Kleismic rank-3 family: Difference between revisions

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**Imported revision 166253029 - Original comment: **
 
Move catakleismoid here
 
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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>2010-09-28 21:42:18 UTC</tt>.<br>
: The original revision id was <tt>166253029</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 11-limit or higher temperaments tempering out the kleisma and an 11-limit comma.


===Enlil===
== Rank-3 kleismic ==
Commas: 15625/15552, 385/384
[[Subgroup]]: 2.3.5.7


Minimax tuning
[[Comma list]]: 15625/15552
[|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;]
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|]
{{Mapping|legend=1| 1 0 1 0 | 0 6 5 0 | 0 0 0 1 }}
Generators: 2, 6/5, 7
: mapping generators: ~2, ~6/5, ~7


[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1659{{c}}, ~6/5 = 317.0504{{c}}, ~7/4 = 968.4936{{c}}
: [[error map]]: {{val| +0.166 +0.347 -0.896 -0.001 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 317.0308{{c}}, ~7/4 = 968.5773{{c}}
: error map: {{val| 0.000 +0.230 -1.160 -0.249 }}


13-limit
[[Minimax tuning]]:
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;,  
|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;]
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|]
[[Badness]] (Sintel): 1.64
Generators: 2, 6/5, 7
 
Edos: 15, 19, 34, 53, 57, 68, 72, 87, 159, 212, 284, 299</pre></div>
=== 2.3.5.7.13 subgroup ===
<h4>Original HTML content:</h4>
Subgroup: 2.3.5.7.13
<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 11-limit or higher temperaments tempering out the kleisma and an 11-limit comma.&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 325/324, 625/624
&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;
 
Commas: 15625/15552, 385/384&lt;br /&gt;
Subgroup-val mapping: {{mapping| 1 0 1 0 0 | 0 6 5 0 14 | 0 0 0 1 0 }}
&lt;br /&gt;
 
Minimax tuning&lt;br /&gt;
Optimal tunings:
[|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;
* WE: ~2 = 1200.1210{{c}}, ~6/5 = 317.1076{{c}}, ~7/4 = 968.5830{{c}}
|3 1/12 0 1/2 -1/2&amp;gt;, |3 1/12 0 -1/2 1/2&amp;gt;]&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0920{{c}}, ~7/4 = 968.6426{{c}}
Eigenmonzos: 2, 4/3, 14/11&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 15, 19, 34d, 49f, 53, 72, 121, 140, 333, 352f, 492f }}
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;
 
Generators: 2, 6/5, 7&lt;br /&gt;
Badness (Sintel): 0.385
&lt;br /&gt;
 
&lt;br /&gt;
== Enlil ==
13-limit&lt;br /&gt;
[[Subgroup]]: 2.3.5.7.11
Commas: 15625/15552, 385/384, 325/324&lt;br /&gt;
 
&lt;br /&gt;
[[Comma list]]: 385/384, 6250/6237
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;
{{Mapping|legend=1| 1 0 1 0 6 | 0 6 5 0 1 | 0 0 0 1 -1 }}
|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;
[[Optimal tuning]]s:  
|28/9 0 0 -14/27 13/27 1/27&amp;gt;, &lt;br /&gt;
* [[WE]]: ~2 = 1200.3806{{c}}, ~6/5 = 317.0420{{c}}, ~7/4 = 967.7397{{c}}
|28/9 0 0 -14/27 -14/27 28/27&amp;gt;]&lt;br /&gt;
: [[error map]]: {{val| +0.381 +0.297 -0.723 -0.325 -0.493 }}
Eigenmonzos: 2, 13/11, 14/11&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 316.9734{{c}}, ~7/4 = 967.4765{{c}}
&lt;br /&gt;
: error map: {{val| 0.000 -0.114 -1.447 -1.349 -1.821 }}
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;
 
Generators: 2, 6/5, 7&lt;br /&gt;
[[Minimax tuning]]:  
Edos: 15, 19, 34, 53, 57, 68, 72, 87, 159, 212, 284, 299&lt;/body&gt;&lt;/html&gt;</pre></div>
* [[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]] (Sintel): 0.959
 
=== 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 tunings:
* WE: ~2 = 1200.3557{{c}}, ~6/5 = 317.1024{{c}}, ~7/4 = 967.7541{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0355{{c}}, ~7/4 = 967.5068{{c}}
 
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=0| 15, 19, 34, 53, 72, 87, 125f, 140, 159, 212, 299, 371df, 511cdef }}
 
