Polymetry

When rhythms overlap

Can you tap a three-beat rhythm with your right hand and a four-beat rhythm with your left? Congratulations, you’ve just played polymeter — a fascinating musical phenomenon that our digital formats struggle to represent.

Where does this article fit in?

Polymeter is one of the musical phenomena that the Bol Processor (I2) handles natively and that MIDI (Musical Instrument Digital Interface, see M1) and MusicXML (see M2) struggle to represent. It is also one of the key motivations for the BP2SC project (BP3 language transpiler to SuperCollider, see I3 and B7).

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Why is it important?

Polymeter is everywhere, but often invisible. It is at the heart of African music that gave birth to jazz and funk. It structures Indian rhythmic cycles. It appears in Chopin, Brahms, Ligeti. Yet, when you open music software, you are limited to a single time signature per measure.

Understanding polymeter means understanding why some music seems to “float,” why the African groove is so difficult to transcribe, and why formats like MIDI or MusicXML fail to represent entire musical traditions.

The idea in one sentence

Polymeter is the simultaneous superposition of several different meters (or pulses), creating a tension between several competing “strong beats.”

Meter vs. Rhythm: what’s the difference?

These two terms are often confused, but they refer to different things:

• Rhythm: the organization of durations in time (a quarter note, then two eighth notes, then a half note…)
• Meter: the underlying regular structure that organizes strong and weak beats (1-2-3-4, 1-2-3-4… in 4/4)

Analogy: rhythm is like a dancer’s steps (free, varied), meter is like the regular beat of the music they dance to.

Polymeter superimposes several different meters (e.g., 3/4 and 4/4 at the same time), not simply different rhythms within the same meter.

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Let’s explain step by step

Example 1: Clapping your hands

Let’s do a concrete experiment. It’s the best way to feel polymeter.

Step 1: Right hand alone — groups of 3 Tap on the table with your right hand. The first beat of each group is a strong beat (tap with a flat palm), the following are weak beats (tap with fingertips, more softly):

PALM - fingers - fingers - PALM - fingers - fingers… (groups of 3)

Step 2: Left hand alone — groups of 2 Same principle, but in groups of 2:

PALM - fingers - PALM - fingers - PALM - fingers… (groups of 2)

Step 3: Both together Start both hands at the same time on the “1”. Be careful, it’s difficult! The palm/fingers distinction helps you feel where the strong beat of each hand falls — and these strong beats do not coincide.

Figure 1 — Polymeter grid 3 against 2. The strong beats (P) of both hands coincide only at positions 1 and 7 (dotted yellow lines). Between these points, each hand “pulls” in its own metric direction.

(P = palm/strong beat, d = fingers/weak beat)

The tension comes from the fact that the strong beats (palms) of both hands only fall together at point 1 and point 7. Between these points, each hand “pulls” attention in its own metric direction.

The LCM (Least Common Multiple)

The two hands meet together every 6 beats — this is the LCM of 2 and 3 (the smallest number divisible by both). Between these meeting points, they “pull” attention in different directions. It is this tension that creates the rhythmic interest of polymeter.

• Right: 12 beats / 3 = 4 cycles of 3
• Left: 12 beats / 2 = 6 cycles of 2
• Resynchronization: every 6 beats

Example 2: Chopin’s Fantaisie-Impromptu

The most famous example in Western classical music is Chopin’s Fantaisie-Impromptu op. 66. The right hand plays groups of 4 notes while the left hand plays groups of 3:

Figure 2 --- Polymeter 4 against 3 in Chopin’s Fantaisie-Impromptu. The right hand plays 3 groups of 4 notes (cyan) and the left hand plays 4 groups of 3 notes (orange) in the same measure of 12 eighth notes. The strong beats coincide only at the beginning.

For a measure of 12 eighth notes:

• The right hand plays 3 groups of 4 = 12 notes
• The left hand plays 4 groups of 3 = 12 notes

The result: the accents of the two hands almost never coincide, creating that characteristic floating sensation that Chopin sought.

Listen to this example

Listen to the first 30 seconds: the right hand “runs” in groups of 4 while the bass “pulses” in groups of 3. The effect is a continuous flow that seems to float above time.

• Valentina Lisitsa — Fantasie-Impromptu (video with hands)
• Krystian Zimerman — Fantasie-Impromptu (studio audio)
• Rousseau — MIDI visualization + hands

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Types of polymeter

Vertical polymeter (simultaneous)

Two (or more) different meters played at the same time. This is the Chopin example above.

Characteristics:

• Cycles have different lengths
• They periodically resynchronize (at the LCM)
• Creates a feeling of floating or ambiguity

Horizontal polymeter (sequential)

Change of meter over time, from one measure to the next. Less spectacular but very common.

Example: a piece that alternates between 4/4 and 7/8.

Hemiola

A very common special case: the superposition of 2 and 3. The word comes from Greek “hemi” (half) + “holos” (whole), meaning “one and a half.”

