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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 '''diminished family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the major diesis a.k.a. diminished comma, [[648/625]], the amount by which four [[6/5]] minor thirds exceed an [[octave]], and so identifies the minor third with the quarter-octave. Hence it has the same 300-cent 6/5-approximations as [[12edo]].
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-05-09 01:09:29 UTC</tt>.<br>
: The original revision id was <tt>332260622</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]]


The dimipent family tempers out the major diesis aka diminished comma, 648/625, the amount by which four 6/5 minor thirds exceed an octave, and so identifies the minor third with the quarter-octave. Hence it has the same 300-cent 6/5-approximations as 12edo.
== Diminished ==
{{Main| Diminished (temperament) }}


=Dimipent=
The [[generator]] of diminished can be taken as a fifth or a semitone, and 12edo, with its excellent fifth, is an obvious tuning, though a flatter fifth might be preferred to go with the flat minor third. Its [[ploidacot]] is tetraploid monocot.
Comma: 648/625


POTE generator: ~3/2 =  699.507
[[Subgroup]]: 2.3.5


Map: [&lt;4 0 3|, &lt;0 1 1|]
[[Comma list]]: 648/625
EDOs: 8, 12
Badness: 0.0472


=Hemidim=
{{Mapping|legend=1| 4 0 3 | 0 1 1 }}
Commas: 49/48, 648/625


POTE generator: ~8/7 = 252.555
[[Optimal tuning]]s:  
* [[WE]]: ~6/5 = 299.6476{{c}}, ~3/2 = 698.6854{{c}} (~25/24 = 99.3903{{c}})
: [[error map]]: {{val| -1.410 -4.679 +9.905 }}
* [[CWE]]: ~6/5 = 300.0000{{c}}, ~3/2 = 698.2660{{c}} (~25/24 = 98.2660{{c}})
: error map: {{val| 0.000 -3.689 +11.952 }}


Map: [&lt;4 0 3 8|, &lt;0 2 2 1|]
{{Optimal ET sequence|legend=1| 4, 8, 12 }}
Wedgie: &lt;&lt;8 8 4 -6 -16 -13||
EDOs: 24, 52d, 76cd
Badness: 0.0864


==11-limit==
[[Badness]] (Sintel): 1.11
Commas: 49/48, 77/75, 243/242


POTE generator: ~8/7 = 251.658
== Septimal diminished ==
{{Main| Diminished (temperament) }}


Map: [&lt;4 0 3 8 -2|, &lt;0 2 2 1 5|]
[[Subgroup]]: 2.3.5.7
EDOs: 24, 76cde
Badness: 0.0566


==13-limit==
[[Comma list]]: 36/35, 50/49
Commas: 49/48, 66/65, 77/75, 648/625


POTE generator: ~8/7 = 252.225
{{Mapping|legend=1| 4 0 3 5 | 0 1 1 1 }}


Map: [&lt;4 0 3 8 -2 -1|, &lt;0 2 2 1 5 5|]
[[Optimal tuning]]s:  
EDOs: 24, 52de, 76cde
* [[WE]]: ~6/5 = 299.0347{{c}}, ~3/2 = 697.2727{{c}} (~21/20 = 99.2032{{c}})
Badness: 0.039
: [[error map]]: {{val| -3.861 -8.543 +4.202 +19.759 }}
* [[CWE]]: ~6/5 = 300.0000{{c}}, ~3/2 = 695.9619{{c}} (~21/20 = 95.9619{{c}})
: error map: {{val| 0.000 -5.993 +9.648 +27.136 }}


=Semidim=
{{Optimal ET sequence|legend=1| 4, 8d, 12 }}
Commas: 245/243, 392/375


POTE generator4: ~3/2 = 707.014
[[Badness]] (Sintel): 0.567


Map: [&lt;8 0 6 -3|, &lt;0 1 1 2|]
=== 11-limit ===
Wedgie: &lt;&lt;8 8 16 -6 3 15||
Subgroup: 2.3.5.7.11
EDOs: 24, 32c, 56c
Badness: 0.1075