Badness (Sintel): 0.636
 
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|legend=1| 1 0 1 0 9 | 0 6 5 0 -21 | 0 0 0 1 0 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.7273{{c}}, ~6/5 = 316.9521{{c}}, ~7/4 = 967.3697{{c}}
: [[error map]]: {{val| +0.727 -0.242 -0.826 -0.002 -0.767 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 316.7598{{c}}, ~7/4 = 967.1919{{c}}
: error map: {{val| 0.000 -1.396 -2.515 -1.634 -3.273 }}
 
{{Optimal ET sequence|legend=1| 19, 34e, …, 53, 72, 197e, 269ce, 341ce, 375cdee, 447cdee, 519bccddee }}
 
[[Badness]] (Sintel): 4.43
 
=== 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 tunings:
* WE: ~2 = 1200.8711{{c}}, ~6/5 = 317.0291{{c}}, ~7/4 = 967.0809{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.8070{{c}}, ~7/4 = 966.8000{{c}}
 
{{Optimal ET sequence|legend=0| 19, 34e, …, 53, 72, 125f, 178def, 197ef, 231eef, 250deff, 303cdeefff }}
 
Badness (Sintel): 2.13
 
== 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]]s:
* [[WE]]: ~2 = 1200.0985{{c}}, ~6/5 = 317.0336{{c}}, ~33/25 = 484.1262{{c}}
: [[error map]]: {{val| +0.099 +0.246 -1.047 -0.376 +1.140 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 317.0222{{c}}, ~33/25 = 484.1546{{c}}
: error map: {{val| 0.000 +0.178 -1.203 -0.517 +0.925 }}
 
[[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]] (Sintel): 1.31
 
=== 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 }}
 
Optimal tunings:
* WE: ~2 = 1200.0373{{c}}, ~6/5 = 317.0967{{c}}, ~33/25 = 484.1601{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0919{{c}}, ~33/25 = 484.1707{{c}}
 
Minimax tuning:  
* 13- and 15-odd-limit
: [{{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
 
{{Optimal ET sequence|legend=0| 15, 23deff, 34d, 49f, 72, 87, 121, 159, 193, 280, 352f }}
 
Badness (Sintel): 0.817
 
== Catakleismoid ==
Catakleismoid is the same as catakleismic in the 2.3.5.7.13 subgroup but with an independent generator for prime 11.
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 225/224, 4375/4374
 
{{Mapping|legend=1| 1 0 1 -3 0 | 0 6 5 22 0 | 0 0 0 0 1 }}
: mapping generators: ~2, ~6/5, ~11
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.5965{{c}}, ~6/5 = 316.8893{{c}}, ~11/8 = 549.5258{{c}}
: [[error map]]: {{val| +0.596 -0.619 -1.271 +0.948 -0.003 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 316.7705{{c}}, ~11/8 = 550.0343{{c}}
: error map: {{val| 0.000 -1.332 -2.461 +0.126 -1.284 }}
 
{{Optimal ET sequence|legend=1| 19, 34d, 53, 72, 197e, 269ce, 341ce }}
 
[[Badness]] (Sintel): 1.53
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 225/224, 325/324
 
Mapping: {{mapping| 1 0 1 -3 0 0 | 0 6 5 22 0 14 | 0 0 0 0 1 0 }}
 
Optimal tunings:
* WE: ~2 = 1200.8238{{c}}, ~6/5 = 316.9478{{c}}, ~11/8 = 548.9611{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.7939{{c}}, ~11/8 = 549.5903{{c}}
 
{{Optimal ET sequence|legend=0| 19, 34d, 53, 72, 125f, 197ef }}
 
Badness (Sintel): 0.853
 
[[Category:Temperament families]]
[[Category:Kleismic rank-3 family| ]] <!-- main article -->
[[Category:Rank 3]]

Latest revision as of 09:42, 11 April 2026

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 tunings:

  • WE: ~2 = 1200.1659 ¢, ~6/5 = 317.0504 ¢, ~7/4 = 968.4936 ¢
error map: ⟨+0.166 +0.347 -0.896 -0.001]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.0308 ¢, ~7/4 = 968.5773 ¢
error map: ⟨0.000 +0.230 -1.160 -0.249]

Minimax tuning:

