Mirkwai family: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Technical data page}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
The '''mirkwai family''' of [[planar temperament]]s tempers out the [[mirkwai comma]], 16875/16807.
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-10-23 17:54:47 UTC</tt>.<br>
: The original revision id was <tt>267688398</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]]


== Mirkwai ==
[[Subgroup]]: 2.3.5.7


The mirkwai family of [[planar temperament]]s tempers out the mirkwai comma, 16875/16807.
[[Comma list]]: 16875/16807 (mirkwai)


=Vital statistics=
[[Mapping]]: [{{val| 1 0 0 0 }}, {{val| 0 1 3 3 }}, {{val| 0 0 -5 -4 }}]
[[Comma]]: mirkwai 16875/16807


7-limit minimax
Mapping generators: ~2, ~3, ~7/5
[|1 0 0 0&gt;, |0 4/7 -4/7 5/7&gt;,  
|0 -3/7 3/7 5/7&gt;, |0 0 0 1&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 8/7, 6/5


9-limit minimax
[[Minimax tuning]]:
[|1 0 0 0&gt;, |0 8/11 -4/11 5/11&gt;,  
* [[7-odd-limit]]
|0 -6/11 3/11 10/11&gt;, |0 0 0 1&gt;]
: [{{monzo| 1 0 0 0 }}, {{monzo| 0 4/7 -4/7 5/7 }}, {{monzo| 0 -3/7 3/7 5/7 }}, {{monzo| 0 0 0 1 }}]
[[Eigenmonzo|Eigenmonzos]]: 2, 10/9, 8/7
: [[Eigenmonzo]]s (unchanged-intervals): 2, 8/7, 6/5
* [[9-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 0 8/11 -4/11 5/11 }}, {{monzo| 0 -6/11 3/11 10/11 }}, {{monzo| 0 0 0 1 }}]
: [[Eigenmonzo]]s (unchanged-intervals): 2, 10/9, 8/7


Map: [&lt;1 0 0 0|, &lt;0 1 3 3|, &lt;0 0 -5 -4|]
{{Optimal ET sequence|legend=1| 31, 41, 72, 152, 224 }}
[[Generator|Generators]]: 2, 3, 7/5
 
[[edo|Edos]]: [[72edo|72]], [[152edo|152]], [[193edo|193]], [[224edo|224]], [[296edo|296]], [[327edo|327]], [[775edo|775]]
[[Badness]]: 0.342 × 10<sup>-3</sup>
Badness: 0.000342


[[Projection pair]]s: 5 84375/16807 7 16875/2401 to 2.3.7/5
[[Projection pair]]s: 5 84375/16807 7 16875/2401 to 2.3.7/5


=Indra=
== Indra ==
[[Comma|Commas]]: 540/539, 1375/1372
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 540/539, 1375/1372
 
[[Mapping]]: [{{val| 1 0 0 0 2 }}, {{val| 0 1 3 3 0 }}, {{val| 0 0 -5 -4 3 }}]
 
Mapping generators: ~2, ~3, ~7/5
 
[[Mapping to lattice]]: [{{val| 0 2 1 2 3 }}, {{val| 0 -1 2 1 -3 }}]
 
Lattice basis:
: 15/14 length = 0.6184, 55/36 length = 0.7585
: Angle (15/14, 55/36) = 68.1048 deg


[[Minimax tuning]]
[[Minimax tuning]]:
[|1 0 0 0 0&gt;, |10 10 -3 0 -5&gt;, |0 0 1 0 0&gt;,  
* [[11-odd-limit]]
|6 6 -1 0 -3&gt;, |20 18 -6 0 -9&gt;]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 10 10 -3 0 -5 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 6 6 -1 0 -3 }}, {{monzo| 20 18 -6 0 -9 }}]
[[Eigenmonzo|Eigenmonzos]]: 2, 11/9, 5/4
: [[Eigenmonzo]]s (unchanged-intervals): 2, 11/9, 5/4


Lattice basis: 15/14 length 0.6184 55/36 length 0.7585
{{Optimal ET sequence|legend=1| 31, 41, 72, 152, 224, 631d, 855d }}
Angle(15/14, 55/36) = 68.1048 cents
Map to lattice: [&lt;0 2 1 2 3|, &lt;0 -1 2 1 -3|]


