Optimal patent val: Difference between revisions
Wikispaces>genewardsmith **Imported revision 258356204 - Original comment: ** |
Wikispaces>mbattaglia1 **Imported revision 258391932 - 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: | : This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2011-09-26 16:15:52 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>258391932</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">[[toc|flat]] | <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. | 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. | ||
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[[Marvel temperaments|Tritonic]]: [[31edo]] 2&29 23.659 | [[Marvel temperaments|Tritonic]]: [[31edo]] 2&29 23.659 | ||
[[Starling temperaments|Casablanca]]: [[31edo]] 31&42 67.291 | [[Starling temperaments|Casablanca]]: [[31edo]] 31&42 67.291 | ||
[[Meantone family# | [[Meantone family#Cuboctahedra|Cuboctahedra]]: [[31edo]] 31&45 56.826 | ||
[[Kleismic family|Keemun]]: [[34edo]] 4&15 27.410 | [[Kleismic family|Keemun]]: [[34edo]] 4&15 27.410 | ||
[[Augmented family|Triforce]]: [[39edo]] 6&9 20.248 | [[Augmented family|Triforce]]: [[39edo]] 6&9 20.248 | ||
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[[Ragismic microtemperaments|Hemiamity]]: [[350edo]] 46&106 31.307 | [[Ragismic microtemperaments|Hemiamity]]: [[350edo]] 46&106 31.307 | ||
[[Ragismic microtemperaments|Semiennealimmal]]: [[441edo]] 72&441 34.196 | [[Ragismic microtemperaments|Semiennealimmal]]: [[441edo]] 72&441 34.196 | ||
[[Mirkwai clan#Eris|Eris]]: [[703edo]]: [[703edo]] 31&193 27.621 | [[Mirkwai clan#Eris|Eris]]: [[703edo]]: [[703edo]] 31&193 27.621 | ||
[[Ragismic microtemperaments|Abigail]]: [[764edo]] 46&132 12.860 | [[Ragismic microtemperaments|Abigail]]: [[764edo]] 46&132 12.860 | ||
[[Breedsmic temperaments|Decoid]]: [[1210edo]] 130&270 18.735 | [[Breedsmic temperaments|Decoid]]: [[1210edo]] 130&270 18.735 | ||
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[[Gamelismic family|Ominous]]: [[149edo]] 5&26&46 75.055 | [[Gamelismic family|Ominous]]: [[149edo]] 5&26&46 75.055 | ||
[[Gamelismic family|Portending]]: [[159edo]] 15&26&46 62.715 | [[Gamelismic family|Portending]]: [[159edo]] 15&26&46 62.715 | ||
[[Gamelismic family|Momentous]]: [[164edo]] 10&31&46 83.202 | [[Gamelismic family|Momentous]]: [[164edo]] 10&31&46 83.202 | ||
[[Marvel family|Hecate]]: [[166edo]] 19&41&53 72.113 | [[Marvel family|Hecate]]: [[166edo]] 19&41&53 72.113 | ||
[[Marvel family|Marvell]]: [[166edo]] 9&31&63 86.160 | [[Marvel family|Marvell]]: [[166edo]] 9&31&63 86.160 | ||
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[[Breed family|Agni]]: [[212edo]] 6&31&72 92.251 | [[Breed family|Agni]]: [[212edo]] 6&31&72 92.251 | ||
[[Hemifamity family|Pele]]: [[232edo]] 29&41&46 70.328 | [[Hemifamity family|Pele]]: [[232edo]] 29&41&46 70.328 | ||
[[Breed family|Jovial]]: [[243edo]] 41&58&72 62.380 | [[Breed family|Jovial]]: [[243edo]] 41&58&72 62.380 | ||
[[The Archipelago|Borneo]]: [[270edo]] 15&72&111 54.937 | [[The Archipelago|Borneo]]: [[270edo]] 15&72&111 54.937 | ||
[[Werckismic temperaments|History]]: [[289edo]] 15&29&43 53.955 | [[Werckismic temperaments|History]]: [[289edo]] 15&29&43 53.955 | ||
