Patent val: Difference between revisions
Wikispaces>genewardsmith **Imported revision 246419827 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 246419929 - 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:genewardsmith|genewardsmith]] and made on <tt>2011-08-17 01: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-08-17 01:31:01 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>246419929</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> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">Given N-edo, the equal division of the octave into N parts, we may for any prime p find a corresponding [[p-limit]] [[val]] in a canonical manner by [[http://en.wikipedia.org/wiki/Scalar_multiplication|scalar multiplying]] <1 [[log2]](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 //patent// comes from the fact that "patent" in one sense of the word is a synonym for "obvious"; the patent val may or may not be the best choice but it's the obvious choice. | <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">[[toc|flat]] | ||
=Definition= | |||
Given N-edo, the equal division of the octave into N parts, we may for any prime p find a corresponding [[p-limit]] [[val]] in a canonical manner by [[http://en.wikipedia.org/wiki/Scalar_multiplication|scalar multiplying]] <1 [[log2]](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 //patent// comes from the fact that "patent" in one sense of the word is a synonym for "obvious"; the patent val may or may not be the best choice but it's the obvious choice. | |||
=A 12 EDO Example= | =A 12 EDO Example= | ||
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< 31 49 72 87 107 115 127 132 |</pre></div> | < 31 49 72 87 107 115 127 132 |</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>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 /> | <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:6:&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:6 --><!-- ws:start:WikiTextTocRule:7: --><a href="#Definition">Definition</a><!-- ws:end:WikiTextTocRule:7 --><!-- ws:start:WikiTextTocRule:8: --> | <a href="#A 12 EDO Example">A 12 EDO Example</a><!-- ws:end:WikiTextTocRule:8 --><!-- ws:start:WikiTextTocRule:9: --> | <a href="#xAn expanded example for 31 EDO=">An expanded example for 31 EDO=</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: --> | ||
<!-- ws:end:WikiTextTocRule:10 --><br /> | |||
<!-- 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 /> | |||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="A 12 EDO Example"></a><!-- ws:end:WikiTextHeadingRule:2 -->A 12 EDO Example</h1> | ||
multiplying 12 times &lt;1 1.585 2.322 2.807 3.459|<br /> | multiplying 12 times &lt;1 1.585 2.322 2.807 3.459|<br /> | ||
yields &lt;12 19.020 27.863 33.688 41.513|,<br /> | yields &lt;12 19.020 27.863 33.688 41.513|,<br /> | ||
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which is the <strong>11-limit patent val for <a class="wiki_link" href="/12edo">12edo</a></strong>.<br /> | which is the <strong>11-limit patent val for <a class="wiki_link" href="/12edo">12edo</a></strong>.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="xAn expanded example for 31 EDO="></a><!-- ws:end:WikiTextHeadingRule:4 -->An expanded example for 31 EDO=</h1> | ||
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The val contains the number of steps it takes to get to a given prime number, in prime number order:<br /> | The val contains the number of steps it takes to get to a given prime number, in prime number order:<br /> | ||