Although the word polyrhythm technically means "many rhythms", it's most commonly used to describe the layering of multiple time signatures. For example, Steve Vai's The Attitude Song, in the main riff, has the guitars playing in 7/8 and the drums in 4/4 simultaneously. How could this work? What happens is this: 4/4 is the same as 8/8. So, when one bar of 7/8 is complete, the bar of 4/4 still has one eighth note remaining, so the "1" beat of the second bar of the 7/8 rhythm falls on the last eighth note of the 4/4 rhythm.
Now, after the second 7/8 bar has finished, the "1" beat of the third 7/8 bar will fall on the second-last eighth note in the second bar of the 4/4 rhythm. What you will notice from the example, in which guitar 1 plays in 7/8, and guitar 2 plays in 4/4, is that the "1" beat of the two rythms become increasingly displaced from one another by one eighth note, until eight bars of 7/8 are complete, at which point the "1" beat of the 7/8 rhythm syncronises with the "1" beat of the 4/4 rhythm again.
Ex1:
D|->------->-----|->------->-----|->-3----->-----|
A|-----2---------|---3-2---------|-1-----------1-|
E|-0-1---3-1-0-1-|-1-----4-3-1-2-|-----1-2-0-1---|
|
D|->------->-------|->------->-------|->------->-------|
A|-----------------|-----------------|-1-1-1-1-1-1-1-1-|
E|-0-0-0-0-0-0-0-0-|-1-1-1-1-1-1-1-1-|-----------------|
etc.Notice the displacement? This causes a shifting effect that sounds cool when done properly. Often, when the two "1" beats re-synchronize, one of the instruments will change time the two instruments are in the same time.
Of course, you could layer any two, or three, time signatures you like. If you could find a musician who can count 128th notes, you could layer a 4/4 rhythm over a 127/128 rhythm, and they'd synchronize exactly every 128 bars! Now, some "homework": write and/or record music using some of these polyrhythmic layers:
4/4, 3/4.
5/4. 2/4
7/4, 4/4
5+4)/4, 6/4
5/8, 3/4
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