Intro to Polyrhythms
A polyrhythm consists of two different rhythms occuring at the same time, often described as "something against something" - "two against three," or "three against four," and so forth. Often, musicians have difficulty conceiving of ways to understand how to count these rhythms, but the answers often can be found with some relatively basic mathematics.
Some polyrhythms, at first glance, don't seem to offer an easy way to count both rhythms simultaneously, but when it comes to some of the simpler, more common polyrhythms, it's simply a matter of finding a common division - a "common denominator" - that can be applied to both rhythms. In this article, we'll take two fairly common counterrhythms, "two against three" and "three against four," and present several different ways of figuring out these polyrhythms mathematically. Once one has figured out the math, and learned to play the polyrhythm, the "feel" of the rhythm can be internalized, and the polyrhythm can be applied to a number of different meters, and notated a number of different ways.
2:3 or 3:2
This polyrhythm consists of simultaneously dividing the same amount of time into two equal parts, and three equal parts. A common way of notating this would be two eighth notes, set against three eighth note triplets - dividing a single quarter note into two and three equal parts, respectively. A common shorthand for polyrhythms is to have one of the numbers, followed by a colon, followed by the second number:
"two against three" = "2:3"
"three against two" = "3:2"
To start, find the lowest number into wish both numbers (two and three) can be divided, yielding a whole number. When dealing with fractions in a mathematics class, this is referred to as the "lowest common denominator." In music, we're finding a division of the beat that allows for both parts of the polyrhythm. For a 2:3 polyrhythm, the lowest number into which one can divide both two and three, is six. (The obvious math: six divided by two equals three; six divided by three equals two.) Usually, especially when smaller numbers are involved, you can get this common denominator by simply multiplying the two numbers together ("two times three equals six"), but this may not be the case if larger numbers are involved.
Setting aside meter and notation for the moment, let's take those six counts, and simply number them one through six. Dividing those six counts into two equal parts (the "two" of "two against three"), and three equal parts (the "three" of the polyrhythm), and you'll get the following:
2:
1 2 3
4 5 6 (every third count)
3:
1 2
3 4
5 6 (every second count)
Then, in order to play the polyrhythm, count all six counts aloud; play one line's highlighted counts with one hand, and the second line's highlighted counts with the other hand. (Obviously, for drum set players, the polyrhythm can be applied in myriad ways to all four limbs, not just the hands.)
Applying the polyrhythm to a meter
In any polyrhythm, either of the basic divisions can be considered the "beat," or the primary
pulse. (In some styles of music,
both can function as the "beat," adding a very interesting rhythmic ambiguity to the song, groove, or composition, giving not only a polyrhythmic but
polymetric quality to the music.) By expressing the same polyrhythm in different meter signatures, the "feel" and application(s) of the polyrhythm can be varied.
The way the 2:3 or (3:2) rhythm has been presented in this article:
2:
1 2 3
4 5 6
3:
1 2
3 4
5 6
...would fit perfectly into a 6/8 meter signature, with each count signifying one eighth note division. However, that's not the only way to count six eighth note counts. 3/4 time also contains six eighth note counts, commonly counted as:
1 & 2 & 3 &
Counting in 3/4 time, the same 2:3 (or 3:2) polyrhythm could be expressed as:
2:
1 & 2
& 3 &
3:
1 &
2 &
3 &
There are also many other ways of notating 2:3 or 3:2, using a variety of note values.
3:4 or 4:3
Using the same methods presented above, here are two ways to count a "three aganst four" or "four against three" polyrhythm.
1) Find the lowest common denominator - three times four equals 12.
2) Simultaneously divide 12 counts into three and four equal parts:
3:
1 2 3 4
5 6 7 8
9 10 11 12 (every fourth count)
4:
1 2 3
4 5 6
7 8 9
10 11 12 (every third count)
3) (optional) Adapt the count to fit a certain meter signature:
12/8 -
3:
1 2 3 4
5 6 7 8
9 10 11 12
4:
1 2 3
4 5 6
7 8 9
10 11 12
4/4 (divided into triplets) -
3:
1 trip let two
trip let three trip
let four trip let
4:
1 trip let
two trip let
three trip let
four trip let
3/4 (divided into sixteenth notes) -
3:
1 e & a
2 e & a
3 e & a (every fourth count)
4:
1 e &
a 2 e
& a 3
e & a (every third count)
Conclusion
With some relatively basic math skills, it is possible to figure out ways to count many polyrhythms, and to find different ways to count the same polyrhythm in different meter signatures. Once one has figured out how to count a given polyrhythm, it becomes easier to internalize - "feel" - the polyrhythm, and for drum set players to perform it using various combinations of one's limbs.
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