Quince clan: 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 '''quince clan''' [[Tempering out|tempers out]] the [[quince comma]] ({{monzo|legend=1| -15 0 -2 7 }}, [[ratio]]: 823543/819200).
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-05-21 15:31:43 UTC</tt>.<br>
: The original revision id was <tt>338028770</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]]


This tempers out quince, the no-threes comma |-15 0 -2 7&gt; = 823543/819200. Quince temperaments include miracle, quincy, octowerck and birds as well as the ones listed below.
== Mercy ==
Mercy is the no-3 version of [[miracle]]. Two generators make [[~]][[8/7]], five generators make ~[[7/5]], and seven make ~[[8/5]]. The [[ploidacot]] for this temperament is omega-heptaseph.  


=Essence=
[[Subgroup]]: 2.5.7
Essence is of interest because of the abundant supply of essentially tempered chords.


Commas: 321489/320000, 823543/820125
[[Comma list]]: 823543/819200


POTE generator: ~243/196 = 372.597
{{Mapping|legend=2| 1 3 3 | 0 -7 -2 }}


Map: [&lt;2 10 27 23|, &lt;0 -11 -36 -28|]
{{Mapping|legend=3| 1 0 3 3 | 0 0 -7 -2 }}
Wedgie: &lt;&lt;22 72 56 63 27 -72||
: mapping generators: ~2, ~343/320
EDOs: 58, 190, 248, 438d, 686d
Badness: 0.1736


==11-limit==
[[Optimal tuning]]s:
Commas: 441/440, 8019/8000, 456533/455625
* [[WE]]: ~2 = 1200.2169{{c}}, ~343/320 = 116.3116{{c}}
: [[error map]]: {{val| +0.217 +0.156 -0.798 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~343/320 = 116.2576{{c}}
: error map: {{val| 0.000 -0.117 -1.341 }}


POTE generator: ~243/196 = 372.599
{{Optimal ET sequence|legend=1| 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd }}


Map: [&lt;2 10 27 23 33|, &lt;0 -11 -36 -28 -42|]
[[Badness]] (Sintel): 0.428
EDOs: 58, 190, 248, 438d
Badness: 0.0464


==13-limit==
=== Overview to extensions ===
Commas: 441/440, 729/728, 847/845, 1001/1000
Temperaments discussed elsewhere are:
* ''[[Casablanca]]'' (+126/125) → [[Starling temperaments #Casablanca|Starling temperaments]]
* [[Miracle]] (+225/224) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* ''[[Slendi]]'' (+4000/3969) → [[Octagar temperaments #Slendi|Octagar temperaments]]
* ''[[Quincy]]'' (+4375/4374) → [[Ragismic microtemperaments #Quincy|Ragismic microtemperaments]]
* ''[[Birds]]'' (+3136/3125) → [[31st-octave temperaments #Birds|31st-octave temperaments]]
* ''[[Octowerck]]'' (+321489/320000 or 420175/419904) → [[Varunismic temperaments #Octowerck|Varunismic temperaments]]
* [[Cotoneum]] (+10976/10935) → [[Garischismic clan #Cotoneum|Garischismic clan]]


POTE generator: ~243/196 = 372.602
Discussed below is countermiracle.  


Map: [&lt;2 10 27 23 33 31|, &lt;0 -11 -36 -28 -42 -38|]
== Countermiracle ==
EDOs: 58, 190, 248, 438d
The countermiracle temperament tempers out the porwell comma ([[6144/6125]]), the trimyna comma ([[50421/50000]]) and the quince comma ([[823543/819200]]). It can be described as the {{nowrap| 31 & 145 }} temperament, and finds the [[3/2|perfect fifth]] -25 generators away (normalized for 2.5.7) as opposed to miracle's +6 generators. It has the [[ploidacot]] signature of 22-sheared 25-cot.  
Badness: 0.0261