Map: [&lt;1 0 0 0 2|, &lt;0 1 3 3 0|, &lt;0 0 -5 -4 3|]
[[Badness]]: 0.266 × 10<sup>-3</sup>
[[Generator|Generators]]: 2, 3, 7/5
[[edo|Edos]]: 31, 41, [[72edo|72]], [[152edo|152]], [[224edo|224]], [[296edo|296]], [[703edo|703]]
Badness: 0.000266


[[Projection pair]]s: 5 84375/16807 7 16875/2401 11 1372/125 to 2.3.7/5
[[Projection pair]]s: 5 84375/16807 7 16875/2401 11 1372/125 to 2.3.7/5
Line 52: Line 53:
Scales: [[indra31]]
Scales: [[indra31]]


==Shibi==
=== Shibi ===
Commas: 540/539, 729/728, 1375/1372
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 729/728, 1375/1372
 
Mapping: [{{val| 1 0 0 0 2 -3 }}, {{val| 0 1 3 3 0 3 }}, {{val| 0 0 -5 -4 4 }}]
 
{{Optimal ET sequence|legend=1| 31f, 39df, 41, 72, 111, 152f, 183, 224, 631d, 855d }}
 
Badness: 0.721 × 10<sup>-3</sup>
 
=== Yajna ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 625/624, 1375/1372
 
Mapping: [{{val| 1 0 0 0 2 -4 }}, {{val| 0 1 3 3 0 11 }}, {{val| 0 0 -5 -4 3 -20 }}]
 
Mapping generators: ~2, ~3, ~7/5
 
Minimax tuning:
* 13-odd-limit
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 4/5 6/5 -4/5 0 0 1/5 }}, {{monzo| 8/5 2/5 -3/5 0 0 2/5 }}, {{monzo| 44/25 26/25 -24/25 0 0 11/25 }}, {{monzo| 62/25 48/25 -27/25 0 0 3/25 }}, {{monzo| 8/5 2/5 -8/5 0 0 7/5 }}]
: [[Eigenmonzo]]s (unchanged-intervals): 2, 10/9, 13/10
* 15-odd-limit minimax
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 0 1 0 0 0 0 }}, {{monzo| 5/4 1/2 0 -5/16 0 5/16 }}, {{monzo| 1 1 0 -1/4 0 1/4 }}, {{monzo| 5/4 3/2 0 3/16 0 -3/16 }}, {{monzo| 1 1 0 -5/4 0 5/4 }}]
: [[Eigenmonzo]]s (unchanged-intervals): 2, 14/13, 4/3
 
{{Optimal ET sequence|legend=1| 31, 49f, 72, 103, 121, 152f, 193, 224 }}
 
Badness: 0.700 × 10<sup>-3</sup>
 
=== Vibhu ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 540/539, 1375/1372


Map: [&lt;1 0 0 0 2 -4 -3|, &lt;0 1 3 3 0 11 3|, &lt;0 0 -5 -4 3 4|]
Mapping: [{{val| 1 0 0 0 2 1 }}, {{val| 0 1 3 3 0 6 }}, {{val| 0 0 -5 -4 3 -14 }}]
EDOs: 41, 72, 111, 183, 224, 703
Badness: 0.000721


==Yajna==
{{Optimal ET sequence|legend=1| 31, 64cdf, 72, 103, 111, 152, 175f, 183, 469f }}
[[Comma|Commas]]: 540/539, 625/624, 1375/1372


13-limit minimax
Badness: 0.826 × 10<sup>-3</sup>
[|1 0 0 0 0 0&gt;, |4/5 6/5 -4/5 0 0 1/5&gt;,
|8/5 2/5 -3/5 0 0 2/5&gt;,
|44/25 26/25 -24/25 0 0 11/25&gt;,
|62/25 48/25 -27/25 0 0 3/25&gt;,
|8/5 2/5 -8/5 0 0 7/5&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 10/9, 13/10


15-limit minimax
=== Vish ===
[|1 0 0 0 0 0&gt;, |0 1 0 0 0 0&gt;,
Subgroup: 2.3.5.7.11.13
|5/4 1/2 0 -5/16 0 5/16&gt;,
|1 1 0 -1/4 0 1/4&gt;,
|5/4 3/2 0 3/16 0 -3/16&gt;,
|1 1 0 -5/4 0 5/4&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 14/13, 4/3


Map: [&lt;1 0 0 0 2 -4|, &lt;0 1 3 3 0 11|, &lt;0 0 -5 -4 3 -20|]
Comma list: 325/324, 364/363, 540/539
[[Generator|Generators]]: 2, 3, 7/5
[[edo|Edos]]: [[31edo|31]], [[72edo|72]], [[103edo|103]], [[193edo|193]], 224, 296, 520, 703
Badness: 0.000700


