Kleismic rank three family: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
Wikispaces>genewardsmith
**Imported revision 303529768 - Original comment: **
Wikispaces>FREEZE
No edit summary
Line 1: Line 1:
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
__FORCETOC__
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>2012-02-20 22:44:05 UTC</tt>.<br>
: The original revision id was <tt>303529768</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|flat]]
 
 
These are the rank three temperaments tempering out the kleisma, 15625/15552. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.
These are the rank three temperaments tempering out the kleisma, 15625/15552. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.


7 and 9 limit minimax
7 and 9 limit minimax
[|1 0 0 0&gt;, |0 1 0 0&gt;, |1 5/6 0 0&gt;, |0 0 0 1&gt;]
[|1 0 0 0&gt;, |0 1 0 0&gt;, |1 5/6 0 0&gt;, |0 0 0 1&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 7/6, 4/3
[[Eigenmonzo|Eigenmonzos]]: 2, 7/6, 4/3


Map: [&lt;1 0 1 0|, &lt;0 6 5 0|, &lt;0 0 0 1|]
Map: [&lt;1 0 1 0|, &lt;0 6 5 0|, &lt;0 0 0 1|]
[[Generator|Generators]]: 2, 6/5, 7
 
[[edo|Edos]]: [[15edo|15]], [[19edo|19]], [[23edo|23]], [[34edo|34]], [[53edo|53]], [[68edo|68]], [[72edo|72]], [[87edo|87]], [[121edo|121]], [[125edo|125]], [[140edo|140]], [[212edo|212]], 333, 405
[[generator|Generators]]: 2, 6/5, 7
 
[[EDO|Edos]]: [[15edo|15]], [[19edo|19]], [[23edo|23]], [[34edo|34]], [[53edo|53]], [[68edo|68]], [[72edo|72]], [[87edo|87]], [[121edo|121]], [[125edo|125]], [[140edo|140]], [[212edo|212]], 333, 405


=Enlil=
=Enlil=
[[Comma|Commas]]: 385/384, 6250/6237
[[Comma|Commas]]: 385/384, 6250/6237


[[Minimax tuning]]
[[Minimax_tuning|Minimax tuning]]
 
[|1 0 0 0 0&gt;, |0 1 0 0 0&gt;, |1 5/6 0 0 0&gt;,  
[|1 0 0 0 0&gt;, |0 1 0 0 0&gt;, |1 5/6 0 0 0&gt;,  
|3 1/12 0 1/2 -1/2&gt;, |3 1/12 0 -1/2 1/2&gt;]
|3 1/12 0 1/2 -1/2&gt;, |3 1/12 0 -1/2 1/2&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 4/3, 14/11
[[Eigenmonzo|Eigenmonzos]]: 2, 4/3, 14/11


Map: [&lt;1 0 1 0 6|, &lt;0 6 5 0 1|, &lt;0 0 0 1 -1|]
Map: [&lt;1 0 1 0 6|, &lt;0 6 5 0 1|, &lt;0 0 0 1 -1|]
[[Generator|Generators]]: 2, 6/5, 7
 
[[generator|Generators]]: 2, 6/5, 7
 
EDOs: 53, 68, 72, 140, 159, 212, 284
EDOs: 53, 68, 72, 140, 159, 212, 284
Badness: 0.000798
Badness: 0.000798


Line 36: Line 36:


13 and 15 limit minimax tuning
13 and 15 limit minimax tuning
[|1 0 0 0 0 0&gt;, |4/3 0 0 -2/9 -2/9 4/9&gt;,  
[|1 0 0 0 0 0&gt;, |4/3 0 0 -2/9 -2/9 4/9&gt;,  
|19/9 0 0 -5/27 -5/27 10/27&gt;,  
|19/9 0 0 -5/27 -5/27 10/27&gt;,  
Line 41: Line 42:
|28/9 0 0 -14/27 13/27 1/27&gt;,  
|28/9 0 0 -14/27 13/27 1/27&gt;,  
|28/9 0 0 -14/27 -14/27 28/27&gt;]
|28/9 0 0 -14/27 -14/27 28/27&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 13/11, 14/11
[[Eigenmonzo|Eigenmonzos]]: 2, 13/11, 14/11


Map: [&lt;1 0 1 0 6 0|, &lt;0 6 5 0 1 14|, &lt;0 0 0 1 -1 0|]
Map: [&lt;1 0 1 0 6 0|, &lt;0 6 5 0 1 14|, &lt;0 0 0 1 -1 0|]
[[Generator|Generators]]: 2, 6/5, 7
 