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<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><!-- ws:start:WikiTextTocRule:14:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --><a href="#x5-limit rank two">5-limit rank two</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> | <a href="#x7-limit rank two">7-limit rank two</a><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --> | <a href="#x7-limit rank three">7-limit rank three</a><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --> | <a href="#x11-limit rank two">11-limit rank two</a><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --> | <a href="#x11-limit rank three">11-limit rank three</a><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --> | <a href="#x13-limit rank two">13-limit rank two</a><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --> | <a href="#x13-limit rank three">13-limit rank three</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --> | <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><!-- ws:start:WikiTextTocRule:14:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --><a href="#x5-limit rank two">5-limit rank two</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> | <a href="#x7-limit rank two">7-limit rank two</a><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --> | <a href="#x7-limit rank three">7-limit rank three</a><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --> | <a href="#x11-limit rank two">11-limit rank two</a><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --> | <a href="#x11-limit rank three">11-limit rank three</a><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --> | <a href="#x13-limit rank two">13-limit rank two</a><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --> | <a href="#x13-limit rank three">13-limit rank three</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --> | ||
<!-- ws:end:WikiTextTocRule:22 --> | <!-- ws:end:WikiTextTocRule:22 -->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 /> | ||
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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<a class="wiki_link" href="/Marvel%20temperaments">Tritonic</a>: <a class="wiki_link" href="/31edo">31edo</a> 2&amp;29 23.659<br /> | <a class="wiki_link" href="/Marvel%20temperaments">Tritonic</a>: <a class="wiki_link" href="/31edo">31edo</a> 2&amp;29 23.659<br /> | ||
<a class="wiki_link" href="/Starling%20temperaments">Casablanca</a>: <a class="wiki_link" href="/31edo">31edo</a> 31&amp;42 67.291<br /> | <a class="wiki_link" href="/Starling%20temperaments">Casablanca</a>: <a class="wiki_link" href="/31edo">31edo</a> 31&amp;42 67.291<br /> | ||
<a class="wiki_link" href="/Meantone%20family# | <a class="wiki_link" href="/Meantone%20family#Cuboctahedra">Cuboctahedra</a>: <a class="wiki_link" href="/31edo">31edo</a> 31&amp;45 56.826<br /> | ||
<a class="wiki_link" href="/Kleismic%20family">Keemun</a>: <a class="wiki_link" href="/34edo">34edo</a> 4&amp;15 27.410<br /> | <a class="wiki_link" href="/Kleismic%20family">Keemun</a>: <a class="wiki_link" href="/34edo">34edo</a> 4&amp;15 27.410<br /> | ||
<a class="wiki_link" href="/Augmented%20family">Triforce</a>: <a class="wiki_link" href="/39edo">39edo</a> 6&amp;9 20.248<br /> | <a class="wiki_link" href="/Augmented%20family">Triforce</a>: <a class="wiki_link" href="/39edo">39edo</a> 6&amp;9 20.248<br /> | ||
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<a class="wiki_link" href="/Ragismic%20microtemperaments">Hemiamity</a>: <a class="wiki_link" href="/350edo">350edo</a> 46&amp;106 31.307<br /> | <a class="wiki_link" href="/Ragismic%20microtemperaments">Hemiamity</a>: <a class="wiki_link" href="/350edo">350edo</a> 46&amp;106 31.307<br /> | ||
<a class="wiki_link" href="/Ragismic%20microtemperaments">Semiennealimmal</a>: <a class="wiki_link" href="/441edo">441edo</a> 72&amp;441 34.196<br /> | <a class="wiki_link" href="/Ragismic%20microtemperaments">Semiennealimmal</a>: <a class="wiki_link" href="/441edo">441edo</a> 72&amp;441 34.196<br /> | ||
