Polyrhythm and polymeter

Clap a steady beat with one hand and, with the other, tap a rhythm that does not follow it. When two rhythms that do not coincide sound together, neither disappears: the ear hears both, and the friction between them becomes a rhythm of its own. That superposition is the subject here.

One rhythm or several

Med calls a single rhythm monorhythm, and the superposition of different rhythms polyrhythm, a plurality of combined rhythms (Med 1996, ch. XIX). He separates two kinds.

In a complementary polyrhythm, the rhythms share the same division and fill each other in, merging into one (Med 1996, ch. XIX). The upper part plays on the beat, the lower one between the beats, and together they make a line of even eighths:

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In a contradictory polyrhythm, each rhythm keeps its own identity, as when groups of two sound against triplets (Med 1996, ch. XIX). That is the one the ear hears as two layers.

Three against two

The most common contradictory polyrhythm puts a triplet against a normal division. Lacerda gives this simultaneity its own name, contra-ritmo: normal groups and tuplets performed at the same time, or different tuplets at the same time (Lacerda 1966, ch. XII).

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The two layers start each beat together and part in the middle: the second note of the triplet comes a little before the second eighth, and the third a little after. The same relation one level up, triplet quarters against quarters, spreads the friction over two beats:

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Four against three

With more notes the layers take longer to meet. A quadruplet of quarters in three-four puts four notes where the meter holds three, and the two parts only coincide again on the next downbeat:

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The hemiola

Polyrhythm does not need two parts. When three equal durations take the place of two within a single line, as in triplets, the relation is called hemiola, and Blatter points to it in many of Tchaikovsky's waltzes and in Leonard Bernstein's "America" from West Side Story (Blatter 2007, part 1, art. 4). In six-eight, a measure of two dotted quarters answered by a measure of three quarters:

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The six eighths are the same in both measures; what changes is how they group, two groups of three and then three groups of two. In three-four the same swap is written the other way round, with ties across the beat (Blatter 2007, part 1, art. 4):

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Polymeter

Polyrhythm superimposes rhythms inside one meter. Polymeter superimposes the meters themselves: measures of different kinds performed at the same time. Med calls them compassos mistos and adds the rule that makes them work: the first beats must coincide (Med 1996, ch. XIX). The name needs care, because other books use "mixed meter" for a single line whose measures add unequal groups, such as five-four as 3 + 2.

Three-four above two-four, the downbeats marked:

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The quarters line up one by one, but the accents do not: the upper part stresses every third quarter and the lower one every second. The two downbeats fall together at the start, drift apart, and meet again after six quarters, the smallest length both meters divide. That meeting is what the ear waits for.

What comes next

Polyrhythm and polymeter are everyday rhythm in much of the world's music, from the cross-rhythms of African and Afro-Brazilian percussion to the polymeters of twentieth-century composers. Listening for the layers, and for the moment they meet, is the first step to playing them.