Patent val: Difference between revisions

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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:guest|guest]] and made on <tt>2011-08-19 23:~~34:34~~ UTC</tt>.<br>

: This revision was by author [[User:guest|guest]] and made on <tt>2011-08-20 14:23:32 UTC</tt>.<br>

: The original revision id was <tt>~~247113399~~</tt>.<br>

: The original revision id was <tt>247216643</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>

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<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 [[p-limit]] [[Vals and Tuning Space|val]] contains the number of steps it takes to get to each prime number up to p, in prime number order:

< [2/1] [3/1] [5/1] [7/1] ... [p/1] |

Given N-EDO, the equal division of the octave into N parts, we may define vals that map a specific number of N-EDO steps to these primes.

For any prime p we can 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.

One way to think of this process is to first ask, "How many 1200-cent steps (octaves) does it take to get to each prime?" It takes one 1200-cent step to get to 2/1, log2(3) steps to get to 3/1, log2(5) to get to 5/1, and so on. This gives us <1 1.585 2.322 2.807 3.459 ... log2(p) |.

Then ask, "How many more N-EDO steps does it take to get to the same places?" One 12-EDO step is 1/12th of an octave, by definition; therefore, you need 12 times as many steps to reach 2/1, 3/1, 5/1... Similarly, one 31-EDO step is 1/31st of an octave, so you need 31 times as many steps to reach 2/1, 3/1, 5/1...

Thus, the way to get the patent val for N-EDO is to multiply <1 1.585 2.322 2.807 ... log2(p) | by N. Then, since you can't take fractional steps in an EDO, you round the results to the nearest integers.

=A 12 EDO Example=

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which is the **11-limit patent val for [[12edo]]**.

=An expanded example for 31 EDO=

=An alternate and 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:

As stated above, the val contains the number of steps it takes to get to a given prime number, in prime number order:

< [2/1] [3/1] [5/1] [7/1] [etc.] |

By definition, for any EDO, the number of steps to 2/1 is the EDO division: 31 for 31 EDO. The 2-limit patent val is < 31 |.

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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>

<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><a href="#Definition">Definition</a> | <a href="#A 12 EDO Example">A 12 EDO Example</a> | <a href="#xAn expanded example for 31 EDO">An expanded example for 31 EDO</a> | <a href="#How this defines a rank 1 temperament">How this defines a rank 1 temperament</a> | <a href="#How this relates to commas">How this relates to commas</a>

<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><a href="#Definition">Definition</a> | <a href="#A 12 EDO Example">A 12 EDO Example</a> | <a href="#xAn alternate and expanded example for 31 EDO">An alternate and expanded example for 31 EDO</a> | <a href="#How this defines a rank 1 temperament">How this defines a rank 1 temperament</a> | <a href="#How this relates to commas">How this relates to commas</a>

<h1 id="toc0"><a name="Definition"></a>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 />

A <a class="wiki_link" href="/p-limit">p-limit</a> <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">val</a> contains the number of steps it takes to get to each prime number up to p, in prime number order:<br />

&lt; [2/1] [3/1] [5/1] [7/1] ... [p/1] |<br />

Given N-EDO, the equal division of the octave into N parts, we may define vals that map a specific number of N-EDO steps to these primes.<br />

<br />

For any prime p we can find a corresponding p-limit val 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 />

One way to think of this process is to first ask, &quot;How many 1200-cent steps (octaves) does it take to get to each prime?&quot; It takes one 1200-cent step to get to 2/1, log2(3) steps to get to 3/1, log2(5) to get to 5/1, and so on. This gives us &lt;1 1.585 2.322 2.807 3.459 ... log2(p) |.<br />

<br />

Then ask, &quot;How many more N-EDO steps does it take to get to the same places?&quot; One 12-EDO step is 1/12th of an octave, by definition; therefore, you need 12 times as many steps to reach 2/1, 3/1, 5/1... Similarly, one 31-EDO step is 1/31st of an octave, so you need 31 times as many steps to reach 2/1, 3/1, 5/1...<br />

<br />

Thus, the way to get the patent val for N-EDO is to multiply &lt;1 1.585 2.322 2.807 ... log2(p) | by N. Then, since you can't take fractional steps in an EDO, you round the results to the nearest integers.<br />

