Euclidean Rhythms

Euclidean rhythms are generated by distributing a number of hits as evenly as possible across a number of steps. This algorithm produces many traditional rhythms from around the world.

The Euclidean Algorithm

Given n hits and k steps, the Euclidean algorithm spaces the hits as evenly as possible. For example, 3 hits over 8 steps produces: x . . x . . x .

This is the Cuban tresillo rhythm, found in many styles of music.

Basic Syntax

Use parentheses after a note to apply Euclidean rhythm:

// note(hits, steps)
[C4(3,8)]             // C4 with 3 hits over 8 steps
[bd(3,8)]             // Kick drum with tresillo rhythm

The note plays on the generated hit positions and rests on the others.

With Rotation

Add a third parameter to rotate the pattern:

// note(hits, steps, rotation)
[C4(3,8,2)]           // Same pattern rotated by 2 steps
[bd(3,8,1)]           // Tresillo starting on second hit

Rotation is a left shift of the hit pattern. All hit/rest positions maintain their relative spacing, but the whole pattern shifts leftward by rotation steps — positions that shift off the left wrap to the right.

Here is (3,8) at different rotations:

Rotation	Pattern	Onset positions
0	x . . x . . x .	0, 3, 6
1	. . x . . x . x	2, 5, 7
2	. x . . x . x .	1, 4, 6
3	x . . x . x . .	0, 3, 5

Tip

Rotation shifts where the downbeat falls relative to the cycle. This lets you derive variations from one base rhythm — same spacing, different feel.

Note

Rotation wraps via modulo, so rotation 10 on 8 steps equals rotation 2.

Layering rotations of the same pattern creates interlocking rhythms:

use "std/instruments" { Sampler, Kit };

let drums = AudioTrack("drums");
drums.load_instrument(Sampler(Kit("cr78")));

// Three tresillo rotations interlocking
drums << [bd(3,8,0), sd(3,8,2), hh(3,8,5)];
PLAY;

Creating Euclidean Patterns

There are three equivalent ways to write Euclidean patterns.

Inline Notation

Use parentheses directly after a note or sample inside []:

[C4(3,8)]             // 3 hits, 8 steps
[bd(5,8,2)]           // 5 hits, 8 steps, rotation 2

The euclid() Function

Create Euclidean patterns with the standalone function:

euclid(3, 8, [C4])            // 3 hits, 8 steps, source pattern
euclid(3, 8, [C4], 2)         // With rotation

The .euclid() Method

Or use it as a method on any pattern:

[C4].euclid(3, 8)              // 3 hits, 8 steps
[C4].euclid(3, 8, 2)           // With rotation

Syntax Comparison

	Inline C4(3,8)	Function euclid(3, 8, [C4])	Method [C4].euclid(3, 8)
Source	Single element	Any pattern	Any pattern
Inside []	Yes	No	No
Multi-note source	No	Yes	Yes
Chaining	With modifiers	Standalone	With methods

Tip

Use inline for single notes or samples inside patterns, the method for chaining with other transformations, and the function for explicitness. All three produce identical output.

These are equivalent:

[C4(3,8)]
euclid(3, 8, [C4])
[C4].euclid(3, 8)

Classic Euclidean Rhythms

Many traditional rhythms are Euclidean:

Pattern	Hit pattern	Name	Origin/Style
(2,3)	x x .	—	Various
(3,4)	x x x .	Cumbia	Colombian
(2,5)	x . x . .	Khafif-e-ramal	Persian
(3,7)	x . x . x . .	Aksak	Turkish
(3,8)	x . . x . . x .	Tresillo	Cuban, many styles
(5,8)	x . x x . x x .	Cinquillo	Cuban, Brazilian
(4,12)	x . . x . . x . . x . .	—	Various
(7,12)	x . x x . x . x x . x .	West African bell	Traditional African
(5,16)	x . . x . . x . . x . . x . . .	Bossa nova	Brazilian
(7,16)	x . . x . x . x . . x . x . x .	Samba	Brazilian

Key Patterns in Detail

Tresillo (3,8)

x . . x . . x .

