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:~~genewardsmith~~|~~genewardsmith~~]] and made on <tt>2011-09-04 10:20:32 UTC</tt>.<br>

: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-09-04 20:49:39 UTC</tt>.<br>

: The original revision id was <tt>~~250603170~~</tt>.<br>

: The original revision id was <tt>250708506</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>

<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">[[toc|flat]]

=Definition=

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:

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These deliberate errors ensure that certain commas get tempered out. The patent vals for both 12 EDO and 31 EDO temper out 81/80. Here are the calculations:

81 = 3*3*3*3. This can also be written as a power of a prime -- 3^4 -- or as a monzo -- | 0 3 >.

81 = 3*3*3*3. This can also be written as a power of a prime -- 3^4 -- or as a monzo -- | 0 4 >.

80 = 2*2*2*2*5. This can also be written as a product of powers of primes -- (2^4)*(5^1) -- or as a monzo -- | 2 0 1 >.

80 = 2*2*2*2*5. This can also be written as a product of powers of primes -- (2^4)*(5^1) -- or as a monzo -- | 4 0 1 >.

Substitute in the values for 81/80 in 12 EDO and get this: (2.9966141538^4) / (2^4)*(5.0396842) = 80.6349472 / 80.6349472 = 1/1.

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You're dividing 81 by 80, so (assuming we're starting at zero, though it works no matter where you start) you add the steps for 81 (+196) and subtract the steps for 80 (-196). 196-196 = 0. This means that it takes zero steps to reach 81/80 -- in other words, 81/80 "vanishes".

=Patent vals from real numbers=

Instead of assuming the patent val for N-edo comes from an integer N, we could define a patent val for X-edo, where X is any real number, in just the same way. For instance, the [[The Riemann Zeta Function and Tuning#The ~~Z function~~|Z-function]] maximum at 48.9451 leads to a 13-limit X-edo val of <49 78 114 137 169 181|, whereas the minimum at 49.1412 leads to an X-edo val of <49 78 114 138 170 182|. Meanwhile, the patent val, which is the X-edo val for X=49.0000 exactly, is <49 78 114 138 170 181|.</pre></div>

Instead of assuming the patent val for N-edo comes from an integer N, we could define a patent val for X-edo, where X is any real number, in just the same way. For instance, the [[The Riemann Zeta Function and Tuning#The%20Z%20function|Z-function]] maximum at 48.9451 leads to a 13-limit X-edo val of <49 78 114 137 169 181|, whereas the minimum at 49.1412 leads to an X-edo val of <49 78 114 138 170 182|. Meanwhile, the patent val, which is the X-edo val for X=49.0000 exactly, is <49 78 114 138 170 181|.</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 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> | <a href="#Patent vals from real numbers">Patent vals from real numbers</a>

~~<br />~~

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

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

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 />

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These deliberate errors ensure that certain commas get tempered out. The patent vals for both 12 EDO and 31 EDO temper out 81/80. Here are the calculations:<br />

<br />

81 = 3*3*3*3. This can also be written as a power of a prime -- 3^4 -- or as a monzo -- | 0 3 &gt;.<br />

81 = 3*3*3*3. This can also be written as a power of a prime -- 3^4 -- or as a monzo -- | 0 4 &gt;.<br />

80 = 2*2*2*2*5. This can also be written as a product of powers of primes -- (2^4)*(5^1) -- or as a monzo -- | 2 0 1 &gt;.<br />

80 = 2*2*2*2*5. This can also be written as a product of powers of primes -- (2^4)*(5^1) -- or as a monzo -- | 4 0 1 &gt;.<br />

<br />

Substitute in the values for 81/80 in 12 EDO and get this: (2.9966141538^4) / (2^4)*(5.0396842) = 80.6349472 / 80.6349472 = 1/1.<br />

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<br />

<h1 id="toc5"><a name="Patent vals from real numbers"></a>Patent vals from real numbers</h1>

Instead of assuming the patent val for N-edo comes from an integer N, we could define a patent val for X-edo, where X is any real number, in just the same way. For instance, the <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#The ~~Z function~~">Z-function</a> maximum at 48.9451 leads to a 13-limit X-edo val of &lt;49 78 114 137 169 181|, whereas the minimum at 49.1412 leads to an X-edo val of &lt;49 78 114 138 170 182|. Meanwhile, the patent val, which is the X-edo val for X=49.0000 exactly, is &lt;49 78 114 138 170 181|.</body></html></pre></div>

Instead of assuming the patent val for N-edo comes from an integer N, we could define a patent val for X-edo, where X is any real number, in just the same way. For instance, the <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#The%20Z%20function">Z-function</a> maximum at 48.9451 leads to a 13-limit X-edo val of &lt;49 78 114 137 169 181|, whereas the minimum at 49.1412 leads to an X-edo val of &lt;49 78 114 138 170 182|. Meanwhile, the patent val, which is the X-edo val for X=49.0000 exactly, is &lt;49 78 114 138 170 181|.</body></html></pre></div>