<br />

<h1 id="toc1"><a name="A 12 EDO Example"></a>A 12 EDO Example</h1>

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which is the <strong>11-limit patent val for <a class="wiki_link" href="/12edo">12edo</a></strong>.<br />

<br />

<h1 id="toc2"><a name="xAn expanded example for 31 EDO"></a>An expanded example for 31 EDO</h1>

<h1 id="toc2"><a name="xAn alternate and expanded example for 31 EDO"></a>An alternate and expanded example for 31 EDO</h1>

~~The~~ val contains the number of steps it takes to get to a given prime number, in prime number order:<br />

As stated above, 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 />

By definition, for any EDO, the number of steps to 2/1 is the EDO division: 31 for 31 EDO. The 2-limit patent val is &lt; 31 |.<br />

@@ Line 1: / Line 1: @@
 <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:guest|guest]] and made on <tt>2011-08-19 23:34:34 UTC</tt>.<br>
+: This revision was by author [[User:guest|guest]] and made on <tt>2011-08-20 14:23:32 UTC</tt>.<br>
-: The original revision id was <tt>247113399</tt>.<br>
+: The original revision id was <tt>247216643</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 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">[[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]] &lt;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 [[p-limit]] [[Vals and Tuning Space|val]] contains the number of steps it takes to get to each prime number up to p, in prime number order:
+&lt; [2/1] [3/1] [5/1] [7/1] ... [p/1] |
+Given N-EDO, the equal division of the octave into N parts, we may define vals that map a specific number of N-EDO steps to these primes.
+For any prime p we can find a corresponding p-limit val in a canonical manner by [[http://en.wikipedia.org/wiki/Scalar_multiplication|scalar multiplying]] &lt;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.
+One way to think of this process is to first ask, "How many 1200-cent steps (octaves) does it take to get to each prime?" It takes one 1200-cent step to get to 2/1, log2(3) steps to get to 3/1, log2(5) to get to 5/1, and so on. This gives us &lt;1 1.585 2.322 2.807 3.459 ... log2(p) |.
+Then ask, "How many more N-EDO steps does it take to get to the same places?" One 12-EDO step is 1/12th of an octave, by definition; therefore, you need 12 times as many steps to reach 2/1, 3/1, 5/1... Similarly, one 31-EDO step is 1/31st of an octave, so you need 31 times as many steps to reach 2/1, 3/1, 5/1...
+Thus, the way to get the patent val for N-EDO is to multiply &lt;1 1.585 2.322 2.807 ... log2(p) | by N. Then, since you can't take fractional steps in an EDO, you round the results to the nearest integers.
 =A 12 EDO Example=
@@ Line 16: / Line 26: @@
 which is the **11-limit patent val for [[12edo]]**.
-=An expanded example for 31 EDO=
+=An alternate and 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:
+As stated above, the val contains the number of steps it takes to get to a given prime number, in prime number order:
 &lt; [2/1] [3/1] [5/1] [7/1] [etc.] |
 By definition, for any EDO, the number of steps to 2/1 is the EDO division: 31 for 31 EDO. The 2-limit patent val is &lt; 31 |.
@@ Line 69: / Line 79: @@
 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>
-<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Patent val&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:10:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;!-- ws:start:WikiTextTocRule:11: --&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:11 --&gt;&lt;!-- ws:start:WikiTextTocRule:12: --&gt; | &lt;a href="#A 12 EDO Example"&gt;A 12 EDO Example&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt; | &lt;a href="#xAn expanded example for 31 EDO"&gt;An expanded example for 31 EDO&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextTocRule:14: --&gt; | &lt;a href="#How this defines a rank 1 temperament"&gt;How this defines a rank 1 temperament&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt; | &lt;a href="#How this relates to commas"&gt;How this relates to commas&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt;
