Optimal 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>2011-02-~~14 21~~:08:23 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-02-15 00:15:00 UTC</tt>.<br>

: The original revision id was <tt>~~201791450~~</tt>.<br>

: The original revision id was <tt>201827910</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">Given any collection of p-limit commas, there is a finite list of p-limit [[Patent val|patent vals]] tempering out the commas. The list is not guaranteed to contain any members, but in most actual circumstances it will. If the list is not empty, then among these patent vals will be found the unique patent val which has the lowest [[Tenney-Euclidean temperament measures|TE error]]; this is the //optimal (TE) patent val// for the temperament defined by the commas. Note that other defintions of error, such as maximum p-limit error, or maximum q-limit error where q is the largest odd number less than the prime above p, lead to different results.

~~Given~~ N-edo, and ~~the~~ odd ~~primes up to p, let us set e = max_q |round(Nlog2(~~q~~))/N - log2(q)|~~; ~~that is~~, e is the ~~maximum~~ absolute value of the ~~error~~ of the ~~patent val for~~ N-edo, ~~measured in octaves, for all of the odd primes q up to p. Then e~~ < 1/2N, ~~since the error cannot be more than half the size of a step of N-edo. From this~~ it follows that N < 1/2e, ~~or if~~ we ~~measure e in cents~~, we ~~can also say~~ N ~~< 600/e~~.

To limit the search range when finding the optimal patent val a useful observation is this: given N-edo, and an odd prime q <= p, if eq is the absolute value in cents of the difference between the tuning of q given by the [[POTE tuning]] and the POTE tuning rounded to the nearest N-edo value, then eq < 600/N, from which it follows that N < 600/eq. If N-edo defines an optimal patent val, then the patent val will be identical to the val obtained by rounding the POTE tuning to the nearest N-edo value. If we take the minimum value for 600/eq for the odd primes up to p, we therefore obtain an upper bound for N.

Below are tabulated some values.

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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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Optimal patent val</title></head><body>Given any collection of p-limit commas, there is a finite list of p-limit <a class="wiki_link" href="/Patent%20val">patent vals</a> tempering out the commas. The list is not guaranteed to contain any members, but in most actual circumstances it will. If the list is not empty, then among these patent vals will be found the unique patent val which has the lowest <a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures">TE error</a>; this is the <em>optimal (TE) patent val</em> for the temperament defined by the commas. Note that other defintions of error, such as maximum p-limit error, or maximum q-limit error where q is the largest odd number less than the prime above p, lead to different results.<br />

<br />

~~Given~~ N-edo, and ~~the~~ odd ~~primes up to p, let us set e = max_q |round(Nlog2(~~q~~))/N - log2(q)|~~; ~~that is~~, e is the ~~maximum~~ absolute value of the ~~error~~ of the ~~patent val for~~ N-edo, ~~measured in octaves, for all of the odd primes q up to p. Then e~~ &lt; 1/2N, ~~since the error cannot be more than half the size of a step of N-edo. From this~~ it follows that N &lt; 1/2e, ~~or if~~ we ~~measure e in cents~~, we ~~can also say~~ N ~~&lt; 600/e~~. <br />

To limit the search range when finding the optimal patent val a useful observation is this: given N-edo, and an odd prime q &lt;= p, if eq is the absolute value in cents of the difference between the tuning of q given by the <a class="wiki_link" href="/POTE%20tuning">POTE tuning</a> and the POTE tuning rounded to the nearest N-edo value, then eq &lt; 600/N, from which it follows that N &lt; 600/eq. If N-edo defines an optimal patent val, then the patent val will be identical to the val obtained by rounding the POTE tuning to the nearest N-edo value. If we take the minimum value for 600/eq for the odd primes up to p, we therefore obtain an upper bound for N.<br />

<br />

Below are tabulated some values.<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:genewardsmith|genewardsmith]] and made on <tt>2011-02-14 21:08:23 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-02-15 00:15:00 UTC</tt>.<br>
-: The original revision id was <tt>201791450</tt>.<br>
+: The original revision id was <tt>201827910</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">Given any collection of p-limit commas, there is a finite list of p-limit [[Patent val|patent vals]] tempering out the commas. The list is not guaranteed to contain any members, but in most actual circumstances it will. If the list is not empty, then among these patent vals will be found the unique patent val which has the lowest [[Tenney-Euclidean temperament measures|TE error]]; this is the //optimal (TE) patent val// for the temperament defined by the commas. Note that other defintions of error, such as maximum p-limit error, or maximum q-limit error where q is the largest odd number less than the prime above p, lead to different results.
-Given N-edo, and the odd primes up to p, let us set e = max_q |round(Nlog2(q))/N - log2(q)|; that is, e is the maximum absolute value of the error of the patent val for N-edo, measured in octaves, for all of the odd primes q up to p. Then e &lt; 1/2N, since the error cannot be more than half the size of a step of N-edo. From this it follows that N &lt; 1/2e, or if we measure e in cents, we can also say N &lt; 600/e.
+To limit the search range when finding the optimal patent val a useful observation is this: given N-edo, and an odd prime q &lt;= p, if eq is the absolute value in cents of the difference between the tuning of q given by the [[POTE tuning]] and the POTE tuning rounded to the nearest N-edo value, then eq &lt; 600/N, from which it follows that N &lt; 600/eq. If N-edo defines an optimal patent val, then the patent val will be identical to the val obtained by rounding the POTE tuning to the nearest N-edo value. If we take the minimum value for 600/eq for the odd primes up to p, we therefore obtain an upper bound for N.
 Below are tabulated some values.
@@ Line 118: / Line 118: @@
 <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;Optimal patent val&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Given any collection of p-limit commas, there is a finite list of p-limit &lt;a class="wiki_link" href="/Patent%20val"&gt;patent vals&lt;/a&gt; tempering out the commas. The list is not guaranteed to contain any members, but in most actual circumstances it will. If the list is not empty, then among these patent vals will be found the unique patent val which has the lowest &lt;a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures"&gt;TE error&lt;/a&gt;; this is the &lt;em&gt;optimal (TE) patent val&lt;/em&gt; for the temperament defined by the commas. Note that other defintions of error, such as maximum p-limit error, or maximum q-limit error where q is the largest odd number less than the prime above p, lead to different results.&lt;br /&gt;
 &lt;br /&gt;
-Given N-edo, and the odd primes up to p, let us set e = max_q |round(Nlog2(q))/N - log2(q)|; that is, e is the maximum absolute value of the error of the patent val for N-edo, measured in octaves, for all of the odd primes q up to p. Then e &amp;lt; 1/2N, since the error cannot be more than half the size of a step of N-edo. From this it follows that N &amp;lt; 1/2e, or if we measure e in cents, we can also say N &amp;lt; 600/e. &lt;br /&gt;
+To limit the search range when finding the optimal patent val a useful observation is this: given N-edo, and an odd prime q &amp;lt;= p, if eq is the absolute value in cents of the difference between the tuning of q given by the &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE tuning&lt;/a&gt; and the POTE tuning rounded to the nearest N-edo value, then eq &amp;lt; 600/N, from which it follows that N &amp;lt; 600/eq. If N-edo defines an optimal patent val, then the patent val will be identical to the val obtained by rounding the POTE tuning to the nearest N-edo value. If we take the minimum value for 600/eq for the odd primes up to p, we therefore obtain an upper bound for N.&lt;br /&gt;
 &lt;br /&gt;
 Below are tabulated some values.&lt;br /&gt;