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?
If straight 2:3 polyrhythm split into two tempos, when you play 3 beats of the “two” tempo followed by 2 beats of “three” tempo you get “son clave” … clave is a linear representation of 2:3 …. 2:3 is also the harmonic interval known as the perfect fifth, the most harmonious interval besides the octave and unison.
Therefore “son clave is the bomb”
Hi Ethan –
I thought that the only maximally even 5 in 16 rhythms were the 5 different rotations of 4-3-3-3-3, but Son Clave is 3-3-4-2-4, which isn’t maximal, according to the Euclidian metrics for evenness. Is Toussaint constructing his 16 examples above using some rules other than maximal evenness/Euclidian rhythms?
Is Son Clave particularly special because it isn’t one of the Euclidian Rhythms, but is really close/shares contours with/acts like those rhythms in lots of contexts?
Yes, son clave is not a Euclidean rhythm. My guess is that since it’s an “almost” maximally even rhythm, that it’s close enough to have a similarly gratifying effect to the really maximally even ones. Maybe it’s the very slight lopsidedness that adds interest.
Consider the structure of the simplest mora in adi tala in karnatic music. WXXXY.WXXXY.WXXX (Y) where W is a strong pulse (TA) X is a minor syllable (lets say, ka, di, mi…etc) and Y is again strong and also long which is why it is followed by a . The W’s form exactly the first half of the son clave when thinking about compressing the 16 beat pattern into the first half of the son clave. This rhythmic distribution of three phrases each starting with a strong stroke and ending with a long stroke followed by a rest is extreemly common in karnatic music and constitutes often the strongest parts of a piece of music. The rhythmic climaxes and cadences. Perhaps just more evidence why the first half of the son clave is so strong. Irefer to 3.2 clave. Of course in karnatic music this concept of a mora, or a trice, is abstracted to all of the talas using many varied phrases.
Hi Elaine,
I do acknowledge that this article imposes frameworks that are not traditionally, a part of the actual music. Standard notation is not only sufficient for representing the relative durations of notes though. It is also sufficient for representing the pattern of attack-points, the subject under discussion. It is as capable of representing sub-Saharan African rhythms and their Diasporic correlatives as it is capable of representing any other rhythms.
For many years, there have been incorrect representations of African rhythms published. The myriad confused representations do not invalidate the notational system. There are plenty of correct representations out there as well. I have encountered some misrepresentations informed by a Euro-centric viewpoint, and some misled by a predisposition to over-exotify the music.
An example of correct representation would be clave written in a single measure of 4/4. There are four main beats under clave.[1] As you demonstrated, writing clave in two measures of 4/4 (eight main beats) is incorrect . Eight main beats would mean that only the second stroke of clave is an offbeat. Your description of how clave is structured over four main beats, with clave’s resolution occurring where the last stroke and the last beat coincide is correct. It also sets clave within a specific metric structure.
You ask “how would you hear the clave if you were not imposing any time signature (and their baggage of beats) on the actual music?” Clave comes with its own inherent “baggage of beats,” whether its written or not. The pattern, Agawu states, “should never be divorced from its main beats and the referential cycle” (2003: 77). This view is held by other African master drummers such as C.K. Ladzekpo, who practice their traditions and are of course, not cultural imperialists. Ladzekpo states “The flavor and energy of a main beat is of prime importance in Anlo-Ewe dance-drumming. A main beat possesses that character of regular energy or accent that runs throughout a composition as a unifying element and gives the cross rhythms as well as the entire composition, a unique quality of logical coherence” (1995: web). In most cases, the steps of the dance, rather than the music, accent the main beats.[2]
The only inherent downbeat of clave occurs on the first stroke of the pattern. The downbeat is the first beat of a measure or of a musical composition. In traditional European music accent indicates meter, whereas in sub-Saharan music accent is counter-metric. The understanding of weak beat and strong beat in European music does not apply to the hierarchy of beats underlying clave. Making this clear would probably help in the correct interpretation of clave written in staff notation.
1. “The recurring bell pattern establishes the basic musical period or time span and the beats divide that span into equal divisions . . . four beats to each cycle of the bell pattern” (Locke 1982: 220-221).
2. “No one hears a [bell pattern] without also hearing – actuality or imaginatively — the movement of feet. And the movement of feet in turn registers directly or indirectly the metrical structure of the dance. Conceptually, then, the music and dance of a given [bell pattern] exist at the same level; the music is not prior to the dance, nor is the dance to the music” (Agawu 2003: 73).
• Agawu, Kofi 2003 Representing African Music: Postcolonial Notes, Queries, Positions. New York: Routledge.
