Polyrhythms

A polyrhythm is what happens when we mix two different time signatures and overlapped them, there are several different ways we can overlap them.

The first "problem" is figuring out how do they overlap in the most basic and direct way.

Let's start with the most common time signatures, 2/4, 3/4 and 4/4.

Let's say I want to overlap a 2/4 and a 3/4. The first thing to do is think in terms of basic math.

The total sum of "grids" on which we can think of this pattern will be the numerator of the first fraction, times the numerator of the second fraction. So two times three will be six grids on which to think of our pattern, and we'll have to have two rows since we're working with two time signatures.

The next step will be to insert the 2/4 on the first row, we'll put a one and a two, and each of them has to be the same size, so here each beat of the time signature will have three squares. Like so:

Now that we have our 2/4 on the grid we need to add our 3/4 on the other row, so place the beats one, two and three in a way that each beat has the same amount of squares:

As one can notice on the 3/4 each beat has two squares, and on the 2/4 each beat has three squares, this is because multiplication is a "fusion" of two or more numbers, the two time signatures are now arranged in a grid that facilitates visually how the thing should be played. We can think of each beat (individual squares) as a quarter note. If we wish to make the pattern more intuition friendly by making it shorter we can then think of it as each square is a quaver, which will make it be two squares a quarter note. So instead of six squares, the pattern will happen in three squares, this will come in handy when we get patterns that can only happen with very big grids (bigger numbers you get a bigger multiplication), for example, a 5/4 over a 4/4, just at first glance one can tell that we will need a grid with twenty squares, sense five times four is twenty, but using the method of reducing to half notes we will get a pattern inside a grid of ten squares, which will be a lot better for the brain to learn the pattern instinctively.

Let's then try out the 5/4 over a 4/4:

First, let's do a twenty square grid:

This pattern is enormous, how can we ever memorize such a long pattern?

Fear not, let's proceed with inserting each time signature, I didn't say yet but it's irrelevant if the time signature goes to the first or second row (depends on what one wishes to do, but it will still be the same polyrhythm just inverted). So this time let's put the 5/4 on the first row:

Again notice how each beat of the 5/4 has four squares so it can fit inside the 4/4.

Next, we will insert the 4/4 and each beat will have five squares:

And now again we come to that point where we think of each square as a quarter note, but because the pattern is too long we better reduce it to half and think of each square as a quaver, and so we will get this grid instead:

Notice for example, that beats number two of each time signature do not align, which means the beat two of the 5/4 is going on the downbeat and the beat two of the 4/4 happens on the upbeat, then on the beat four of the 4/4, we are on the upbeat of the eighth grid and immediately we hear the beat five of the 5/4 on the downbeat of the ninth grid and then the tenth grid is silent.

We can do the same downsizing for the 2/4 over 3/4, instead of six grids we will get three grids, like so:

Last one is a 4/4 over a 3/4:

Here four times three is twelve, reducing the grid to half will get us six squares: