The Trigonometry of Samba: An Analysis of Brazilian Rhythm

As I carefully cut through the tape enveloping the cardboard box, I knew at once that I’d made what one might call an error of pragmatism. By the end of the day, I concluded that I was an idiot. It was a stupid mistake, all the more so because I was by no means unfamiliar with it; I’d done it many, many times before. It pulled me away from my work more times than I could count, and filled certain periods of my life with the acute dread of procrastination. And I already knew that I was pretty assured to repeat the same actions in the near–if not immediate–future.

I got a new instrument and compulsively became obsessed with music that has nothing to with what I study.

I frequently reach this slightly cynical conclusion, but then again, I eventually tell myself, if music doesn’t feel like a mistake you never learn from, you’re probably not doing it right.

However, this week’s iteration of my hilariously self-sabotaging behavioral cycle was at least unique in terms of the musical traditions and the instrument it concerned itself with: the pandeiro, the national instrument of Brazil, and the microcosm of the batería that powers the groove of samba, beloved by all the world.

It was a kind of Christmas gift to myself–if lobbying your parents to get you one after previously advocating that you no longer qualified for the receiving of presents (due to your identity as a tax-paying citizen) can be called such a thing. It began with a certain jazz piano professor telling me that knowing to play drums was an informal graduation requirement, further developed by my persistent state of financial insolvency, and reached its conclusion in the realization that there was no way at 21 that I could persuade the purchase of a full drum kit.

So I went with the cheap, “ethnic” hand drum, and the awareness of the rich culture and history from which it sprang came tethered to it. It quickly got out of hand.

Jesse Featherstone Pandeiro Head An Analysis of Brazilian Rhythm

The closest I’ll come to ever being photobombed.

Essentially a tambourine with the jingles–or pratinelas–turned inwards instead of out (to muffle the sound), the pandeiro is nonetheless a multi-dimensionally deceptive instrument, not least in the Herculean effort involved in getting someone like me to pronounce it properly (“Pon-DEY-roo”). More importantly, however, is my sneaking suspicion that when the average person stateside learns that I dabble in playing a “Brazilian tambourine,” this seems to be what comes to mind:

Mr. Tambourine Man

Courtesy of Evan Dorkin.


Yeah, not exactly. A bit more like this:

A wild Serginho appeared!

Enter Sergio Krakowski, known in some circles by his alter-ego: Serginho do Pandeiro. A prominent member of the Brazilian music diaspora residing in New York City, Sergio picked up the pandeiro at age 15, subsequently spending years being ensconced in and carrying out the instrument’s tradition in Rio de Janeiro (everyone’s favorite samba party town, and his home). However, as per the common narrative of the archetypical ground-breaking musician, he’s done more than his fair share of pushing the pandeiro’s sonic and functional capabilities in both solo and ensemble context. He is, in sum, a bona-fide Brazilian, futurist, master panderista for the 21st century, and I have–completely by dumb circumstance–the humbling privilege of getting schooled by him on a pretty-much-weekly basis.

Notwithstanding, here’s where the story defies expectation: Sergio also has a PhD in Mathematics and Computer Music from New York University (that was very much a curveball). In conversation with him, it doesn’t rear its head often, but math and I have a history, to say the least–a brief love affair in college, nothing more–so it would seem that we step out of the number crunchers’ closet when in each other’s company.

It was in one of these sessions of heated musical discussion that we haplessly stumbled across an analysis of Brazilian rhythm that neither of us had ever considered before.

For clarity’s sake, what I’m groping at here is what many musicians and layman refer to as the music’s “feel,” or in the words of a former teacher of mine, the esteemed Carlos Valdez: “You gotta figgah out where dat shit sits.”

Brazilian music sits like Cubist painting; it’s relaxed, slumping, but angular, the features jarringly asymmetrical, like a chair that Picasso and Van Gogh may have traded notes on. It harbors the potential to string the listener along as they float, like a fishing rod in ether, and simultaneously slam them back down to the ground with an off-beat slap (or pah, to borrow from Sergio’s pneumonic system for hand-drum sounds). And on the pandeiro, it sounds like this:

Already, we can easily conclude without much contemplation that this isn’t exactly your midnight jam session at Fat Cat. It is certainly a swing, but even the clinical way of outlining the North American swing feel has more consistency than this: “Dah dah-de-DAH, dah-de-DAH,” all day long, which can really be broken down further to–specifics of “feel” aside–a way of phrasing eighth-note triplets as eighth notes.

There’s no analogous cell to be found in Brazilian feel; hence, the aforementioned asymmetry. But as Sergio and I accidentally illuminated, there is in fact a way of thinking about and depicting it that goes beyond the intuitive. It begins by accepting that Brazilian feel attempts to bridge the gap between two meters at once: 2/4 and 6/8.