• Ladzekpo, C.K. 1995 “Main Beat Schemes” Foundation Course: Rhythmic Principles. Web.
• Locke, David 1982 “Principles of Off-Beat Timing and Cross-Rhythm in Southern Ewe Dance Drumming,” Society for Ethnomusicology Journal, November, p. 217.
David,
You misunderstand my question about downbeat and time signatures. Yes, Western notation is mostly sufficient for representing the relative durations of notes (and is equally as foreign to the development of Irish jigs as African drumming, since both were originally aural, not written, traditions.)
My point is that the assignment of where the downbeat occurs, and the choice of time signature, offers additional cultural overhead.
This article advances a theory to explain why Son clave is popular, parts of which can only be understood by imposing a subjective framework on top of it that is not part of the actual music. The fact that that perspective is Euro-centric, whereas the music is not, should be taken into consideration when evaluating its conclusions.
To illustrate how much the imposition of a subjective time signature changes the way you hear the actual music, consider the difference between the examples in this article, which are in 4/4, as compared to 2/2. While 2/2 and 4/4 are mathematically indistinguishable, musically they impose different rhythms on top of whatever is being played, which in this case is either 8 beats (as done in this article) or 4 beats (if you choose 2/2.)
Check out how the notes in the 3-2 clave line up against these beats. If you count in 4/4, the pattern is: on – off – on | on – on. Which is to say, it is a mostly regular pattern where only the second note is heard as syncopated. However, if you count in 2/2, the pattern becomes: on – off – off | off – on. The same exact pattern all of a sudden becomes more syncopated than it is regular.
On the one hand, I think that “hearing” the clave in 2/2 more accurately describes the feel of the clave than in 4/4, and does a better job of explaining why it sounds “tense” at the end of the first bar (it ends on a syncopated beat) and sounds “resolved” at the end of the second bar (since you finally hear another note on the beat.)
But more importantly, how would you hear the clave if you were not imposing any time signature (and their baggage of beats) on the actual music? Maybe the right way to hear it is simply “short short long, short long”? As Westerners, we sometimes need to unlearn our imposition of meter if we are to understand rhythm more organically.
Rationalizing non-Western music into Western structures may help provide insights, but we need to be aware that by doing so, we change the experience of hearing them. It is especially problematic to assert that, only by applying Western technology, we can figure something out that we otherwise could not. That smacks of cultural imperialism, and we need to be be very careful about its application.
Elaine:
Concerning your first point, standard staff notation is as precise a means for representing African rhythms as it is a for representing Irish jigs. Ghanaian master drummer and scholar Kofi Agawu points out that the standard notation system is regularly used by African scholars for the purpose of writing African music: “It is noteworthy that the debate about the appropriateness of staff notation for African music is a subject of particular interest to outsiders, not insiders. African scholars from Kyagambiddwa to Kongo have for the most part accepted the conventions—and limitations—of staff notation and gone on to produce transcriptions in order to inform and to make possible a higher level of discussion and debate” (‘Representing African Music’ 2003: 52).
Concerning your second point, the 3-2/2-3 clave concept and terminology is a North American invention. It is not used in Cuba. 3-2/2-3 refers to the side of clave a chord progression begins on.
Clave has only one rhythmic progression: three-side – two-side, or, tension – release. It’s not release – tension.
When a chord progression begins on the two-side of clave (a 2-3 chord progression), the harmonic progression and the rhythmic progression are opposed, which is part of the appeal of 2-3 songs.
Ethan:
The problem is that you use ‘beat’ to mean both pattern and time unit in the same essay. It makes an already complex blog unnecessarily confusing.
Thanks for this fascinating exploration of rhythmic patterns.
However, I wanted to raise some issues with the conclusions regarding why Son Clave is the bomb. (And, to be clear, I agree that it surely is the bomb.)
1) The concepts of downbeat and time signatures are Western concepts and cannot be used to accurately explain non-Western musical practices and aesthetics.
2) Son Clave comes in both 3-2 and 2-3 versions. Which means that anything you might assert about how the beat pattern emphasizes the beginning of a phrase in 3-2 is completely inapplicable and backwards in 2-3.
3) The actual incarnation of “shadow rhythm” is unlikely to occur by dividing the time between notes in half, and more likely to be a fixed duration before each note.
I could say a lot more about each of these, but am wondering if you have considered them already?
Thanks again.
Hi Elaine. I have given these things some thought.
1) You’re right that the concepts of downbeat and time signatures are specific to our culture. However, they’re still useful for understanding the music of other cultures. Whether or not drummers around the world are thinking of clave as being in 4/4 or 6/8 or whatever, we can still meaningfully count through them that way, and the same sounds comes out when we do. Other cultures are free to analyze our music using their own vocabulary. I’ve been recording some music with a friend who’s a Carnatic tabla player. He has almost no knowledge of western music theory, and it’s fun to hear how he conceptualizes the stuff I play for him.