</pre></div>
Like miracle, it is naturally an 11-limit temperament with the generator representing [[77/72]], but here the generator does not represent [[15/14]] or [[16/15]].
<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;Quince clan&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:6:&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:6 --&gt;&lt;!-- ws:start:WikiTextTocRule:7: --&gt;&lt;a href="#Essence"&gt;Essence&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:7 --&gt;&lt;!-- ws:start:WikiTextTocRule:8: --&gt;&lt;!-- ws:end:WikiTextTocRule:8 --&gt;&lt;!-- ws:start:WikiTextTocRule:9: --&gt;&lt;!-- ws:end:WikiTextTocRule:9 --&gt;&lt;!-- ws:start:WikiTextTocRule:10: --&gt;
[[Subgroup]]: 2.3.5.7
&lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;br /&gt;
 
This tempers out quince, the no-threes comma |-15 0 -2 7&amp;gt; = 823543/819200. Quince temperaments include miracle, quincy, octowerck and birds as well as the ones listed below.&lt;br /&gt;
[[Comma list]]: 6144/6125, 50421/50000
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Essence"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Essence&lt;/h1&gt;
{{Mapping|legend=1| 1 -21 -4 1 | 0 25 7 2 }}
Essence is of interest because of the abundant supply of essentially tempered chords.&lt;br /&gt;
: mapping generators: ~2, ~625/336
&lt;br /&gt;
 
Commas: 321489/320000, 823543/820125&lt;br /&gt;
[[Optimal tuning]]s:  
&lt;br /&gt;
* [[WE]]: ~2 = 1199.7944{{c}}, ~625/336 = 1083.8973{{c}} (~343/320 = 115.8970{{c}})
POTE generator: ~243/196 = 372.597&lt;br /&gt;
: [[error map]]: {{val| -0.206 -0.203 +1.790 -1.237 }}
&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~625/336 = 1084.0787{{c}} (~343/320 = 115.9213{{c}})
Map: [&amp;lt;2 10 27 23|, &amp;lt;0 -11 -36 -28|]&lt;br /&gt;
: error map: {{val| 0.000 +0.013 +2.237 -0.669 }}
Wedgie: &amp;lt;&amp;lt;22 72 56 63 27 -72||&lt;br /&gt;
 
EDOs: 58, 190, 248, 438d, 686d&lt;br /&gt;
{{Optimal ET sequence|legend=1| 31, 114, 145, 176 }}
Badness: 0.1736&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]] (Sintel): 2.59
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Essence-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;11-limit&lt;/h2&gt;
 
Commas: 441/440, 8019/8000, 456533/455625&lt;br /&gt;
=== 11-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
POTE generator: ~243/196 = 372.599&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 441/440, 3388/3375, 6144/6125
Map: [&amp;lt;2 10 27 23 33|, &amp;lt;0 -11 -36 -28 -42|]&lt;br /&gt;
 
EDOs: 58, 190, 248, 438d&lt;br /&gt;
Mapping: {{mapping| 1 -21 -4 1 -39 | 0 25 7 2 47 }}
Badness: 0.0464&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Essence-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;13-limit&lt;/h2&gt;
* WE: ~2 = 1199.7942{{c}}, ~144/77 = 1083.8983{{c}} (~77/72 = 115.8959{{c}})
Commas: 441/440, 729/728, 847/845, 1001/1000&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.0798{{c}} (~77/72 = 115.9202{{c}})
&lt;br /&gt;
 
POTE generator: ~243/196 = 372.602&lt;br /&gt;
{{Optimal ET sequence|legend=0| 31, 114e, 145, 176 }}
&lt;br /&gt;
 
Map: [&amp;lt;2 10 27 23 33 31|, &amp;lt;0 -11 -36 -28 -42 -38|]&lt;br /&gt;
Badness (Sintel): 1.29
EDOs: 58, 190, 248, 438d&lt;br /&gt;
 
Badness: 0.0261&lt;/body&gt;&lt;/html&gt;</pre></div>
==== Countermiraculous ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 196/195, 352/351, 1001/1000, 6144/6125
 
Mapping: {{mapping| 1 -21 -4 1 -39 29 | 0 25 7 2 47 -28 }}
 
Optimal tunings:  
* WE: ~2 = 1199.4465{{c}}, ~144/77 = 1083.6196{{c}} (~77/72 = 115.8268{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.1148{{c}} (~77/72 = 115.8852{{c}})
 