==11-limit==
Comma list: 36/35, 50/49, 56/55
Commas: 56/55, 77/75, 245/243


POTE generator: ~3/2 = 706.645
Mapping: {{mapping| 4 0 3 5 14 | 0 1 1 1 0 }}


Map: [&lt;8 0 6 -3 15|, &lt;0 1 1 2 1|]
Optimal tunings:  
EDOs: 24, 32c, 56c
* WE: ~6/5 = 297.8458{{c}}, ~3/2 = 703.9277{{c}} (~15/14 = 108.2361{{c}})
Badness: 0.0476
* CWE: ~6/5 = 300.0000{{c}}, ~3/2 = 703.5558{{c}} (~15/14 = 103.5558{{c}})


==13-limit==
{{Optimal ET sequence|legend=0| 4, 8d, 12, 32cddee, 44cddeee }}
Commas: 56/55, 66/65, 77/75, 507/500


POTE generator: ~3/2 = 707.376
Badness (Sintel): 0.732


Map: [&lt;8 0 6 -3 15 17|, &lt;0 1 1 2 1 1|]
==== 13-limit ====
EDOs: 24, 32cf, 56cf
Subgroup: 2.3.5.7.11.13
Badness: 0.0306


Comma list: 36/35, 40/39, 50/49, 66/65


Mapping: {{mapping| 4 0 3 5 14 15 | 0 1 1 1 0 0 }}


Optimal tunings:
* WE: ~6/5 = 297.2520{{c}}, ~3/2 = 707.2352{{c}} (~15/14 = 112.7312{{c}})
* CWE: ~6/5 = 300.0000{{c}}, ~3/2 = 708.4648{{c}} (~15/14 = 108.4648{{c}})


{{Optimal ET sequence|legend=0| 4, 8d, 12f, 20cdef, 32cddeefff }}


Badness (Sintel): 0.806


=== Demolished ===
Subgroup: 2.3.5.7.11


</pre></div>
Comma list: 36/35, 45/44, 50/49
<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;Dimipent family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:14:&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:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt;&lt;a href="#Dimipent"&gt;Dimipent&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt; | &lt;a href="#Hemidim"&gt;Hemidim&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&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;a href="#Semidim"&gt;Semidim&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;!-- ws:start:WikiTextTocRule:20: --&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt;
Mapping: {{mapping| 4 0 3 5 -5 | 0 1 1 1 3 }}
&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;br /&gt;
 
The dimipent family tempers out the major diesis aka diminished comma, 648/625, the amount by which four 6/5 minor thirds exceed an octave, and so identifies the minor third with the quarter-octave. Hence it has the same 300-cent 6/5-approximations as 12edo. &lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~6/5 = 299.6308{{c}}, ~3/2 = 689.0322{{c}} (~21/20 = 89.7707{{c}})
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Dimipent"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Dimipent&lt;/h1&gt;
* CWE: ~6/5 = 300.0000{{c}}, ~3/2 = 688.9304{{c}} (~21/20 = 88.9304{{c}})
Comma: 648/625&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 4e, 8dee, 12, 28 }}
POTE generator: ~3/2 = 699.507&lt;br /&gt;
 
&lt;br /&gt;
Badness (Smith): 0.879
Map: [&amp;lt;4 0 3|, &amp;lt;0 1 1|]&lt;br /&gt;
 
EDOs: 8, 12&lt;br /&gt;
=== Cohedim ===
Badness: 0.0472&lt;br /&gt;
This extension has been documented in Graham Breed's temperament finder as ''hemidim'', the same name as [[#Hemidim|11-limit 4e & 24 and 13-limit 4ef & 24]]. For the 11-limit 8bce & 12 temperament, ''cohedim'' arguably makes more sense. Its ploidacot is tetraploid alpha-dicot.
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Hemidim"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Hemidim&lt;/h1&gt;
Subgroup: 2.3.5.7.11
Commas: 49/48, 648/625&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 36/35, 50/49, 125/121
POTE generator: ~8/7 = 252.555&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 4 1 4 6 6 | 0 2 2 2 3 }}
Map: [&amp;lt;4 0 3 8|, &amp;lt;0 2 2 1|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;8 8 4 -6 -16 -13||&lt;br /&gt;
: mapping generators: ~6/5, ~11/7
EDOs: 24, 52d, 76cd&lt;br /&gt;
 
