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Aberschismic temperaments: 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>
This is a collection of [[rank-2 temperament|rank-2]] '''aberschismic temperaments''', which [[tempering out|temper out]] the [[aberschisma]] ({{monzo|legend=1| 10 -6 1 -1 }}, [[ratio]]: 5120/5103). These temperaments divide an exact or approximate septimal quartertone, [[36/35]] into two equal steps, each representing [[81/80]][[~]][[64/63]], the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same [[chain of fifths]] inflected by the syntonic~septimal comma to the opposite sides. In addition we may identify [[10/7]] by the augmented fourth and [[50/49]] by the [[Pythagorean comma]].  
: This revision was by author [[User:Natebedell|Natebedell]] and made on <tt>2011-09-06 12:36:35 UTC</tt>.<br>
: The original revision id was <tt>251220552</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">The hemifamity temperaments temper out the hemifamity comma, |10 -6 1 -1&gt; = 5120/5103. Belonging to it and considered below are buzzard, undecental, leapday, mystery and quanic. Other hemifamity temperaments are dominant, garibaldi, hemififths, amity, misty, rodan, countercata and kwai.


[[toc|flat]]
Temperaments belonging to this category and generated by the fifth are dominant, garibaldi, kwai, undecental, and leapday. Dominant has 5/4 mapped to M3. Garibaldi has 5/4 mapped to d4. Kwai has 5/4 mapped to 4A7. Undecental has 5/4 mapped to 5d7. Leapday has 5/4 mapped to 3A1.
=Buzzard=
Commas: 1728/1715, 5120/5103


[[POTE tuning|POTE generator]]: 475.636
Diaschismic is generated by the fifth with a semi-octave period. Hemififths has the fifth sliced into two and 5/4 mapped to the hemififth + Pyth. comma. Hemidromeda has the fourth sliced into two and 5/4 mapped to the hemifourth + 3d4. Rodan has the fifth sliced into three as does slendric. Alphatrimot has the twelfth sliced into three as does alphatricot. Monkey has the fifth sliced into four as does tetracot. Buzzard has the twelfth sliced into four as does vulture. Misty is generated by the fifth with a 1/3-octave period. Supers has the fifth sliced into three with a semi-octave period. Undim is generated by the fifth with a 1/4-octave period. Quinticosiennic and quintakwai have the fourth sliced into five. Amity has the eleventh sliced into five. Countercata has the twelfth sliced into six as does hanson. Warrior has the 6th harmonic sliced into seven as does sensi. Finally, alphaquarter has the fourth sliced into nine as does escapade.  


Map: [&lt;1 0 -6 4|, &lt;0 4 21 -3|]
Temperaments discussed elsewhere are:
Wedgie: &lt;&lt;4 21 -3 24 -16 -66||
* [[Dominant (temperament)|Dominant]] (+36/35) → [[Meantone family #Dominant|Meantone family]]
EDOs: 48, 53, 111, 164, 275
* [[Garibaldi]] (+225/224) → [[Schismatic family #Garibaldi|Schismatic family]]
* [[Diaschismic]] (+126/125) → [[Diaschismic family #Septimal diaschismic|Diaschismic family]]
* [[Hemififths]] (+2401/2400) → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]]
* [[Rodan]] (+245/243) → [[Gamelismic clan #Rodan|Gamelismic clan]]
* ''[[Alphatrimot]]'' (+2430/2401) → [[Alphatricot family #Alphatrimot|Alphatricot family]]
* [[Misty]] (+3136/3125) → [[Misty family #Misty|Misty family]]
* [[Monkey]] (+875/864) → [[Tetracot family #Monkey|Tetracot family]]
* [[Buzzard]] (+1728/1715) → [[Buzzardsmic clan #Buzzard|Buzzardsmic clan]]
* ''[[Undim]]'' (+390625/388962) → [[Undim family #Septimal undim|Undim family]]
* ''[[Quinticosiennic]]'' (+395136/390625) → [[Quintaleap family #Quinticosiennic|Quintaleap family]]
* ''[[Quintakwai]]'' (+9765625/9680832) → [[Quindromeda family #Quintakwai|Quindromeda family]]
* [[Amity]] (+4375/4374) → [[Amity family #Septimal amity|Amity family]]
* ''[[Countercata]]'' (+15625/15552) → [[Kleismic family #Countercata|Kleismic family]]
* ''[[Abergravity]]'' (+177147/175000) → [[Gravity family #Abergravity|Gravity family]]
* ''[[Supers]]'' (+118098/117649) → [[Stearnsmic clan #Supers|Stearnsmic clan]]
* ''[[Warrior]]'' (+78732/78125) → [[Sensipent family #Warrior|Sensipent family]]
* ''[[Alphaquarter]]'' (+29360128/29296875) → [[Escapade family #Alphaquarter|Escapade family]]