@@ 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:genewardsmith|genewardsmith]] and made on <tt>2011-09-04 10:20:32 UTC</tt>.<br>
+: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-09-04 20:49:39 UTC</tt>.<br>
-: The original revision id was <tt>250603170</tt>.<br>
+: The original revision id was <tt>250708506</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>
 <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">[[toc|flat]]
 =Definition=
 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:
@@ Line 76: / Line 75: @@
 These deliberate errors ensure that certain commas get tempered out. The patent vals for both 12 EDO and 31 EDO temper out 81/80. Here are the calculations:
-= 3*3*3*3. This can also be written as a power of a prime -- 3^4 -- or as a monzo -- | 0 3 &gt;.
+= 3*3*3*3. This can also be written as a power of a prime -- 3^4 -- or as a monzo -- | 0 4 &gt;.
-= 2*2*2*2*5. This can also be written as a product of powers of primes -- (2^4)*(5^1) -- or as a monzo -- | 2 0 1 &gt;.
+= 2*2*2*2*5. This can also be written as a product of powers of primes -- (2^4)*(5^1) -- or as a monzo -- | 4 0 1 &gt;.
 Substitute in the values for 81/80 in 12 EDO and get this: (2.9966141538^4) / (2^4)*(5.0396842) = 80.6349472 / 80.6349472 = 1/1.
@@ Line 94: / Line 93: @@
 You're dividing 81 by 80, so (assuming we're starting at zero, though it works no matter where you start) you add the steps for 81 (+196) and subtract the steps for 80 (-196). 196-196 = 0. This means that it takes zero steps to reach 81/80 -- in other words, 81/80 "vanishes".
 =Patent vals from real numbers=
-Instead of assuming the patent val for N-edo comes from an integer N, we could define a patent val for X-edo, where X is any real number, in just the same way. For instance, the [[The Riemann Zeta Function and Tuning#The Z function|Z-function]] maximum at 48.9451 leads to a 13-limit X-edo val of &lt;49 78 114 137 169 181|, whereas the minimum at 49.1412 leads to an X-edo val of &lt;49 78 114 138 170 182|. Meanwhile, the patent val, which is the X-edo val for X=49.0000 exactly, is &lt;49 78 114 138 170 181|.</pre></div>
+Instead of assuming the patent val for N-edo comes from an integer N, we could define a patent val for X-edo, where X is any real number, in just the same way. For instance, the [[The Riemann Zeta Function and Tuning#The%20Z%20function|Z-function]] maximum at 48.9451 leads to a 13-limit X-edo val of &lt;49 78 114 137 169 181|, whereas the minimum at 49.1412 leads to an X-edo val of &lt;49 78 114 138 170 182|. Meanwhile, the patent val, which is the X-edo val for X=49.0000 exactly, is &lt;49 78 114 138 170 181|.</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:12:&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:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextTocRule:14: --&gt; | &lt;a href="#A 12 EDO Example"&gt;A 12 EDO Example&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&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:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&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:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt; | &lt;a href="#How this relates to commas"&gt;How this relates to commas&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt; | &lt;a href="#Patent vals from real numbers"&gt;Patent vals from real numbers&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;
-&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;br /&gt;
+&lt;!-- ws:end:WikiTextTocRule:19 --&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;
-&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;
   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;
@@ Line 167: / Line 165: @@
   These deliberate errors ensure that certain commas get tempered out. The patent vals for both 12 EDO and 31 EDO temper out 81/80. Here are the calculations:&lt;br /&gt;
 &lt;br /&gt;
-= 3*3*3*3. This can also be written as a power of a prime -- 3^4 -- or as a monzo -- | 0 3 &amp;gt;.&lt;br /&gt;
+= 3*3*3*3. This can also be written as a power of a prime -- 3^4 -- or as a monzo -- | 0 4 &amp;gt;.&lt;br /&gt;
-= 2*2*2*2*5. This can also be written as a product of powers of primes -- (2^4)*(5^1) -- or as a monzo -- | 2 0 1 &amp;gt;.&lt;br /&gt;
+= 2*2*2*2*5. This can also be written as a product of powers of primes -- (2^4)*(5^1) -- or as a monzo -- | 4 0 1 &amp;gt;.&lt;br /&gt;
 &lt;br /&gt;
 Substitute in the values for 81/80 in 12 EDO and get this: (2.9966141538^4) / (2^4)*(5.0396842) = 80.6349472 / 80.6349472 = 1/1.&lt;br /&gt;
@@ Line 186: / Line 184: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Patent vals from real numbers"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Patent vals from real numbers&lt;/h1&gt;
-Instead of assuming the patent val for N-edo comes from an integer N, we could define a patent val for X-edo, where X is any real number, in just the same way. For instance, the &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#The Z function"&gt;Z-function&lt;/a&gt; maximum at 48.9451 leads to a 13-limit X-edo val of &amp;lt;49 78 114 137 169 181|, whereas the minimum at 49.1412 leads to an X-edo val of &amp;lt;49 78 114 138 170 182|. Meanwhile, the patent val, which is the X-edo val for X=49.0000 exactly, is &amp;lt;49 78 114 138 170 181|.&lt;/body&gt;&lt;/html&gt;</pre></div>
+ Instead of assuming the patent val for N-edo comes from an integer N, we could define a patent val for X-edo, where X is any real number, in just the same way. For instance, the &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#The%20Z%20function"&gt;Z-function&lt;/a&gt; maximum at 48.9451 leads to a 13-limit X-edo val of &amp;lt;49 78 114 137 169 181|, whereas the minimum at 49.1412 leads to an X-edo val of &amp;lt;49 78 114 138 170 182|. Meanwhile, the patent val, which is the X-edo val for X=49.0000 exactly, is &amp;lt;49 78 114 138 170 181|.&lt;/body&gt;&lt;/html&gt;</pre></div>