Badness: 0.0864&lt;br /&gt;
Optimal tunings:
&lt;br /&gt;
* WE: ~6/5 = 298.7799{{c}}, ~11/7 = 795.0744{{c}} (~12/11 = 101.2653{{c}})
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Hemidim-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;11-limit&lt;/h2&gt;
* CWE: ~6/5 = 300.0000{{c}}, ~11/7 = 796.0102{{c}} (~12/11 = 103.9898{{c}})
Commas: 49/48, 77/75, 243/242&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 8bce, 12 }}
POTE generator: ~8/7 = 251.658&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.82
Map: [&amp;lt;4 0 3 8 -2|, &amp;lt;0 2 2 1 5|]&lt;br /&gt;
 
EDOs: 24, 76cde&lt;br /&gt;
==== 13-limit ====
Badness: 0.0566&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Hemidim-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;13-limit&lt;/h2&gt;
Comma list: 36/35, 50/49, 66/65, 125/121
Commas: 49/48, 66/65, 77/75, 648/625&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 4 1 4 6 6 7 | 0 2 2 2 3 3 }}
POTE generator: ~8/7 = 252.225&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:
Map: [&amp;lt;4 0 3 8 -2 -1|, &amp;lt;0 2 2 1 5 5|]&lt;br /&gt;
* WE: ~6/5 = 298.4646{{c}}, ~11/7 = 793.6185{{c}} (~12/11 = 101.7754{{c}})
EDOs: 24, 52de, 76cde&lt;br /&gt;
* CWE: ~6/5 = 300.0000{{c}}, ~11/7 = 794.7323{{c}} (~12/11 = 105.2677{{c}})
Badness: 0.039&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 8bcef, 12f }}
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Semidim"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Semidim&lt;/h1&gt;
 
Commas: 245/243, 392/375&lt;br /&gt;
Badness (Sintel): 1.72
&lt;br /&gt;
 
POTE generator4: ~3/2 = 707.014&lt;br /&gt;
== Hemidim ==
&lt;br /&gt;
Hemidim tempers out 49/48 and may be described as the {{nowrap| 4 & 24 }} temperament. Its ploidcot is tetraploid dicot.
Map: [&amp;lt;8 0 6 -3|, &amp;lt;0 1 1 2|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;8 8 16 -6 3 15||&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
EDOs: 24, 32c, 56c &lt;br /&gt;
 
Badness: 0.1075&lt;br /&gt;
[[Comma list]]: 49/48, 648/625
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Semidim-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;11-limit&lt;/h2&gt;
{{Mapping|legend=1| 4 0 3 8 | 0 2 2 1 }}
Commas: 56/55, 77/75, 245/243&lt;br /&gt;
 
&lt;br /&gt;
: mapping generators: ~6/5, ~7/4
POTE generator: ~3/2 = 706.645&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]]s:
Map: [&amp;lt;8 0 6 -3 15|, &amp;lt;0 1 1 2 1|]&lt;br /&gt;
* [[WE]]: ~6/5 = 300.5053{{c}}, ~7/4 = 949.0409{{c}} (~36/35 = 47.5250{{c}})
EDOs: 24, 32c, 56c&lt;br /&gt;
: [[error map]]: {{val| +2.021 -3.873 +13.284 -15.743 }}
Badness: 0.0476&lt;br /&gt;
* [[CWE]]: ~6/5 = 300.0000{{c}}, ~7/4 = 948.2575{{c}} (~36/35 = 48.2575{{c}})
&lt;br /&gt;
: error map: {{val| 0.000 -5.440 +10.201 -20.568 }}
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Semidim-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;13-limit&lt;/h2&gt;
 
Commas: 56/55, 66/65, 77/75, 507/500&lt;br /&gt;
{{Optimal ET sequence|legend=1| 4, …, 20c, 24 }}
&lt;br /&gt;
 
POTE generator: ~3/2 = 707.376&lt;br /&gt;
[[Badness]] (Sintel): 2.19
&lt;br /&gt;
 
Map: [&amp;lt;8 0 6 -3 15 17|, &amp;lt;0 1 1 2 1 1|]&lt;br /&gt;
=== 11-limit ===
EDOs: 24, 32cf, 56cf&lt;br /&gt;
Subgroup: 2.3.5.7.11
Badness: 0.0306&lt;/body&gt;&lt;/html&gt;</pre></div>
 
