Polyrhythm Simulator: Cross-Rhythms Explained

simulator intermediate ~10 min
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3:2 polyrhythm — the foundation of cross-rhythm

Three beats against two creates the fundamental cross-rhythm heard in African music, Latin genres, and classical piano. The two patterns align only on beat 1, creating rhythmic tension that resolves at each cycle.

Formula

LCM(a, b) = (a × b) / GCD(a, b)
Coincidence points per cycle = GCD(beats_a, beats_b)
Event density = (beats_a + beats_b − GCD(a,b)) / LCM(a,b) events per beat

When Rhythms Collide

A polyrhythm occurs when two independent rhythmic cycles play simultaneously over the same time span. The simplest — three beats against two — is so fundamental that it appears in virtually every musical tradition on earth. West African djembe ensembles layer 3:2 and 4:3 patterns into intricate textures. Chopin wrote the same ratios into his piano nocturnes. This simulation lets you see and hear how different rhythmic ratios create emergent patterns.

The Mathematics of Cross-Rhythm

Every polyrhythm can be described by its ratio — 3:2, 4:3, 5:4, and so on. The key mathematical concept is the Least Common Multiple (LCM): a 3:2 pattern has an LCM of 6, meaning you can subdivide one cycle into 6 equal parts to locate every beat. The Greatest Common Divisor (GCD) tells you how many times per cycle the two rhythms coincide. When GCD is 1, they meet only on the downbeat — maximum rhythmic tension.

Swing and Groove

Straight rhythms divide time into equal parts. Swing shifts alternate beats later, creating a triplet feel — the lilting bounce of jazz and blues. This simulator lets you add swing to both rhythm layers, transforming a mechanical pattern into something that grooves. The swing parameter moves from 50/50 (straight) toward 67/33 (full triplet swing), showing how a small timing shift completely changes the rhythmic character.

From Drumming Circles to Electronic Music

Polyrhythms are experiencing a renaissance in modern music. Electronic producers use 5:4 and 7:4 patterns to create hypnotic, evolving textures. Prog-metal bands like Meshuggah build entire songs on polymeter. Meanwhile, neuroscientists study polyrhythm perception to understand how the brain tracks multiple temporal streams — making this both an artistic and cognitive frontier.

FAQ

What is a polyrhythm?

A polyrhythm is two or more independent rhythmic patterns played simultaneously. The most common is 3:2 (three beats against two), where the two patterns share a downbeat but subdivide time differently, creating a rich, interlocking texture.

How do you count a 3 against 2 polyrhythm?

A common mnemonic is 'nice cup of tea' — the syllable stresses naturally fall on the 3:2 pattern. Mathematically, in a 6-unit cycle, rhythm A hits at positions 0, 2, 4 and rhythm B hits at 0, 3, creating the characteristic cross-rhythm feel.

Where are polyrhythms used in music?

Polyrhythms are central to West African drumming, Afro-Cuban music, and Brazilian samba. They appear in jazz (Elvin Jones), classical music (Chopin, Ligeti), progressive rock, and electronic music. Hip-hop frequently layers triplets over straight beats.

What is the difference between polyrhythm and polymeter?

Polyrhythm fits different beat divisions into the same time span (e.g., 3 and 2 beats in one measure). Polymeter uses different time signatures simultaneously (e.g., 3/4 and 4/4), so barlines align only at the LCM. Both create rhythmic complexity but in different ways.

Sources

Embed

<iframe src="https://homo-deus.com/lab/music-theory/rhythm-patterns/embed" width="100%" height="400" frameborder="0"></iframe>
View source on GitHub