==11-limit==
Considered below are septiquarter, kwai, ketchup, undecental, leapday, mystery, hemidromeda, countriton, artoneutral, quanic and jorgensen, in the order of increasing [[TE logflat badness]].
Commas: 176/175, 540/539, 5120/5103


[[POTE tuning|POTE generator]]: 475.700
== Septiquarter ==
Septiquarter tempers out [[420175/419904]] and may be described as the {{nowrap| 94 & 99 }} temperament. Its [[ploidacot]] is epsilon-heptacot. [[99edo]] makes for an excellent tuning, and [[292edo]] an even better one. [[94edo]] and [[104edo]] in the 104c val are also among the possibilities.  


Map: [&lt;1 0 -6 4 -12|, &lt;0 4 21 -3 39|]
[[Subgroup]]: 2.3.5.7
EDOs: 53, 58, 111, 391


==13-limit==
[[Comma list]]: 5120/5103, 420175/419904
Commas: 176/175, 351/350, 540/539, 676/675


[[POTE tuning|POTE generator]]: 475.697
{{Mapping|legend=1| 1 -4 -28 6 | 0 7 38 -4 }}
: mapping generators: ~2, ~243/140


Map: [&lt;1 0 -6 4 -12 -7|, &lt;0 4 21 -3 39 27|]
[[Optimal tuning]]s:
EDOs: 53, 58, 111, 391, 502
* [[WE]]: ~2 = 1199.7212{{c}}, ~243/140 = 957.3250{{c}}
: [[error map]]: {{val| -0.279 +0.435 -0.158 +0.201 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~243/140 = 957.5424{{c}}
: error map: {{val| 0.000 +0.842 +0.298 +1.004 }}


==17-limit==
{{Optimal ET sequence|legend=1| 94, 99, 292, 391, 881bd, 1272bcd }}
Commas: 176/175, 256/255, 351/350, 442/441, 540/539


[[POTE tuning|POTE generator]]: 475.692
[[Badness]] (Sintel): 1.36


Map: [&lt;1 0 -6 4 -12 -7 14|, &lt;0 4 21 -3 39 27 -25|]
=== Semiseptiquarter ===
EDOs: 53, 58, 111, 613
Subgroup: 2.3.5.7.11


=Undecental=
Comma list: 5120/5103, 9801/9800, 14641/14580
Commas: 5120/5103, 235298/234375


[[POTE tuning|POTE generator]]: 703.039
Mapping: {{mapping| 2 -8 -56 12 -25 | 0 7 38 -4 20 }}


Map: [&lt;1 0 61 71|, &lt;0 1 -37 -43|]
Optimal tunings:  
Wedgie: &lt;&lt;1 -37 -43 -61 -71 4||
* WE: ~99/70 = 599.8953{{c}}, ~210/121 = 957.3819{{c}}
EDOs: 12, 29, 70, 99
* CWE: ~99/70 = 600.0000{{c}}, ~210/121 = 957.5449{{c}}


