One of the best discoveries I made while researching the Groove Pizza is the mathematician Godfried Toussaint. While the bookshelves groan with mathematical analyses of Western harmony, Toussaint is the rare scholar who uses the same tools to understand Afro-Cuban rhythms. He’s especially interested in the rhythm known to Latin musicians as 3-2 son clave, to Ghanaians as the kpanlogo bell pattern, and to rock musicians as the Bo Diddley beat. Toussaint calls it “The Rhythm that Conquered the World” in his paper of the same name. Here it is as programmed by me on a drum machine:

The image behind the SoundCloud player is my preferred circular notation for son clave. Here are eight more conventional representations as rendered by Toussaint:

Son clave probably traveled from West Africa to Cuba with the slave trade. It may have arrived in its present form, or it could have evolved from a similar 12/8 pattern called fume-fume. It has a long history–I was delighted to learn from Toussaint that son clave appears under the name “al-thaqil al-awwal” in the Kitāb al-Adwār, a manuscript written in Baghdad in the middle of the Thirteenth Century by the music scholar Safi al-Din al-Urmawi. The whole book is extraordinarily beautiful; click through the image below to see many more.

There’s no way to know how old son clave is, but I would guess that it’s probably very ancient. A forty-thousand-year-old bone flute was found in Germany that plays the major pentatonic scale. Rhythm is probably vastly older than harmony, and for all we know, hominids were chipping away at their stone axes to a son clave beat millions of years ago.

Wherever son clave came from, it’s incredibly popular. Toussaint observes that the beat “is heard in all corners of the world, in almost any type of music, including rhythm and blues, salsa, rockabilly, rock, soukous, jazz, house, and the fusion pop music of scores of countries.” So what makes this rhythm so special? Toussaint has a series of intriguing mathematical explanations.

By and large, people prefer rhythms that are “maximally even,” meaning that they’re spaced more or less equally in time. Son clave is one of many widely-used beats consisting of five hits per sixteen-step cycle (one measure 4/4 time counted in sixteenth notes, or two measures counted in eighth notes.) Think of the sixteen steps as sixteen cubbyholes, each of which can hold one “object,” that is, one drum hit. Sixteen doesn’t divide by five evenly, so there are several different possible ways to distribute the five hits among the sixteen cubbyholes to make a maximally even beat. Toussaint lists them all, and labels the ones that are in common usage.

Several of these rhythms are rotations of each other, like different modes of the same scale. Rhythms 5 and 11 are “modes” of son clave; rhythms 1, 9, 15, and 16 are “modes” of bossa nova; and rhythm 3 is a “mode” of the rumba. Cool!

Toussaint asks why this combination of five beats distributed across a sixteen-step cycle should be so popular:

Why not eleven [beats], thirteen, or seventeen for example? And what is it that is so singular about five onsets? Why not four, six, or nine? These two numbers, the number of pulses in the cycle of a timeline, and the number of these pulses that are sounded, vary widely among different cultures around the world. It is quite common for the number of pulses in the cycle to be as little as four. In Bulgarian music it may go as high as 33, and in the talas of Indian classical art music it may be as long as 128. The answers to these questions are essentially physiological and psychological; they lie to a large extent in the nature of the mental and physical constraints imposed by the human brain and body. Fundamentally, to be popular a rhythm should not be so complex that it becomes difficult to grasp by the masses, and at the same time it should not be so simple that it quickly becomes boring. Furthermore, to serve well as a timeline for dancing, its realization should not take much more than about two seconds, the duration of our conscious sense of the present. Rhythms with an even number of pulses that is also a power of two are, for most people of the world, easier to assimilate than other rhythms. These constraints are already sufficient to bring the workable number of pulses down to small values that are powers of two, such as eight or sixteen. As for the number of onsets, for a timeline to afford a rich enough structure, five appears to be a good choice. However, a cycle of eight pulses does not provide enough room (in the sense of time) for five onsets to be distributed so as to create interesting patterns. Thus we are left with sixteen pulses and five onsets as the most feasible candidates for creating a timeline that has a sufficiently rich structure.

Why, then, out of the sixteen patterns above, is son clave so much more popular than the others? Toussaint attributes it to son clave’s “rhythmic oddity,” meaning that there are no pairs of hits located directly across from each other across the circle. If there were, the pair would tend to divide the pattern in half, making you hear two simpler eight-step patterns rather than one more complex sixteen-step pattern. Because of its oddness, son clave can’t be broken down into smaller symmetrical pieces.

Okay, so son clave has desirable rhythmic oddity. But so do many other beats. What else does son clave have? Toussaint points to some special symmetries hidden in the beat. Any rhythm comes with a “shadow rhythm” with an implicit hit in between each of the actual ones. When you’re drumming, your hands or sticks reach their maximum height at the onsets of the shadow rhythm, so while you may not hear it, you feel it, and both you and your listeners can see it.

So far I’ve been talking exclusively about the so-called “three-side” version of the clave. There’s also the “two-side” version, where the two-hit pattern comes first, followed by the three-hit pattern. You can switch from one to the other by moving the downbeat from the top of the circle to the bottom. You might notice that the shadow rhythm of three-side clave bears a strong resemblance to two-side clave, and conversely, the shadow rhythm of two-side clave resembles three-side. (Thanks to Roberto Thais for this observation.)

So here’s where it gets interesting. We tend to hear rhythms as patterns of short-long time intervals, rather than perceiving the length of the time intervals directly. Toussaint calls the pattern of long and short intervals the “rhythmic contour.” Son clave and fume-fume feel like “the same” rhythm because they have the same rhythmic contour, even though they’re in two different time signatures.

So here’s the magic: if you take son clave’s “shadow” and rotate it 180 degrees around the circle, it has the same rhythmic contour as son clave itself. In other words, son clave sounds “the same” as its own shadow backwards. It’s the only one of the sixteen-step rhythms listed above to have this property. We might not be able to perceive this bit of symmetry consciously, but it must act on us somehow or we wouldn’t be so wild about the beat.

Son clave shares its special qualities with all of its own rotations, the beats you get treating each of the five onset as the downbeat. So why do we prefer the downbeat that we do? Toussaint thinks it has to do with son clave’s metrical ambiguity. Those first three hits strongly imply triple meter, which is at odds with the underlying 4/4. The last two hits confirm the 4/4 feeling, but without hitting the second downbeat, the one at the bottom of the circle. You, the listener, have to involve your musical intelligence to make sense of all this ambiguity. It’s this invitation to your own imaginative participation in the beat that ultimately makes son clave so much more popular than all of its close rhythmic cousins. Math! Who says it has to be boring?

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