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**Imported revision 166305559 - Original comment: **
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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:Mirkwai family]] <!-- main article -->
[[Category:Rank 3]]

Revision as of 12:31, 27 November 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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