[[edo|Edos]]: [[15edo|15]], [[19edo|19]], [[34edo|34]], [[53edo|53]], [[57edo|57]], [[68edo|68]], [[72edo|72]], [[87edo|87]], [[159edo|159]], [[212edo|212]], [[284edo|284]], [[299edo|299]]
[[generator|Generators]]: 2, 6/5, 7
 
[[EDO|Edos]]: [[15edo|15]], [[19edo|19]], [[34edo|34]], [[53edo|53]], [[57edo|57]], [[68edo|68]], [[72edo|72]], [[87edo|87]], [[159edo|159]], [[212edo|212]], [[284edo|284]], [[299edo|299]]
 
Badness: 0.000680
Badness: 0.000680


Scales: [[enlil19_13]]
Scales: [[enlil19_13|enlil19_13]]


=Enki=
=Enki=
[[Comma|Commas]]: 3025/3024, 4375/4356
[[Comma|Commas]]: 3025/3024, 4375/4356


[[Minimax tuning]]
[[Minimax_tuning|Minimax tuning]]
 
[|1 0 0 0 0&gt;, |-18/13 0 6/13 -6/13 12/13&gt;,  
[|1 0 0 0 0&gt;, |-18/13 0 6/13 -6/13 12/13&gt;,  
|-2/13 0 5/13 -5/13 10/13&gt;,  
|-2/13 0 5/13 -5/13 10/13&gt;,  
|-2/13 0 -8/13 8/13 10/13&gt;, |0 0 0 0 1&gt;]
|-2/13 0 -8/13 8/13 10/13&gt;, |0 0 0 0 1&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 11/8, 7/5
[[Eigenmonzo|Eigenmonzos]]: 2, 11/8, 7/5


Map: [&lt;1 0 1 0 1|, &lt;0 6 5 0 4|, &lt;0 0 0 2 1|]
Map: [&lt;1 0 1 0 1|, &lt;0 6 5 0 4|, &lt;0 0 0 2 1|]
[[Generator|Generators]]: 2, 6/5, 66/25
 
[[edo|Edos]]: [[15edo|15]], [[49edo|49]], [[57edo|57]], [[72edo|72]], [[87edo|87]], [[159edo|159]], [[193edo|193]], [[231|231]], 265, 280, 337, 352, 439  
[[generator|Generators]]: 2, 6/5, 66/25
 
[[EDO|Edos]]: [[15edo|15]], [[49edo|49]], [[57edo|57]], [[72edo|72]], [[87edo|87]], [[159edo|159]], [[193edo|193]], [[231|231]], 265, 280, 337, 352, 439  
 
Badness: 0.00109
Badness: 0.00109


Line 68: Line 78:


13 and 15 limit
13 and 15 limit
[|1 0 0 0 0 0&gt;, |2/3 0 -2/3 0 0 2/3&gt;,  
[|1 0 0 0 0 0&gt;, |2/3 0 -2/3 0 0 2/3&gt;,  
|14/9 0 -5/9 0 0 5/9&gt;, |0 0 0 1 0 0&gt;,  
|14/9 0 -5/9 0 0 5/9&gt;, |0 0 0 1 0 0&gt;,  
|13/9 0 -4/9 1/2 0 4/9&gt;,  
|13/9 0 -4/9 1/2 0 4/9&gt;,  
|14/9 0 -14/9 0 0 14/9&gt;]
|14/9 0 -14/9 0 0 14/9&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 8/7 13/10
[[Eigenmonzo|Eigenmonzos]]: 2, 8/7 13/10


Map: [&lt;1 0 1 0 1 0|, &lt;0 6 5 0 4 14|, &lt;0 0 0 2 1 0|]
Map: [&lt;1 0 1 0 1 0|, &lt;0 6 5 0 4 14|, &lt;0 0 0 2 1 0|]
[[Generator|Generators]]: 2, 6/5, 66/25
 
[[edo|Edos]]: [[15edo|15]], [[72edo|72]], [[87edo|87]], [[159edo|159]], [[193edo|193]], 280, 439
[[generator|Generators]]: 2, 6/5, 66/25
 
[[EDO|Edos]]: [[15edo|15]], [[72edo|72]], [[87edo|87]], [[159edo|159]], [[193edo|193]], 280, 439
 
Badness: 0.000874
Badness: 0.000874


Line 83: Line 98:


Map: [&lt;1 0 1 0 9|, &lt;0 6 5 0 -21|, &lt;0 0 0 1 0|]
Map: [&lt;1 0 1 0 9|, &lt;0 6 5 0 -21|, &lt;0 0 0 1 0|]
EDOs: 19, 53, 72, 197e, 269ce, 341ce, 375cde, 447cde, 519bcde
EDOs: 19, 53, 72, 197e, 269ce, 341ce, 375cde, 447cde, 519bcde
Badness: 0.003690
Badness: 0.003690


