Patent val: Difference between revisions
Wikispaces>guest **Imported revision 247102257 - Original comment: ** |
Wikispaces>guest **Imported revision 247113399 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
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:guest|guest]] and made on <tt>2011-08-19 | : This revision was by author [[User:guest|guest]] and made on <tt>2011-08-19 23:34:34 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>247113399</tt>.<br> | ||
: The revision comment was: <tt></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> | 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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Note that these are the same answers you would get if you multiplied 31 times <1 1.585 2.322 2.807 3.459 | and rounded the result. | Note that these are the same answers you would get if you multiplied 31 times <1 1.585 2.322 2.807 3.459 | and rounded the result. | ||
= | =How this defines a rank 1 temperament= | ||
A val defines a | A val defines a rank 1 temperament by defining the deliberate introduction of an error into one or more primes. In 12 EDO, for instance, the perfect fifth (ratio 3/2, or exactly 1.5) is mapped to 700 cents, which is actually just barely flat: a ratio of 2^(700/1200), or 1.4983070769. | ||
As stated above, the 2/1 in the patent val is perfect. The patent val for 12 EDO, <12 19 28 34 42 (etc) |, implies that it takes 12 steps to get the octave. The patent val for 31 EDO, <31 49 72 87 107 (etc) |, implies that it takes 31 steps to get to the octave. | As stated above, the 2/1 in the patent val is perfect. The patent val for 12 EDO, <12 19 28 34 42 (etc) |, implies that it takes 12 steps to get the octave. The patent val for 31 EDO, <31 49 72 87 107 (etc) |, implies that it takes 31 steps to get to the octave. | ||
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The lesson here is that even though the errors in the primes are different for each EDO + patent val in these cases, 81/80 is still tempered out. However, that's not true for all commas; for instance, 12 EDO tempers out 128/125, the diesis, while 31 EDO does not; and 31 EDO tempers out 393216/390625, the Wuerschmidt comma, while 12 EDO does not.</pre></div> | The lesson here is that even though the errors in the primes are different for each EDO + patent val in these cases, 81/80 is still tempered out. However, that's not true for all commas; for instance, 12 EDO tempers out 128/125, the diesis, while 31 EDO does not; and 31 EDO tempers out 393216/390625, the Wuerschmidt comma, while 12 EDO does not.</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>Patent val</title></head><body><!-- ws:start:WikiTextTocRule:10:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --><a href="#Definition">Definition</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> | <a href="#A 12 EDO Example">A 12 EDO Example</a><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --> | <a href="#xAn expanded example for 31 EDO">An expanded example for 31 EDO</a><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --> | <a href="# | <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>Patent val</title></head><body><!-- ws:start:WikiTextTocRule:10:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --><a href="#Definition">Definition</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> | <a href="#A 12 EDO Example">A 12 EDO Example</a><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --> | <a href="#xAn expanded example for 31 EDO">An expanded example for 31 EDO</a><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --> | <a href="#How this defines a rank 1 temperament">How this defines a rank 1 temperament</a><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --> | <a href="#How this relates to commas">How this relates to commas</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> | ||
<!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definition</h1> | <!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definition</h1> | ||
Given N-edo, the equal division of the octave into N parts, we may for any prime p find a corresponding <a class="wiki_link" href="/p-limit">p-limit</a> <a class="wiki_link" href="/val">val</a> in a canonical manner by <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Scalar_multiplication" rel="nofollow">scalar multiplying</a> &lt;1 <a class="wiki_link" href="/log2">log2</a>(3) log2(5) ... log(p)| by N and rounding to the nearest integer. In general this is not guaranteed to be the most accurate available val, but if N-edo has enough relative accuracy in the p-limit, it will be. The name <em>patent</em> comes from the fact that &quot;patent&quot; in one sense of the word is a synonym for &quot;obvious&quot;; the patent val may or may not be the best choice but it's the obvious choice.<br /> | Given N-edo, the equal division of the octave into N parts, we may for any prime p find a corresponding <a class="wiki_link" href="/p-limit">p-limit</a> <a class="wiki_link" href="/val">val</a> in a canonical manner by <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Scalar_multiplication" rel="nofollow">scalar multiplying</a> &lt;1 <a class="wiki_link" href="/log2">log2</a>(3) log2(5) ... log(p)| by N and rounding to the nearest integer. In general this is not guaranteed to be the most accurate available val, but if N-edo has enough relative accuracy in the p-limit, it will be. The name <em>patent</em> comes from the fact that &quot;patent&quot; in one sense of the word is a synonym for &quot;obvious&quot;; the patent val may or may not be the best choice but it's the obvious choice.<br /> | ||
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Note that these are the same answers you would get if you multiplied 31 times &lt;1 1.585 2.322 2.807 3.459 | and rounded the result.<br /> | Note that these are the same answers you would get if you multiplied 31 times &lt;1 1.585 2.322 2.807 3.459 | and rounded the result.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name=" | <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="How this defines a rank 1 temperament"></a><!-- ws:end:WikiTextHeadingRule:6 -->How this defines a rank 1 temperament</h1> | ||
A val defines a | A val defines a rank 1 temperament by defining the deliberate introduction of an error into one or more primes. In 12 EDO, for instance, the perfect fifth (ratio 3/2, or exactly 1.5) is mapped to 700 cents, which is actually just barely flat: a ratio of 2^(700/1200), or 1.4983070769.<br /> | ||
<br /> | <br /> | ||
As stated above, the 2/1 in the patent val is perfect. The patent val for 12 EDO, &lt;12 19 28 34 42 (etc) |, implies that it takes 12 steps to get the octave. The patent val for 31 EDO, &lt;31 49 72 87 107 (etc) |, implies that it takes 31 steps to get to the octave.<br /> | As stated above, the 2/1 in the patent val is perfect. The patent val for 12 EDO, &lt;12 19 28 34 42 (etc) |, implies that it takes 12 steps to get the octave. The patent val for 31 EDO, &lt;31 49 72 87 107 (etc) |, implies that it takes 31 steps to get to the octave.<br /> | ||