Kleismic rank three family: Difference between revisions

Latest revision as of 00:30, 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 kleismic rank-3 family of temperaments 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.

Rank-3 kleismic

Subgroup: 2.3.5.7

Comma list: 15625/15552

Mapping: [⟨1 0 1 0], ⟨0 6 5 0], ⟨0 0 0 1]]

mapping generators: ~2, ~6/5, ~7

Optimal tuning (CTE): ~2 = 1\1, ~6/5 = 317.0586, ~7/4 = 968.8259

Minimax tuning:

7- and 9-odd-limit

[[1 0 0 0⟩, [0 1 0 0⟩, [1 5/6 0 0⟩, [0 0 0 1⟩]

unchanged-interval (eigenmonzo) basis: 2.3.7

Optimal ET sequence: 15, 19, 34d, 49, 53, 72, 140, 212, 617c, 829c

Badness: 0.363 × 10^-3

2.3.5.7.13 subgroup

Subgroup: 2.3.5.7.13

Comma list: 325/324, 625/624

Sval 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: 15, 19, 34d, 49f, 53, 72, 121, 140, 333, 352f, 492f

Badness: 0.290 × 10^-3

Enlil

Subgroup: 2.3.5.7.11

Comma list: 385/384, 6250/6237

Mapping: [⟨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

[[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⟩]

unchanged-interval (eigenmonzo) basis: 2.3.11/7

Optimal ET sequence: 15, 19, 34, 38, 53, 68, 72, 140, 159, 212, 284, 496ce, 583cde, 655cde, 867ccdee, 939ccddee

Badness: 0.798 × 10^-3

13-limit (sumatra)

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 625/624

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

[[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⟩]

unchanged-interval (eigenmonzo) basis: 2.11/7.13/7

Optimal ET sequence: 15, 19, 34, 53, 72, 87, 125f, 140, 159, 212, 299, 371df, 511cdef

Badness: 0.680 × 10^-3

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

Scales: enlil19_13

Isimud

Subgroup: 2.3.5.7.11

Comma list: 4125/4096, 8019/8000

Mapping: [⟨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: 19, 34e, 53, 72, 197e, 269ce, 341ce, 375cdee, 447cdee, 519bccddee

Badness: 3.690 × 10^-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 625/624, 1287/1280

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: 19, 34e, 53, 72, 125f, 178def, 197ef, 231eef, 250deff, 303cdeeff

Badness: 2.273 × 10^-3

Enki

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4356

Mapping: [⟨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

[[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⟩]

unchanged-interval (eigenmonzo) basis: 2.7/5.11

Optimal ET sequence: 15, 23de, 34d, 49, 72, 159, 193, 265, 352, 424

Badness: 1.09 × 10^-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 625/624

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

Optimal tuning (CTE): ~2 = 1\1, ~6/5 = 317.1387, ~33/25 = 483.3828

Minimax tuning:

13- and 15-odd-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⟩]

unchanged-interval (eigenmonzo) basis: 2.7.13/5

Optimal ET sequence: 34d, 49f, 72, 87, 159, 193, 280

Badness: 0.874 × 10^-3

Kleismic rank three family: Difference between revisions

Latest revision as of 00:30, 24 June 2025

Contents

Rank-3 kleismic

2.3.5.7.13 subgroup

Enlil

13-limit (sumatra)

Isimud

13-limit

Enki

13-limit

Navigation menu

@@ Line 1: / Line 1: @@
-<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>2011-10-24 00:46:27 UTC</tt>.<br>
-: The original revision id was <tt>267788536</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]]
+== Rank-3 kleismic ==
+[[Subgroup]]: 2.3.5.7
-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.
+[[Comma list]]: 15625/15552
-and 9 limit minimax
+{{Mapping|legend=1| 1 0 1 0 | 0 6 5 0 | 0 0 0 1 }}
-[|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
-Map: [&lt;1 0 1 0|, &lt;0 6 5 0|, &lt;0 0 0 1|]
+: mapping generators: ~2, ~6/5, ~7
-[[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=
+[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6/5 = 317.0586, ~7/4 = 968.8259
-[[Comma|Commas]]: 385/384, 6250/6237
-[[Minimax tuning]]
+[[Minimax tuning]]:
-[|1 0 0 0 0&gt;, |0 1 0 0 0&gt;, |1 5/6 0 0 0&gt;,
+* [[7-odd-limit|7-]] and [[9-odd-limit]]
-|3 1/12 0 1/2 -1/2&gt;, |3 1/12 0 -1/2 1/2&gt;]
+: {{monzo list| 1 0 0 0 | 0 1 0 0 | 1 5/6 0 0 | 0 0 0 1 }}
-[[Eigenmonzo|Eigenmonzos]]: 2, 4/3, 14/11
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7
-Map: [&lt;1 0 1 0 6|, &lt;0 6 5 0 1|, &lt;0 0 0 1 -1|]
+{{Optimal ET sequence|legend=1| 15, 19, 34d, 49, 53, 72, 140, 212, 617c, 829c }}
-[[Generator|Generators]]: 2, 6/5, 7
-EDOs: 53, 68, 72, 140, 159, 212, 284
-Badness: 0.000798
-==13-limit==
+[[Badness]]: 0.363 × 10<sup>-3</sup>
-[[Comma|Commas]]: 325/324, 385/384, 625/624
-and 15 limit minimax tuning
+=== 2.3.5.7.13 subgroup ===
-[|1 0 0 0 0 0&gt;, |4/3 0 0 -2/9 -2/9 4/9&gt;,
+Subgroup: 2.3.5.7.13
-|19/9 0 0 -5/27 -5/27 10/27&gt;,
-|28/9 0 0 13/27 -14/27 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;]
-[[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|]
+Comma list: 325/324, 625/624
-[[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]]
+Sval mapping: {{mapping| 1 0 1 0 0 | 0 6 5 0 14 | 0 0 0 1 0 }}
-Badness: 0.000680
+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
 Scales: [[enlil19_13]]
-=Enki=
+== Isimud ==
-[[Comma|Commas]]: 3025/3024, 4375/4356
+[[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 }}]
-Badness: 0.00109
+: unchanged-interval (eigenmonzo) basis: 2.7.13/5
-==13-limit==
+{{Optimal ET sequence|legend=1| 34d, 49f, 72, 87, 159, 193, 280 }}
-[[Comma|Commas]]: 325/324, 364/363, 625/624
-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
+[[Category:Kleismic rank-3 family| ]] <!-- main article -->
-Badness: 0.000874</pre></div>
+[[Category:Rank 3]]
-<h4>Original HTML 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;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:8:&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:8 --&gt;&lt;!-- ws:start:WikiTextTocRule:9: --&gt;&lt;a href="#Enlil"&gt;Enlil&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:9 --&gt;&lt;!-- ws:start:WikiTextTocRule:10: --&gt;&lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;!-- ws:start:WikiTextTocRule:11: --&gt; | &lt;a href="#Enki"&gt;Enki&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:11 --&gt;&lt;!-- ws:start:WikiTextTocRule:12: --&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;
-&lt;!-- ws:end:WikiTextTocRule:13 --&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;
-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;
-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;
-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;/body&gt;&lt;/html&gt;</pre></div>

Kleismic rank three family: Difference between revisions

Latest revision as of 00:30, 24 June 2025

Rank-3 kleismic

2.3.5.7.13 subgroup

Enlil

13-limit (sumatra)

Isimud

13-limit

Enki

13-limit

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