Optimal patent val: Difference between revisions
Wikispaces>genewardsmith **Imported revision 207595086 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 209300348 - 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>2011-03- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-03-10 13:20:32 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>209300348</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> | ||
<h4>Original Wikitext content:</h4> | <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">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 definitions 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">[toc|flat] | ||
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 definitions 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. | |||
By tempering a JI scale using N-edo found on the list below we automatically temper it to the corresponding temperament. This can be done in [[http://www.huygens-fokker.org/scala/|Scala]] using the Quantize command: either type in "Quantize/consistent N" on the bottom, or use the pull-down menu under "Modify", check the box saying "Consistent" and type N (without a decimal point) into "Resolution". | By tempering a JI scale using N-edo found on the list below we automatically temper it to the corresponding temperament. This can be done in [[http://www.huygens-fokker.org/scala/|Scala]] using the Quantize command: either type in "Quantize/consistent N" on the bottom, or use the pull-down menu under "Modify", check the box saying "Consistent" and type N (without a decimal point) into "Resolution". | ||
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Below are tabulated some values. In each case an identifier which uniquely identifies the temperament in question is given. In the codimension one case, where the temperament is defined by a single comma, the comma is given and used as a name. In other cases, for a temperament of rank n, n independent [[Patent val|patent vals]] are given. Normally this is by way of integers conjoined by ampersands, such as 2&10 for 7-limit pajara. This tells us we can use the 7-limit patent vals for 2 and 10 to define the temperament. In case n independent patent vals cannot be found, the [[Normal lists|normal val list]] is given. Note that this differs from the method of [[Graham Breed]]'s [[http://x31eq.com/temper/|regular temperament finder]], which does not assume the numbers refer to patent vals. | Below are tabulated some values. In each case an identifier which uniquely identifies the temperament in question is given. In the codimension one case, where the temperament is defined by a single comma, the comma is given and used as a name. In other cases, for a temperament of rank n, n independent [[Patent val|patent vals]] are given. Normally this is by way of integers conjoined by ampersands, such as 2&10 for 7-limit pajara. This tells us we can use the 7-limit patent vals for 2 and 10 to define the temperament. In case n independent patent vals cannot be found, the [[Normal lists|normal val list]] is given. Note that this differs from the method of [[Graham Breed]]'s [[http://x31eq.com/temper/|regular temperament finder]], which does not assume the numbers refer to patent vals. | ||
=5-limit rank two= | |||
Comma: Optimal patent val: 1000 * badness | Comma: Optimal patent val: 1000 * badness | ||
16/15: [[8edo]] 14.884 | 16/15: [[8edo]] 14.884 | ||
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7629394531250/7625597484987: [[3501edo]] 17.191 | 7629394531250/7625597484987: [[3501edo]] 17.191 | ||
=7-limit rank two= | |||
Name: Optimal patent val: Val name: 1000*badness | Name: Optimal patent val: Val name: 1000*badness | ||
[[Trienstonic clan|Father]]: [[5edo]] [<1 0 4 -2|, <0 1 -1 3|] 21.312 | [[Trienstonic clan|Father]]: [[5edo]] [<1 0 4 -2|, <0 1 -1 3|] 21.312 | ||
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[[Ragismic microtemperaments|Supermajor]]: [[6214edo]] 80&171 10.836 | [[Ragismic microtemperaments|Supermajor]]: [[6214edo]] 80&171 10.836 | ||
