Toussaint’s Maximally Spaced Rhythms
In this article, I explored Godfried Toussaint’s concept of maximally even rhythms and how they underpin so many grooves across cultures. I’ve always been fascinated by this intersection of math and music, so when I came across a video by a French mathematician diving into Toussaint’s research with fresh twists, I had to dig in. I ended up taking notes directly from the subtitles and transcribed all of the rhythms he demonstrated.
What follows is a deep dive into the video’s ideas, Toussaint’s framework, and how certain rhythmic “recipes” show up again and again in musical traditions worldwide.
Why Are Some Rhythms More Popular?
The video begins with a simple question: between two basic rhythms, which one is more common? Most of us can tell intuitively which pattern feels more “musical,” but why?
Godfried Toussaint (1936–2019), a computer scientist and musicologist, asked exactly this question. In his book The Geometry of Musical Rhythm: What Makes a “Good” Rhythm Good?, he proposed that rhythms can be studied mathematically by placing their onsets around a circle, turning them into polygons. The spacing and symmetry of those polygons reveal why certain rhythms “work” across cultures.
The Six Classic 5-Note Rhythms in 16 Subdivisions
Out of the thousands of possible rhythms within a 16-pulse cycle, only a handful dominate traditional and popular music. Toussaint and later researchers identified six especially important 5-note rhythms, many with Afro-Cuban or African roots:
- Son:
|1--&--4-|--2-3---| - Bossa Nova:
|1--&--4-|--2--&--| - Soukous:
|1--&--4-|--2&----| - Shiko:
|1---3-4-|--2-3---| - Rumba:
|1--&---&|--2-3---| - Gahu:
|1--&--4-|--2---4-|
These six rhythms are not just mathematical curiosities. They represent centuries of cultural migration and exchange: African roots blending with Cuban music during colonial times, later influencing jazz, salsa, and global popular music.
Mathematical Properties of “Good” Rhythms
The French video (building on Toussaint’s work and a 2024 paper co-written with Isabelle Bloch of Sorbonne University) proposes scoring rhythms according to seven properties:
- Even Spacing (Euclidean) – Notes are spread as evenly as possible.
- Balance – The rhythm’s geometric center matches the circle’s center.
- Area – The polygon covers as much space as possible.
- Asymmetry – Avoiding rhythms that are too regular or mirror-split.
- Opening / Closing – Early weak-beat notes create tension; strong-beat endings give resolution.
- Symmetry – Some rhythms have axes of reflection, others don’t.
- Combination – Scoring all properties together.
The beauty of this system is that it both explains why familiar rhythms feel satisfying and allows us to generate new rhythms with similar qualities.
High-Scoring Rhythms: 4 in 8
When tested on rhythms with 4 notes in 8 subdivisions, the following emerge with the best scores:
|1&-&--4-||1-2&-&--||1--&-&4-||1-2--&-&|
At first glance, these may seem plain compared to Afro-Cuban claves. But they reveal the underlying principle: rhythms that balance space, asymmetry, and closure tend to feel intuitively strong.
The Twelve Best 5-in-16 Rhythms
Expanding Toussaint’s idea, the researchers ranked all 1,365 possible 5-note rhythms in 16 subdivisions. The top twelve are:
|1--&--4-|--2-3---||1--&--4-|-&--3---||1--&-&--|-&--3---||1--&---&|-&--3---||1--&---&|--2-3---||1--&-&--|--2-3---||1-2---4-|-&--3---||1----&-&|--2-3---||1&----4-|--2-3---||1----&4-|--2-3---||1--&---&|--2---4-||1----&-&|-&----4-|
Notice how closely they resemble the six “classic” ones earlier. The math validates what cultures discovered organically: some rhythms just have the right mix of balance, variation, and forward motion.
Beyond 5-in-16: Other Subdivision Cases
The scoring system works not just for 5-in-16, but across many note counts and grids. Let’s look at a few examples:
3 Notes in 8 Subdivisions
- Tresillo:
|1--&--4-| |1--&-&--||1-2--&--|
The tresillo is famous in Latin and Afro-Cuban music, forming the skeleton of countless grooves.
5 Notes in 12 Subdivisions
|1-a-&--&-4--||1-a--a-&-4--||1&--&--&a---|
The first of these is the fouetté, often described as the ternary cousin of the son clave. It naturally emerges if you “swing” or ternarize the son.
7 Notes in 16 Subdivisions
|1--&-&-&|--2-3-4-||1--&-&-&|-&--3-4-||1&-&--4&|--2-3---|
These patterns resemble samba and other Brazilian grooves, once again showing how math aligns with musical practice.
Applications and Broader Implications
What’s powerful here is not only that math can describe rhythms, but that it can also generate new ones. By scoring properties, composers can discover patterns that feel as natural as traditional ones but with fresh twists.
For example:
- African bell patterns often maximize evenness while avoiding symmetry.
- Afro-Cuban claves emphasize opening/closing properties.
- Brazilian samba blends balance with asymmetry, creating forward drive.
Modern producers and composers can use these algorithms to generate beats that feel alive—avoiding the stiffness of computer-generated loops by embedding natural mathematical “groove.”
Conclusion
At first, it might seem strange to think of rhythm as geometry. But the more you explore Toussaint’s ideas—and extensions like those in this French mathematician’s video—the more you realize how universal these structures are.
Whether it’s the son clave in Havana, the tresillo in West Africa, or a samba in Rio, rhythms that thrive share certain mathematical DNA: balance, asymmetry, closure, and evenness.
For me, digging into these ideas is both intellectually fascinating and musically inspiring. They remind me that math and art are not opposites, but partners in discovery.
And if you want to explore for yourself, check out the free interactive tool mentioned in the video: the Cercle Rythmique, which lets you rotate, deform, and combine rhythms visually before exporting them to MIDI. It’s a playground for rhythm geeks and musicians alike.
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