Tag: maximally even

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!

Maximally Even Rhythms Part 1

Maximally Even Rhythms Part 1

Today I’m investigating the simple maths behind some of the deepest rhythms of groove music!

The notes of a looped rhythm can be imagined as dividing up the loop into sections.

Not Displaced
A 1-bar cycle, 6 8th notes in length, evenly divided into 3 groups of 2 8th notes’ length each.

(There is no distinguishing feature in this cycle that makes it 6/8, by the way… unlike the rhythms we’ll be focusing on later which do have a shape. This one could equally be in 4/4 or 1/4.)

This version does the same thing, except displaced an 8th note from the pulse.


This rhythm is even. Each of its 3 parts are equal in duration (2 8th notes). But, many divisions can’t be done evenly, for example dividing 8 8th notes into 3 sections.

In arithmetic, this division would either result in a fraction 8 / 3, with the value approx. 2.667, or need a remainder: 2 remainder 2.

ME 2 Even
Fraction: 3 groups, but the length of each group does not fit the 8th note subdivision.
ME 3 Remainder
Remainder: There are now 4 groups due to the need for a remainder of 2.

The only way to have three groups of whole 8th notes in this cycle is to have different-sized groups:

ME 4 224 125
Division of 8 notes into groups of 2 2 4 or 1 2 5

Other solutions would be 1 3 4 or 3 3 2 or 1 1 6.One of these mixed-sizes solutions has special properties, though. It is “maximally even”, meaning it’s the solution most similar to equal-sized groups. For divisions, like 8/3, that can’t be done evenly, the maximally even solution requires groupings of two different sizes. In this case, groupings of 2 and 3. So the maximally even solution to dividing 8 notes into 3 groups is…


ME 5 3 3 2
3 3 2 grouping AKA Cuban triplet AKA tresillo

It makes intuitive sense that the division most similar to 2.667 2.667 2.667 would involve a mixture of 2s and 3s – the whole numbers most similar to 2.667.

Is there anything musically special about this division? Well, yes. It is used almost universally in rock, pop, and dance music. Its closeness to the even division of 3:2 can confuse, and I’ve seen student musicians write down one while meaning the other. You can often hear rock, reggae or folk bands hesitate between these two rhythms, perhaps playing something between the two. More intentionally, Afro-Cuban musicians make use of this ambiguity regularly.

These examples show it’s possible to lean or warp between rhythms containing the same amount of notes per beat or bar, even if they have different subdivisions. This is because West African cultures view rhythms as divisions, not additions.

By contrast, Middle Eastern or Indian musicians would be more likely to view a Cuban triplet rhythm as a group of 3 units, followed by 3 units, followed by 2 units. That’s called additive rhythm because, conceptually, different groupings are added on to each other to form sequences. From an additive perspective, the analogue of a Cuban triplet in a triple subdivision would be use the same sequence: group of 3 notes, a group of 3, and a group of 2:

ME 6 Karnatic Style

From the West African perspective, though, the closest thing to a Cuban triplet in a triple subdivision would be the rhythm that divides 2 beats into 3 parts:

ME 1 3 over 2

So, whereas South Indian musicians excel at playing the same sequence at different speeds against a pulse (like the first example), African and African Diaspora musicians are adept at warping rhythms into a different subdivision, creating tension between the resemblance of the rhythmic shape (same average rate of notes) and the change in the flow of the subdivision (e.g. triplets feeling more rolling/circular than 16ths).

To move on: one important thing about every “maximally even” rhythm is that they are cyclical – there is no particular start or end. Like modes of a musical scale, any note can be imagined as the start of the pattern:

ME 7 3 Variants
Modes of the Cuban triplet: 3 3 2 (original), 3 2 3, 2 3 3

But unlike modes of a scale which must have a root note, cyclical rhythms needn’t have a note on beat 1, which opens up 5 more variations:

ME 8 5 Variants
5 modes of Cuban triplet that don’t land on 1.

As I unfortunately don’t have all week to write each weekly post here, I’m gonna spend the rest of today’s piece focusing only on these rhythms. In part 2 I’ll cover maximally even rhythms over 12, 16 and 20 notes, including Afro-Cuban, African and Brazilian rhythms. For now, let’s find applications for the variations of 3 3 2, and maybe make some general observations.

All of these rhythms are short, and so when I’m composing or improvising, I find they work well as a sort of basic pulsation within the groove. In a 16th-note-based style like say techno or hip hop, one or more of these rhythms can underlie all the other rhythmic activity.