Line 90: Line 107:


Map: [&lt;1 0 1 0 9 0|, &lt;0 6 5 0 -21 14|, &lt;0 0 0 1 0 0|]
Map: [&lt;1 0 1 0 9 0|, &lt;0 6 5 0 -21 14|, &lt;0 0 0 1 0 0|]
EDOs: 19, 53, 72, 178def, 197ef, 231ef, 250def, 303cdef
EDOs: 19, 53, 72, 178def, 197ef, 231ef, 250def, 303cdef
Badness: 0.002273
Badness: 0.002273
 
[[Category:family]]
 
[[Category:kleismic]]
</pre></div>
[[Category:rank_3]]
<h4>Original HTML content:</h4>
[[Category:theory]]
<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;Kleismic rank three family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:12:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;&lt;a href="#Enlil"&gt;Enlil&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextTocRule:14: --&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt; | &lt;a href="#Enki"&gt;Enki&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt;&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt; | &lt;a href="#Isimud"&gt;Isimud&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;
&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;br /&gt;
&lt;br /&gt;
These are the rank three temperaments tempering out the kleisma, 15625/15552. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.&lt;br /&gt;
&lt;br /&gt;
7 and 9 limit minimax&lt;br /&gt;
[|1 0 0 0&amp;gt;, |0 1 0 0&amp;gt;, |1 5/6 0 0&amp;gt;, |0 0 0 1&amp;gt;]&lt;br /&gt;
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 7/6, 4/3&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 1 0|, &amp;lt;0 6 5 0|, &amp;lt;0 0 0 1|]&lt;br /&gt;
&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 6/5, 7&lt;br /&gt;
&lt;a class="wiki_link" href="/edo"&gt;Edos&lt;/a&gt;: &lt;a class="wiki_link" href="/15edo"&gt;15&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19&lt;/a&gt;, &lt;a class="wiki_link" href="/23edo"&gt;23&lt;/a&gt;, &lt;a class="wiki_link" href="/34edo"&gt;34&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53&lt;/a&gt;, &lt;a class="wiki_link" href="/68edo"&gt;68&lt;/a&gt;, &lt;a class="wiki_link" href="/72edo"&gt;72&lt;/a&gt;, &lt;a class="wiki_link" href="/87edo"&gt;87&lt;/a&gt;, &lt;a class="wiki_link" href="/121edo"&gt;121&lt;/a&gt;, &lt;a class="wiki_link" href="/125edo"&gt;125&lt;/a&gt;, &lt;a class="wiki_link" href="/140edo"&gt;140&lt;/a&gt;, &lt;a class="wiki_link" href="/212edo"&gt;212&lt;/a&gt;, 333, 405&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="Enlil"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Enlil&lt;/h1&gt;
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 385/384, 6250/6237&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;, |0 1 0 0 0&amp;gt;, |1 5/6 0 0 0&amp;gt;, &lt;br /&gt;
|3 1/12 0 1/2 -1/2&amp;gt;, |3 1/12 0 -1/2 1/2&amp;gt;]&lt;br /&gt;
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 4/3, 14/11&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 1 0 6|, &amp;lt;0 6 5 0 1|, &amp;lt;0 0 0 1 -1|]&lt;br /&gt;
&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 6/5, 7&lt;br /&gt;
EDOs: 53, 68, 72, 140, 159, 212, 284&lt;br /&gt;
Badness: 0.000798&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Enlil-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;13-limit&lt;/h2&gt;
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 325/324, 385/384, 625/624&lt;br /&gt;
&lt;br /&gt;
13 and 15 limit minimax tuning&lt;br /&gt;
[|1 0 0 0 0 0&amp;gt;, |4/3 0 0 -2/9 -2/9 4/9&amp;gt;, &lt;br /&gt;
|19/9 0 0 -5/27 -5/27 10/27&amp;gt;, &lt;br /&gt;
|28/9 0 0 13/27 -14/27 1/27&amp;gt;, &lt;br /&gt;
|28/9 0 0 -14/27 13/27 1/27&amp;gt;, &lt;br /&gt;
|28/9 0 0 -14/27 -14/27 28/27&amp;gt;]&lt;br /&gt;
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 13/11, 14/11&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 1 0 6 0|, &amp;lt;0 6 5 0 1 14|, &amp;lt;0 0 0 1 -1 0|]&lt;br /&gt;
&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 6/5, 7&lt;br /&gt;