Comma list: 49/48, 77/75, 243/242
 
Mapping: {{mapping| 4 0 3 8 -2 | 0 2 2 1 5 }}
 
Optimal tunings:
* WE: ~6/5 = 300.4282{{c}}, ~7/4 = 949.6958{{c}} (~36/35 = 48.4112{{c}})
* CWE: ~6/5 = 300.0000{{c}}, ~7/4 = 948.9065{{c}} (~36/35 = 48.9065{{c}})
 
{{Optimal ET sequence|legend=0| 4e, 20ce, 24 }}
 
Badness (Sintel): 1.87
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 49/48, 66/65, 77/75, 243/242
 
Mapping: {{mapping| 4 0 3 8 -2 -1 | 0 2 2 1 5 5 }}
 
Optimal tunings:
* WE: ~6/5 = 300.4282{{c}}, ~7/4 = 949.2440{{c}} (~36/35 = 47.8487{{c}})
* CWE: ~6/5 = 300.0000{{c}}, ~7/4 = 948.3581{{c}} (~36/35 = 48.3581{{c}})
 
{{Optimal ET sequence|legend=0| 4ef, 24 }}
 
Badness (Sintel): 1.61
 
== Octonion ==
Octonion tempers out 245/243, and may be described as the {{nowrap| 8d & 24 }} temperament. Its ploidacot is octoploid monocot.
 
It was formerly known as ''semidim'' but renamed to avoid confusion with another temperament of the same name.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 245/243, 392/375
 
{{Mapping|legend=1| 8 0 6 -3 | 0 1 1 2 }}
 
: mapping generators: ~15/14, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~15/14 = 149.6673{{c}}, ~3/2 = 705.4455{{c}} (~36/35 = 42.8910{{c}})
: [[error map]]: {{val| -2.662 +0.828 +14.474 -12.260 }}
* [[CWE]]: ~15/14 = 150.0000{{c}}, ~3/2 = 704.9636{{c}} (~36/35 = 45.0364{{c}})
: error map: {{val| 0.000 +3.008 +18.650 -8.899 }}
 
{{Optimal ET sequence|legend=1| 8d, 16d, 24, 32c }}
 
[[Badness]] (Sintel): 2.72
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 56/55, 77/75, 245/243
 
Mapping: {{mapping| 8 0 6 -3 15 | 0 1 1 2 1 }}
 
Optimal tunings:
* WE: ~12/11 = 149.7102{{c}}, ~3/2 = 705.2799{{c}} (~36/35 = 43.2712{{c}})
* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 704.9285{{c}} (~36/35 = 45.0715{{c}})
 
{{Optimal ET sequence|legend=0| 8d, 16d, 24, 32c }}
 
Badness (Sintel): 1.57
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 56/55, 66/65, 77/75, 507/500
 
Mapping: {{mapping| 8 0 6 -3 15 17 | 0 1 1 2 1 1 }}
 
Optimal tunings:
* WE: ~12/11 = 149.6311{{c}}, ~3/2 = 705.6367{{c}} (~36/35 = 42.5188{{c}})
* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 705.2777{{c}} (~36/35 = 44.7223{{c}})
 
{{Optimal ET sequence|legend=0| 8d, 16d, 24, 32cf }}
 
Badness (Sintel): 1.26
 
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Diminished family| ]] <!-- main article -->
[[Category:Diminished| ]] <!-- key article -->
[[Category:Rank 2]]

Latest revision as of 12:13, 21 August 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 diminished family of temperaments tempers out the major diesis a.k.a. diminished comma, 648/625, the amount by which four 6/5 minor thirds exceed an octave, and so identifies the minor third with the quarter-octave. Hence it has the same 300-cent 6/5-approximations as 12edo.

Diminished

Main article: Diminished (temperament)

The generator of diminished can be taken as a fifth or a semitone, and 12edo, with its excellent fifth, is an obvious tuning, though a flatter fifth might be preferred to go with the flat minor third. Its ploidacot is tetraploid monocot.