{{Optimal ET sequence|legend=0| 31, 83e, 114e, 145, 321ceff }}
 
Badness (Sintel): 1.62
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 196/195, 256/255, 352/351, 1001/1000, 1225/1224
 
Mapping: {{mapping| 1 -21 -4 1 -39 29 33 | 0 25 7 2 47 -28 -32 }}
 
Optimal tunings:  
* WE: ~2 = 1199.4013{{c}}, ~144/77 = 1083.5835{{c}} (~77/72 = 115.8178{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.1219{{c}} (~77/72 = 115.8781{{c}})
 
{{Optimal ET sequence|legend=0| 31, 83e, 114e, 145 }}
 
Badness (Sintel): 1.50
 
==== Counterbenediction ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 441/440, 3146/3125, 3584/3575
 
Mapping: {{mapping| 1 -21 -4 1 -39 29 57 | 0 25 7 2 47 -59 }}
 
Optimal tunings:  
* WE: ~2 = 1199.9460{{c}}, ~144/77 = 1084.0178{{c}} (~77/72 = 115.9283{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.0663{{c}} (~77/72 = 115.9337{{c}})
 
{{Optimal ET sequence|legend=0| 31, 145f, 176, 207 }}
 
Badness (Sintel): 1.88
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 351/350, 441/440, 561/560, 1632/1625, 3146/3125
 
Mapping: {{mapping| 1 -21 -4 1 -39 29 57 61 | 0 25 7 2 47 -59 -63 }}
 
Optimal tunings:
* WE: ~2 = 1199.9950{{c}}, ~144/77 = 1084.0564{{c}} (~77/72 = 115.9386{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.0609{{c}} (~77/72 = 115.9391{{c}})
 
{{Optimal ET sequence|legend=0| 31, 176, 207 }}
 
Badness (Sintel): 1.85
 
==== Countermanna ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 364/363, 441/440, 3388/3375, 6144/6125
 
Mapping: {{mapping| 1 -21 -4 1 -39 -102 | 0 25 7 2 47 117 }}
 
Optimal tunings:  
* WE: ~2 = 1199.6687{{c}}, ~144/77 = 1083.8108{{c}} (~77/72 = 115.8578{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.1065{{c}} (~77/72 = 115.8935{{c}})
 
{{Optimal ET sequence|legend=0| 31f, 145, 176, 321ce }}
 
Badness (Sintel): 2.21
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 364/363, 441/440, 595/594, 1632/1625, 3388/3375
 
Mapping: {{mapping| 1 -21 -4 1 -39 -102 -98 | 0 25 7 2 47 117 113 }}
 
Optimal tunings:
* WE: ~2 = 1199.6458{{c}}, ~144/77 = 1083.7968{{c}} (~77/72 = 115.8490{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.1132{{c}} (~77/72 = 115.8868{{c}})
 
{{Optimal ET sequence|legend=0| 31fg, 145, 321ce }}
 
Badness (Sintel): 2.08
 
=== Counterrevelation ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 176/175, 50421/50000
 
Mapping: {{mapping| 1 -21 -4 1 -11 | 0 25 7 2 16 }}
 
Optimal tunings:
* WE: ~2 = 1200.1897{{c}}, ~275/147 = 1084.2521{{c}} (~343/320 = 115.9375{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~275/147 = 1084.0850{{c}} (~343/320 = 115.9150{{c}})
 
{{Optimal ET sequence|legend=0| 31, 114, 145e }}
 
Badness (Sintel): 2.12
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 121/120, 176/175, 196/195, 13750/13689
 
Mapping: {{mapping| 1 -21 -4 1 -11 29 | 0 25 7 2 16 -28 }}
 
Optimal tunings:  
* WE: ~2 = 1199.7651{{c}}, ~220/117 = 1083.9253{{c}} (~117/110 = 115.8398{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~220/117 = 1084.1357{{c}} (~117/110 = 115.8643{{c}})
 