==Vibhu==
Mapping: [{{val| 1 0 0 0 2 2 }}, {{val| 0 1 3 3 0 -2 }}, {{val| 0 0 -5 -4 3 10 }}]
[[Comma|Commas]]: 540/539, 1375/1372, 351/350


Map: [&lt;1 0 0 0 2 1|, &lt;0 1 3 3 0 6|, &lt;0 0 -5 -4 3 -14|]
{{Optimal ET sequence|legend=1| 31f, 41, 72, 121, 193 }}
EDOs: 31, 72, 103, 111, 152, 183
Badness: 0.000826


==Vish==
Badness: 0.930 × 10<sup>-3</sup>
[[Comma|Commas]]: 325/324, 364/363, 540/539


Map: [&lt;1 0 0 0 2 2], &lt;0 1 3 3 0 -2], &lt;0 0 -5 -4 3 10|]
[[Category:Temperament families]]
EDOs: 41, 72, 121, 193
[[Category:Pages with mostly numerical content]]
Badness: 0.000930</pre></div>
[[Category:Mirkwai family]] <!-- main article -->
<h4>Original HTML content:</h4>
[[Category:Rank 3]]
<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;Mirkwai family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:12:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Vital statistics"&gt;Vital statistics&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextTocRule:14: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Indra"&gt;Indra&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Indra-Shibi"&gt;Shibi&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Indra-Yajna"&gt;Yajna&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Indra-Vibhu"&gt;Vibhu&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Indra-Vish"&gt;Vish&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;br /&gt;
&lt;br /&gt;
The mirkwai family of &lt;a class="wiki_link" href="/planar%20temperament"&gt;planar temperament&lt;/a&gt;s tempers out the mirkwai comma, 16875/16807.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Vital statistics"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Vital statistics&lt;/h1&gt;
&lt;a class="wiki_link" href="/Comma"&gt;Comma&lt;/a&gt;: mirkwai 16875/16807&lt;br /&gt;
&lt;br /&gt;
7-limit minimax&lt;br /&gt;
[|1 0 0 0&amp;gt;, |0 4/7 -4/7 5/7&amp;gt;, &lt;br /&gt;
|0 -3/7 3/7 5/7&amp;gt;, |0 0 0 1&amp;gt;]&lt;br /&gt;
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 8/7, 6/5&lt;br /&gt;
&lt;br /&gt;
9-limit minimax&lt;br /&gt;
[|1 0 0 0&amp;gt;, |0 8/11 -4/11 5/11&amp;gt;, &lt;br /&gt;
|0 -6/11 3/11 10/11&amp;gt;, |0 0 0 1&amp;gt;]&lt;br /&gt;
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 10/9, 8/7&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 0|, &amp;lt;0 1 3 3|, &amp;lt;0 0 -5 -4|]&lt;br /&gt;
&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 3, 7/5&lt;br /&gt;
&lt;a class="wiki_link" href="/edo"&gt;Edos&lt;/a&gt;: &lt;a class="wiki_link" href="/72edo"&gt;72&lt;/a&gt;, &lt;a class="wiki_link" href="/152edo"&gt;152&lt;/a&gt;, &lt;a class="wiki_link" href="/193edo"&gt;193&lt;/a&gt;, &lt;a class="wiki_link" href="/224edo"&gt;224&lt;/a&gt;, &lt;a class="wiki_link" href="/296edo"&gt;296&lt;/a&gt;, &lt;a class="wiki_link" href="/327edo"&gt;327&lt;/a&gt;, &lt;a class="wiki_link" href="/775edo"&gt;775&lt;/a&gt;&lt;br /&gt;
Badness: 0.000342&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/Projection%20pair"&gt;Projection pair&lt;/a&gt;s: 5 84375/16807 7 16875/2401 to 2.3.7/5&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Indra"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Indra&lt;/h1&gt;
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 540/539, 1375/1372&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/Minimax%20tuning"&gt;Minimax tuning&lt;/a&gt;&lt;br /&gt;
[|1 0 0 0 0&amp;gt;, |10 10 -3 0 -5&amp;gt;, |0 0 1 0 0&amp;gt;, &lt;br /&gt;
|6 6 -1 0 -3&amp;gt;, |20 18 -6 0 -9&amp;gt;]&lt;br /&gt;
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 11/9, 5/4&lt;br /&gt;
&lt;br /&gt;
Lattice basis: 15/14 length 0.6184 55/36 length 0.7585&lt;br /&gt;