=Leapday=
{{Optimal ET sequence|legend=0| 94, 198, 292, 490 }}
Commas: 686/675, 5120/5103


[[POTE tuning|POTE generator]]: 704.263
Badness (Sintel): 2.12


Map: [&lt;1 0 -31 -21|, &lt;0 1 21 15|]
==== 13-limit ====
Wedgie: &lt;&lt;1 21 15 31 21 -24||
Subgroup: 2.3.5.7.11.13
EDOs: 29, 46, 305


==11-limit==
Comma list: 352/351, 847/845, 1716/1715, 14641/14580
Commas: 121/120, 441/440, 686/675


[[POTE tuning|POTE generator]]: 704.250
Mapping: {{mapping| 2 -8 -56 12 -25 9 | 0 7 38 -4 20 -1 }}


Map: [&lt;1 0 -31 -21 -14|, &lt;0 1 21 15 11|]
Optimal tunings:  
EDOs: 29, 46, 259
* WE: ~99/70 = 599.8565{{c}}, ~210/121 = 957.3261{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~210/121 = 957.5508{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 94, 198, 490f }}
Commas: 91/90, 121/120, 169/168, 441/440


[[POTE tuning|POTE generator]]: 704.214
Badness (Sintel): 1.44


Map: [&lt;1 0 -31 -21 -14 -9|, &lt;0 1 21 15 11 8|]
== Kwai ==
EDOs: 29, 46, 167, 213, 380
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Kwai]].''


=Mystery=
Named by [[Gene Ward Smith]] in 2004 for its "bridgeability"<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10766.html Yahoo! Tuning Group | ''Kwai'']</ref>, kwai is generated by a [[3/2|perfect fifth]], and can be described as {{nowrap| 41 & 70 }}.
Commas: 5120/5103, 50421/50000


[[POTE tuning|POTE generator]]: 388.646
[[Subgroup]]: 2.3.5.7


Map: [&lt;29 46 0 14|, &lt;0 0 1 1|]
[[Comma list]]: 5120/5103, 16875/16807
Wedgie: &lt;&lt;0 29 29 46 46 -14||
EDOs: 29, 58, 87, 145
Badness: 0.1037


==11-limit==
{{Mapping|legend=1| 1 0 -50 -40 | 0 1 33 27 }}
Commas: 441/440, 896/891, 3388/3375
: mapping generators: ~2, ~3


[[POTE tuning|POTE generator]]: 388.460
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7337{{c}}, ~3/2 = 702.4600{{c}}
: [[error map]]: {{val| -0.266 +0.239 -0.607 +1.055 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.6085{{c}}
: error map: {{val| 0.000 +0.653 -0.234 +1.603 }}


Map: [&lt;29 46 0 14 33|, &lt;0 0 1 1 1|]
{{Optimal ET sequence|legend=1| 41, 111, 152, 345, 497d }}
EDOs: 29, 58, 87, 145
Badness: 0.0343


==13-limit==
[[Badness]] (Sintel): 1.38
Commas: 196/195, 352/351, 364/363, 676/675


[[POTE tuning|POTE generator]]: 388.354
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;29 46 0 14 33 40|, &lt;0 0 1 1 1 1|]
Comma list: 540/539, 1375/1372, 5120/5103
EDOs: 29, 58, 87, 145, 377
Badness: 0.0186


=Quanic=
Mapping: {{mapping| 1 0 -50 -40 32 | 0 1 33 27 -18 }}
Commas: 5120/5103, 5832000/5764801


POTE generator: ~160/147 = 140.493
Optimal tunings:  
* WE: ~2 = 1199.6672{{c}}, ~3/2 = 702.4282{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6189{{c}}


Map: [&lt;1 1 -4 0|, &lt;0 5 54 24|]
{{Optimal ET sequence|legend=0| 41, 111, 152, 497de, 649dde }}
EDOs: 94, 111, 205
Badness: 0.1795


