Lesson 4

As even as possible

A question with no reference to music in it: spread k things around a circle of n positions as evenly as they will go. The answer turns out to be playing in a great many places, and to be the same answer another course already arrived at from a completely different direction.

One question, asked twice

Harmony lesson 2 asked which seven of the twelve pitch classes to use, and found that the diatonic set is the maximally even choice: the one whose gaps are as close to equal as seven into twelve permits.

Ask the identical question about time. Given n steps in a cycle and k onsets to place, put them as evenly as possible. There is a neat algorithm for it, published by Bjorklund for a problem in particle accelerator timing, and it is Euclid's algorithm for the greatest common divisor with the intermediate steps kept as structure instead of discarded. Toussaint noticed in 2005 that its output keeps coinciding with traditional timelines, and the name stuck.

The test for maximal evenness is simple enough to apply by eye, and the figure applies it to the output rather than trusting the procedure: the gaps between onsets must use only two sizes, the floor and the ceiling of n divided by k.

E(3, 8)

3 onsets spread across 8 steps as evenly as they will go.

Every gap is 2 or 3, which is what maximally even means: no gap can be made more equal without making another less. This one is played as the Tresillo in Cuban, and most of the Americas.

Steps8

Onsets3

Tempo104 bpm

Pattern: x..x..x.
Gap sizes used: 2 and 3
Maximally even: yes
Known as: Tresillo

Only two gap sizes ever appear, and they always differ by one. That is forced: if n divides by k the spacing is exact, and if it does not, the remainder has to go somewhere, so some gaps are one step longer than the rest and the algorithm distributes those as evenly as it distributes everything else. Try E(3,8), E(5,8), E(4,12) and E(7,12), each of which is named below as a timeline somebody plays, then try E(5,13). It is exactly as evenly spread as the others and it sounds nothing like a groove, which is worth noticing before the next figure: evenness is clearly not sufficient on its own.

The one that closes the loop

Set that figure to seven onsets in twelve steps. The gaps come out 2 1 2 2 1 2 2. Read the same cycle starting from its third onset instead and they read 2 2 1 2 2 2 1, which is how the bell pattern of a great deal of West African music is usually transcribed. It is the row named Bembe bell in the next figure.

Rotation is the right equivalence for a cycle, and not merely a convenience adopted for the occasion. A timeline repeats forever; where a transcriber chose to put the barline is a fact about the transcription and not about the music, which is why the audit below tests every rotation rather than the written one.

Which is the major scale. Tone, tone, semitone, tone, tone, tone, semitone. Not an analogy: the same twelve positions, the same seven chosen, the same object. Harmony's maximally even seven of twelve and Rhythm's E(7,12) are rotations of one another, and this site reaches them through two separate modules that share no code - one selecting pitch classes by an evenness criterion, one running Bjorklund on a step count. They agree because the problem is the same, not because the answer was copied across.

It is worth being precise about what that does and does not mean. There is no claim here that the bell pattern sounds like a major scale, or that anybody derived one from the other. The claim is narrower and stranger: two traditions, working on two different axes with no contact between them, both landed on the solution to the same combinatorial problem.

A tempting overreach to avoid. Maximal evenness explains why these particular sets are special among all the sets you could pick, and says nothing at all about why evenness should be desirable. E(5,13) is exactly as maximally even as the tresillo, and a thirteen-step cycle is not something you will meet often. Evenness is a property the useful patterns turn out to share; it is not a reason, and the construction offers no account of why twelve and sixteen steps are everywhere and thirteen is not.

Now the audit

The Euclidean-rhythm result gets repeated a great deal, usually in the form that traditional rhythms around the world are Euclidean. That is the sort of claim this course is supposed to check rather than pass along, so here are ten transcribed timelines with the test applied to each.

Which of these are actually Euclidean

Ten transcribed timelines, each tested rather than assumed. Click to hear one.

Timelines tested: 10
Euclidean: 8
Not Euclidean: 2
Tested up to: rotation

Eight of ten come out Euclidean, which is a striking hit rate for a construction nobody involved had heard of. The two that fail are the son and rumba claves, and they fail for a reason you can read off the gap column: five onsets in sixteen steps can only be maximally even with gaps of 3 and 4, and both claves contain a gap of 2. The bossa nova, one row above, has the same five onsets in the same sixteen steps and is Euclidean. So the claves are not failing on a technicality of transcription. Two of the most widely played timelines in the world are simply not the thing the theory says they should be.

Eight out of ten, and which two

Eight are Euclidean. For a construction that none of the musicians involved had any reason to know about, that is a genuinely surprising hit rate and the result deserves its reputation.

The two failures are the interesting part, because they are not obscure. The son clave and the rumba clave are among the most widely played rhythmic figures on earth, and neither is Euclidean. The reason is visible in the gap column: five onsets in sixteen steps can only be maximally even using gaps of three and four, and both claves contain a gap of two.

The control is sitting one row above them. The bossa nova has the same five onsets in the same sixteen steps and is Euclidean. So the claves are not failing because of an awkward transcription or a borderline case. They are a different shape, deliberately, and the theory does not reach them.

Which is the honest summary of the whole result. Maximal evenness accounts for a surprising amount of the world's timelines and not all of them, and the exceptions include the ones a Cuban musician would name first. Both halves of that are worth carrying.