# Ratio math

This page describes the way we manipulate ratios arithmetically. Some of it is just math, and some of it is specific to music.

I am not trying to format this particularly nicely, since a lot of these kinds of discussions come up on Facebook and similar places, and you may as well see it ugly here so you understand it when it's ugly there.

## About Ratios

Ratios are just one way of writing various kinds of numbers.

1/1, 2/1, 3/1, 4/1, 5/1, and so on are the same as 1, 2, 3, 4, 5, and so on. Put differently, when there is no bottom number (denominator), it's as if there was a denominator equal to 1.

Ratios can be reduced by dividing both top and bottom by the same numbers until you can't reduce them anymore. So:

54/24 = 27 / 12 [dividing top and bottom by 2] = 9 / 4 [dividing top and bottom by 3]

...so 54/24 can be reduced to 9/4.

(You can do this faster if you get used to prime factorization:

54/24 = 3*3*3*2 / 3*2*2*2 = 3*3 / 2*2 [by dividing top and bottom by 3*2] = 9/4.)

All of these numbers mean the same thing: 2, 2/1, 4/2, 8/4, 16/8. The top (numerator) is double the bottom (denominator).

It's legitimate to write things like 2/1 = 2 or 54/24 = 9/4

## Stacking ratios and finding their differences

When we "add intervals", or stack them, we're actually multiplying their ratios.

Multiplying ratios means multiplying the tops (numerators) and the bottoms (denominators).

(x/y) * (a/b) = xa / yb

So if we took a 5/4 major third and stacked a 6/5 minor third on top of it, the outer interval would be

(5/4) * (6/5) = 5*6 / 4*5 = 30 / 20 = 3/2.

3/2 is a perfect fifth, so that should make sense: The major third C-E with a minor third E-G stacked on top of it encompasses the perfect fifth C-G.

When we're finding the difference between intervals, we're actually dividing their ratios.

Remember that dividing by a ratio is the same as multiplying by its reciprocal (flip, then multiply).

(x/y) / (a/b) = (x/y) * (b/a) = xb / ya

When we take the difference between the perfect fifth, 3/2, and the major third, 5/4, we should get a minor third.

(3/2) / (5/4) = (3/2) * (4/5) = 12/10 = 6/5.

## Octave Reduction

When a number is larger than the octave, which is 2/1 (or equivalently just "2"), we octave-reduce it by dividing by 2/1.

9/4 is greater than 8/4, so it's greater than 2.

To octave-reduce, divide by 2.

(9/4) / (2/1) = (9/4) * (1/2) = 9*1 / 4*2 = 9/8.

Let's do it again, using 8/3.

(8/3) / (2/1) = (8/3) * (1/2) = (8*1) / (2*3) = 8/6 = 4/3.

As you can see, you can divide the top number in half (8/3 goes to 4/3) or double the bottom number (9/4 goes to 9/8) to have the same effect and avoid the annoying arithmetic.

Same kind of thing when a number is smaller than 1: Multiply by 2 until you get somewhere between 1 (the root) and 2 (the octave).

2/3 is less than 1.

2/3 * 2/1 = 2*2 / 3*1 = 4/3.

As you can see, you could also just double the numerator (top).

Let's do it again, using 5/12.

5/12 * 2 = 10/12, which is still less than 1.

10/12 * 2 = 20/12, which is between 1 (12/12) and 2 (24/12).

Reduce 20/12 to get 5/3.

As you can see, you could just cut the denominator in half.

Whichever way you're going, keep going until the ratio is between 1 and 2.