Polyrhythms are a way of introducing some freshness and novelty in the way you combine different rhythms played by different or a single instrument. Also referred to as cross-rhythm, a polyrhythm is essentially a combination of two or more different rhythms that follow the same basic pulse reference but subdivide it in different ways.
When played together they form an overall rhythmic phrase that is perceived as one. That said, normally one of the rhythmic parts being played is based on irrational rhythmic subdivision (using tuplets) against another that divides the pulse in multiples of two – although this is not mandatory as you can have a triplet played against a quintuplet (3:5).
What is not considered to be a polyrhythm is a subdivision of 4:8 because we would just end up having a 4 and another multiple of 4 that divides the beat in the exact same way but in half. The following polyrhythm example is the composite rhythm of three beats against two (3:2):
But you can combine other subdivisions as well like a 5:4 polyrhythm, or even have multiple layers of polyrhythms like 5:4:3:
Each will give you a distinct rhythmic feel that you can follow or be inspired by and, obviously, apply it to a drum/percussion part or to a melodic/harmonic instrument:
A 4:3 polyrhythm in the percussion while the main melody and chord changes are based on a quintuplet subdivision, making an overall 5:4:3 polyrhythm example
Traditional African or Indian music make extensive use of polyrhythm and by now this shouldn’t come as a surprise if you consider the influence of the harmonic series in timbre, pitch, rhythm, and how it influenced the way we organize and use our musical materials in cultures around the world.
When I wrote about the harmonic series, I mentioned that rhythm and pitch are the same thing only at different speeds. Because now the focus is on rhythm, we are interested in lower frequencies that can be perceived as beats or pulses.
The reproduction of the 3:2 polyrhythm can be based on relationships found in the harmonic series in the sense that we could play the rhythm of a fifth interval based on the 2nd and 3rd partial of the series:
That said, if you wish to play the rhythm of a major chord, the resulting polyrhythm would be something based on a 4:5:6 or a 3:4:5, as shown in previous examples.
When working with polyrhythms you should focus on the overall rhythmic feel and then build from there. Then you should be free from the regular polyrhythmic subdivision and, while influenced by it, proceed to a more natural rhythmic phrasing spread across the instruments.
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