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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>2010-09-29 04:03:56 UTC</tt>.<br>
: The original revision id was <tt>166305559</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">===Vital statistics===
[[Comma]]: mirkwai 16875/16807


7-limit minimax
== Mirkwai ==
[|1 0 0 0&gt;, |0 4/7 -4/7 5/7&gt;,
[[Subgroup]]: 2.3.5.7
|0 -3/7 3/7 5/7&gt;, |0 0 0 1&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 8/7, 6/5


9-limit minimax
[[Comma list]]: 16875/16807 (mirkwai)
[|1 0 0 0&gt;, |0 8/11 -4/11 5/11&gt;,
|0 -6/11 3/11 10/11&gt;, |0 0 0 1&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 10/9, 8/7


Map: [&lt;1 0 0 0|, &lt;0 1 3 3|, &lt;0 0 -5 -4|]
[[Mapping]]: [{{val| 1 0 0 0 }}, {{val| 0 1 3 3 }}, {{val| 0 0 -5 -4 }}]
[[Generator|Generators]]: 2, 3, 7/5
[[edo|Edos]]: [[72edo|72]], [[152edo|152]], [[193edo|193]], [[224edo|224]], [[296edo|296]], [[327edo|327]], [[775edo|775]]


==11-limit children==
Mapping generators: ~2, ~3, ~7/5


===Indra===
[[Minimax tuning]]:
[[Comma|Commas]]: 540/539, 1375/1372
* [[7-odd-limit]]
: [{{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]]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


[[Minimax tuning]]
{{Optimal ET sequence|legend=1| 31, 41, 72, 152, 224 }}
[|1 0 0 0 0&gt;, |10 10 -3 0 -5&gt;, |0 0 1 0 0&gt;,  
|6 6 -1 0 -3&gt;, |20 18 -6 0 -9&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 11/9, 5/4


Lattice basis: 15/14 length 0.6184 55/36 length 0.7585
[[Badness]]: 0.342 × 10<sup>-3</sup>
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|]
[[Projection pair]]s: 5 84375/16807 7 16875/2401 to 2.3.7/5
[[Generator|Generators]]: 2, 3, 7/5
[[edo|Edos]]: [[72edo|72]], [[152edo|152]], [[224edo|224]], [[296edo|296]]


13-limit
== Indra ==
[[Subgroup]]: 2.3.5.7.11


====Yajna====
[[Comma list]]: 540/539, 1375/1372
Commas: 540/539, 1375/1372, 625/624


====Shibi====
[[Mapping]]: [{{val| 1 0 0 0 2 }}, {{val| 0 1 3 3 0 }}, {{val| 0 0 -5 -4 3 }}]
Commas: 540/539, 1375/1372, 729/728


====Vibhu====
Mapping generators: ~2, ~3, ~7/5
Commas: 540/539, 1375/1372, 351/350


====Vish====
[[Mapping to lattice]]: [{{val| 0 2 1 2 3 }}, {{val| 0 -1 2 1 -3 }}]
Commas: 540/539, 1375/1372, 325/324
 
</pre></div>
Lattice basis:  
<h4>Original HTML content:</h4>
: 15/14 length = 0.6184, 55/36 length = 0.7585
<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:WikiTextHeadingRule:0:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc0"&gt;&lt;a name="x--Vital statistics"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Vital statistics&lt;/h3&gt;
: Angle (15/14, 55/36) = 68.1048 deg
&lt;a class="wiki_link" href="/Comma"&gt;Comma&lt;/a&gt;: mirkwai 16875/16807&lt;br /&gt;
 
&lt;br /&gt;
[[Minimax tuning]]:
7-limit minimax&lt;br /&gt;
* [[11-odd-limit]]
[|1 0 0 0&amp;gt;, |0 4/7 -4/7 5/7&amp;gt;, &lt;br /&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 }}]
|0 -3/7 3/7 5/7&amp;gt;, |0 0 0 1&amp;gt;]&lt;br /&gt;
: [[Eigenmonzo]]s (unchanged-intervals): 2, 11/9, 5/4
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 8/7, 6/5&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 31, 41, 72, 152, 224, 631d, 855d }}
9-limit minimax&lt;br /&gt;
 