<a class="wiki_link" href="/Mirkwai%20clan#Eris">Eris</a>: <a class="wiki_link" href="/703edo">703edo</a>: <a class="wiki_link" href="/703edo">703edo</a> 31&amp;193 27.621 <br /> | <a class="wiki_link" href="/Mirkwai%20clan#Eris">Eris</a>: <a class="wiki_link" href="/703edo">703edo</a>: <a class="wiki_link" href="/703edo">703edo</a> 31&amp;193 27.621<br /> | ||
<a class="wiki_link" href="/Ragismic%20microtemperaments">Abigail</a>: <a class="wiki_link" href="/764edo">764edo</a> 46&amp;132 12.860<br /> | <a class="wiki_link" href="/Ragismic%20microtemperaments">Abigail</a>: <a class="wiki_link" href="/764edo">764edo</a> 46&amp;132 12.860<br /> | ||
<a class="wiki_link" href="/Breedsmic%20temperaments">Decoid</a>: <a class="wiki_link" href="/1210edo">1210edo</a> 130&amp;270 18.735<br /> | <a class="wiki_link" href="/Breedsmic%20temperaments">Decoid</a>: <a class="wiki_link" href="/1210edo">1210edo</a> 130&amp;270 18.735<br /> | ||
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<a class="wiki_link" href="/Gamelismic%20family">Ominous</a>: <a class="wiki_link" href="/149edo">149edo</a> 5&amp;26&amp;46 75.055<br /> | <a class="wiki_link" href="/Gamelismic%20family">Ominous</a>: <a class="wiki_link" href="/149edo">149edo</a> 5&amp;26&amp;46 75.055<br /> | ||
<a class="wiki_link" href="/Gamelismic%20family">Portending</a>: <a class="wiki_link" href="/159edo">159edo</a> 15&amp;26&amp;46 62.715<br /> | <a class="wiki_link" href="/Gamelismic%20family">Portending</a>: <a class="wiki_link" href="/159edo">159edo</a> 15&amp;26&amp;46 62.715<br /> | ||
<a class="wiki_link" href="/Gamelismic%20family">Momentous</a>: <a class="wiki_link" href="/164edo">164edo</a> 10&amp;31&amp;46 83.202 <br /> | <a class="wiki_link" href="/Gamelismic%20family">Momentous</a>: <a class="wiki_link" href="/164edo">164edo</a> 10&amp;31&amp;46 83.202<br /> | ||
<a class="wiki_link" href="/Marvel%20family">Hecate</a>: <a class="wiki_link" href="/166edo">166edo</a> 19&amp;41&amp;53 72.113<br /> | <a class="wiki_link" href="/Marvel%20family">Hecate</a>: <a class="wiki_link" href="/166edo">166edo</a> 19&amp;41&amp;53 72.113<br /> | ||
<a class="wiki_link" href="/Marvel%20family">Marvell</a>: <a class="wiki_link" href="/166edo">166edo</a> 9&amp;31&amp;63 86.160<br /> | <a class="wiki_link" href="/Marvel%20family">Marvell</a>: <a class="wiki_link" href="/166edo">166edo</a> 9&amp;31&amp;63 86.160<br /> | ||
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<a class="wiki_link" href="/Breed%20family">Agni</a>: <a class="wiki_link" href="/212edo">212edo</a> 6&amp;31&amp;72 92.251<br /> | <a class="wiki_link" href="/Breed%20family">Agni</a>: <a class="wiki_link" href="/212edo">212edo</a> 6&amp;31&amp;72 92.251<br /> | ||
<a class="wiki_link" href="/Hemifamity%20family">Pele</a>: <a class="wiki_link" href="/232edo">232edo</a> 29&amp;41&amp;46 70.328<br /> | <a class="wiki_link" href="/Hemifamity%20family">Pele</a>: <a class="wiki_link" href="/232edo">232edo</a> 29&amp;41&amp;46 70.328<br /> | ||
<a class="wiki_link" href="/Breed%20family">Jovial</a>: <a class="wiki_link" href="/243edo">243edo</a> 41&amp;58&amp;72 62.380 <br /> | <a class="wiki_link" href="/Breed%20family">Jovial</a>: <a class="wiki_link" href="/243edo">243edo</a> 41&amp;58&amp;72 62.380<br /> | ||
<a class="wiki_link" href="/The%20Archipelago">Borneo</a>: <a class="wiki_link" href="/270edo">270edo</a> 15&amp;72&amp;111 54.937<br /> | <a class="wiki_link" href="/The%20Archipelago">Borneo</a>: <a class="wiki_link" href="/270edo">270edo</a> 15&amp;72&amp;111 54.937<br /> | ||
<a class="wiki_link" href="/Werckismic%20temperaments">History</a>: <a class="wiki_link" href="/289edo">289edo</a> 15&amp;29&amp;43 53.955<br /> | <a class="wiki_link" href="/Werckismic%20temperaments">History</a>: <a class="wiki_link" href="/289edo">289edo</a> 15&amp;29&amp;43 53.955<br /> | ||