&lt;a class="wiki_link" href="/edo"&gt;Edos&lt;/a&gt;: &lt;a class="wiki_link" href="/15edo"&gt;15&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19&lt;/a&gt;, &lt;a class="wiki_link" href="/34edo"&gt;34&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53&lt;/a&gt;, &lt;a class="wiki_link" href="/57edo"&gt;57&lt;/a&gt;, &lt;a class="wiki_link" href="/68edo"&gt;68&lt;/a&gt;, &lt;a class="wiki_link" href="/72edo"&gt;72&lt;/a&gt;, &lt;a class="wiki_link" href="/87edo"&gt;87&lt;/a&gt;, &lt;a class="wiki_link" href="/159edo"&gt;159&lt;/a&gt;, &lt;a class="wiki_link" href="/212edo"&gt;212&lt;/a&gt;, &lt;a class="wiki_link" href="/284edo"&gt;284&lt;/a&gt;, &lt;a class="wiki_link" href="/299edo"&gt;299&lt;/a&gt;&lt;br /&gt;
Badness: 0.000680&lt;br /&gt;
&lt;br /&gt;
Scales: &lt;a class="wiki_link" href="/enlil19_13"&gt;enlil19_13&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Enki"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Enki&lt;/h1&gt;
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 3025/3024, 4375/4356&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;, |-18/13 0 6/13 -6/13 12/13&amp;gt;, &lt;br /&gt;
|-2/13 0 5/13 -5/13 10/13&amp;gt;, &lt;br /&gt;
|-2/13 0 -8/13 8/13 10/13&amp;gt;, |0 0 0 0 1&amp;gt;]&lt;br /&gt;
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 11/8, 7/5&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 1 0 1|, &amp;lt;0 6 5 0 4|, &amp;lt;0 0 0 2 1|]&lt;br /&gt;
&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 6/5, 66/25&lt;br /&gt;
&lt;a class="wiki_link" href="/edo"&gt;Edos&lt;/a&gt;: &lt;a class="wiki_link" href="/15edo"&gt;15&lt;/a&gt;, &lt;a class="wiki_link" href="/49edo"&gt;49&lt;/a&gt;, &lt;a class="wiki_link" href="/57edo"&gt;57&lt;/a&gt;, &lt;a class="wiki_link" href="/72edo"&gt;72&lt;/a&gt;, &lt;a class="wiki_link" href="/87edo"&gt;87&lt;/a&gt;, &lt;a class="wiki_link" href="/159edo"&gt;159&lt;/a&gt;, &lt;a class="wiki_link" href="/193edo"&gt;193&lt;/a&gt;, &lt;a class="wiki_link" href="/231"&gt;231&lt;/a&gt;, 265, 280, 337, 352, 439 &lt;br /&gt;
Badness: 0.00109&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="Enki-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;13-limit&lt;/h2&gt;
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 325/324, 364/363, 625/624&lt;br /&gt;
&lt;br /&gt;
13 and 15 limit&lt;br /&gt;
[|1 0 0 0 0 0&amp;gt;, |2/3 0 -2/3 0 0 2/3&amp;gt;, &lt;br /&gt;
|14/9 0 -5/9 0 0 5/9&amp;gt;, |0 0 0 1 0 0&amp;gt;, &lt;br /&gt;
|13/9 0 -4/9 1/2 0 4/9&amp;gt;, &lt;br /&gt;
|14/9 0 -14/9 0 0 14/9&amp;gt;]&lt;br /&gt;
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 8/7 13/10&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 1 0 1 0|, &amp;lt;0 6 5 0 4 14|, &amp;lt;0 0 0 2 1 0|]&lt;br /&gt;
&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 6/5, 66/25&lt;br /&gt;
&lt;a class="wiki_link" href="/edo"&gt;Edos&lt;/a&gt;: &lt;a class="wiki_link" href="/15edo"&gt;15&lt;/a&gt;, &lt;a class="wiki_link" href="/72edo"&gt;72&lt;/a&gt;, &lt;a class="wiki_link" href="/87edo"&gt;87&lt;/a&gt;, &lt;a class="wiki_link" href="/159edo"&gt;159&lt;/a&gt;, &lt;a class="wiki_link" href="/193edo"&gt;193&lt;/a&gt;, 280, 439&lt;br /&gt;
Badness: 0.000874&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Isimud"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Isimud&lt;/h1&gt;
Commas: 4125/4096, 8019/8000&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 1 0 9|, &amp;lt;0 6 5 0 -21|, &amp;lt;0 0 0 1 0|]&lt;br /&gt;
EDOs: 19, 53, 72, 197e, 269ce, 341ce, 375cde, 447cde, 519bcde&lt;br /&gt;
Badness: 0.003690&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="Isimud-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;13-limit&lt;/h2&gt;
Commas: 325/324, 625/624, 1287/1280&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 1 0 9 0|, &amp;lt;0 6 5 0 -21 14|, &amp;lt;0 0 0 1 0 0|]&lt;br /&gt;
EDOs: 19, 53, 72, 178def, 197ef, 231ef, 250def, 303cdef&lt;br /&gt;
Badness: 0.002273&lt;/body&gt;&lt;/html&gt;</pre></div>

