Tempering out: Difference between revisions
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'''Tempering out''' is what a [[Regular_temperament|regular temperament]], including the "rank one" temperaments derived from a [[EDO|EDO]]s, does to a small interval like a [[Comma|comma]]: it makes it disappear. | |||
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 as a ratio to "disappear", it must become equal to 1/1, so that multiplying by the ratio doesn't change anything. | ||
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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. | 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. | ||
=Example= | |||
=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 > . | 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: | 19 EDO tempers out 81/80. (Technically, we should say that 19 EDO tempers out 81/80 when you use the [[Patent_val|patent val]].) You can see this in several ways: | ||
==1. Counting steps of the val== | ==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. | 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. | ||
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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". | 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== | ==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. | 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: | 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 | 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) | (2.98751792330896 ^ 4) / (4.97877035785607)*(2^4) | ||
= 79.66032573 / (4.97877035785607 * 16) | = 79.66032573 / (4.97877035785607 * 16) | ||
= 79.66032573 / 79.66032573 | = 79.66032573 / 79.66032573 | ||
= 1/1. | = 1/1. | ||
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Revision as of 00:00, 17 July 2018
Tempering out is what a regular temperament, including the "rank one" temperaments derived from a EDOs, does to a small interval like a comma: it makes it disappear.
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.
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.
| 2.98751792330896 |