which re-derives those two cases entirely from their nine rhythms. Whether those nine are the only six-onset rhythms is a separate question, and an exhaustive sweep of all 464,306,843 six-element subsets of the cycle is running as this page goes up. It is not finished, so this page does not claim its result; when it lands the lab and this line will say what it found.