==11-limit==
Badness (Sintel): 0.867
Commas: 540/539, 1331/1323, 5120/5103


POTE generator: ~88/81 = 140.489
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 1 -4 0 1|, &lt;0 5 54 24 21|]
Comma list: 352/351, 540/539, 729/728, 1375/1372
EDOs: 94, 111, 205
Badness: 0.0587


==13-limit==
Mapping: {{mapping| 1 0 -50 -40 32 27 | 0 1 33 27 -18 -21 }}
Commas: 352/351, 540/539, 729/728, 1331/1323


POTE generator: ~13/12 = 140.496
Optimal tunings:  
* WE: ~2 = 1199.4772{{c}}, ~3/2 = 702.3379{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6409{{c}}


Map: [&lt;1 1 -4 0 1 3|, &lt;0 5 54 24 21 6|]
{{Optimal ET sequence|legend=0| 41, 111, 152f, 415dff }}
EDOs: 94, 111, 205
 
Badness: 0.0325</pre></div>
Badness (Sintel): 1.01
<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;Hemifamity temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The hemifamity temperaments temper out the hemifamity comma, |10 -6 1 -1&amp;gt; = 5120/5103. Belonging to it and considered below are buzzard, undecental, leapday, mystery and quanic. Other hemifamity temperaments are dominant, garibaldi, hemififths, amity, misty, rodan, countercata and kwai.&lt;br /&gt;
===== 17-limit =====
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17
&lt;!-- ws:start:WikiTextTocRule:28:&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:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt;&lt;a href="#Buzzard"&gt;Buzzard&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt;&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt;&lt;!-- ws:end:WikiTextTocRule:31 --&gt;&lt;!-- ws:start:WikiTextTocRule:32: --&gt;&lt;!-- ws:end:WikiTextTocRule:32 --&gt;&lt;!-- ws:start:WikiTextTocRule:33: --&gt; | &lt;a href="#Undecental"&gt;Undecental&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:33 --&gt;&lt;!-- ws:start:WikiTextTocRule:34: --&gt; | &lt;a href="#Leapday"&gt;Leapday&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:34 --&gt;&lt;!-- ws:start:WikiTextTocRule:35: --&gt;&lt;!-- ws:end:WikiTextTocRule:35 --&gt;&lt;!-- ws:start:WikiTextTocRule:36: --&gt;&lt;!-- ws:end:WikiTextTocRule:36 --&gt;&lt;!-- ws:start:WikiTextTocRule:37: --&gt; | &lt;a href="#Mystery"&gt;Mystery&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:37 --&gt;&lt;!-- ws:start:WikiTextTocRule:38: --&gt;&lt;!-- ws:end:WikiTextTocRule:38 --&gt;&lt;!-- ws:start:WikiTextTocRule:39: --&gt;&lt;!-- ws:end:WikiTextTocRule:39 --&gt;&lt;!-- ws:start:WikiTextTocRule:40: --&gt; | &lt;a href="#Quanic"&gt;Quanic&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:40 --&gt;&lt;!-- ws:start:WikiTextTocRule:41: --&gt;&lt;!-- ws:end:WikiTextTocRule:41 --&gt;&lt;!-- ws:start:WikiTextTocRule:42: --&gt;&lt;!-- ws:end:WikiTextTocRule:42 --&gt;&lt;!-- ws:start:WikiTextTocRule:43: --&gt;
 
