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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | '''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. |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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| : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-10-29 20:58:34 UTC</tt>.<br>
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| : The original revision id was <tt>269909960</tt>.<br>
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| : The revision comment was: <tt></tt><br>
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| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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| <h4>Original Wikitext content:</h4>
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| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">**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.
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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. | | 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 > . |
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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 [[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: |
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| ==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". |
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| ==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. |
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| 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: |
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| 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 |
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| (2.98751792330896 ^ 4) / (4.97877035785607)*(2^4) | | (2.98751792330896 ^ 4) / (4.97877035785607)*(2^4) |
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| = 79.66032573 / (4.97877035785607 * 16) | | = 79.66032573 / (4.97877035785607 * 16) |
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| = 79.66032573 / 79.66032573 | | = 79.66032573 / 79.66032573 |
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| = 1/1. | | = 1/1. |
| <span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: -25px; width: 1px;">
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| || 2.98751792330896 ||
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| </span></pre></div>
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| <h4>Original HTML content:</h4>
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| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><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 &quot;rank one&quot; 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 &quot;disappear&quot;, 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 &quot;disappear&quot;, 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 />
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| <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Example"></a><!-- ws:end:WikiTextHeadingRule:0 -->Example</h1>
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| The syntonic comma is 81/80. That's 3*3*3*3 / 5*2*2*2*2 or, in monzo form, | -4 4 -1 &gt; .<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 />
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| <!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="Example-1. Counting steps of the val"></a><!-- ws:end:WikiTextHeadingRule:2 -->1. Counting steps of the val</h2>
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| 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 &lt; 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 />
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| 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 />
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| Applying the monzo to the val (also called getting the &quot;homomorphism&quot;) 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 &quot;tempered out&quot;.<br />
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| <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Example-2. Painstakingly doing the math"></a><!-- ws:end:WikiTextHeadingRule:4 -->2. Painstakingly doing the math</h2>
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| 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 />
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| 81/80 = 3*3*3*3 / 5*2*2*2*2 = (3^4) / (5)*(2^4). Substitute our values and you get<br />
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| (2.98751792330896 ^ 4) / (4.97877035785607)*(2^4) <br />
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| = 79.66032573 / (4.97877035785607 * 16) <br />
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| = 79.66032573 / 79.66032573<br />
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| = 1/1.<br />
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| <td>2.98751792330896<br />
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| </span></body></html></pre></div>
| | {| class="wikitable" |
| | |- |
| | | | 2.98751792330896 |
| | |} |
| | </span> [[Category:comma]] |
| | [[Category:method]] |
| | [[Category:term]] |
| | [[Category:theory]] |