Onsets at positions 0, 3, 6 — grouping as 3+3+2 eighth notes (long-long-short). This asymmetric grouping is the rhythmic backbone of Cuban son, Afro-Cuban jazz, pop, EDM, and reggaeton. The tresillo is one of the most widely used rhythmic cells in music.

let drums = AudioTrack("drums");
drums.load_instrument(Sampler(Kit("cr78")));

drums << [bd(3,8)];              // Tresillo kick
PLAY;

Cinquillo (5,8)

x . x x . x x .

Onsets at positions 0, 2, 3, 5, 6. The cinquillo is a denser complement of the tresillo — it fills in two of the tresillo’s gaps while keeping the same 8-step frame. Found in Cuban contradanza and Brazilian samba patterns.

let drums = AudioTrack("drums");
drums.load_instrument(Sampler(Kit("cr78")));

drums << [rs(5,8)];              // Cinquillo on rim shot
PLAY;

Samba (7,16)

x . . x . x . x . . x . x . x .

Seven hits across 16 sixteenth notes — a driving, intricate pattern with onsets at positions 0, 3, 5, 7, 10, 12, 14.

Tip

Compare (5,16) bossa nova (x . . x . . x . . x . . x . . .) with (7,16) samba. The bossa nova is sparser and laid-back; the samba is denser and driving. Both are Brazilian but have very different feels.

let drums = AudioTrack("drums");
drums.load_instrument(Sampler(Kit("cr78")));

// Samba feel
drums << [bd(7,16), hh*16];
PLAY;

Multiple Euclidean Patterns

Combine different Euclidean rhythms for polyrhythmic textures:

// Different euclidean per voice
[C4(3,8) D4(5,8)]     // Two notes with different rhythms

// Mix euclidean and plain notes
[C4(3,8) E4 G4]       // C4 has euclidean, E4/G4 are plain

Multiple Notes in Source Pattern

The source pattern is queried each cycle and its notes are distributed across the Euclidean hit positions. If there are more hits than notes, the notes wrap around:

euclid(3, 8, [C4 E4 G4])      // 1:1 mapping — each note gets one hit
euclid(5, 8, [C4 E4])          // Wraps: C4 E4 C4 E4 C4

// Alternating source changes per cycle
[<C4 E4 G4>].euclid(3, 8)

Drum Patterns

Euclidean rhythms create interesting drum patterns:

let drums = AudioTrack("drums");
drums.load_instrument(Sampler(Kit("cr78")));

// Classic polyrhythm
drums << [bd(3,8) sd(2,8) hh(5,8)];

// 3 against 4 against 5
drums << [bd(3,16), sd(4,16), hh(5,16)];

// Afrobeat-style
drums << [bd(3,8), sd(2,8), hh(7,8)];

Euclidean with Compact Notation

Combine Euclidean patterns with other modifiers:

[bd(3,8)*2]           // Euclidean pattern sped up 2x
[C4(5,8) D4@2]        // Euclidean C4, weighted D4

Modifier Order

The order matters — modifiers applied before or after Euclidean give different results:

[C4(3,8)*2]                    // Euclid then fast: 3 hits played twice as fast
euclid(3, 8, [C4 E4].fast(2)) // Fast before euclid: source doubled, then 3 hits picked

Musical Examples

Afro-Cuban Feel

let drums = AudioTrack("drums");
drums.load_instrument(Sampler(Kit("cr78")));

// Cascara pattern approximation
drums << [rs(5,8), bd(3,8), hh*8];
PLAY;

Minimal Techno

let drums = AudioTrack("drums");
drums.load_instrument(Sampler(Kit("cr78")));

// Sparse, hypnotic
drums << [bd(3,16), hh(7,16), sd(2,16,4)];
PLAY;

Melodic Euclidean

let melody = MidiTrack(1);

// Euclidean arpeggio
melody << [C4(3,8) E4(5,8) G4(7,8)];
PLAY;

See Also

• Pattern Basics — Pattern fundamentals
• Compact Notation — Other pattern modifiers
• Nested & Parallel Patterns — Layering patterns