Maximally Even Rhythms Part 2

Maximally Even Rhythms Part 2

In the first part of this series, I defined “maximally even” rhythms. I explored what could be done with one such rhythm, the 3 3 2 grouping or “Cuban triplet”: warping it into 3:2, using different modes (different placements against the downbeat, also called rotations), and creating larger patterns from it. I also went into detail on the mathematical nature of the rhythm.


This formula gives the total amount of notes and the amount of inner groups for all the divisions I’ll discuss, including the Cuban triplet.

x = 8
8/((8/2)-1) = 8/(4-1) = 8/3 (Cuban triplet groups 8 units into 3 sections)

The satisfactory divisions obtained from this formula require x to be a multiple of 4. So today we’ll be looking at:

x = 12
Division: 12/5 (6/8 clave groups 12 units into 5 sections)

x = 16
Division: 16/7 (Partido alto groups 16 units into 7 sections)

x = 20
Division: 20/9 (20 units grouped in 9 sections)

Sorry about all the numbers. Let’s start with 12/5. It has two deceptively similar solutions, one of which, 2 2 2 3 3, is lopsided because all the short notes are clumped together (and so the long ones are too, at the end). Better to intersperse the 2s and 3s, arriving at the maximally even solution: 2 3 2 2 3.

ME 1 6 8
Sectioning 12 8th notes (two bars of 6/8 time) into 5 maximally even groups

This rhythm is known to jazz musicians as 3:2 6/8 Afro-Cuban clave. The most important word there is “clave” meaning “key” rhythm of Cuban music. This rhythm can be traced back to West Africa where it is a hugely important structure in many complex drumming/dancing/music traditions. I know nothing about these, but articles like this show how musicologists have tried to describe them. Willie Anku is an important figure in this research.


6/8 clave can be warped from a triplet to a sixteenth grid (subdivision) to create son clave and rhumba clave. That is, the relationship of the attacks to the pulse is approximated using a new subdivision of each beat. So, notes that were on the beat (beats 1 and 4) stay on the beat.

M E 2 Warp Son
Warping (similar to quantising in a MIDI program) from 6/8 clave to son clave
M E 3 Warp Rhy.png
From 6/8 clave to son clave

Rhumba clave is closer to the 6/8 original than son clave, but this is only a mathematical detail – both versions are fundamental to Cuban music, with son clave’s more stable placement of the third attack proving crucial to how it is used. Son clave and rhumba clave, by the way, are also present in West African music. They are not maximally even divisions but they keep the strength of 6/8 clave, and indeed music built off them tends to strongly reference the triplet feel of the 6/8 clave and to warp between triplets and sixteenths.

Gerhard Kubik, who I quoted extensively in my blues discussion, pointed out that the presence of rhythms such as 6/8 clave anywhere in the world indicates a connection to Africa. This is because the mathematical nature of the pattern cannot be altered without it losing its (maximally even) properties. So, unlike words, gestures, melodies, lyrics, etc., these rhythms spread between cultures without changing in any way!

Actually I’m glad I pulled out my copy of Africa and the Blues to check that, because in it Kubik lists all the rhythms I’m discussing today as unambiguously African. His name for them is “asymmetric time-line patterns” – asymmetric because each rhythm breaks into two unequal halves, e.g. 3 and 5 for Cuban triplet or 5 and 7 for 6/8 clave.

M E 4 Asy.png

Okay, here’s a riff I’m working on at the moment with my band Mescalito, which has just reunited! (First gig on March 24th in the Opium Rooms, come on down!)

After composing the riff, I discovered that it is derived from a mode of 6/8 clave (actually its bell pattern version, bembe, where the 3s are filled in with (2 1)s), modulated into 3/4 (so 3 beats of 16ths rather than 4 beats of triplets). This modulation does not affect the cyclic strength of the pattern because it still comes back to the beat at the top of each cycle. There is no polyrhythmic off-and-then-on energy.

M E Riff 1.png
The numbers show where the original 6/8 clave fits over my riff

Let’s move on quickly to the last two rhythms in the series. The maximally even division of 16/7 comes out like this:

M E 5 Part
And this is a crucial rhythm in Afro-Brazilian music. It is called partido alto and can feature as a guitar comping pattern or as a guide for bass and drum accents. I detected a rotation of partido alto in another new Mescalito riff:


M E Riff 2.png
This is rotation shifts the rhythm earlier in time by one 16th – I mark where it starts with a nifty little arrow above.

Note how using mostly only two pitches mimics the low-high/kick-snare energy of a drum line. Also, one nice thing about this riff is the placement of the lower B notes. They form a neat clave-like syncopated rhythm of their own.

Partido alto can be simplified to a grouping of 6 5 5 by hitting only some of its notes.

M E 6 (5 5 6).png
Isolating a 5 5 6 rhythm in partido alto

5 5 6 is the maximally even solution of the division 16/3 – a very important result for techno, funk and other genres that might want to spread a motif three times over a bar of 4/4 in sixteenth notes. I often use it. It’s also the distinctive comping pattern of Tito Puente’s “Oye Como Va”, mapping out the first of each pair of hits.

M E 7 (5 5 6).png
Chord hits for Oye Como Va


Okay, to finish today’s post, let’s look at the last one in the series. I’ve never heard anyone play this rhythm, although Gerhard Kubik mentions it. I think it carries a lot of the balance and power of the others. It’s the maximally even division of 20/9.

M E 8 Quint.png

The 20 units can accommodate different pulses – this example uses a 4 beats of quintuplets.

But 5 beats of sixteenth notes is also possible.

What have we found in all the rhythms so far? They’re asymmetric, made of a pair of 3s separated by an odd number of 2s, and we’ve seen them all in a rotation in which the only beats landed on are 1 and 4. This relates to what Steve Coleman calls “dynamic balance”. All of these rhythms have points of rest and, at the opposite “pole” of the cycle, tension.  There is an elegant alternation of rest and tension that expresses itself in the forward motion of the rhythm.

Okay better sign off soon, though there are obviously many avenues opening up from this kind of analysis. One cool idea I want to explore more myself is creating new bell patterns by simply crashing together all or part of these rhythms. The new rhythms mightn’t be maximally even but they could retain the flow of the originals even in strange time signatures.

Last thing I’ll say is a warning. This kind of analysis is incredibly reductive because it leaves out the cultural/political/social/historical meanings of these musical structures. I’m not informed enough to deal with those, and I apologise for that! Actually, I intend to study Afro-Cuban music for a project I’m envisaging, based on exploring very large, interlocking rhythmic forms. Hopefully you’ll see some of that research in future posts (writing that should motivate me to do it!).

Thanks for reading! Please follow the blog if you like it!


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