[[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 (Sintel): 1.64

2.3.5.7.13 subgroup

Subgroup: 2.3.5.7.13

Comma list: 325/324, 625/624

Subgroup-val mapping: [⟨1 0 1 0 0], ⟨0 6 5 0 14], ⟨0 0 0 1 0]]

Optimal tunings:

  • WE: ~2 = 1200.1210 ¢, ~6/5 = 317.1076 ¢, ~7/4 = 968.5830 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.0920 ¢, ~7/4 = 968.6426 ¢

Optimal ET sequence: 15, 19, 34d, 49f, 53, 72, 121, 140, 333, 352f, 492f

Badness (Sintel): 0.385

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 tunings:

  • WE: ~2 = 1200.3806 ¢, ~6/5 = 317.0420 ¢, ~7/4 = 967.7397 ¢
error map: ⟨+0.381 +0.297 -0.723 -0.325 -0.493]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.9734 ¢, ~7/4 = 967.4765 ¢
error map: ⟨0.000 -0.114 -1.447 -1.349 -1.821]

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⟩]
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 (Sintel): 0.959

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 tunings:

  • WE: ~2 = 1200.3557 ¢, ~6/5 = 317.1024 ¢, ~7/4 = 967.7541 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.0355 ¢, ~7/4 = 967.5068 ¢

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 (Sintel): 0.636

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 tunings:

  • WE: ~2 = 1200.7273 ¢, ~6/5 = 316.9521 ¢, ~7/4 = 967.3697 ¢
error map: ⟨+0.727 -0.242 -0.826 -0.002 -0.767]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.7598 ¢, ~7/4 = 967.1919 ¢
error map: ⟨0.000 -1.396 -2.515 -1.634 -3.273]

Optimal ET sequence: 19, 34e, …, 53, 72, 197e, 269ce, 341ce, 375cdee, 447cdee, 519bccddee

Badness (Sintel): 4.43

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 tunings:

  • WE: ~2 = 1200.8711 ¢, ~6/5 = 317.0291 ¢, ~7/4 = 967.0809 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.8070 ¢, ~7/4 = 966.8000 ¢

Optimal ET sequence: 19, 34e, …, 53, 72, 125f, 178def, 197ef, 231eef, 250deff, 303cdeefff

Badness (Sintel): 2.13

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 tunings:

  • WE: ~2 = 1200.0985 ¢, ~6/5 = 317.0336 ¢, ~33/25 = 484.1262 ¢
error map: ⟨+0.099 +0.246 -1.047 -0.376 +1.140]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.0222 ¢, ~33/25 = 484.1546 ¢
error map: ⟨0.000 +0.178 -1.203 -0.517 +0.925]

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⟩]
unchanged-interval (eigenmonzo) basis: 2.7/5.11

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

Badness (Sintel): 1.31

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 tunings:

  • WE: ~2 = 1200.0373 ¢, ~6/5 = 317.0967 ¢, ~33/25 = 484.1601 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.0919 ¢, ~33/25 = 484.1707 ¢

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: 15, 23deff, 34d, 49f, 72, 87, 121, 159, 193, 280, 352f

Badness (Sintel): 0.817

Catakleismoid

Catakleismoid is the same as catakleismic in the 2.3.5.7.13 subgroup but with an independent generator for prime 11.

Subgroup: 2.3.5.7.11

Comma list: 225/224, 4375/4374

Mapping: [⟨1 0 1 -3 0], ⟨0 6 5 22 0], ⟨0 0 0 0 1]]

mapping generators: ~2, ~6/5, ~11

Optimal tunings:

  • WE: ~2 = 1200.5965 ¢, ~6/5 = 316.8893 ¢, ~11/8 = 549.5258 ¢
error map: ⟨+0.596 -0.619 -1.271 +0.948 -0.003]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.7705 ¢, ~11/8 = 550.0343 ¢
error map: ⟨0.000 -1.332 -2.461 +0.126 -1.284]

Optimal ET sequence: 19, 34d, 53, 72, 197e, 269ce, 341ce

Badness (Sintel): 1.53

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 325/324

Mapping: [⟨1 0 1 -3 0 0], ⟨0 6 5 22 0 14], ⟨0 0 0 0 1 0]]

Optimal tunings:

  • WE: ~2 = 1200.8238 ¢, ~6/5 = 316.9478 ¢, ~11/8 = 548.9611 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.7939 ¢, ~11/8 = 549.5903 ¢

Optimal ET sequence: 19, 34d, 53, 72, 125f, 197ef

Badness (Sintel): 0.853

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