M5) Polymetry

When Meters 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 polymetry — a fascinating musical phenomenon that our digital formats struggle to represent.

Where Does This Article Fit In?

Polymetry 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 (a transpiler from the BP3 language to SuperCollider, see I3 and B7).

Why Is It Important?

Polymetry 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 works by Chopin, Brahms, Ligeti. Yet, when you open music software, you are constrained to a single time signature per measure.

Understanding polymetry 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

Polymetry is the simultaneous superposition of several different meters (or pulses), creating a tension between multiple 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 (or metre): 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.

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

Let’s Explain Step by Step

Example 1: Clapping Your Hands

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

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 your palm flat), the subsequent ones are weak beats (tap with your 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 “1”. Be careful, it’s difficult! The palm/fingers distinction helps you feel where the strong beat falls for each hand — and these strong beats do not coincide.

 

Right (3)  : P   d   d   P   d   d   P   d   d   P   d   d
Left (2)   : P   d   P   d   P   d   P   d   P   d   P   d
Position   : 1   2   3   4   5   6   7   8   9   10  11  12
             ↑               ↑               ↑
         P+P together    P+d conflict    P+P together

 

(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 polymetry.

  • Right: 12 beats / 3 = 4 cycles of 3
  • Left: 12 beats / 2 = 6 cycles of 2
  • Re-synchronization: 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:

 

Main droite : [do ré mi fa] [sol la si do] [ré mi fa sol]...
              ----4 notes---- ----4 notes---- ----4 notes----

Main gauche : [do mi sol] [do mi sol] [do mi sol] [do mi sol]...
              ---3 notes-- ---3 notes-- ---3 notes-- ---3 notes--

 

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 quality 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 the beat.

Types of Polymetry

Vertical Polymetry (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 Polymetry (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”.

 

Hemiola 3:2 — 6 beats can be divided into:
- 3 groups of 2 : | x . x . x . |
- 2 groups of 3 : | x . . x . . |

 

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

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 the downbeat) is often felt but not explicitly played, which can disorient 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 can be 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 Polymetry

India: The Tala System

In classical Indian 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

 

Decoding 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
  • The 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 within a 16-beat cycle), creating temporary polymetry that always resolves to the sam — often with a tihai (a triple cadential repetition).

Listen to Indian Talas

Ligeti: Polymetry Pushed to the Extreme

In his Études pour piano (1985-2001), György Ligeti (Hungarian composer, 1923-2006) explored polymetry 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 performing polymetries impossible for human hands.

Listen to Ligeti

How to Notate Polymetry?

Traditional Western Notation

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

 

      3        3        3        3
    ┌─┴─┐    ┌─┴─┐    ┌─┴─┐    ┌─┴─┐
    ♪ ♪ ♪    ♪ ♪ ♪    ♪ ♪ ♪    ♪ ♪ ♪    (triplets = 3 in the time of 2)

    ♪   ♪    ♪   ♪    ♪   ♪    ♪   ♪    (normal eighth notes)

 

Problems:

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

BP3: An Elegant Solution

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

The Ratio — Compress N elements into one beat:

{3, do re mi}     -- 3 notes in the time of one
{4, fa sol la si} -- 4 notes in the time of 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} = both voices played in parallel, like a pianist’s two hands.

Example of 3 against 4 polymetry 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.

Why MIDI and MusicXML Fail

MIDI: No Concept of Meter

MIDI only knows point-in-time events with timestamps (time markers). 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. Polymetry simply does not exist.

MusicXML: One Measure at a Time

MusicXML represents traditional notation. It allows for:

  • 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? Directly impossible. 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 illegible and loses the musical intent.

The Comparative Table

Concept MIDI MusicXML BP3
Polymetry 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 = one cycle max Recursive grammar
Metric intent Lost Approximate Preserved

Key Takeaways

  • Polymetry 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), omnipresent in jazz, funk, flamenco.
  • MIDI cannot represent it because it only knows point-in-time events, with no notion of meter.
  • MusicXML approximates it via tuplets in a unified meter, but loses the musical intent.
  • BP3 represents it natively with its ratio {n, ...} and parallel {..., ...} syntaxes (see B5 for formalism details).

To Go Further

  • Agawu, K. (2003). Representing African Music. Routledge. — On African polymetry.
  • 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 polymetry.

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 polymetry.
  • 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: A family of two-headed cylindrical drums (dununba, sangban, kenkeni) playing patterns in West Africa.
  • Hemiola: A specific polymetry where 2 and 3 are superimposed (6 beats = 2×3 = 3×2).
  • Khali: In Indian music, an “empty” beat marked by an open hand gesture.
  • Meter: The regular organization of beats into groups with a hierarchy of accents (strong/weak beats).
  • LCM: Least Common Multiple — in polymetry, the point where two cycles resynchronize (e.g., LCM of 2 and 3 = 6).
  • Polymetry: Superposition of several different meters (e.g., 3 against 4).
  • Polyrhythm: A term sometimes synonymous with polymetry, 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).

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

Next article: M6 — Hierarchical Structure in Music: GTTM Demystified