Angle(15/14, 55/36) = 68.1048 cents&lt;br /&gt;
Map to lattice: [&amp;lt;0 2 1 2 3|, &amp;lt;0 -1 2 1 -3|]&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 0 2|, &amp;lt;0 1 3 3 0|, &amp;lt;0 0 -5 -4 3|]&lt;br /&gt;
&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 3, 7/5&lt;br /&gt;
&lt;a class="wiki_link" href="/edo"&gt;Edos&lt;/a&gt;: 31, 41, &lt;a class="wiki_link" href="/72edo"&gt;72&lt;/a&gt;, &lt;a class="wiki_link" href="/152edo"&gt;152&lt;/a&gt;, &lt;a class="wiki_link" href="/224edo"&gt;224&lt;/a&gt;, &lt;a class="wiki_link" href="/296edo"&gt;296&lt;/a&gt;, &lt;a class="wiki_link" href="/703edo"&gt;703&lt;/a&gt;&lt;br /&gt;
Badness: 0.000266&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/Projection%20pair"&gt;Projection pair&lt;/a&gt;s: 5 84375/16807 7 16875/2401 11 1372/125 to 2.3.7/5&lt;br /&gt;
&lt;br /&gt;
Scales: &lt;a class="wiki_link" href="/indra31"&gt;indra31&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Indra-Shibi"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Shibi&lt;/h2&gt;
Commas: 540/539, 729/728, 1375/1372&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 0 2 -4 -3|, &amp;lt;0 1 3 3 0 11 3|, &amp;lt;0 0 -5 -4 3 4|]&lt;br /&gt;
EDOs: 41, 72, 111, 183, 224, 703&lt;br /&gt;
Badness: 0.000721&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Indra-Yajna"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Yajna&lt;/h2&gt;
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 540/539, 625/624, 1375/1372&lt;br /&gt;
&lt;br /&gt;
13-limit minimax&lt;br /&gt;
[|1 0 0 0 0 0&amp;gt;, |4/5 6/5 -4/5 0 0 1/5&amp;gt;, &lt;br /&gt;
|8/5 2/5 -3/5 0 0 2/5&amp;gt;, &lt;br /&gt;
|44/25 26/25 -24/25 0 0 11/25&amp;gt;, &lt;br /&gt;
|62/25 48/25 -27/25 0 0 3/25&amp;gt;, &lt;br /&gt;
|8/5 2/5 -8/5 0 0 7/5&amp;gt;]&lt;br /&gt;
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 10/9, 13/10&lt;br /&gt;
&lt;br /&gt;
15-limit minimax&lt;br /&gt;
[|1 0 0 0 0 0&amp;gt;, |0 1 0 0 0 0&amp;gt;, &lt;br /&gt;
|5/4 1/2 0 -5/16 0 5/16&amp;gt;,&lt;br /&gt;
|1 1 0 -1/4 0 1/4&amp;gt;, &lt;br /&gt;
|5/4 3/2 0 3/16 0 -3/16&amp;gt;, &lt;br /&gt;
|1 1 0 -5/4 0 5/4&amp;gt;]&lt;br /&gt;
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 14/13, 4/3&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 0 2 -4|, &amp;lt;0 1 3 3 0 11|, &amp;lt;0 0 -5 -4 3 -20|]&lt;br /&gt;
&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 3, 7/5&lt;br /&gt;
&lt;a class="wiki_link" href="/edo"&gt;Edos&lt;/a&gt;: &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/72edo"&gt;72&lt;/a&gt;, &lt;a class="wiki_link" href="/103edo"&gt;103&lt;/a&gt;, &lt;a class="wiki_link" href="/193edo"&gt;193&lt;/a&gt;, 224, 296, 520, 703&lt;br /&gt;
Badness: 0.000700&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Indra-Vibhu"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Vibhu&lt;/h2&gt;
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 540/539, 1375/1372, 351/350&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 0 2 1|, &amp;lt;0 1 3 3 0 6|, &amp;lt;0 0 -5 -4 3 -14|]&lt;br /&gt;
EDOs: 31, 72, 103, 111, 152, 183&lt;br /&gt;
Badness: 0.000826&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Indra-Vish"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Vish&lt;/h2&gt;
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 325/324, 364/363, 540/539&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 0 2 2], &amp;lt;0 1 3 3 0 -2], &amp;lt;0 0 -5 -4 3 10|]&lt;br /&gt;
EDOs: 41, 72, 121, 193&lt;br /&gt;
Badness: 0.000930&lt;/body&gt;&lt;/html&gt;</pre></div>

Latest revision as of 00:28, 24 June 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The mirkwai family of planar temperaments tempers out the mirkwai comma, 16875/16807.

