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Kleismic rank three 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 '''kleismic rank-3 family''' of temperaments [[tempering out|tempers out]] the kleisma, [[15625/15552]]. If nothing else is tempered out in the 7-limit, we have a [[planar temperament]]. With an 11-limit comma we get an 11-limit temperament, and so forth.
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-10-08 22:58:44 UTC</tt>.<br>
: The original revision id was <tt>169075471</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">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
== Rank-3 kleismic ==
[|1 0 0 0&gt;, |0 1 0 0&gt;, |1 5/6 0 0&gt;, |0 0 0 1&gt;]
[[Subgroup]]: 2.3.5.7
[[Eigenmonzo|Eigenmonzos]]: 2, 7/6, 4/3


Map: [&lt;1 0 1 0|, &lt;0 6 5 0|, &lt;0 0 0 1|]
[[Comma list]]: 15625/15552
[[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===
{{Mapping|legend=1| 1 0 1 0 | 0 6 5 0 | 0 0 0 1 }}
[[Comma|Commas]]: 15625/15552, 385/384


[[Minimax tuning]]
: mapping generators: ~2, ~6/5, ~7
[|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;]
[[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|]
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6/5 = 317.0586, ~7/4 = 968.8259
[[Generator|Generators]]: 2, 6/5, 7


13-limit
[[Minimax tuning]]:
[[Comma|Commas]]: 15625/15552, 385/384, 325/324
* [[7-odd-limit|7-]] and [[9-odd-limit]]
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 1 5/6 0 0 | 0 0 0 1 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7


13 and 15 limit minimax tuning
{{Optimal ET sequence|legend=1| 15, 19, 34d, 49, 53, 72, 140, 212, 617c, 829c }}
[|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;,  
[[Badness]]: 0.363 × 10<sup>-3</sup>
|28/9 0 0 13/27 -14/27 1/27&gt;,  
 
|28/9 0 0 -14/27 13/27 1/27&gt;,  
=== 2.3.5.7.13 subgroup ===
|28/9 0 0 -14/27 -14/27 28/27&gt;]
Subgroup: 2.3.5.7.13
[[Eigenmonzo|Eigenmonzos]]: 2, 13/11, 14/11
 
Comma list: 325/324, 625/624
 
Sval mapping: {{mapping| 1 0 1 0 0 | 0 6 5 0 14 | 0 0 0 1 0 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~6/5 = 317.1110, ~7/4 = 968.8259
 
{{Optimal ET sequence|legend=1| 15, 19, 34d, 49f, 53, 72, 121, 140, 333, 352f, 492f }}
 
Badness: 0.290 × 10<sup>-3</sup>
 
== Enlil ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 385/384, 6250/6237
 
{{Mapping|legend=1| 1 0 1 0 6 | 0 6 5 0 1 | 0 0 0 1 -1 }}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6/5 = 317.0667, ~7/4 = 967.6041
 
[[Minimax tuning]]:
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 0 1 0 0 0 }}, {{monzo| 1 5/6 0 0 0 }}, {{monzo| 3 1/12 0 1/2 -1/2 }}, {{monzo| 3 1/12 0 -1/2 1/2 }}]
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.11/7
 
{{Optimal ET sequence|legend=1| 15, 19, 34, 38, 53, 68, 72, 140, 159, 212, 284, 496ce, 583cde, 655cde, 867ccdee, 939ccddee }}
 
[[Badness]]: 0.798 × 10<sup>-3</sup>
 
=== 13-limit (sumatra) ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 385/384, 625/624
 
Mapping: {{mapping| 1 0 1 0 6 0 | 0 6 5 0 1 14 | 0 0 0 1 -1 0 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~6/5 = 317.1156, ~7/4 = 967.6235
 
Minimax tuning:
* 13- and 15-odd-limit
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 4/3 0 0 -2/9 -2/9 4/9 }}, {{monzo| 19/9 0 0 -5/27 -5/27 10/27 }}, {{monzo| 28/9 0 0 13/27 -14/27 1/27 }}, {{monzo| 28/9 0 0 -14/27 13/27 1/27 }}, {{monzo| 28/9 0 0 -14/27 -14/27 28/27 }}]
: unchanged-interval (eigenmonzo) basis: 2.11/7.13/7
 
{{Optimal ET sequence|legend=1| 15, 19, 34, 53, 72, 87, 125f, 140, 159, 212, 299, 371df, 511cdef }}
 
Badness: 0.680 × 10<sup>-3</sup>
 
Complexity spectrum: 6/5, 18/13, 15/13, 5/4, 4/3, 10/9, 13/12, 13/10, 16/15, 9/8, 8/7, 11/8, 11/10, 7/5, 12/11, 7/6, 16/13, 15/11, 15/14, 11/9, 9/7, 13/11, 14/13, 14/11