Subgroup: 2.3.5

Comma list: 648/625

Mapping: [⟨4 0 3], ⟨0 1 1]]

Optimal tunings:

  • WE: ~6/5 = 299.6476 ¢, ~3/2 = 698.6854 ¢ (~25/24 = 99.3903 ¢)
error map: ⟨-1.410 -4.679 +9.905]
  • CWE: ~6/5 = 300.0000 ¢, ~3/2 = 698.2660 ¢ (~25/24 = 98.2660 ¢)
error map: ⟨0.000 -3.689 +11.952]

Optimal ET sequence: 4, 8, 12

Badness (Sintel): 1.11

Septimal diminished

Main article: Diminished (temperament)

Subgroup: 2.3.5.7

Comma list: 36/35, 50/49

Mapping: [⟨4 0 3 5], ⟨0 1 1 1]]

Optimal tunings:

  • WE: ~6/5 = 299.0347 ¢, ~3/2 = 697.2727 ¢ (~21/20 = 99.2032 ¢)
error map: ⟨-3.861 -8.543 +4.202 +19.759]
  • CWE: ~6/5 = 300.0000 ¢, ~3/2 = 695.9619 ¢ (~21/20 = 95.9619 ¢)
error map: ⟨0.000 -5.993 +9.648 +27.136]

Optimal ET sequence: 4, 8d, 12

Badness (Sintel): 0.567

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 50/49, 56/55

Mapping: [⟨4 0 3 5 14], ⟨0 1 1 1 0]]

Optimal tunings:

  • WE: ~6/5 = 297.8458 ¢, ~3/2 = 703.9277 ¢ (~15/14 = 108.2361 ¢)
  • CWE: ~6/5 = 300.0000 ¢, ~3/2 = 703.5558 ¢ (~15/14 = 103.5558 ¢)

Optimal ET sequence: 4, 8d, 12, 32cddee, 44cddeee

Badness (Sintel): 0.732

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 36/35, 40/39, 50/49, 66/65

Mapping: [⟨4 0 3 5 14 15], ⟨0 1 1 1 0 0]]

Optimal tunings:

  • WE: ~6/5 = 297.2520 ¢, ~3/2 = 707.2352 ¢ (~15/14 = 112.7312 ¢)
  • CWE: ~6/5 = 300.0000 ¢, ~3/2 = 708.4648 ¢ (~15/14 = 108.4648 ¢)

Optimal ET sequence: 4, 8d, 12f, 20cdef, 32cddeefff

Badness (Sintel): 0.806

Demolished

Subgroup: 2.3.5.7.11

Comma list: 36/35, 45/44, 50/49

Mapping: [⟨4 0 3 5 -5], ⟨0 1 1 1 3]]

Optimal tunings:

  • WE: ~6/5 = 299.6308 ¢, ~3/2 = 689.0322 ¢ (~21/20 = 89.7707 ¢)
  • CWE: ~6/5 = 300.0000 ¢, ~3/2 = 688.9304 ¢ (~21/20 = 88.9304 ¢)

Optimal ET sequence: 4e, 8dee, 12, 28

Badness (Smith): 0.879

Cohedim

This extension has been documented in Graham Breed's temperament finder as hemidim, the same name as 11-limit 4e & 24 and 13-limit 4ef & 24. For the 11-limit 8bce & 12 temperament, cohedim arguably makes more sense. Its ploidacot is tetraploid alpha-dicot.

Subgroup: 2.3.5.7.11

Comma list: 36/35, 50/49, 125/121

Mapping: [⟨4 1 4 6 6], ⟨0 2 2 2 3]]

mapping generators: ~6/5, ~11/7

Optimal tunings:

  • WE: ~6/5 = 298.7799 ¢, ~11/7 = 795.0744 ¢ (~12/11 = 101.2653 ¢)
  • CWE: ~6/5 = 300.0000 ¢, ~11/7 = 796.0102 ¢ (~12/11 = 103.9898 ¢)

Optimal ET sequence: 8bce, 12

Badness (Sintel): 1.82

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 36/35, 50/49, 66/65, 125/121

Mapping: [⟨4 1 4 6 6 7], ⟨0 2 2 2 3 3]]

Optimal tunings:

  • WE: ~6/5 = 298.4646 ¢, ~11/7 = 793.6185 ¢ (~12/11 = 101.7754 ¢)
  • CWE: ~6/5 = 300.0000 ¢, ~11/7 = 794.7323 ¢ (~12/11 = 105.2677 ¢)

Optimal ET sequence: 8bcef, 12f

Badness (Sintel): 1.72

Hemidim

Hemidim tempers out 49/48 and may be described as the 4 & 24 temperament. Its ploidcot is tetraploid dicot.