Revision as of 00:00, 17 July 2018

These are the rank three temperaments tempering out the kleisma, 15625/15552. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.

7 and 9 limit minimax

[|1 0 0 0>, |0 1 0 0>, |1 5/6 0 0>, |0 0 0 1>]

Eigenmonzos: 2, 7/6, 4/3

Map: [<1 0 1 0|, <0 6 5 0|, <0 0 0 1|]

Generators: 2, 6/5, 7

Edos: 15, 19, 23, 34, 53, 68, 72, 87, 121, 125, 140, 212, 333, 405

Enlil

Commas: 385/384, 6250/6237

Minimax tuning

[|1 0 0 0 0>, |0 1 0 0 0>, |1 5/6 0 0 0>, |3 1/12 0 1/2 -1/2>, |3 1/12 0 -1/2 1/2>]

Eigenmonzos: 2, 4/3, 14/11

Map: [<1 0 1 0 6|, <0 6 5 0 1|, <0 0 0 1 -1|]

Generators: 2, 6/5, 7

EDOs: 53, 68, 72, 140, 159, 212, 284

Badness: 0.000798

13-limit

Commas: 325/324, 385/384, 625/624

13 and 15 limit minimax tuning

[|1 0 0 0 0 0>, |4/3 0 0 -2/9 -2/9 4/9>, |19/9 0 0 -5/27 -5/27 10/27>, |28/9 0 0 13/27 -14/27 1/27>, |28/9 0 0 -14/27 13/27 1/27>, |28/9 0 0 -14/27 -14/27 28/27>]

Eigenmonzos: 2, 13/11, 14/11

Map: [<1 0 1 0 6 0|, <0 6 5 0 1 14|, <0 0 0 1 -1 0|]

Generators: 2, 6/5, 7

Edos: 15, 19, 34, 53, 57, 68, 72, 87, 159, 212, 284, 299

Badness: 0.000680

Scales: enlil19_13

Enki

Commas: 3025/3024, 4375/4356

Minimax tuning

[|1 0 0 0 0>, |-18/13 0 6/13 -6/13 12/13>, |-2/13 0 5/13 -5/13 10/13>, |-2/13 0 -8/13 8/13 10/13>, |0 0 0 0 1>]

Eigenmonzos: 2, 11/8, 7/5

Map: [<1 0 1 0 1|, <0 6 5 0 4|, <0 0 0 2 1|]

Generators: 2, 6/5, 66/25

Edos: 15, 49, 57, 72, 87, 159, 193, 231, 265, 280, 337, 352, 439

Badness: 0.00109

13-limit

Commas: 325/324, 364/363, 625/624

13 and 15 limit

[|1 0 0 0 0 0>, |2/3 0 -2/3 0 0 2/3>, |14/9 0 -5/9 0 0 5/9>, |0 0 0 1 0 0>, |13/9 0 -4/9 1/2 0 4/9>, |14/9 0 -14/9 0 0 14/9>]

Eigenmonzos: 2, 8/7 13/10

Map: [<1 0 1 0 1 0|, <0 6 5 0 4 14|, <0 0 0 2 1 0|]

Generators: 2, 6/5, 66/25

Edos: 15, 72, 87, 159, 193, 280, 439

Badness: 0.000874

Isimud

Commas: 4125/4096, 8019/8000

Map: [<1 0 1 0 9|, <0 6 5 0 -21|, <0 0 0 1 0|]

EDOs: 19, 53, 72, 197e, 269ce, 341ce, 375cde, 447cde, 519bcde

Badness: 0.003690

13-limit

Commas: 325/324, 625/624, 1287/1280

Map: [<1 0 1 0 9 0|, <0 6 5 0 -21 14|, <0 0 0 1 0 0|]

EDOs: 19, 53, 72, 178def, 197ef, 231ef, 250def, 303cdef

Badness: 0.002273

Retrieved from "https://en.xen.wiki/index.php?title=Kleismic_rank_three_family&oldid=2337"