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]], 284, 299
Scales: [[enlil19_13]]
Scales: [[enlil19_13]]


===Enki===
== Isimud ==
[[Comma|Commas]]: 15625/15552, 3025/3024
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 4125/4096, 8019/8000
 
{{Mapping|legend=1| 1 0 1 0 9 | 0 6 5 0 -21 | 0 0 0 1 0 }}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6/5 = 316.7584, ~7/4 = 968.8259
 
{{Optimal ET sequence|legend=1| 19, 34e, 53, 72, 197e, 269ce, 341ce, 375cdee, 447cdee, 519bccddee }}
 
[[Badness]]: 3.690 × 10<sup>-3</sup>
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 625/624, 1287/1280
 
Mapping: {{mapping| 1 0 1 0 9 0 | 0 6 5 0 -21 14 | 0 0 0 1 0 0 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~6/5 = 316.8446, ~7/4 = 968.8259
 
{{Optimal ET sequence|legend=1| 19, 34e, 53, 72, 125f, 178def, 197ef, 231eef, 250deff, 303cdeeff }}
 
Badness: 2.273 × 10<sup>-3</sup>
 
== Enki ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 3025/3024, 4375/4356
 
{{Mapping|legend=1| 1 0 1 0 1 | 0 6 5 0 4 | 0 0 0 2 1 }}
 
: mapping generators: ~2, ~6/5, ~66/25
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6/5 = 317.0396, ~33/25 = 484.2358
 
[[Minimax tuning]]:
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| -18/13 0 6/13 -6/13 12/13 }}, {{monzo| -2/13 0 5/13 -5/13 10/13 }}, {{monzo| -2/13 0 -8/13 8/13 10/13 }}, {{monzo| 0 0 0 0 1 }}]
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11
 
{{Optimal ET sequence|legend=1| 15, 23de, 34d, 49, 72, 159, 193, 265, 352, 424 }}
 
[[Badness]]: 1.09 × 10<sup>-3</sup>
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 364/363, 625/624
 
Mapping: {{mapping| 1 0 1 0 1 0 | 0 6 5 0 4 14 | 0 0 0 2 1 0 }}


[[Minimax tuning]]
Optimal tuning (CTE): ~2 = 1\1, ~6/5 = 317.1387, ~33/25 = 483.3828
[|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 -8/13 8/13 10/13&gt;, |0 0 0 0 1&gt;]
[[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|]
Minimax tuning:
[[Generator|Generators]]: 2, 6/5, 66/25
* 13- and 15-odd-limit
[[edo|Edos]]: [[15edo|15]], [[49edo|49]], [[57edo|57]], [[72edo|72]], [[87edo|87]], [[159edo|159]], [[193edo|193]], [[231|231]], 265, 280, 337, 352, 439
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 2/3 0 -2/3 0 0 2/3 }}, {{monzo| 14/9 0 -5/9 0 0 5/9 }}, {{monzo| 0 0 0 1 0 0 }}, {{monzo| 13/9 0 -4/9 1/2 0 4/9 }}, {{monzo| 14/9 0 -14/9 0 0 14/9 }}]
: unchanged-interval (eigenmonzo) basis: 2.7.13/5


13-limit
{{Optimal ET sequence|legend=1| 34d, 49f, 72, 87, 159, 193, 280 }}
[[Comma|Commas]]: 15625/15552, 3025/3024, 325/324


13 and 15 limit
Badness: 0.874 × 10<sup>-3</sup>
[|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;,
|13/9 0 -4/9 1/2 0 4/9&gt;,
|14/9 0 -14/9 0 0 14/9&gt;]
[[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|]
[[Category:Temperament families]]
[[Generator|Generators]]: 2, 6/5, 66/25
[[Category:Pages with mostly numerical content]]
[[edo|Edos]]: [[15edo|15]], [[72edo|72]], [[87edo|87]], [[159edo|159]], [[193edo|193]], 280, 439</pre></div>
[[Category:Kleismic rank-3 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;Kleismic rank three family&lt;/title&gt;&lt;/head&gt;&lt;body&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;h3&amp;gt; --&gt;&lt;h3 id="toc0"&gt;&lt;a name="x--Enlil"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Enlil&lt;/h3&gt;
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 15625/15552, 385/384&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;
&lt;br /&gt;
13-limit&lt;br /&gt;
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 15625/15552, 385/384, 325/324&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;, 284, 299&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:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x--Enki"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Enki&lt;/h3&gt;
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 15625/15552, 3025/3024&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;
&lt;br /&gt;
13-limit&lt;br /&gt;
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 15625/15552, 3025/3024, 325/324&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;/body&gt;&lt;/html&gt;</pre></div>
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