{{Optimal ET sequence|legend=0| 31, 83, 114 }}
 
Badness (Sintel): 2.38
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 121/120, 154/153, 176/175, 196/195, 10647/10625
 
Mapping: {{mapping| 1 -21 -4 1 -11 29 33 | 0 25 7 2 16 -28 -32 }}
 
Optimal tunings:
* WE: ~2 = 1199.6935{{c}}, ~170/91 = 1083.8704{{c}} (~91/85 = 115.8231{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~170/91 = 1084.1464{{c}} (~91/85 = 115.8536{{c}})
 
{{Optimal ET sequence|legend=0| 31, 83, 114 }}
 
Badness (Sintel): 2.24
 
== Subgroup extensions ==
=== Mercy (2.5.7.13) ===
This extension may be thought of as a way to conceptualize the 2.5.7.13.17.19 subgroup of [[31edo]].
 
Subgroup: 2.5.7.13
 
Comma list: 343/338, 640/637
 
Subgroup-val mapping: {{mapping| 1 3 3 4 | 0 -7 -2 -3 }}
 
Gencom mapping: {{mapping| 1 0 3 3 0 4 | 0 0 -7 -2 0 -3 }}
 
Optimal tunings:
* WE: ~2 = 1199.2166{{c}}, ~14/13 = 116.0181{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 116.2099{{c}}
 
{{Optimal ET sequence|legend=0| 10, 21, 31 }}
 
Badness (Sintel): 0.546
 
==== 2.5.7.13.17 subgroup ====
Subgroup: 2.5.7.13.17
 
Comma list: 170/169, 224/221, 343/338
 
Subgroup-val mapping: {{mapping| 1 3 3 4 4 | 0 -7 -2 -3 1 }}
 
Gencom mapping: {{mapping| 1 0 3 3 0 4 4 | 0 0 -7 -2 0 -3 1 }}
 
Optimal tunings:  
* WE: ~2 = 1198.5476{{c}}, ~14/13 = 115.6294{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 115.9132{{c}}
 
{{Optimal ET sequence|legend=0| 10, 21, 31, 83fg }}
 
Badness (Sintel): 0.451
 
==== 2.5.7.13.17.19 subgroup ====
Subgroup: 2.5.7.13.17.19
 
Comma list: 170/169, 224/221, 343/338, 476/475
 
Subgroup-val mapping: {{mapping| 1 3 3 4 4 3 | 0 -7 -2 -3 1 13 }}
 
Gencom mapping: {{mapping| 1 0 3 3 0 4 4 3 | 0 0 -7 -2 0 -3 1 13 }}
 
Optimal tunings:
* WE: ~2 = 1198.4839{{c}}, ~14/13 = 115.5700{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 115.7299{{c}}
 
{{Optimal ET sequence|legend=0| 10h, 21, 31, 52f, 83fg }}
 
Badness (Sintel): 0.759
 
[[Category:Temperament clans]]
[[Category:Quince clan| ]] <!-- main article -->
[[Category:Rank 2]]

Latest revision as of 11:19, 13 March 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 quince clan tempers out the quince comma (monzo: [-15 0 -2 7⟩, ratio: 823543/819200).

Mercy

Mercy is the no-3 version of miracle. Two generators make ~8/7, five generators make ~7/5, and seven make ~8/5. The ploidacot for this temperament is omega-heptaseph.

Subgroup: 2.5.7

Comma list: 823543/819200

Subgroup-val mapping: [⟨1 3 3], ⟨0 -7 -2]]

Gencom mapping: [⟨1 0 3 3], ⟨0 0 -7 -2]]

mapping generators: ~2, ~343/320

Optimal tunings:

  • WE: ~2 = 1200.2169 ¢, ~343/320 = 116.3116 ¢
error map: ⟨+0.217 +0.156 -0.798]
  • CWE: ~2 = 1200.0000 ¢, ~343/320 = 116.2576 ¢
error map: ⟨0.000 -0.117 -1.341]

Optimal ET sequence: 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd

Badness (Sintel): 0.428

Overview to extensions

Temperaments discussed elsewhere are:

Discussed below is countermiracle.