+<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Patent val&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:10:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;!-- ws:start:WikiTextTocRule:11: --&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:11 --&gt;&lt;!-- ws:start:WikiTextTocRule:12: --&gt; | &lt;a href="#A 12 EDO Example"&gt;A 12 EDO Example&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt; | &lt;a href="#xAn alternate and expanded example for 31 EDO"&gt;An alternate and expanded example for 31 EDO&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextTocRule:14: --&gt; | &lt;a href="#How this defines a rank 1 temperament"&gt;How this defines a rank 1 temperament&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt; | &lt;a href="#How this relates to commas"&gt;How this relates to commas&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt;
 &lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Definition&lt;/h1&gt;
-  Given N-edo, the equal division of the octave into N parts, we may for any prime p find a corresponding &lt;a class="wiki_link" href="/p-limit"&gt;p-limit&lt;/a&gt; &lt;a class="wiki_link" href="/val"&gt;val&lt;/a&gt; in a canonical manner by &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Scalar_multiplication" rel="nofollow"&gt;scalar multiplying&lt;/a&gt; &amp;lt;1 &lt;a class="wiki_link" href="/log2"&gt;log2&lt;/a&gt;(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 &lt;em&gt;patent&lt;/em&gt; comes from the fact that &amp;quot;patent&amp;quot; in one sense of the word is a synonym for &amp;quot;obvious&amp;quot;; the patent val may or may not be the best choice but it's the obvious choice.&lt;br /&gt;
+  A &lt;a class="wiki_link" href="/p-limit"&gt;p-limit&lt;/a&gt; &lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&gt;val&lt;/a&gt; contains the number of steps it takes to get to each prime number up to p, in prime number order:&lt;br /&gt;
+&amp;lt; [2/1] [3/1] [5/1] [7/1] ... [p/1] |&lt;br /&gt;
+Given N-EDO, the equal division of the octave into N parts, we may define vals that map a specific number of N-EDO steps to these primes.&lt;br /&gt;
+&lt;br /&gt;
+For any prime p we can find a corresponding p-limit val in a canonical manner by &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Scalar_multiplication" rel="nofollow"&gt;scalar multiplying&lt;/a&gt; &amp;lt;1 &lt;a class="wiki_link" href="/log2"&gt;log2&lt;/a&gt;(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 &lt;em&gt;patent&lt;/em&gt; comes from the fact that &amp;quot;patent&amp;quot; in one sense of the word is a synonym for &amp;quot;obvious&amp;quot;; the patent val may or may not be the best choice but it's the obvious choice.&lt;br /&gt;
+&lt;br /&gt;
+One way to think of this process is to first ask, &amp;quot;How many 1200-cent steps (octaves) does it take to get to each prime?&amp;quot; It takes one 1200-cent step to get to 2/1, log2(3) steps to get to 3/1, log2(5) to get to 5/1, and so on. This gives us &amp;lt;1 1.585 2.322 2.807 3.459 ... log2(p) |.&lt;br /&gt;
+&lt;br /&gt;
+Then ask, &amp;quot;How many more N-EDO steps does it take to get to the same places?&amp;quot; One 12-EDO step is 1/12th of an octave, by definition; therefore, you need 12 times as many steps to reach 2/1, 3/1, 5/1... Similarly, one 31-EDO step is 1/31st of an octave, so you need 31 times as many steps to reach 2/1, 3/1, 5/1...&lt;br /&gt;
+&lt;br /&gt;
+Thus, the way to get the patent val for N-EDO is to multiply &amp;lt;1 1.585 2.322 2.807 ... log2(p) | by N. Then, since you can't take fractional steps in an EDO, you round the results to the nearest integers.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="A 12 EDO Example"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;A 12 EDO Example&lt;/h1&gt;
@@ Line 79: / Line 99: @@
 which is the &lt;strong&gt;11-limit patent val for &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;&lt;/strong&gt;.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="xAn expanded example for 31 EDO"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;An expanded example for 31 EDO&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="xAn alternate and expanded example for 31 EDO"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;An alternate and expanded example for 31 EDO&lt;/h1&gt;
-  The val contains the number of steps it takes to get to a given prime number, in prime number order:&lt;br /&gt;
+  As stated above, the val contains the number of steps it takes to get to a given prime number, in prime number order:&lt;br /&gt;
 &amp;lt; [2/1] [3/1] [5/1] [7/1] [etc.] |&lt;br /&gt;
 By definition, for any EDO, the number of steps to 2/1 is the EDO division: 31 for 31 EDO. The 2-limit patent val is &amp;lt; 31 |.&lt;br /&gt;