Tempering out
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**Tempering out** is what a [[regular temperament]], including the "rank one" temperaments derived from a [[EDO]]s, 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. <span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: -25px; width: 1px;"> || 2.98751792330896 || </span>
Original HTML content:
<html><head><title>tempering out</title></head><body><strong>Tempering out</strong> is what a <a class="wiki_link" href="/regular%20temperament">regular temperament</a>, including the "rank one" temperaments derived from a <a class="wiki_link" href="/EDO">EDO</a>s, does to a small interval like a <a class="wiki_link" href="/comma">comma</a>: it makes it disappear.<br />
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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.<br />
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For a tone measured in cents to "disappear", it must become 0 cents, so that adding it doesn't change anything.<br />
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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.<br />
<!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Example"></a><!-- ws:end:WikiTextHeadingRule:0 -->Example</h1>
The syntonic comma is 81/80. That's 3*3*3*3 / 5*2*2*2*2 or, in monzo form, | -4 4 -1 > .<br />
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19 EDO tempers out 81/80. (Technically, we should say that 19 EDO tempers out 81/80 when you use the <a class="wiki_link" href="/patent%20val">patent val</a>.) You can see this in several ways:<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="Example-1. Counting steps of the val"></a><!-- ws:end:WikiTextHeadingRule:2 -->1. Counting steps of the val</h2>
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.<br />
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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.<br />
<br />
Getting to 81 is 3*3*3*3, or, with 19 EDO steps, 30+30+30+30 = 120 steps of 19 EDO.<br />
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Getting to 80 is 5*2*2*2*2, or, with 19 EDO steps, 44+19+19+19+19 = 120 steps of 19 EDO.<br />
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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.<br />
<br />
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.<br />
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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".<br />
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<!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="Example-2. Painstakingly doing the math"></a><!-- ws:end:WikiTextHeadingRule:4 -->2. Painstakingly doing the math</h2>
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.<br />
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If we plug in these values into 81/80, we see that 81/80 is tempered out:<br />
81/80 = 3*3*3*3 / 5*2*2*2*2 = (3^4) / (5)*(2^4). Substitute our values and you get<br />
(2.98751792330896 ^ 4) / (4.97877035785607)*(2^4) <br />
= 79.66032573 / (4.97877035785607 * 16) <br />
= 79.66032573 / 79.66032573<br />
= 1/1.<br />
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