Countermiracle

The countermiracle temperament tempers out the porwell comma (6144/6125), the trimyna comma (50421/50000) and the quince comma (823543/819200). It can be described as the 31 & 145 temperament, and finds the perfect fifth -25 generators away (normalized for 2.5.7) as opposed to miracle's +6 generators. It has the ploidacot signature of 22-sheared 25-cot.

Like miracle, it is naturally an 11-limit temperament with the generator representing 77/72, but here the generator does not represent 15/14 or 16/15.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 50421/50000

Mapping: [⟨1 -21 -4 1], ⟨0 25 7 2]]

mapping generators: ~2, ~625/336

Optimal tunings:

  • WE: ~2 = 1199.7944 ¢, ~625/336 = 1083.8973 ¢ (~343/320 = 115.8970 ¢)
error map: ⟨-0.206 -0.203 +1.790 -1.237]
  • CWE: ~2 = 1200.0000 ¢, ~625/336 = 1084.0787 ¢ (~343/320 = 115.9213 ¢)
error map: ⟨0.000 +0.013 +2.237 -0.669]

Optimal ET sequence: 31, 114, 145, 176

Badness (Sintel): 2.59

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3388/3375, 6144/6125

Mapping: [⟨1 -21 -4 1 -39], ⟨0 25 7 2 47]]

Optimal tunings:

  • WE: ~2 = 1199.7942 ¢, ~144/77 = 1083.8983 ¢ (~77/72 = 115.8959 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~144/77 = 1084.0798 ¢ (~77/72 = 115.9202 ¢)

Optimal ET sequence: 31, 114e, 145, 176

Badness (Sintel): 1.29

Countermiraculous

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 1001/1000, 6144/6125

Mapping: [⟨1 -21 -4 1 -39 29], ⟨0 25 7 2 47 -28]]

Optimal tunings:

  • WE: ~2 = 1199.4465 ¢, ~144/77 = 1083.6196 ¢ (~77/72 = 115.8268 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~144/77 = 1084.1148 ¢ (~77/72 = 115.8852 ¢)

Optimal ET sequence: 31, 83e, 114e, 145, 321ceff

Badness (Sintel): 1.62

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 196/195, 256/255, 352/351, 1001/1000, 1225/1224

Mapping: [⟨1 -21 -4 1 -39 29 33], ⟨0 25 7 2 47 -28 -32]]

Optimal tunings:

  • WE: ~2 = 1199.4013 ¢, ~144/77 = 1083.5835 ¢ (~77/72 = 115.8178 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~144/77 = 1084.1219 ¢ (~77/72 = 115.8781 ¢)

Optimal ET sequence: 31, 83e, 114e, 145

Badness (Sintel): 1.50

Counterbenediction

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 441/440, 3146/3125, 3584/3575

Mapping: [⟨1 -21 -4 1 -39 29 57], ⟨0 25 7 2 47 -59]]

Optimal tunings:

  • WE: ~2 = 1199.9460 ¢, ~144/77 = 1084.0178 ¢ (~77/72 = 115.9283 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~144/77 = 1084.0663 ¢ (~77/72 = 115.9337 ¢)

Optimal ET sequence: 31, 145f, 176, 207

Badness (Sintel): 1.88

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 441/440, 561/560, 1632/1625, 3146/3125

Mapping: [⟨1 -21 -4 1 -39 29 57 61], ⟨0 25 7 2 47 -59 -63]]

Optimal tunings:

  • WE: ~2 = 1199.9950 ¢, ~144/77 = 1084.0564 ¢ (~77/72 = 115.9386 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~144/77 = 1084.0609 ¢ (~77/72 = 115.9391 ¢)

Optimal ET sequence: 31, 176, 207

Badness (Sintel): 1.85

Countermanna

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 3388/3375, 6144/6125

Mapping: [⟨1 -21 -4 1 -39 -102], ⟨0 25 7 2 47 117]]

Optimal tunings:

  • WE: ~2 = 1199.6687 ¢, ~144/77 = 1083.8108 ¢ (~77/72 = 115.8578 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~144/77 = 1084.1065 ¢ (~77/72 = 115.8935 ¢)