Ah yes, the classic African polyrhythm, a prominent concept in Yoruba and Arará drumming traditions from regions which most of the slaves that found themselves in, say, Cuba, Brazil, and Peru, derived. And true to form, the religions of Cuban Santería and Brazilian Candomblé in which these drumming patterns found a religious outlet across the Atlantic can be thought of as regionally specific revisions of the West African religions their ancestors brought with them, most of them concerning the revered Orishas, in all of their many different guises.

What separates Brazilian feel from a relatively analogous tradition like Cuban music is that, rather than have the parts in the ensemble play in either exclusively 4/4 or 6/8 at any given time (and in Cuba it is very liable to change from moment to moment), the pandeiro groove attempts to imply both meters at the same time.

Before all of our gringo heads explode, I’ve supplied here some handy mathematical diagrams of several different feels to illustrate this concept, all derived from my conversations with Sergio. To begin, let’s look at a perfectly straight beat in 2/4:

Trigonometry of Samba An analysis of Brazilian Rhythm

Here, I’ve depicted a single quarter note in 2/4, divided into four sixteenth notes, on a flat plane, each sixteenth represented by an angle 45° apart from the adjacent notes on either side (I’ve supplied the handy jazz pneumonic device four keeping track of sixteenths, “one-ee-and-a,” to make it easy to follow). The movement from left to right represents the horizontal passage of time, and the four congruent angles give us a sum of 180°, by which point we’ve made it to the next beat in the bar, and as the arrow indicates, the cycle begins again.

Similarly, here’s an illustration of an eighth-note triplet in 2/4 occurring in the same amount of time:

Trigonometry of Samba 2 An Analysis of Brazilian Rhythm Notice here that the notes of the triplet are evenly divided into 60° angles (with a complimentary “tri-pu-let” pneumonic device), three of which add up, of course to 180°, confirming that the lengths of these two groupings of note values–four sixteenths and an eighth-note triplet, respectively–are equal.

Now, here’s where the Brazilian tradition made its hallmark adjustment. If you listen the pandeiro recording supplied earlier in this article (or to really any samba or choro from the nation’s época de ouro), you’ll notice that the most prominent feature of the Brazilian feel is the seemingly delayed transition from the fourth sixteenth note to the down beat of the next grouping. Though people simplify the feel by writing them off as all sixteenth notes, the aural reality is that the last sixteenth sounds more like the last note of an eighth-note triplet than anything.

As a result, we reasoned that mathematically we must depict the last sixteenth before the next beat as a 60° degree angle. Then, working backwards, we subdivided the remaining 120° into three 40° angles in order to shoehorn the remaining three notes, like so:

Trigonometry of Samba 3 An Analysis of Brazilian Rhythm

When we finally compared the diagrams, we were shaken with surprise by something that would appear, on paper, to be so obvious: when all of the adjustments in the placement of notes in the Brazilian feel are made, it’s still remarkable to see how close the second notes in the straight sixteenth note and Brazilian feel diagrams are to each other, in terms of when they are played: 45° and 40°, respectively. By the same token, the third beats of both are also relatively close-by, 90° versus 80°.

This is really how microscopically the infectious Brazilian groove changes a familiar framework; it’s a matter of milliseconds, but the effect is unmistakable: Brazilian music compromises the symmetry of both 2/4 and 6/8 in order to imply the presence of both simultaneously from the ground up.

Still, there remains one final change to be made. Though there is without question a down beat in Brazilian music–and everyone playing is aware of –it is not the note that is accented most prominently. That pedigree in fact belongs to the second note of this rhythmic cell, as depicted above. Below the numbers, I’ve supplied my own little pneumonic device of “Ma-ra-ca-tú,” taken from the name of yet another great Brazilian drumming tradition in the Northeastern Sertão. However, instead of starting the device on the one, as expected, we instead start on the two, with a slight accent on the next one (or “tú”) to compensate: “*one* MA-ra-ca-tú MA-ra-ca-tú MA-ra-ca-tú MA-ra-ca-tú…”

Of course, all of this analysis is merely the beginning of sittin’ with this shit. There remains much to be said for critical listening, intuition, and put baldly, number of reps. But as musicians so removed geographically and culturally from this music, we almost can’t help devising an explicit way of approaching a tradition, the structure of which goes unspoken for its practitioners; except, of course, if you’re one of those who aspire to bring its richness into the context of the here and now.

To conclude, here is a document of one of Serginho’s many efforts to accomplish just that. In the meantime, I’ll be scanning my old Calc textbook for phat beats.


This entry was posted in Analysis, Personal. Bookmark the permalink.