In this song that my sister happened to play as I’m writing, the underlying cell is the Cuban triplet 3 3 2, but it is developed into 2-bar patterns by substituting two rests or two quarter notes.


ME 2 Fixed
Different rhythms in “Desire” with the times they enter

The third of those 2-bar patterns has been used in countless dance and pop-dance tunes.

In these contexts, the cymbal (and usually a 4/4 kick) provides a strong skeleton of 8th notes that the syncopated rhythms can interlock with. Interlocking is, I think, another essential component of groove music. It’s a rather large topic to try and define, but I would say that when two cyclical rhythms have some notes together and other notes a 16th note apart, they will feel interlocked. Here’s an example using a riff from my rock band, Mescalito.

Methuselah Riff
Interlocking rhythms. Notes that are together are indicated by the lines.

Onto the other variants; here’s a 3 2 3 division. This might be the least common of three variants that hit the downbeat. This example is by the Ben Prevo Band, with me on bass and Dominic Mullan playing the pattern on drums. The song is Ben Prevo’s composition, “An Udder Blue”.



ME 11 Prevo
The drum pattern on “An Udder Blue”

Check out how this example is over 4 beats of swung 8ths rather than 2 beats of straight 16ths – still 8 notes in all, divided into 3 groups. It’s important to be able to recognise fundamental rhythms no matter that they might be notated differently or felt with a half- or double-time pulse, or swung. The next example is also over 4 beats.


ME 12 Wilkinson
The basic accents of the drumline in “Too Close”

The main accents in this d’n’b tune’s drumline (0:47) are the 2 3 3 grouping, in 8th notes. But the drumline as a whole is filled with many 16th notes. So, the energy of the maximally even division operates on one frame (8ths) with other rhythmic information in a denser frame (16ths). Take a moment if you like to feel how those interact in the song. To me, there’s a floatiness caused by the powerful but slow 1-bar cycle of the 2 3 3 (which suggests a half-time feeling, actually, and is used by itself to introduce half time at 1:53) mixed with the twitchy intricacy of the 16ths.

There’s a basic transformation that can be applied to all the maximally even rhythms I’ll talk about today and in part 2. I think of it as making a “bell pattern” out of the rhythm, because it is the technique used to turn 6/8 clave into bembe, the Cuban 6/8 bell pattern. However this is probably confusing use of language as all of these rhythms can be played on a bell. A proper name for this rhythm is cinquillo. Quite simply, the 3s in the rhythm are filled in to become (2 1). So 3 3 2 becomes 2 1 2 1 2. This is also a maximally even division of 8 into 5 parts.

ME 13 Bell
Turning the Cuban triplet into cinquillo.

Notice that this is only one note off from being a 3:2 son clave.

ME 14 Son in 16ths
The only note different between the bell pattern and son clave (notated in 16ths)

We’ll see more of how maximally even rhythms can be transformed into each other in part 2.


In Megadeth’s new track “The Threat Is Real”, the kick drum line at 0:59 is the same as cinquillo: 2 1 2 2 1. (The guitar chug follows this line too, with one extra note where the snare hit is.)

ME 15 Megadeth
Drumline at 0:59 of “The Threat Is Real”

I won’t try hunt up examples of all the other variants, because I think you get the idea. The main conclusions we can draw are: these rhythms can exist at half or double speed against a given pulse or subdivision; they can be warped into similar rhythms in different subdivisions (even the swung 8ths rhythm above is arguably warping, from straight 8ths into a triplet grid); they can be constituents of longer patterns like the dance-pop bassline grouping 4 4 3 3 2; and they are a very rich source because they can be spun around in all their modes, filled out and interlocked with other rhythms.

To finish, here’s a spontaneously improvised maximally even division of 8 into 3 groups – one of those that doesn’t fall on beat 1. This is from a bootleg of Mescalito playing live a few years back. I’ve included the build-up because I like how the pattern slowly asserts itself in my bassline, fully emerging at 0:48. Like the d’n’b example, this pattern is in 8ths but the rest of the band play 16ths.


ME 16 Mescalito Improv
Improvised bassline off mode of cinquillo

Thanks for reading! I think next week I’ll get back to my discussion of negative dominants and alternate paths, but stay tuned for a part 2 of this article where I’ll get into more maximally even rhythms in meters up to 5/4. As always, feel free to comment below!