2) You’re also correct to point out the thing about 2-3 clave. I left that out for the sake of simplicity. Just about all of the symmetries and patterns in three-side clave are present in two-side, except for the stuff about the location of the downbeat.
3) Toussaint has written elsewhere about how shadow rhythm gets used in actual practice — there’s some conga pattern where one hand plays the shadow of the other hand, I forget what it’s called. In that case, absolutely, everything gets quantized to the closest beat. But more abstractly, we’re perfectly capable of feeling events in between the grid lines.
Most likely those five strokes appearing in the Arabic rhythm ‘Wahda wa nus’ are the result the African slave trade. That is, Africans brought with them rhythmic motifs that were reinterpreted in their new homeland. Similarly, what we call ‘tresillo’ was brought to North Africa via the trans-Saharan slave trade, and then spread eastward across a long geographic belt (probably with the spread of Islam), as far as present-day Indonesia.
The difference is that in sub-Saharan Africa, these patterns are perceived as forms of divisive rhythm, whereas in the Middle Eats and Asia, they are perceived as forms of additive rhythm.
It appears that you are using the term ‘beat’ (a unit of musical time) for ‘rhythm’ or ‘pattern.’
I know that technically a beat isn’t the same thing as a rhythmic pattern, but it’s common colloquial usage to use the word that way, I’m hardly the only person who does.
Thank you for this great explanation and thanks to all who posted comments, a lot of good info.
I realized that in Arabic drumming the son clave is used in one beat that we call in Arabic; Wahda wa nus. it has the exact 5 strokes placement within the sixteenth. The only difference is that the son clave is imbedded in the beat sense the Wahda wa nus beat is a whole beat and not a part of a poly rhythmic cycle, meaning; the drummer plays everything together. When you listen closely you will hear clave happening inside that beat.
One thing I’ve noticed is how awake it makes you feel. Very useful, as well as hugely enjoyable, when I’m driving and have a long journey.
When I am in plane or a bus and a baby cries nonstop I play the clave with my hands or my mouth and they always stop crying. It is like magic. Always work. Thank you for giving me here a possible explanation of it. I beleived that there is a deep link between the clave and the blood beats: the music of the body. Thank you for another awsome note in your blog.
I see son clave as almost the perfect tension and release rhythm. The three side builds tension due to the fact that it is played across the pulse, then the two side lands you right back on the pulse ready to start again.
Tension and release is a feature of many Cuban and Brazilian rhythms, where the first half is played off or across the beat and the second is played on the beat (for example this popular tamborim pattern |.x.x.xx.x.x.xx.x|).
IMO son clave is special due to the space either side of the two hits that resolve the tension, which allow those two hits to take center stage in the ear of the listener.
When I teach son clave I explain that the “two” is the answer to the question the “three” part asks. Even the most rhythmically inept can respond relatively in time to the call of the tresillo. In New Orleans some of the cats will play 3/2 with their feet and 2/3 with their hands at the same time since each half fits so nicely into the other.
PS, fumi fumi also uses the 4/4 “son clave” pattern (‘Traditional Dances of Ghana v.1 Rancis Kofi 1997: 42). Both the 4/4 and 12/8 forms of what we call son clave and rumba clave exist in both sub-Saharan Africa and Cuba.
The second stroke of tresillo is called bombo. Bomba, with an ‘a’ is a Puerto Rican rhythm. The third stroke of tresillo is ponche, which does translate as punch. Tresillo is the rhythmic cell that generates clave and most other 4/4 African bell patterns. The second half (two-side) of both son and rumba clave are the strokes of a displaced tresillo. So, clave is a cell of prime tresillo answered by a diametrically opposed tresillo. Tresillo sounds the first three strokes of the four-over-three cross rhythm. It is the 4/4 correlative of the 12/8 cross-rhythmic cell of three-over-two. More here:
https://en.wikipedia.org/wiki/Tresillo_(rhythm)
An even more popular, or at the very lest just as popular rhythm is the first half of the son clave repeated in the second half. The first three notes |x..x..x.| known as “Tresillo” in the Afro-Cuban world, Baiao in Brazil and Dembow in Jamaica. This rhythm is symmetrical across a circle of 16 subdivisions, as it will occur twice. In Afro-Cuban music the first note is the down beat, the second the bomba and the third the punch. Often in Afro-Cuban music the bass player will play the bomba and the punch without the down beat. This is what happens in the “Bobbaleo” referenced by Gordon Harvey.