&lt;!-- ws:end:WikiTextTocRule:43 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Buzzard"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Buzzard&lt;/h1&gt;
Comma list: 256/255, 352/351, 540/539, 715/714, 1089/1088
Commas: 1728/1715, 5120/5103&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 0 -50 -40 32 27 58 | 0 1 33 27 -18 -21 -34 }}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 475.636&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:
Map: [&amp;lt;1 0 -6 4|, &amp;lt;0 4 21 -3|]&lt;br /&gt;
* WE: ~2 = 1199.3537{{c}}, ~3/2 = 702.2850{{c}}
Wedgie: &amp;lt;&amp;lt;4 21 -3 24 -16 -66||&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6589{{c}}
EDOs: 48, 53, 111, 164, 275&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 41, 70, 111, 152fg, 263dfg }}
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Buzzard-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;11-limit&lt;/h2&gt;
 
Commas: 176/175, 540/539, 5120/5103&lt;br /&gt;
Badness (Sintel): 1.12
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 475.700&lt;br /&gt;
===== 19-limit =====
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17.19
Map: [&amp;lt;1 0 -6 4 -12|, &amp;lt;0 4 21 -3 39|]&lt;br /&gt;
 
EDOs: 53, 58, 111, 391&lt;br /&gt;
Comma list: 256/255, 352/351, 400/399, 456/455, 715/714, 847/845
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Buzzard-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;13-limit&lt;/h2&gt;
Mapping: {{mapping| 1 0 -50 -40 32 27 58 -56 | 0 1 33 27 -18 -21 -34 38 }}
Commas: 176/175, 351/350, 540/539, 676/675&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 475.697&lt;br /&gt;
* WE: ~2 = 1199.3401{{c}}, ~3/2 = 702.2705{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6548{{c}}
Map: [&amp;lt;1 0 -6 4 -12 -7|, &amp;lt;0 4 21 -3 39 27|]&lt;br /&gt;
 
EDOs: 53, 58, 111, 391, 502&lt;br /&gt;
{{Optimal ET sequence|legend=0| 41, 70h, 111, 152fg, 263dfgh }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Buzzard-17-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;17-limit&lt;/h2&gt;
Badness (Sintel): 1.03
Commas: 176/175, 256/255, 351/350, 442/441, 540/539&lt;br /&gt;
 
&lt;br /&gt;
==== Hemikwai ====
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 475.692&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
Map: [&amp;lt;1 0 -6 4 -12 -7 14|, &amp;lt;0 4 21 -3 39 27 -25|]&lt;br /&gt;
Comma list: 540/539, 676/675, 1375/1372, 5120/5103
EDOs: 53, 58, 111, 613&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 0 -50 -40 32 -51 | 0 2 66 54 -36 69 }}
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Undecental"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Undecental&lt;/h1&gt;
: mapping generators: ~2, ~26/15
Commas: 5120/5103, 235298/234375&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 703.039&lt;br /&gt;
* WE: ~2 = 1199.6968{{c}}, ~26/15 = 951.0740{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3123{{c}}
Map: [&amp;lt;1 0 61 71|, &amp;lt;0 1 -37 -43|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;1 -37 -43 -61 -71 4||&lt;br /&gt;
{{Optimal ET sequence|legend=0| 82, 111, 193, 304d }}
EDOs: 12, 29, 70, 99&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.82
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Leapday"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Leapday&lt;/h1&gt;
 
Commas: 686/675, 5120/5103&lt;br /&gt;
===== 17-limit =====
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 704.263&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 442/441, 540/539, 676/675, 715/714, 5120/5103
Map: [&amp;lt;1 0 -31 -21|, &amp;lt;0 1 21 15|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;1 21 15 31 21 -24||&lt;br /&gt;
Mapping: {{mapping| 1 0 -50 -40 32 -51 -30 | 0 2 66 54 -36 69 43 }}
EDOs: 29, 46, 305&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Leapday-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;11-limit&lt;/h2&gt;
* WE: ~2 = 1199.6861{{c}}, ~26/15 = 951.0654{{c}}
Commas: 121/120, 441/440, 686/675&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3120{{c}}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 704.250&lt;br /&gt;
{{Optimal ET sequence|legend=0| 82, 111, 193, 304d }}
&lt;br /&gt;
 