Optimal ET sequence: 31f, 145, 176, 321ce

Badness (Sintel): 2.21

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 441/440, 595/594, 1632/1625, 3388/3375

Mapping: [⟨1 -21 -4 1 -39 -102 -98], ⟨0 25 7 2 47 117 113]]

Optimal tunings:

  • WE: ~2 = 1199.6458 ¢, ~144/77 = 1083.7968 ¢ (~77/72 = 115.8490 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~144/77 = 1084.1132 ¢ (~77/72 = 115.8868 ¢)

Optimal ET sequence: 31fg, 145, 321ce

Badness (Sintel): 2.08

Counterrevelation

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 50421/50000

Mapping: [⟨1 -21 -4 1 -11], ⟨0 25 7 2 16]]

Optimal tunings:

  • WE: ~2 = 1200.1897 ¢, ~275/147 = 1084.2521 ¢ (~343/320 = 115.9375 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~275/147 = 1084.0850 ¢ (~343/320 = 115.9150 ¢)

Optimal ET sequence: 31, 114, 145e

Badness (Sintel): 2.12

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 176/175, 196/195, 13750/13689

Mapping: [⟨1 -21 -4 1 -11 29], ⟨0 25 7 2 16 -28]]

Optimal tunings:

  • WE: ~2 = 1199.7651 ¢, ~220/117 = 1083.9253 ¢ (~117/110 = 115.8398 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~220/117 = 1084.1357 ¢ (~117/110 = 115.8643 ¢)

Optimal ET sequence: 31, 83, 114

Badness (Sintel): 2.38

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 121/120, 154/153, 176/175, 196/195, 10647/10625

Mapping: [⟨1 -21 -4 1 -11 29 33], ⟨0 25 7 2 16 -28 -32]]

Optimal tunings:

  • WE: ~2 = 1199.6935 ¢, ~170/91 = 1083.8704 ¢ (~91/85 = 115.8231 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~170/91 = 1084.1464 ¢ (~91/85 = 115.8536 ¢)

Optimal ET sequence: 31, 83, 114

Badness (Sintel): 2.24

Subgroup extensions

Mercy (2.5.7.13)

This extension may be thought of as a way to conceptualize the 2.5.7.13.17.19 subgroup of 31edo.

Subgroup: 2.5.7.13

Comma list: 343/338, 640/637

Subgroup-val mapping: [⟨1 3 3 4], ⟨0 -7 -2 -3]]

Gencom mapping: [⟨1 0 3 3 0 4], ⟨0 0 -7 -2 0 -3]]

Optimal tunings:

  • WE: ~2 = 1199.2166 ¢, ~14/13 = 116.0181 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/13 = 116.2099 ¢

Optimal ET sequence: 10, 21, 31

Badness (Sintel): 0.546

2.5.7.13.17 subgroup

Subgroup: 2.5.7.13.17

Comma list: 170/169, 224/221, 343/338

Subgroup-val mapping: [⟨1 3 3 4 4], ⟨0 -7 -2 -3 1]]

Gencom mapping: [⟨1 0 3 3 0 4 4], ⟨0 0 -7 -2 0 -3 1]]

Optimal tunings:

  • WE: ~2 = 1198.5476 ¢, ~14/13 = 115.6294 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/13 = 115.9132 ¢

Optimal ET sequence: 10, 21, 31, 83fg

Badness (Sintel): 0.451

2.5.7.13.17.19 subgroup

Subgroup: 2.5.7.13.17.19

Comma list: 170/169, 224/221, 343/338, 476/475

Subgroup-val mapping: [⟨1 3 3 4 4 3], ⟨0 -7 -2 -3 1 13]]

Gencom mapping: [⟨1 0 3 3 0 4 4 3], ⟨0 0 -7 -2 0 -3 1 13]]

Optimal tunings:

  • WE: ~2 = 1198.4839 ¢, ~14/13 = 115.5700 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/13 = 115.7299 ¢

Optimal ET sequence: 10h, 21, 31, 52f, 83fg

Badness (Sintel): 0.759

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