=7-limit rank three= | |||
Comma: Optimal patent val: 10^6 * badness | Comma: Optimal patent val: 10^6 * badness | ||
36/35: [[12edo]] 63.675 | 36/35: [[12edo]] 63.675 | ||
| Line 148: | Line 150: | ||
78125000/78121827: [[101654edo]] 20.457 | 78125000/78121827: [[101654edo]] 20.457 | ||
=11-limit rank two= | |||
Name: Optimal patent val: Val name: 1000*badness | Name: Optimal patent val: Val name: 1000*badness | ||
[[Marvel temperaments|Negric]]: [[9edo]] 1&9 30.617 | [[Marvel temperaments|Negric]]: [[9edo]] 1&9 30.617 | ||
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[[Ragismic microtemperaments|Hemiennealimmal]]: [[1566edo]] 72&198 6.283 | [[Ragismic microtemperaments|Hemiennealimmal]]: [[1566edo]] 72&198 6.283 | ||
=11-limit rank three= | |||
Name: Optimal patent val: Val name: 10^6 * badness | Name: Optimal patent val: Val name: 10^6 * badness | ||
[[Archytas family|Vulcan]]: [[15edo]] 2&3&10 577.113 | [[Archytas family|Vulcan]]: [[15edo]] 2&3&10 577.113 | ||
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[[Kalismic temperaments|Odin]]: [[7464edo]] 12&42&72 116.151 | [[Kalismic temperaments|Odin]]: [[7464edo]] 12&42&72 116.151 | ||
=13-limit rank two= | |||
Name: Optimal patent val: Val name: 1000*badness | Name: Optimal patent val: Val name: 1000*badness | ||
[[Quartonic temperaments|Progression]]: [[9edo]] [<1 1 2 2 3 3|, <0 5 3 7 4 6|] 18.158 | [[Quartonic temperaments|Progression]]: [[9edo]] [<1 1 2 2 3 3|, <0 5 3 7 4 6|] 18.158 | ||
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[[Ragismic microtemperaments|Abigail]]: [[1258edo]] 46&178 8.856 | [[Ragismic microtemperaments|Abigail]]: [[1258edo]] 46&178 8.856 | ||
=13-limit rank three= | |||
Name: Optimal patent val: Val name: 10^5 * badness | Name: Optimal patent val: Val name: 10^5 * badness | ||
[[Starling family|Mockingbird]]: [[15edo]] 3&4&15 85.942 | [[Starling family|Mockingbird]]: [[15edo]] 3&4&15 85.942 | ||
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[[Breed family|Freya]]: [[2615edo]] 10&31&229 85.515</pre></div> | [[Breed family|Freya]]: [[2615edo]] 10&31&229 85.515</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 definitions 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>[toc|flat]<br /> | ||
<br /> | |||
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 definitions 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 /> | ||
By tempering a JI scale using N-edo found on the list below we automatically temper it to the corresponding temperament. This can be done in <a class="wiki_link_ext" href="http://www.huygens-fokker.org/scala/" rel="nofollow">Scala</a> using the Quantize command: either type in &quot;Quantize/consistent N&quot; on the bottom, or use the pull-down menu under &quot;Modify&quot;, check the box saying &quot;Consistent&quot; and type N (without a decimal point) into &quot;Resolution&quot;.<br /> | By tempering a JI scale using N-edo found on the list below we automatically temper it to the corresponding temperament. This can be done in <a class="wiki_link_ext" href="http://www.huygens-fokker.org/scala/" rel="nofollow">Scala</a> using the Quantize command: either type in &quot;Quantize/consistent N&quot; on the bottom, or use the pull-down menu under &quot;Modify&quot;, check the box saying &quot;Consistent&quot; and type N (without a decimal point) into &quot;Resolution&quot;.<br /> | ||
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Below are tabulated some values. In each case an identifier which uniquely identifies the temperament in question is given. In the codimension one case, where the temperament is defined by a single comma, the comma is given and used as a name. In other cases, for a temperament of rank n, n independent <a class="wiki_link" href="/Patent%20val">patent vals</a> are given. Normally this is by way of integers conjoined by ampersands, such as 2&amp;10 for 7-limit pajara. This tells us we can use the 7-limit patent vals for 2 and 10 to define the temperament. In case n independent patent vals cannot be found, the <a class="wiki_link" href="/Normal%20lists">normal val list</a> is given. Note that this differs from the method of <a class="wiki_link" href="/Graham%20Breed">Graham Breed</a>'s <a class="wiki_link_ext" href="http://x31eq.com/temper/" rel="nofollow">regular temperament finder</a>, which does not assume the numbers refer to patent vals.