Subgroup: 2.3.5.7

Comma list: 49/48, 648/625

Mapping: [⟨4 0 3 8], ⟨0 2 2 1]]

mapping generators: ~6/5, ~7/4

Optimal tunings:

  • WE: ~6/5 = 300.5053 ¢, ~7/4 = 949.0409 ¢ (~36/35 = 47.5250 ¢)
error map: ⟨+2.021 -3.873 +13.284 -15.743]
  • CWE: ~6/5 = 300.0000 ¢, ~7/4 = 948.2575 ¢ (~36/35 = 48.2575 ¢)
error map: ⟨0.000 -5.440 +10.201 -20.568]

Optimal ET sequence: 4, …, 20c, 24

Badness (Sintel): 2.19

11-limit

Subgroup: 2.3.5.7.11

Comma list: 49/48, 77/75, 243/242

Mapping: [⟨4 0 3 8 -2], ⟨0 2 2 1 5]]

Optimal tunings:

  • WE: ~6/5 = 300.4282 ¢, ~7/4 = 949.6958 ¢ (~36/35 = 48.4112 ¢)
  • CWE: ~6/5 = 300.0000 ¢, ~7/4 = 948.9065 ¢ (~36/35 = 48.9065 ¢)

Optimal ET sequence: 4e, 20ce, 24

Badness (Sintel): 1.87

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 66/65, 77/75, 243/242

Mapping: [⟨4 0 3 8 -2 -1], ⟨0 2 2 1 5 5]]

Optimal tunings:

  • WE: ~6/5 = 300.4282 ¢, ~7/4 = 949.2440 ¢ (~36/35 = 47.8487 ¢)
  • CWE: ~6/5 = 300.0000 ¢, ~7/4 = 948.3581 ¢ (~36/35 = 48.3581 ¢)

Optimal ET sequence: 4ef, 24

Badness (Sintel): 1.61

Octonion

Octonion tempers out 245/243, and may be described as the 8d & 24 temperament. Its ploidacot is octoploid monocot.

It was formerly known as semidim but renamed to avoid confusion with another temperament of the same name.

Subgroup: 2.3.5.7

Comma list: 245/243, 392/375

Mapping: [⟨8 0 6 -3], ⟨0 1 1 2]]

mapping generators: ~15/14, ~3

Optimal tunings:

  • WE: ~15/14 = 149.6673 ¢, ~3/2 = 705.4455 ¢ (~36/35 = 42.8910 ¢)
error map: ⟨-2.662 +0.828 +14.474 -12.260]
  • CWE: ~15/14 = 150.0000 ¢, ~3/2 = 704.9636 ¢ (~36/35 = 45.0364 ¢)
error map: ⟨0.000 +3.008 +18.650 -8.899]

Optimal ET sequence: 8d, 16d, 24, 32c

Badness (Sintel): 2.72

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 77/75, 245/243

Mapping: [⟨8 0 6 -3 15], ⟨0 1 1 2 1]]

Optimal tunings:

  • WE: ~12/11 = 149.7102 ¢, ~3/2 = 705.2799 ¢ (~36/35 = 43.2712 ¢)
  • CWE: ~12/11 = 150.0000 ¢, ~3/2 = 704.9285 ¢ (~36/35 = 45.0715 ¢)

Optimal ET sequence: 8d, 16d, 24, 32c

Badness (Sintel): 1.57

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 66/65, 77/75, 507/500

Mapping: [⟨8 0 6 -3 15 17], ⟨0 1 1 2 1 1]]

Optimal tunings:

  • WE: ~12/11 = 149.6311 ¢, ~3/2 = 705.6367 ¢ (~36/35 = 42.5188 ¢)
  • CWE: ~12/11 = 150.0000 ¢, ~3/2 = 705.2777 ¢ (~36/35 = 44.7223 ¢)

Optimal ET sequence: 8d, 16d, 24, 32cf

Badness (Sintel): 1.26

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