If there is one term that causes confusion among the musical community like no other, it is the term "polyrhythm": the definitions of polyrhythm, polymeter, and hemiola are mixed and matched so often, that they've all but become synonymous for the three different distinct musical ideas that they are actually meant to represent. Hopefully this article will help in clearing up the misconceptions and falsities perpetuated by the ill informed musician ;)
Polyrhythm (sometimes referred to as a cross rhythm) - two or more different rhythms that occur in the same amount of space played against each other
Note: The prefix poly- means multiple thus a tuplet is NOT a polyrhythm. A tuplet played against a different tuplet (or some non tuplet rhythm), however, would be a polyrhythm
A septuplet that takes up a whole 4/4 measure played against half-notes would be called a 7:2 polyrhythm
Also note: polyrhythms are denoted with a colon
So now he we have the general idea, but how do you actually build a polyrhythm? To understand how the notes fit together, and what the polyrhythm should sound like, we use some of that useless baloney that we call "math" which we learnt back in third grade.
Let's say we want a 2:3 polyrhythm: first we take the lowest common multiple of those two numbers (6)
In 3, the measure is divided up as such: 1 2 3 4 5 6
In 2, the measure is divided up as such: 1 2 3 4 5 6
Now tap the part in the three with one hand, and with the other hand tap the part in 2
You're now playing a 2:3 polyrhythm
Now what you need to do is memorize the exact sound of that polyrhythm, and internalize it, so that if you need to play it at a faster tempo than is possible to count in 6, you can do it.
Obviously, for complex polyrhythms this method is ineffective (counting out 72 beats for a 8:9 polyrhythm for instance).
Another method would be to take two metronomes, and put them at the right ratio to each other (for instance, for a 3:4 polyrhythm, if one metronome is at 90 bpm and is the three, then the other metronome would be at 120 bpm and would be the four. 90:120 simplifies to 3:4.) The tricky part, however is having to start them at exactly the same time, which may be difficult to pull off correctly.
The last method for internalizing these rhythms would be to buy a metronome that can produce these sorts of rhythms. Since I'd rather not spend the money to buy one of them, however, I use the first link at the bottom of the page (it includes many different rhythms created by a computer metronome)
Now, when I say internalize them, I mean be able to say them/tap them you fingers/play them with your instrument using either rhythm as the base. Which base, you ask? The base that are belong to us? No, I mean which rhythm you are thinking of as the meter, and which you think of as the rhythm on top. If given a tempo from a metronome, and told to make a 4:3 polyrhythm, you could use the metronome tempo as the 3 or the 4. You need to be able think about the rhythm as if its in 3/4 with quadruplets on top AND as if it were in 4/4 with half note triplets. Ideally, you should be able to think about the rhythm and hear it as both, but scientific studies have shown that the mind has trouble juggling those two ideas at once, and would rather pick one out as the background/base and one as the foreground. It's the same idea as looking at following picture and seeing either two faces or a chalice: rather than see both pictures (or hear both rhythms as the meter) the mind picks one out as the background (the underlying meter).
So how might one use polyrhythms in music? Well let's say you have a piece of music with the guitar part already written out doing quarter notes in 3/4 time, though you'd like to spice the song up and make it more interesting. What you could do is have the bass, rather than play quarter notes to back up the guitar, play quarter note quadruplets (four notes that up the space of three quarter notes). Thus between the guitar and the bass you have a 4:3 polyrhythm
Another idea could be to have the polyrhythm played only by one instrument: for instance the whole band is playing in 5 while the drums play a 5:4 polyrhythm. With four limbs and extensive rhythmic sense and ability, the drums are the ideal instrument to have play polyrhythms, though other instruments have the ability to play them as well (albeit not to the extent that a drummer can). For instance, a guitarist could fingerpick a melody of triplets while playing a bass line on the lower strings in quarter notes, once again making that 3:4 polyrhythm
Polymeter - two different time signatures being played at the same time at the same tempo
A guitar playing a part in 3/4 and a bass playing a part in 4/4 at the same time so that they match up after the twelfth beat
Polymeter gives the effect of going "in and out of phase." By this I mean that the two voices playing the polymeter seem to "line up" at first, as if they are just playing a harmony part, but then seem to go "out of sync." A splendid example of this is King Crimson's "Frame by Frame:" at about 4:21 two guitarists start playing the same 7/8 part but then at 4:31 one guitar drops a note and starts playing in 6/8. When the guitar in 7/8 is on the sixth note, the other guitar starts a new measure. In the next measure of 7 the other guitar starts his measure of 6 on the second to last note of the measure in 7. The effect is that they seem to go "out of sync" for a bit but after 42 beats (the lowest common multiple of 6 and 7) they line up again.
To use it in your own music, just give one instrument a part in one meter, and another instrument a part in another meter. A simple example would be to take a melody you've written, and add a repeating harmony part played in a different time signature.
Now to make these incredibly complex, let's say we have a 4:5 polyrhythm between two guitars. What we could say is that the first guitar is in 4/4, but just for the sake of making things complex we could say the other guitar is in 5/5 (we could just write it in 4/4 with quintuplets, but this is more fun). What we could do then, however, is give the second guitar, which is in 5/5, a repeating phrase of seven notes. Thus we have a polymetric polyrhythm in 7/5 and 4/4!
Hemiola - a musical figure in which two groups of three are replaced by three groups of two to give the effect of shifting between a duple and triple meter
These musical figures are fairly common and are found in all types of music. There really isn't much to them though: it's basically just taking a 6/8 measure and instead of counting it 1 2 3 4 5 6, counting it 1 2 3 4 5 6 (or vice versa). It's just shifting from a three feel to a two feel, or a two feel to a three feel: the meter itself, however, does not change.
To find examples of these all these different rhythms start listening to some progressive rock/metal, jazz, and classical music (more of the modern type) bands such as Tool, Yes, King Crimson, Meshuggah, Frank Zappa and composers such as Steve Reich and Rachmaninoff are a good place to start.
Also, take a look at West African music (such as of the Yoruba and Ibo peoples) as well as classical South Indian Carnatic music.
Lastly, 6/8 Afro-Cuban though not explicitly a 2:3 polyrhythm, often has a feel which implies both 3 and 2 at the same time.
Hopefully, this article has helped you on the way to understanding and incorporating these complex musical figures into your own work. Now get out there and write some crazy polyrhythmic, polymetric, hemiola filled music!
Polyrhythm Sound Files.