<br /> | Below are tabulated some values. In each case an identifier which uniquely identifies the temperament in question is given. In the codimension one case, where the temperament is defined by a single comma, the comma is given and used as a name. In other cases, for a temperament of rank n, n independent <a class="wiki_link" href="/Patent%20val">patent vals</a> are given. Normally this is by way of integers conjoined by ampersands, such as 2&amp;10 for 7-limit pajara. This tells us we can use the 7-limit patent vals for 2 and 10 to define the temperament. In case n independent patent vals cannot be found, the <a class="wiki_link" href="/Normal%20lists">normal val list</a> is given. Note that this differs from the method of <a class="wiki_link" href="/Graham%20Breed">Graham Breed</a>'s <a class="wiki_link_ext" href="http://x31eq.com/temper/" rel="nofollow">regular temperament finder</a>, which does not assume the numbers refer to patent vals.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt; | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x5-limit rank two"></a><!-- ws:end:WikiTextHeadingRule:0 -->5-limit rank two</h1> | ||
Comma: Optimal patent val: 1000 * badness<br /> | |||
16/15: <a class="wiki_link" href="/8edo">8edo</a> 14.884<br /> | 16/15: <a class="wiki_link" href="/8edo">8edo</a> 14.884<br /> | ||
648/625: <a class="wiki_link" href="/12edo">12edo</a> 47.231<br /> | 648/625: <a class="wiki_link" href="/12edo">12edo</a> 47.231<br /> | ||
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7629394531250/7625597484987: <a class="wiki_link" href="/3501edo">3501edo</a> 17.191<br /> | 7629394531250/7625597484987: <a class="wiki_link" href="/3501edo">3501edo</a> 17.191<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt; | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="x7-limit rank two"></a><!-- ws:end:WikiTextHeadingRule:2 -->7-limit rank two</h1> | ||
Name: Optimal patent val: Val name: 1000*badness<br /> | |||
<a class="wiki_link" href="/Trienstonic%20clan">Father</a>: <a class="wiki_link" href="/5edo">5edo</a> [&lt;1 0 4 -2|, &lt;0 1 -1 3|] 21.312<br /> | <a class="wiki_link" href="/Trienstonic%20clan">Father</a>: <a class="wiki_link" href="/5edo">5edo</a> [&lt;1 0 4 -2|, &lt;0 1 -1 3|] 21.312<br /> | ||
<a class="wiki_link" href="/Archytas%20clan">Mother</a>: <a class="wiki_link" href="/5edo">5edo</a> 2&amp;3 24.152<br /> | <a class="wiki_link" href="/Archytas%20clan">Mother</a>: <a class="wiki_link" href="/5edo">5edo</a> 2&amp;3 24.152<br /> | ||
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<a class="wiki_link" href="/Ragismic%20microtemperaments">Supermajor</a>: <a class="wiki_link" href="/6214edo">6214edo</a> 80&amp;171 10.836<br /> | <a class="wiki_link" href="/Ragismic%20microtemperaments">Supermajor</a>: <a class="wiki_link" href="/6214edo">6214edo</a> 80&amp;171 10.836<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt; | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="x7-limit rank three"></a><!-- ws:end:WikiTextHeadingRule:4 -->7-limit rank three</h1> | ||
Comma: Optimal patent val: 10^6 * badness<br /> | |||
36/35: <a class="wiki_link" href="/12edo">12edo</a> 63.675<br /> | 36/35: <a class="wiki_link" href="/12edo">12edo</a> 63.675<br /> | ||
49/48: <a class="wiki_link" href="/19edo">19edo</a> 115.514<br /> | 49/48: <a class="wiki_link" href="/19edo">19edo</a> 115.514<br /> | ||
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78125000/78121827: <a class="wiki_link" href="/101654edo">101654edo</a> 20.457<br /> | 78125000/78121827: <a class="wiki_link" href="/101654edo">101654edo</a> 20.457<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt; | <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="x11-limit rank two"></a><!-- ws:end:WikiTextHeadingRule:6 -->11-limit rank two</h1> | ||
