Tempering out: Difference between revisions
Wikispaces>FREEZE No edit summary |
reworked. BTW: interesting article |
||
| Line 1: | Line 1: | ||
'''Tempering out''' is what a [[ | '''Tempering out''' is what a [[regular temperament]] (including "rank one" temperaments like [[EDO]]s) 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 as a ratio to "disappear", it must become equal to 1/1, so that multiplying by the ratio doesn't change anything. | ||
| Line 5: | Line 7: | ||
For a tone measured in cents to "disappear", it must become 0 cents, so that adding it 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. | 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 > . | 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 [[ | 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. | 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. | ||
| Line 27: | Line 31: | ||
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 | <pre> | ||
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. | ||
</pre> | |||
[[Category:Bomma]] | |||
[[Category:Method]] | |||
[[Category:Term]] | |||
[[Category:Theory]] | |||
[[Category: | |||
[[Category: | |||
[[Category: | |||