Equal divisions.  Let's start with twelve beats.  There are many ways (six, to be exact) that these twelve beats can be divided into equal notes.  They are
 

There are six ways to divide twelve beats because the number 12 has six divisors--1, 2, 3, 4, 6, and 12.  Remember that a divisor of 12 is a number that divides 12 with no remainder.

How many ways can 10 beats be divided equally?  First, list the divisors of 10: __________________
Then, write out the patterns.
 

Here is a pattern that occurs often in Latin and African drumming.  Drum 1 and drum 2 are playing at the same time.  In this case, they divide up six beats.
 

Polyrhythms.  A polyrhythm occurs when two (or more) instruments simultaneously divide a fixed number of beats into equal notes in different ways.  The polyrhythm in the previous example is called three against two.  That means one drum (Drum 2) divides the six beats into two notes and the other (Drum 1) divides it into three notes.  Note that 6 is divisible by both 2 and 3.

In order to write out a different polyrhythm, for example, two against five, we will need to find a number that is divisible by both 2 and 5.  What is the smallest number divisible by 2 and 5? _______.
Here is the pattern.  Fill in the abcdrums notation, and try it!
 

For each of the given polyrhythms, write the smallest number of beats that can be used to write out the rhythm.
 

What is the mathematical relationship between the polyrhythm and the smallest number of beats needed???
 

True polyrhythms.  Use abcdrums to compare the polyrhythm 2 against 3 to the polyrhythm 2 against 6.  Both need 6 beats, but the first one is a heck of a lot more interesting!  The problem with the second rhythm is that 2 divides 6, so you're not adding anything interesting rhythmically.  A true polyrhythm of the form m against n occurs when neither n nor m divides the other.  Which of the following are true polyrhythms?

2 against 5
6 against 8
6 against 2
6 against 3
7 against 15
1 against anything
2 against any odd number
2 against any even number

Reducible polyrhythms.  On a separate piece of paper, write out 6 against 8 and compare it to 3 against 4.  You will see that 6 against 8 is just 3 against 4 played twice!  A polyrhythm is reducible if it is composed of repeats of smaller polyrhythms.  For example, 2 against 10 is reducible because it's just 1 against 5 played twice.  Which of the polyrhythms above are reducible?  If m against n is not reducible, what is the mathematical relationship between m and n?

A prime number is a whole number (greater than 1) whose only divisors are 1 and itself.  For example, 7, 2, and 5 are prime numbers but 6 is not, because 2 and 3 divide it.  Musically, prime numbers of beats cannot be divided to form true polyrhythms.  Can you explain why?

It's also true that numbers that can be expressed as a prime number to some power have no true polyrhythms.  For example, you can't divide 16 beats to get a true polyrhythm, because 16 equals 2⁴.  To verify this, write out the possible divisions of 16 beats on your paper.

Fun with polyrhythms.  Write a rhythm cycle that includes a true polyrhythm.  For extra fun, write a rhythm cycle that has multiple polyrhythms (for example, 2 against 3 against 4) or even multiple true polyrhythms (for example, 2 against 3 against 5)!