Figure 3 --- Hemiola 3:2. Six beats can be divided into 3 groups of 2 (cyan, top) or 2 groups of 3 (orange, bottom). The strong beats coincide only at the beginning.

Hemiola is ubiquitous in flamenco, jazz, and baroque music.

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Cultural examples

West Africa: the foundation of groove

Traditional African music is intrinsically polymetric. A typical percussion ensemble might include:

• Djembe solo: free phrases, improvisation
• Djembe accompaniment: 4-beat pattern
• Dununba (large bass, cylindrical drum): 3-beat pattern
• Sangban (medium bass): 6-beat pattern
• Kenkeni (high bass): constant pulse, the reference “click”

These patterns interlock in a complex way. The “one” (the strong beat, called downbeat) is often felt but not explicitly played, which disorients Western musicians accustomed to a clearly marked strong beat.

The “3 against 2” pattern is so fundamental that it has a name: cross-rhythm. It traveled with the African diaspora and is found in:

• Jazz (swing is essentially a 3:2 tension)
• Funk (James Brown’s groove)
• Salsa and Caribbean music
• Rock (the blues shuffle, a ternary rhythm based on triplets)

Listen to African polymeter

• Foli (documentary, 2010) — 10 minutes of pure Guinean polymeter, without commentary
• Doudou N’Diaye Rose — Djabote — master of Senegalese sabar drums
• Babatunde Olatunji — Drums of Passion — foundational album (1960) that introduced African polyrhythms to the West

India: the tala system

In Indian classical music (North Indian Hindustani and South Indian Carnatic traditions), the tala defines the rhythmic cycle. Unlike Western measures (2, 3, or 4 beats), a tala can be very long (up to 128 beats) and have a complex internal structure.

Example: Tintal (16 beats) — the most common tala in Hindustani music

Structure : 4 + 4 + 4 + 4 (four groups of 4 beats)
Accents  : X (sam)  2  0 (khali)  3
           dha dhin dhin dha | dha dhin dhin dha | dha tin  tin  ta  | ta  dhin dhin dha
           1   2    3    4     5   6    7    8     9   10   11   12    13  14   15   16

Decode the notation

• sam (X): beat 1, the absolute anchor point where everything resolves
• khali (0): “empty” beat, marked by an open hand gesture (no strike)
• dha, dhin, ta, tin: onomatopoeic syllables called bols (from Hindi bol, “speech”) representing tabla strokes
• Numbers 2, 3 indicate other secondary strong beats

The sam is the anchor point where everything resolves. Improvisers can create phrases that span several cycles (for example, a 19-beat phrase in a 16-beat cycle), creating a temporary polymeter that always resolves to the sam — often with a tihai (cadential triple repetition).

Listen to Indian talas

• Zakir Hussain — tabla solo — tabla virtuoso, improvisation in Tintal
• Shakti — John McLaughlin + Zakir Hussain — jazz-Indian fusion with extreme polymeter
• Trichy Sankaran — mridangam solo — master of Carnatic percussion

Ligeti: polymeter pushed to the extreme

In his Études for Piano (1985-2001), György Ligeti (Hungarian composer, 1923-2006) explored polymeter with systematic rigor. Étude No. 6 “Automne à Varsovie” superimposes:

• Right hand: groups of 3 notes
• Left hand: groups of 5 notes
• Result: a continuous flow where no clear pulse emerges

Ligeti explicitly drew inspiration from African polyrhythms (he had studied the Aka Pygmies) and the player piano of Conlon Nancarrow (American-Mexican composer, 1912-1997), who composed for player pianos capable of playing polymeters impossible for human hands.

Listen to Ligeti

• Étude No. 6 “Automne à Varsovie” — Pierre-Laurent Aimard — the reference interpreter
• Étude No. 4 “Fanfares” — Pierre-Laurent Aimard — polymeter 3:2:5, more accessible
• The effect is dizzying: the brain searches for a pulse and finds none, then eventually “lets go” and appreciates the flow.

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How to notate polymeter?

Traditional Western notation

Standard notation can represent polymeter, but in a limited way:

Figure 4 --- Traditional polymeter notation: triplets (3 notes in the time of 2, in cyan) are the only way to express metric superposition, as exceptions to the regular subdivision of eighth notes (orange).

Problems:

• Only one time signature per measure
• Polymetric voices must be expressed as “exceptions” (triplets, quintuplets)
• Illegible when polymeter is complex

BP3: an elegant solution

Bernard Bel (French CNRS researcher) designed BP3 (Bol Processor 3) in the 1990s to natively represent polymeter, especially for Indian music. Two key syntaxes:

The ratio — Compress N elements into one beat:

{3, do re mi}     -- 3 notes in the time of a single one
{4, fa sol la si} -- 4 notes in the time of a single one

How to read {3, do re mi}?

Curly braces { } indicate a special group in BP3.

• The first element (3) is the ratio: how many time units this group should occupy
• The following elements (do re mi) are the content

So {3, do re mi} means: “play do, re, mi in the space of 3 time units” If each note normally lasts 1 unit, then {3, do re mi} = 3 notes in 3 units = normal duration But {2, do re mi} = 3 notes in 2 units = accelerated (like a triplet)

The parallel — Superimpose voices:

{voix1, voix2}    -- voice1 and voice2 played simultaneously

Beware of the comma!

