It seems like when I start talking about polyrhythms & polymeters in clinics and workshops I get some blank stares, which makes sense if you haven't spent too much time thinking about them, or studying them. I just wanted to get a simple explanation out there with an audio example and a written music example so you could come at the concept from a few different angles and perspectives. The bottom line is, any complex relationships require a deep understanding of what something sounds like, what it feels like to play, and also what it looks like written down.
A polymeter is two or more meters happening at the same time. These meters share a common subdivision, so in essence a polymeter is different groupings of the same note values being played alongside one another. For example, 3 sixteenth notes being played over and over at the same time as 4 sixteenth notes are being played over and over. They begin together, but then grow further apart in their starting note, until finally (3 beats later in this case) they re-align and begin the cycle again.
A polyrhythm is two or more evenly spaced note values with different subdivisions resolving within the same amount of time. For example, four quarter notes being played in the same amount of time as 3 half note triplets.
In the audio example you will hear four cymbal notes in the same amount of time that you hear 3 snare drum notes. You can practice shifting your perception as you listen to the track.
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Here are some ways you can change your perception as you listen to the the track, which is same thing over and over...
First you will most likely hear the snare drum as the main pulse, and you will probably also organize it as 3/4 time. With this establishment of the time you will perceive the cymbal as playing a dotted eighth pattern. This is also called a hemiola. Practice counting aloud with this pattern by saying 1,2,3,1,2,3,etc. This is a simple resolution of the pattern you are hearing. It resolves every measure. (before even trying to play these patterns, simply listen and count).
Say this sentence as you hear the rhythm - "KICK the Sil-ver Buck-et"
Secondly, we can start counting the complex resolution of the same pattern. To do this, you will count to four over and over with the snare drum pulse. This will require 3 measures of 4/4 time to resolve back to one. 1,2,3,4 1,2,3,4 1,2,3,4. Really let this settle into your inner core of time. Feel the groove of the pattern as you hear the dance of the cymbal carry you across the barlines.
We will still use "kick the silver bucket" but it will now be:
"KICK the Sil-ver Buck-et Kick the SIL-ver Buck-et Kick the Sil-ver BUCK-et"
Go back and fourth counting along with the snare drum - 1,2,3,1,2,3 to 1,2,3,4,1,2,3,4,1,2,3,4. This will help you establish the groove, and not just the sound. The groove is the bottom line. It has to feel good, not just sound interesting...
Third. We can now turn the polymeter into a polyrhythm by shifting our attention to the cymbal. This will become the new pulse foundation rather than the snare. Jump in and begin counting to 4 over and over with the cymbal. You will notice the snare is now in a triplet feel. You are now perceiving this pattern as a polyrhythm rather than a polymeter. You did it! So count with the cymbal 1,2,3,4 1,2,3,4, etc. and really feel what the snare is doing. You will sense a time signature that might be 12/8, or you may just feel it in 8th note triplets in 4/4 time. Your choice, your perception ;-) You are counting in a simple resolution of this polyrhythm. 4/4 time in this case allows the rhythms to resolve every measure.
Say this sentence as you listen to the rhythm - "THIS Rhy-thm Is a Trip"
Fourthly, we can move into the complex resolution of the polyrhythm 4:3. Continue to count with the ride but now you will only count to three over and over. 1,2,3 1,2,3 1,2,3, 1,2,3, etc. This is certainly the most complex of the four perceptions of this pattern. Over the bar line, triplet based, long resolution, etc. makes it a tricky one to keep track of. Stay with it. Once this one truly sinks in your playing will definitely take on a new shape through a new found freedom in your perception of time.
Now the sentence becomes "THIS Rhy-thm Is a TRIP This Rhy-thm IS a Trip This RHY-thm is a trip"
Now that you have a handle on listening to, and perceiving what is happening in a variety of ways you can begin to coordinate your hands to actually play the pattern and then apply the four counting approaches (aloud) again while playing it. This is a great workout. Don't only limit the coordination to your hands. Try it with all two limb combinations: RH/LH, RH/LF, RH/RF, LH/RF, LH/LF, RF/LF (and then switch the role of each limb...).