[|1 0 0 0&amp;gt;, |0 8/11 -4/11 5/11&amp;gt;, &lt;br /&gt;
[[Badness]]: 0.266 × 10<sup>-3</sup>
|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;
[[Projection pair]]s: 5 84375/16807 7 16875/2401 11 1372/125 to 2.3.7/5
&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;
Scales: [[indra31]]
&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;
=== Shibi ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-11-limit children"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;11-limit children&lt;/h2&gt;
 
&lt;br /&gt;
Comma list: 540/539, 729/728, 1375/1372
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x-11-limit children-Indra"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Indra&lt;/h3&gt;
 
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 540/539, 1375/1372&lt;br /&gt;
Mapping: [{{val| 1 0 0 0 2 -3 }}, {{val| 0 1 3 3 0 3 }}, {{val| 0 0 -5 -4 4 }}]
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/Minimax%20tuning"&gt;Minimax tuning&lt;/a&gt;&lt;br /&gt;
{{Optimal ET sequence|legend=1| 31f, 39df, 41, 72, 111, 152f, 183, 224, 631d, 855d }}
[|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;
Badness: 0.721 × 10<sup>-3</sup>
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 11/9, 5/4&lt;br /&gt;
 
&lt;br /&gt;
=== Yajna ===
Lattice basis: 15/14 length 0.6184 55/36 length 0.7585&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
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;
Comma list: 540/539, 625/624, 1375/1372
&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;
Mapping: [{{val| 1 0 0 0 2 -4 }}, {{val| 0 1 3 3 0 11 }}, {{val| 0 0 -5 -4 3 -20 }}]
&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="/224edo"&gt;224&lt;/a&gt;, &lt;a class="wiki_link" href="/296edo"&gt;296&lt;/a&gt;&lt;br /&gt;
Mapping generators: ~2, ~3, ~7/5
&lt;br /&gt;
 
13-limit&lt;br /&gt;
Minimax tuning:
&lt;br /&gt;
* 13-odd-limit
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc3"&gt;&lt;a name="x-11-limit children-Indra-Yajna"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Yajna&lt;/h4&gt;
: [{{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 }}]
Commas: 540/539, 1375/1372, 625/624&lt;br /&gt;
: [[Eigenmonzo]]s (unchanged-intervals): 2, 10/9, 13/10
&lt;br /&gt;
* 15-odd-limit minimax
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc4"&gt;&lt;a name="x-11-limit children-Indra-Shibi"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Shibi&lt;/h4&gt;
: [{{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 }}]
Commas: 540/539, 1375/1372, 729/728&lt;br /&gt;
: [[Eigenmonzo]]s (unchanged-intervals): 2, 14/13, 4/3
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc5"&gt;&lt;a name="x-11-limit children-Indra-Vibhu"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Vibhu&lt;/h4&gt;
{{Optimal ET sequence|legend=1| 31, 49f, 72, 103, 121, 152f, 193, 224 }}
Commas: 540/539, 1375/1372, 351/350&lt;br /&gt;
 
&lt;br /&gt;
Badness: 0.700 × 10<sup>-3</sup>
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc6"&gt;&lt;a name="x-11-limit children-Indra-Vish"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Vish&lt;/h4&gt;
 
Commas: 540/539, 1375/1372, 325/324&lt;/body&gt;&lt;/html&gt;</pre></div>
=== Vibhu ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 540/539, 1375/1372
 
Mapping: [{{val| 1 0 0 0 2 1 }}, {{val| 0 1 3 3 0 6 }}, {{val| 0 0 -5 -4 3 -14 }}]
 
{{Optimal ET sequence|legend=1| 31, 64cdf, 72, 103, 111, 152, 175f, 183, 469f }}
 
Badness: 0.826 × 10<sup>-3</sup>
 
=== Vish ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 364/363, 540/539
 
Mapping: [{{val| 1 0 0 0 2 2 }}, {{val| 0 1 3 3 0 -2 }}, {{val| 0 0 -5 -4 3 10 }}]
 
{{Optimal ET sequence|legend=1| 31f, 41, 72, 121, 193 }}
 
Badness: 0.930 × 10<sup>-3</sup>
 
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Mirkwai family]] <!-- main article -->
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Mirkwai_family"