Tresillo works with the son and rumba clave as well as functioning as a clave (rhythmic key) of it’s own. You can find this rhythm all over the globe. You have a couple of genres based solely on this rhythm Dance Hall Reggae and Reggatone with the Dembow beat. In this context you often hear the bass drum playing the down beats and the the snare playing the bomba and the punch, sometimes all three. Tresillo is a main stay in modern Hip Hop and Electronic Dance Music.
@ Rieux, if you want to get “I Want Candy” out your head when you hear the son clave try this.
Good point. Toussaint talks a lot about tresillo, and yeah, it is everywhere. I didn’t know that each hit in the pattern has a name. That’s really useful, I’ll update the post to include them.
I like Rumba Clave much better. Son is boring in its predictability; Bossa is fine except that, having grown up with it, it sounds like old news. That 3rd beat on Rumba plays off the obviousness of Son in a nice way.
I agree with you, but Toussaint is making an argument for why son clave is so popular, not why it’s the coolest clave.
I agree, Rumba clave swings far more intensely. Cutting a path of organic groove 2nd to none.
Both the 12/8 and 4/4 forms of son clave are generated through cross-rhythmic ratios. If you write both forms within a metric scheme of four main beats (not eight as you have done), they both sound 1, 1a, 2&, 3&, 4. You may want to check out my book ‘The Clave Matrix,’ which Toussaint has cited in his writings.
http://www.amazon.com/The-Clave-Matrix-Afro-Cuban-Principles/dp/1478299479
The simpler version of this is fitting 3 notes into 8:
X–X–X- baiao (appears in all kinds of music)
-x–X–x rock snare drum
-X–X-X- maracatu clave (brazilian rhythm)
It all makes sense now.
Fernando Ortiz wrote a small essay entitled La clave xilofónica de la música cubana that might be of interest to many herein.
Forgive me if I’m wrong, but there seems to be an important one Toussaint missed. Using his grid, the notes would fall on beats 0, 3, 6, 11 and 14. For me, it reaches its zenith of excitement when, after the first instance at least, the player (usually on bass)doesn’t bother with the 0. A good example is Fania All-Stars’ Bamboleo at http://www.youtube.com/watch?v=CLayNnVm9zI. It’s made even more fun by starting the introductory guitars on an off-beat, messing with our expectations.
Or is this not considered a clave?
Toussaint’s list doesn’t cover all the clave patterns; it’s a list of maximally even five-beat patterns over a sixteen-beat cycle. The beat you’re talking about has that longer space between steps 6 and 11, so it isn’t maximally even. It’s still a rad pattern though.
Is it weird that I find Clave Bossa-Nova much more attractive than Clave Son? Somehow the final two drum-hits/onsets are just too square, over-cute, precious. I can barely think about the rhythm without hearing “I Want Candy”–https://www.youtube.com/watch?v=G6Vw9RGm1tM .
I s’pose Clave Bossa-Nova analogously suggests (to me) “Break on Through” (https://www.youtube.com/watch?v=cJQwnAhXnBk ), among the million other works it’s been used in. (I gather from the post that Clave Son has been used even more–which surprises me, because I can’t think of more than one or two.)
Anyway, the non-downbeatyness makes the bossa-nova permutation much easier to listen to, to my mind.
I feel you on the bossa thing — it has that extra syncopation to it. I assume that Toussaint would say that bossa is a step too complex for most people, which is why it isn’t as common as son. And you’re right, because of Bo Diddley and Buddy Holly, American rock has really run son clave into the ground. Still, it is a bit more versatile — bossa always just sounds like bossa, but son blends into everything.
Certainly the Bossa Clave is more sophisticated than the Son. It flows much more. If you play in real fast tempo it almost sounds like 5./.8, a large quintuplet, similar to the Son Nr. 15. The Son sounds more elegant if you begin it with the second bar and has not this kind of “interrupt” and the same with the Rumba Clave.
“According to conversations I’ve had with Brazilian drummer and colleague Duduka da Fonseca, . . . it was something António Carlos Jobim developed as a basic rhythmic motif.
He also told me that Jobim later regretted it because Latino musicians confused it with the Cuban concept of clave, which was not his intention and is not utilized in Brazilian music”—Bobby Sanabria.
The above quote refers to the so called “bossa nova clave.”
Love the shadow rhythm concept. Music can be so visually appealing. I love watching Evelyn Glennie perform – her intensity when she strikes down a mallet or a stick regardless of the volume of the note created.
Evelyn Glennie is the coolest, we got to see her do an all-improvised set with Zeena Parkins and it was pretty magical.