Patent val: Difference between revisions
Wikispaces>jdfreivald **Imported revision 246416533 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 246419827 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<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: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-08-17 01:30:11 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>246419827</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> | ||
| Line 8: | Line 8: | ||
<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">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= | ||
multiplying 12 times <1 1.585 2.322 2.807 3.459| | multiplying 12 times <1 1.585 2.322 2.807 3.459| | ||
yields <12 19.020 27.863 33.688 41.513|, | yields <12 19.020 27.863 33.688 41.513|, | ||
| Line 14: | Line 14: | ||
which is the **11-limit patent val for [[12edo]]**. | which is the **11-limit patent val for [[12edo]]**. | ||
= | =An expanded example for 31 EDO== | ||
The val contains the number of steps it takes to get to a given prime number, in prime number order: | The val contains the number of steps it takes to get to a given prime number, in prime number order: | ||
| Line 26: | Line 24: | ||
3/1 is 1901.96 in cents. | 3/1 is 1901.96 in cents. | ||
1901.96 cents / 38.70967742 cents/step = 49.13383752 steps. | 1901.96 cents / 38.70967742 cents/step = 49.13383752 steps. | ||
This is an EDO, | This is an EDO, but we can't take 0.13383752 steps. So we round. This is clearly closer to 49 steps, so that's the "obvious" or "patent" choice. The 3-limit patent val is < 31 49 | | ||
The 3-limit patent val is < 31 49 | | |||
Doing the same thing up through 17, and we get an 17-limit patent val of < 31 49 72 87 107 115 127 | | |||
< 31 49 72 87 107 115 127 | | |||
To | To see how to extend from one limit to another, we may look at what to do for 19/1 and use that to go from the 17-limit to the 19-limit. | ||
19/1 = 5097.51 cents, 5097.51 / 38.70967742 cents/step = 131.6857529 steps. Round to get 132. | |||
The 19-limit patent val is | 19/1 = 5097.51 cents, 5097.51 / 38.70967742 cents/step = 131.6857529 steps. Round to get 132. The 19-limit patent val is | ||
< 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>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 /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt; | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="A 12 EDO Example"></a><!-- ws:end:WikiTextHeadingRule:0 -->A 12 EDO Example</h1> | ||
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 /> | ||
rounded to &lt;12 19 28 34 42|,<br /> | rounded to &lt;12 19 28 34 42|,<br /> | ||
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 /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt; | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="xAn expanded example for 31 EDO="></a><!-- ws:end:WikiTextHeadingRule:2 -->An expanded example for 31 EDO=</h1> | ||
<br /> | |||
<br /> | |||
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 /> | ||
&lt; [2/1] [3/1] [5/1] [7/1] [etc.] |<br /> | &lt; [2/1] [3/1] [5/1] [7/1] [etc.] |<br /> | ||
| Line 57: | Line 51: | ||
3/1 is 1901.96 in cents.<br /> | 3/1 is 1901.96 in cents.<br /> | ||
1901.96 cents / 38.70967742 cents/step = 49.13383752 steps.<br /> | 1901.96 cents / 38.70967742 cents/step = 49.13383752 steps.<br /> | ||
This is an EDO, | This is an EDO, but we can't take 0.13383752 steps. So we round. This is clearly closer to 49 steps, so that's the &quot;obvious&quot; or &quot;patent&quot; choice. The 3-limit patent val is &lt; 31 49 |<br /> | ||
<br /> | |||
Doing the same thing up through 17, and we get an 17-limit patent val of &lt; 31 49 72 87 107 115 127 |<br /> | |||
<br /> | <br /> | ||
To see how to extend from one limit to another, we may look at what to do for 19/1 and use that to go from the 17-limit to the 19-limit.<br /> | |||
<br /> | <br /> | ||
19/1 = 5097.51 cents, 5097.51 / 38.70967742 cents/step = 131.6857529 steps. Round to get 132. The 19-limit patent val is<br /> | |||
19/1 = 5097.51 cents, 5097.51 / 38.70967742 cents/step = 131.6857529 steps. Round to get 132. | |||
The 19-limit patent val is<br /> | |||
&lt; 31 49 72 87 107 115 127 132 |</body></html></pre></div> | &lt; 31 49 72 87 107 115 127 132 |</body></html></pre></div> | ||