Mirkwai

Subgroup: 2.3.5.7

Comma list: 16875/16807 (mirkwai)

Mapping: [⟨1 0 0 0], ⟨0 1 3 3], ⟨0 0 -5 -4]]

Mapping generators: ~2, ~3, ~7/5

Minimax tuning:

[[1 0 0 0⟩, [0 4/7 -4/7 5/7⟩, [0 -3/7 3/7 5/7⟩, [0 0 0 1⟩]
Eigenmonzos (unchanged-intervals): 2, 8/7, 6/5
[[1 0 0 0⟩, [0 8/11 -4/11 5/11⟩, [0 -6/11 3/11 10/11⟩, [0 0 0 1⟩]
Eigenmonzos (unchanged-intervals): 2, 10/9, 8/7

Optimal ET sequence: 31, 41, 72, 152, 224

Badness: 0.342 × 10-3

Projection pairs: 5 84375/16807 7 16875/2401 to 2.3.7/5

Indra

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372

Mapping: [⟨1 0 0 0 2], ⟨0 1 3 3 0], ⟨0 0 -5 -4 3]]

Mapping generators: ~2, ~3, ~7/5

Mapping to lattice: [⟨0 2 1 2 3], ⟨0 -1 2 1 -3]]

Lattice basis:

15/14 length = 0.6184, 55/36 length = 0.7585
Angle (15/14, 55/36) = 68.1048 deg

Minimax tuning:

[[1 0 0 0 0⟩, [10 10 -3 0 -5⟩, [0 0 1 0 0⟩, [6 6 -1 0 -3⟩, [20 18 -6 0 -9⟩]
Eigenmonzos (unchanged-intervals): 2, 11/9, 5/4

Optimal ET sequence: 31, 41, 72, 152, 224, 631d, 855d

Badness: 0.266 × 10-3

Projection pairs: 5 84375/16807 7 16875/2401 11 1372/125 to 2.3.7/5

Scales: indra31

Shibi

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 1375/1372

Mapping: [⟨1 0 0 0 2 -3], ⟨0 1 3 3 0 3], ⟨0 0 -5 -4 4]]

Optimal ET sequence: 31f, 39df, 41, 72, 111, 152f, 183, 224, 631d, 855d

Badness: 0.721 × 10-3

Yajna

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624, 1375/1372

Mapping: [⟨1 0 0 0 2 -4], ⟨0 1 3 3 0 11], ⟨0 0 -5 -4 3 -20]]

Mapping generators: ~2, ~3, ~7/5

Minimax tuning:

  • 13-odd-limit
[[1 0 0 0 0 0⟩, [4/5 6/5 -4/5 0 0 1/5⟩, [8/5 2/5 -3/5 0 0 2/5⟩, [44/25 26/25 -24/25 0 0 11/25⟩, [62/25 48/25 -27/25 0 0 3/25⟩, [8/5 2/5 -8/5 0 0 7/5⟩]
Eigenmonzos (unchanged-intervals): 2, 10/9, 13/10
  • 15-odd-limit minimax
[[1 0 0 0 0 0⟩, [0 1 0 0 0 0⟩, [5/4 1/2 0 -5/16 0 5/16⟩, [1 1 0 -1/4 0 1/4⟩, [5/4 3/2 0 3/16 0 -3/16⟩, [1 1 0 -5/4 0 5/4⟩]
Eigenmonzos (unchanged-intervals): 2, 14/13, 4/3

Optimal ET sequence: 31, 49f, 72, 103, 121, 152f, 193, 224

Badness: 0.700 × 10-3

Vibhu

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 1375/1372

Mapping: [⟨1 0 0 0 2 1], ⟨0 1 3 3 0 6], ⟨0 0 -5 -4 3 -14]]

Optimal ET sequence: 31, 64cdf, 72, 103, 111, 152, 175f, 183, 469f

Badness: 0.826 × 10-3

Vish

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 540/539

Mapping: [⟨1 0 0 0 2 2], ⟨0 1 3 3 0 -2], ⟨0 0 -5 -4 3 10]]

Optimal ET sequence: 31f, 41, 72, 121, 193

Badness: 0.930 × 10-3

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