Keenan Pepper's explanation of vals: Difference between revisions
Wikispaces>keenanpepper **Imported revision 288348732 - Original comment: ** |
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This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-12- | : This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-12-24 12:47:22 UTC</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> | 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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Temperaments other than equal temperaments (that is, rank-2 and higher) can be constructed out of vals. This operation can be denoted "v1 ^ v2" or "v1 & v2". One of the many possible ways to think about this operation is that the resulting temperament tempers out only those commas common to both vals. | Temperaments other than equal temperaments (that is, rank-2 and higher) can be constructed out of vals. This operation can be denoted "v1 ^ v2" or "v1 & v2". One of the many possible ways to think about this operation is that the resulting temperament tempers out only those commas common to both vals. | ||
For example, consider the statement "5-limit meantone is 12p & 19p"...</pre></div> | For example, consider the statement "5-limit meantone is 12p & 19p". Here's a list of the simplest commas tempered out of those two 5-limit equal temperaments: | ||
* 12p: 81/80, 128/125, 648/625, 2048/2025, 6561/6400... | |||
* 19p: 81/80, 3125/3072, 6561/6400, 15625/15552... | |||
In the 12p equal temperament, all of the commas in the first list vanish (are mapped to 0). In 19p, all of the commas in the second list vanish. In the temperament you get from combining them, "12p & 19p", only the commas common to both lists are tempered out. These are 81/80, 6561/6400 = (81/80)^2, (81/80)^3, (81/80)^4... In other words, it works out that 81/80 is the only basic, fundamental comma that vanishes in both 12p and 19p - all the other commas are powers of 81/80, so they automatically vanish if 81/80 vanishes. So we say that the 5-limit temperament "12p & 19p" is the same thing as "the 81/80 temperament" or "meantone". | |||
In practice, the easy way to find information about temperaments like this is to go to Graham Breed's temperament finder, http://x31eq.com/temper/net.html , type "12p 19p" into the equal temperaments field, and type "5" into the limit field. It tells you that the resulting temperament is called "meantone", it has 81/80 as its only "unison vector" (aka comma), and other information you might find useful. | |||
</pre></div> | |||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<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>Keenan's explanation of vals</title></head><body>A val is a function that assigns a whole number to a JI interval in a <strong>consistent</strong> way. Here's an example of what &quot;consistent&quot; means: if v is a val and x and y are JI intervals, then v(x+y) = v(x) + v(y), where x+y means the composition of the intervals x and y (so you add their cents values, which means you <strong>multiply</strong> their ratios). For example, if v is any val, then v(3/2) + v(4/3) = v(2/1). The mathematical term for this is a &quot;homomorphism&quot;.<br /> | <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>Keenan's explanation of vals</title></head><body>A val is a function that assigns a whole number to a JI interval in a <strong>consistent</strong> way. Here's an example of what &quot;consistent&quot; means: if v is a val and x and y are JI intervals, then v(x+y) = v(x) + v(y), where x+y means the composition of the intervals x and y (so you add their cents values, which means you <strong>multiply</strong> their ratios). For example, if v is any val, then v(3/2) + v(4/3) = v(2/1). The mathematical term for this is a &quot;homomorphism&quot;.<br /> | ||
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Temperaments other than equal temperaments (that is, rank-2 and higher) can be constructed out of vals. This operation can be denoted &quot;v1 ^ v2&quot; or &quot;v1 &amp; v2&quot;. One of the many possible ways to think about this operation is that the resulting temperament tempers out only those commas common to both vals.<br /> | Temperaments other than equal temperaments (that is, rank-2 and higher) can be constructed out of vals. This operation can be denoted &quot;v1 ^ v2&quot; or &quot;v1 &amp; v2&quot;. One of the many possible ways to think about this operation is that the resulting temperament tempers out only those commas common to both vals.<br /> | ||
<br /> | <br /> | ||
For example, consider the statement &quot;5-limit meantone is 12p &amp; 19p&quot;...</body></html></pre></div> | For example, consider the statement &quot;5-limit meantone is 12p &amp; 19p&quot;. Here's a list of the simplest commas tempered out of those two 5-limit equal temperaments:<br /> | ||
<br /> | |||
<ul><li>12p: 81/80, 128/125, 648/625, 2048/2025, 6561/6400...</li><li>19p: 81/80, 3125/3072, 6561/6400, 15625/15552...</li></ul><br /> | |||
In the 12p equal temperament, all of the commas in the first list vanish (are mapped to 0). In 19p, all of the commas in the second list vanish. In the temperament you get from combining them, &quot;12p &amp; 19p&quot;, only the commas common to both lists are tempered out. These are 81/80, 6561/6400 = (81/80)^2, (81/80)^3, (81/80)^4... In other words, it works out that 81/80 is the only basic, fundamental comma that vanishes in both 12p and 19p - all the other commas are powers of 81/80, so they automatically vanish if 81/80 vanishes. So we say that the 5-limit temperament &quot;12p &amp; 19p&quot; is the same thing as &quot;the 81/80 temperament&quot; or &quot;meantone&quot;.<br /> | |||
<br /> | |||
In practice, the easy way to find information about temperaments like this is to go to Graham Breed's temperament finder, <!-- ws:start:WikiTextUrlRule:32:http://x31eq.com/temper/net.html --><a class="wiki_link_ext" href="http://x31eq.com/temper/net.html" rel="nofollow">http://x31eq.com/temper/net.html</a><!-- ws:end:WikiTextUrlRule:32 --> , type &quot;12p 19p&quot; into the equal temperaments field, and type &quot;5&quot; into the limit field. It tells you that the resulting temperament is called &quot;meantone&quot;, it has 81/80 as its only &quot;unison vector&quot; (aka comma), and other information you might find useful.</body></html></pre></div> | |||