Name: Optimal patent val: Val name: 1000*badness<br /> | |||
<a class="wiki_link" href="/Marvel%20temperaments">Negric</a>: <a class="wiki_link" href="/9edo">9edo</a> 1&amp;9 30.617<br /> | <a class="wiki_link" href="/Marvel%20temperaments">Negric</a>: <a class="wiki_link" href="/9edo">9edo</a> 1&amp;9 30.617<br /> | ||
<a class="wiki_link" href="/Jubilismic%20clan">Diminished</a>: <a class="wiki_link" href="/12edo">12edo</a> 4&amp;12 22.132<br /> | <a class="wiki_link" href="/Jubilismic%20clan">Diminished</a>: <a class="wiki_link" href="/12edo">12edo</a> 4&amp;12 22.132<br /> | ||
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<a class="wiki_link" href="/Ragismic%20microtemperaments">Hemiennealimmal</a>: <a class="wiki_link" href="/1566edo">1566edo</a> 72&amp;198 6.283<br /> | <a class="wiki_link" href="/Ragismic%20microtemperaments">Hemiennealimmal</a>: <a class="wiki_link" href="/1566edo">1566edo</a> 72&amp;198 6.283<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:8:&lt; | <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="x11-limit rank three"></a><!-- ws:end:WikiTextHeadingRule:8 -->11-limit rank three</h1> | ||
Name: Optimal patent val: Val name: 10^6 * badness<br /> | Name: Optimal patent val: Val name: 10^6 * badness<br /> | ||
<a class="wiki_link" href="/Archytas%20family">Vulcan</a>: <a class="wiki_link" href="/15edo">15edo</a> 2&amp;3&amp;10 577.113<br /> | <a class="wiki_link" href="/Archytas%20family">Vulcan</a>: <a class="wiki_link" href="/15edo">15edo</a> 2&amp;3&amp;10 577.113<br /> | ||
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<a class="wiki_link" href="/Kalismic%20temperaments">Odin</a>: <a class="wiki_link" href="/7464edo">7464edo</a> 12&amp;42&amp;72 116.151<br /> | <a class="wiki_link" href="/Kalismic%20temperaments">Odin</a>: <a class="wiki_link" href="/7464edo">7464edo</a> 12&amp;42&amp;72 116.151<br /> | ||
<br /> | <br /> | ||
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Name: Optimal patent val: Val name: 1000*badness<br /> | |||
<a class="wiki_link" href="/Quartonic%20temperaments">Progression</a>: <a class="wiki_link" href="/9edo">9edo</a> [&lt;1 1 2 2 3 3|, &lt;0 5 3 7 4 6|] 18.158<br /> | <a class="wiki_link" href="/Quartonic%20temperaments">Progression</a>: <a class="wiki_link" href="/9edo">9edo</a> [&lt;1 1 2 2 3 3|, &lt;0 5 3 7 4 6|] 18.158<br /> | ||
<a class="wiki_link" href="/Marvel%20temperaments">Negric</a>: <a class="wiki_link" href="/9edo">9edo</a> 1&amp;9 20.205<br /> | <a class="wiki_link" href="/Marvel%20temperaments">Negric</a>: <a class="wiki_link" href="/9edo">9edo</a> 1&amp;9 20.205<br /> | ||
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<a class="wiki_link" href="/Ragismic%20microtemperaments">Abigail</a>: <a class="wiki_link" href="/1258edo">1258edo</a> 46&amp;178 8.856<br /> | <a class="wiki_link" href="/Ragismic%20microtemperaments">Abigail</a>: <a class="wiki_link" href="/1258edo">1258edo</a> 46&amp;178 8.856<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:12:&lt; | <!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="x13-limit rank three"></a><!-- ws:end:WikiTextHeadingRule:12 -->13-limit rank three</h1> | ||
Name: Optimal patent val: Val name: 10^5 * badness<br /> | |||
<a class="wiki_link" href="/Starling%20family">Mockingbird</a>: <a class="wiki_link" href="/15edo">15edo</a> 3&amp;4&amp;15 85.942<br /> | <a class="wiki_link" href="/Starling%20family">Mockingbird</a>: <a class="wiki_link" href="/15edo">15edo</a> 3&amp;4&amp;15 85.942<br /> | ||
<a class="wiki_link" href="/Starling%20family">Catbird</a>: <a class="wiki_link" href="/19edo">19edo</a> [&lt;1 0 0 -1 2 0|, &lt;0 1 0 -2 -2 -5|, &lt;0 0 1 3 2 5|] 90.475<br /> | <a class="wiki_link" href="/Starling%20family">Catbird</a>: <a class="wiki_link" href="/19edo">19edo</a> [&lt;1 0 0 -1 2 0|, &lt;0 1 0 -2 -2 -5|, &lt;0 0 1 3 2 5|] 90.475<br /> | ||