Tempering out

Tempering out is what a regular temperament (including "rank one" temperaments like EDOs) does to a small interval like a comma: it makes it disappear.

Overview

For a tone measured as a ratio to "disappear", it must become equal to 1/1, so that multiplying by the ratio doesn't change anything.

For a tone measured in cents to "disappear", it must become 0 cents, so that adding it doesn't change anything.

In both cases, that implies that we're introducing some error into our tunings: Where we would use 3, for instance, we use a number slightly larger or smaller than 3. You can introduce error into any prime, and when tempering out a single comma you can choose to leave any given prime pure. In practice, many people leave 2 pure to achieve pure octaves. Systems that also temper out the octave are called TOP tuning.

Example

The syntonic comma is 81/80. That's 3*3*3*3 / 5*2*2*2*2 or, in monzo form, | -4 4 -1 > .

19 EDO tempers out 81/80. (Technically, we should say that 19 EDO tempers out 81/80 when you use the patent val.) You can see this in several ways:

1. Counting steps of the val

Because there are no primes larger than 5 in 81/80, we say it's a 5-limit comma. The 5-limit patent val for 19 EDO is < 19 30 44 |. That means that you add 19 steps of 19 EDO to get to 2/1, 30 steps to get closest to 3/1, and 44 steps to get closest to 5/1.

Note that, because this is an EDO, 19 steps gets you precisely to 2/1. We say that 30 steps of 19 EDO gets you to 3/1, but that's only an approximation. Same with 5/1, etc. This is where the error in the primes gets introduced. Don't worry, though, it's very useful error.

Getting to 81 is 3*3*3*3, or, with 19 EDO steps, 30+30+30+30 = 120 steps of 19 EDO.

Getting to 80 is 5*2*2*2*2, or, with 19 EDO steps, 44+19+19+19+19 = 120 steps of 19 EDO.

Getting to 81/80 means adding the steps needed to get to 81, and subtracting the steps needed to get to 80. 120 steps - 120 steps = 0 steps.

Applying the monzo to the val (also called getting the "homomorphism") is easier. Multiply the first number in the monzo (which represents the number of 2/1s in the comma) and by the first number in the val (which represents the number of steps it takes to get to 2/1), then multiply the second number in the monzo by the second number in the val, then the third by the third, and add them all together: (-4 * 19) + (4 * 30) + (-1 * 44) = 0 steps.

Therefore, adding 81/80 to any interval in 19 EDO means adding 0 steps of 19 EDO to it. In other words, 81/80 is effectively zero: 81/80 is "tempered out".

2. Painstakingly doing the math

We say that 30 steps of 19 EDO gets you to 3/1, but, as we say above, that's an error. One step of 19 EDO is the 19th root of 2, or 2^(1/19), or approximately 1.03715504445. (That's 63.15789474 cents.) If you multiply that by itself 19 times, you get exactly 2. But if you multiply that by itself 30 times, you don't get 3: You get 2.98751792330896. Similarly, multiplying it by 44 steps gets you 4.97877035785607 instead of 5.

If we plug in these values into 81/80, we see that 81/80 is tempered out:

81/80 = 3*3*3*3 / 5*2*2*2*2 = (3^4) / (5)*(2^4). // Substitute our values and you get

(2.98751792330896 ^ 4) / (4.97877035785607)*(2^4) 
= 79.66032573 / (4.97877035785607 * 16) 
= 79.66032573 / 79.66032573
= 1/1.