Keenan Pepper's explanation of vals: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:~~clumma~~|~~clumma~~]] and made on <tt>2011-12-~~27 17~~:54:32 UTC</tt>.<br>

: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-12-28 00:12:17 UTC</tt>.<br>

: The original revision id was <tt>~~288601092~~</tt>.<br>

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The three whole numbers are the values of the val applied to the first three primes (that is, 2, 3, and 5). This is enough information to figure out the value of the val applied to any 5-limit JI interval. For higher-limit JI intervals, its value is undefined.

Every EDO has a particular val associated to it called the "patent val". This is the val where you simply round each prime to the nearest number of steps of the EDO. For example, the 5-limit patent val of 12edo ~~(denoted "h12")~~ is <12 19 28|. Now consider the interval 625/512. This JI interval is about 345 cents, so it's closer to 3 steps of 12edo than to 4 steps. However, 625/512 = (5/4)^4/(2/1), so in order to be consistent with the ~~h12~~ mapping of 5/4, it must be represented as 4 steps, even though it is not the best direct approximation of 625/512 available in 12edo. In other words, the result of tempering with a val is that primes are consistently represented as the best possible approximations, at the cost of sometimes using a worse approximation for composite intervals.

Every EDO has a particular val associated to it called the "patent val". This is the val where you simply round each prime to the nearest number of steps of the EDO. For example, the 5-limit patent val of 12edo is <12 19 28|. Now consider the interval 625/512. This JI interval is about 345 cents, so it's closer to 3 steps of 12edo than to 4 steps. However, 625/512 = (5/4)^4/(2/1), so in order to be consistent with the mapping of 5/4, it must be represented as 4 steps, even though it is not the best direct approximation of 625/512 available in 12edo. In other words, the result of tempering with a val is that primes are consistently represented as the best possible approximations, at the cost of sometimes using a worse approximation for composite intervals.

==Vals supporting temperaments==

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The three whole numbers are the values of the val applied to the first three primes (that is, 2, 3, and 5). This is enough information to figure out the value of the val applied to any 5-limit JI interval. For higher-limit JI intervals, its value is undefined.<br />

<br />

Every EDO has a particular val associated to it called the &quot;patent val&quot;. This is the val where you simply round each prime to the nearest number of steps of the EDO. For example, the 5-limit patent val of 12edo ~~(denoted &quot;h12&quot;)~~ is &lt;12 19 28|. Now consider the interval 625/512. This JI interval is about 345 cents, so it's closer to 3 steps of 12edo than to 4 steps. However, 625/512 = (5/4)^4/(2/1), so in order to be consistent with the ~~h12~~ mapping of 5/4, it must be represented as 4 steps, even though it is not the best direct approximation of 625/512 available in 12edo. In other words, the result of tempering with a val is that primes are consistently represented as the best possible approximations, at the cost of sometimes using a worse approximation for composite intervals.<br />

Every EDO has a particular val associated to it called the &quot;patent val&quot;. This is the val where you simply round each prime to the nearest number of steps of the EDO. For example, the 5-limit patent val of 12edo is &lt;12 19 28|. Now consider the interval 625/512. This JI interval is about 345 cents, so it's closer to 3 steps of 12edo than to 4 steps. However, 625/512 = (5/4)^4/(2/1), so in order to be consistent with the mapping of 5/4, it must be represented as 4 steps, even though it is not the best direct approximation of 625/512 available in 12edo. In other words, the result of tempering with a val is that primes are consistently represented as the best possible approximations, at the cost of sometimes using a worse approximation for composite intervals.<br />

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<h2 id="toc0"><a name="x-Vals supporting temperaments"></a>Vals supporting temperaments</h2>

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 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:clumma|clumma]] and made on <tt>2011-12-27 17:54:32 UTC</tt>.<br>
+: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-12-28 00:12:17 UTC</tt>.<br>
-: The original revision id was <tt>288601092</tt>.<br>
+: The original revision id was <tt>288620398</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 14: / Line 14: @@
 The three whole numbers are the values of the val applied to the first three primes (that is, 2, 3, and 5). This is enough information to figure out the value of the val applied to any 5-limit JI interval. For higher-limit JI intervals, its value is undefined.
-Every EDO has a particular val associated to it called the "patent val". This is the val where you simply round each prime to the nearest number of steps of the EDO. For example, the 5-limit patent val of 12edo (denoted "h12") is &lt;12 19 28|. Now consider the interval 625/512. This JI interval is about 345 cents, so it's closer to 3 steps of 12edo than to 4 steps. However, 625/512 = (5/4)^4/(2/1), so in order to be consistent with the h12 mapping of 5/4, it must be represented as 4 steps, even though it is not the best direct approximation of 625/512 available in 12edo. In other words, the result of tempering with a val is that primes are consistently represented as the best possible approximations, at the cost of sometimes using a worse approximation for composite intervals.
+Every EDO has a particular val associated to it called the "patent val". This is the val where you simply round each prime to the nearest number of steps of the EDO. For example, the 5-limit patent val of 12edo is &lt;12 19 28|. Now consider the interval 625/512. This JI interval is about 345 cents, so it's closer to 3 steps of 12edo than to 4 steps. However, 625/512 = (5/4)^4/(2/1), so in order to be consistent with the mapping of 5/4, it must be represented as 4 steps, even though it is not the best direct approximation of 625/512 available in 12edo. In other words, the result of tempering with a val is that primes are consistently represented as the best possible approximations, at the cost of sometimes using a worse approximation for composite intervals.
 ==Vals supporting temperaments==
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 The three whole numbers are the values of the val applied to the first three primes (that is, 2, 3, and 5). This is enough information to figure out the value of the val applied to any 5-limit JI interval. For higher-limit JI intervals, its value is undefined.&lt;br /&gt;
 &lt;br /&gt;
-Every EDO has a particular val associated to it called the &amp;quot;patent val&amp;quot;. This is the val where you simply round each prime to the nearest number of steps of the EDO. For example, the 5-limit patent val of 12edo (denoted &amp;quot;h12&amp;quot;) is &amp;lt;12 19 28|. Now consider the interval 625/512. This JI interval is about 345 cents, so it's closer to 3 steps of 12edo than to 4 steps. However, 625/512 = (5/4)^4/(2/1), so in order to be consistent with the h12 mapping of 5/4, it must be represented as 4 steps, even though it is not the best direct approximation of 625/512 available in 12edo. In other words, the result of tempering with a val is that primes are consistently represented as the best possible approximations, at the cost of sometimes using a worse approximation for composite intervals.&lt;br /&gt;
+Every EDO has a particular val associated to it called the &amp;quot;patent val&amp;quot;. This is the val where you simply round each prime to the nearest number of steps of the EDO. For example, the 5-limit patent val of 12edo is &amp;lt;12 19 28|. Now consider the interval 625/512. This JI interval is about 345 cents, so it's closer to 3 steps of 12edo than to 4 steps. However, 625/512 = (5/4)^4/(2/1), so in order to be consistent with the mapping of 5/4, it must be represented as 4 steps, even though it is not the best direct approximation of 625/512 available in 12edo. In other words, the result of tempering with a val is that primes are consistently represented as the best possible approximations, at the cost of sometimes using a worse approximation for composite intervals.&lt;br /&gt;
 &lt;br /&gt;
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