In BP3, the comma in {a, b} means simultaneity (a and b played at the same time). The space in a b means sequence (a then b).

So {voix1, voix2} = the two voices played in parallel, like a pianist’s two hands.

Example of 3 against 4 polymeter in BP3 (on generative grammars applied to music, see M4):

S → {voix_haute, voix_basse}
voix_haute → {3, do re mi} {3, fa sol la} {3, si do re} {3, mi fa sol}
voix_basse → {4, do mi sol do} {4, re fa la re} {4, mi sol si mi}

The two voices last the same total time, but with different subdivisions. BP3 automatically calculates the exact durations so that everything remains synchronized.

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Why MIDI and MusicXML fail

MIDI: no concept of meter

MIDI only knows point events with timestamps. It has no notion of:

• Measure
• Strong/weak beat
• Rhythmic grouping

For MIDI, a note that falls on beat 1 and a note that falls on beat 2 are just two events with different timestamps. Polymeter simply does not exist.

MusicXML: one measure at a time

MusicXML represents traditional notation. It allows:

• One time signature per measure
• Triplets and other tuplets (irregular groups: triplets, quintuplets, etc.)
• Multiple voices per staff

But it cannot represent two structurally different meters in parallel. A voice in 3/4 and a voice in 4/4 simultaneously? Impossible directly. It would require:

1. Finding a common denominator (12/8)
2. Writing both voices in this single meter
3. Using ties and tuplets to simulate the effect

The result is unreadable and loses the musical intention.

The comparative table

Concept	MIDI	MusicXML	BP3
Polymeter 3:2	Timestamps only	Tuplets in single meter	{3,a b c} vs {2,x y}
Two parallel meters	Impossible	Workaround (common measure)	{voix1, voix2}
Long cycle (tala 16 beats)	No cycle	One measure = max one cycle	Recursive grammar
Metric intention	Lost	Approximate	Preserved

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Key takeaways

• Polymeter is the superposition of several different meters or pulses.
• It is fundamental in many traditions: West Africa, India, Western classical music (Chopin, Brahms, Ligeti).
• The simplest case is hemiola (3 against 2), ubiquitous in jazz, funk, flamenco.
• MIDI cannot represent it because it only knows point events, with no notion of meter.
• MusicXML approximates it via tuplets in a unified meter, but loses the musical intention.
• BP3 represents it natively with its ratio {n, ...} and parallel {..., ...} syntaxes (see B5 for formalism details).

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To go further

• Agawu, K. (2003). Representing African Music. Routledge. — On African polymeter.
• Clayton, M. (2000). Time in Indian Music. Oxford University Press. — On the tala system.
• London, J. (2012). Hearing in Time. Oxford University Press. — Cognitive theory of meter.
• Bel, B. (1992). “Symbolic and sonic representations of sound-object structures”. — On BP3 and polymeter.

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Glossary

• Bol: In Indian music, an onomatopoeic syllable representing a percussion stroke (dha, tin, ta, etc.).
• BP3 (Bol Processor 3): Bernard Bel’s musical grammar software, designed to represent polymeter.
• Cross-rhythm: A rhythmic pattern where the accents of one voice systematically fall between those of another.
• Djembe: A goblet-shaped drum originating from West Africa, played with the hands.
• Downbeat: Strong beat, usually the first beat of a measure.
• Dunun: Family of cylindrical double-headed drums (dununba, sangban, kenkeni) playing patterns in West Africa.
• Hemiola: Specific polymeter where 2 and 3 are superimposed (6 beats = 2x3 = 3x2).
• Khali: In Indian music, an “empty” beat marked by an open hand gesture.
• Meter: Regular organization of beats into groups with a hierarchy of accents (strong/weak beats).
• LCM: Least Common Multiple — in polymeter, the point where two cycles resynchronize (e.g., LCM of 2 and 3 = 6).
• Polymeter: Superposition of several different meters (e.g., 3 against 4).
• Polyrhythm: Term sometimes synonymous with polymeter, sometimes distinguished (polyrhythm = different accents on the same meter).
• Sam: In Indian music, the first beat of the cycle (tala), the point of absolute resolution.
• Shuffle: A ternary rhythm in blues and rock, based on a triplet subdivision.
• Tala: Indian rhythmic cycle, structured in groups with specific strokes (e.g., Tintal = 16 beats).
• Tihai: In Indian music, a final cadence where a motif is repeated three times to fall on the sam.
• Tuplet: In Western notation, a group of compressed notes (triplet = 3 notes in the time of 2).

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Prerequisites: M1 — MIDI under the formal microscope, I2 — Bol Processor Reading time: 10 min Tags: #rhythm #polymeter #midi #bp3 #africa #india #chopin #ligeti

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Next article: M6 — Hierarchical structure in music: GTTM demystified

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