Optimal patent val: Difference between revisions
Wikispaces>genewardsmith **Imported revision 201709182 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 201734222 - 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>2011-02-14 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-02-14 17:14:00 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>201734222</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> | ||
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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. | <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 | 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. | ||
Below are tabulated some values. | Below are tabulated some values. | ||
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[[Gamera]]: [[422edo]] | [[Gamera]]: [[422edo]] | ||
==7-limit rank three== | |||
1029/1000: [[19edo]] | |||
36/35: [[12edo]] | |||
525/512: [[45edo]] | |||
49/48: [[19edo]] | |||
50/49: [[48edo]] | |||
686/675: [[46edo]] | |||
64/63: [[27edo]] | |||
875/864: [[41edo]] | |||
3125/8087: [[94edo]] | |||
2430/2401: [[137edo]] | |||
245/243: [[196edo]] | |||
</pre></div> | </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>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 /> | <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 /> | <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 | 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 /> | ||
<br /> | <br /> | ||
Below are tabulated some values.<br /> | Below are tabulated some values.<br /> | ||
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<a class="wiki_link" href="/Rodan">Rodan</a>: <a class="wiki_link" href="/128edo">128edo</a><br /> | <a class="wiki_link" href="/Rodan">Rodan</a>: <a class="wiki_link" href="/128edo">128edo</a><br /> | ||
<a class="wiki_link" href="/Mothra">Mothra</a>: <a class="wiki_link" href="/31edo">31edo</a><br /> | <a class="wiki_link" href="/Mothra">Mothra</a>: <a class="wiki_link" href="/31edo">31edo</a><br /> | ||
<a class="wiki_link" href="/Gamera">Gamera</a>: <a class="wiki_link" href="/422edo">422edo</a></body></html></pre></div> | <a class="wiki_link" href="/Gamera">Gamera</a>: <a class="wiki_link" href="/422edo">422edo</a><br /> | ||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x-7-limit rank three"></a><!-- ws:end:WikiTextHeadingRule:4 -->7-limit rank three</h2> | |||
1029/1000: <a class="wiki_link" href="/19edo">19edo</a><br /> | |||
36/35: <a class="wiki_link" href="/12edo">12edo</a><br /> | |||
525/512: <a class="wiki_link" href="/45edo">45edo</a><br /> | |||
49/48: <a class="wiki_link" href="/19edo">19edo</a><br /> | |||
50/49: <a class="wiki_link" href="/48edo">48edo</a><br /> | |||
686/675: <a class="wiki_link" href="/46edo">46edo</a><br /> | |||
64/63: <a class="wiki_link" href="/27edo">27edo</a><br /> | |||
875/864: <a class="wiki_link" href="/41edo">41edo</a><br /> | |||
3125/8087: <a class="wiki_link" href="/94edo">94edo</a><br /> | |||
2430/2401: <a class="wiki_link" href="/137edo">137edo</a><br /> | |||
245/243: <a class="wiki_link" href="/196edo">196edo</a></body></html></pre></div> | |||