Tempering out

Tempering out is what a regular temperament (including rank-1 temperaments aka equal temperaments) does to a small interval like a comma: it makes it disappear, or as some authors put it, vanish.

Overview

For a tone measured as a ratio to "disappear", it must become equal to 1/1, so that multiplying by the ratio does not change anything. For a tone measured in cents to "disappear", it must become 0 cents, so that adding it does not change anything.

In both cases, that implies that we are 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. There are also options to temper the octave, such as TOP tuning.

Tempering together

Two or more chords or intervals are said to be tempered together if the commas that relate all of their corresponding steps are tempered out. If two chords are tempered together, then the representation of each of those chords in the temperament is identical.

For example, in meantone, which tempers out 81/80, the chords 54:64:81 (with steps 32/27, 81/64) and 10:12:15 (with steps 6/5, 5/4) are tempered together. Since 32⁄27 ⋅ 81⁄80 = 6⁄5 and 81⁄64 = 5⁄4 ⋅ 81⁄80, equating 81/80 with 1/1 also equates 32/27 with 6/5, and 81/64 with 5/4.

Example

The syntonic comma is 81/80. That is (3×3×3×3) / (5×2×2×2×2) or, in monzo form, [-4 4 -1⟩.

19edo tempers out 81/80. (Technically, we should say that 19edo 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 is a 5-limit comma. The 5-limit patent val for 19edo is ⟨19 30 44]. That means that you add 19 steps of 19edo 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 19edo gets you to 3/1, but that is only an approximation. Same with 5/1, etc. This is where the error in the primes gets introduced. Don't worry, though, it is very useful error.

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

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

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 2/1's 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 19edo means adding 0 steps of 19edo 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 19edo gets you to 3/1, but, as we say above, that is an error. One step of 19edo is the 19th root of 2, or 2^1/19, or approximately 1.03715504445. (That is 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 do not 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.

See also

• Fudging – or virtual tempering