A polyrhythm is created between two or more rhythmic phrases when each phrase is aligned to its own (separate) timeline. As these phrases coexist with each other, a mathematical ratio can describe their relationship.

With polyrhythms, the ratio formed between the voices should be irregular, (i.e., division between the ratio’s terms should be imperfect, like 2:3, 3:4, 3:5 etc.). Think about it as measuring a distance using both centimeters and inches (but not using centimeters and millimeters, as they’re directly related).

This occurs when, for instance, one voice plays 8th note (quaver) triplets, and another plays straight eighths - the ratio would then be 3:2 which means that for every 3 sounds in one voice, there will be 2 sounds in another voice:

3: 1.2.3.1.2.3.1.2.3. etc.
2: 1..2..1..2..1..2..

In the above example, notice that

• at the top voice, sounds are spaced out every other beat (’beat’ here implies step), whereas
• at the bottom voice, sounds occur every 3 beats

which is directly related to the least common multiple of the two terms (6 in our case). Notice also that voices share a common starting point (i.e., the ‘1’) and the polyrhythmic phrasing is essentially ‘circular’ - voices start on ‘1’, play x sounds each, and return back to ‘1’ together.

So, if we want to create a polyrhythm of, say, 3:5, we need to

• find the least common multiple (3 x 5 = 15)
• place 3 sounds every 5 steps
• place 5 sounds every 3 steps

Here’s a visual representation:

5: 1..2..3..4..5..1..2..3..4..5..
3: 1....2....3....1....2....3.... etc.

Lounge Lizzards’ Voice of Chunk is in 6/8. At the very beginning, you can hear a polyrhythm of 2:3 (hi-hats and woodblocks).

Darrell Fitton’s (aka Bola) Forcasa 3 exhibits the ratio 3:4 between the hi-hat and bass drum.

Conlon Nancarrow composed pieces with incredible polyrhythmic juxtapositions - he even used ratios of pi : e (e.g., 3.14 : 2.71).

Gyorgy Ligeti’s Poeme Symphonique is written for 100 metronomes playing at different speeds.

This track by Pan Sonic’s Mika Vainio is generally weird and can be considered polyrhythmic - but don’t ask for a ratio, can’t be bothered counting the bleeps..