Map: [&amp;lt;1 0 -31 -21 -14|, &amp;lt;0 1 21 15 11|]&lt;br /&gt;
Badness (Sintel): 1.31
EDOs: 29, 46, 259&lt;br /&gt;
 
&lt;br /&gt;
===== 19-limit =====
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Leapday-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;13-limit&lt;/h2&gt;
Subgroup: 2.3.5.7.11.13.17.19
Commas: 91/90, 121/120, 169/168, 441/440&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 400/399, 442/441, 540/539, 676/675, 715/714, 1445/1444
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 704.214&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 0 -50 -40 32 -51 -30 -56 | 0 2 66 54 -36 69 43 76 }}
Map: [&amp;lt;1 0 -31 -21 -14 -9|, &amp;lt;0 1 21 15 11 8|]&lt;br /&gt;
 
EDOs: 29, 46, 167, 213, 380&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1199.6718{{c}}, ~26/15 = 951.0526{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Mystery"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Mystery&lt;/h1&gt;
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3103{{c}}
Commas: 5120/5103, 50421/50000&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 82, 111, 193, 304dh }}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 388.646&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.16
Map: [&amp;lt;29 46 0 14|, &amp;lt;0 0 1 1|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;0 29 29 46 46 -14||&lt;br /&gt;
== Ketchup ==
EDOs: 29, 58, 87, 145&lt;br /&gt;
Ketchup may be described as the {{nowrap| 46 & 94 }} temperament. It has a semi-octave period and a generator for a syntonic~septimal comma, four of which plus a period gives the perfect fifth; its [[ploidacot]] is diploid gamma-tetracot. [[140edo]] is an obvious tuning for this temperament.
Badness: 0.1037&lt;br /&gt;
 
&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Mystery-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;11-limit&lt;/h2&gt;
 
Commas: 441/440, 896/891, 3388/3375&lt;br /&gt;
[[Comma list]]: 5120/5103, 1071875/1062882
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 388.460&lt;br /&gt;
{{Mapping|legend=1| 2 3 4 6 | 0 4 15 -9 }}
&lt;br /&gt;
: mapping generators: ~1225/864, ~64/63
Map: [&amp;lt;29 46 0 14 33|, &amp;lt;0 0 1 1 1|]&lt;br /&gt;
 
EDOs: 29, 58, 87, 145&lt;br /&gt;
[[Optimal tuning]]s:  
Badness: 0.0343&lt;br /&gt;
* [[WE]]: ~1225/864 = 599.9685{{c}}, ~64/63 = 25.7181{{c}}
&lt;br /&gt;
: [[error map]]: {{val| -0.063 +0.823 -0.668 -0.478 }}
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Mystery-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;13-limit&lt;/h2&gt;
* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~64/63 = 25.7181{{c}}
Commas: 196/195, 352/351, 364/363, 676/675&lt;br /&gt;
: error map: {{val| 0.000 +0.917 -0.543 -0.288 }}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 388.354&lt;br /&gt;
{{Optimal ET sequence|legend=1| 46, 94, 140 }}
&lt;br /&gt;
 
Map: [&amp;lt;29 46 0 14 33 40|, &amp;lt;0 0 1 1 1 1|]&lt;br /&gt;
[[Badness]] (Sintel): 2.14
EDOs: 29, 58, 87, 145, 377&lt;br /&gt;
 
Badness: 0.0186&lt;br /&gt;
=== 11-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc11"&gt;&lt;a name="Quanic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;Quanic&lt;/h1&gt;
 
Commas: 5120/5103, 5832000/5764801&lt;br /&gt;
Comma list: 385/384, 1331/1323, 2200/2187
&lt;br /&gt;
 
POTE generator: ~160/147 = 140.493&lt;br /&gt;
Mapping: {{mapping| 2 3 4 6 7 | 0 4 15 -9 -2 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 1 -4 0|, &amp;lt;0 5 54 24|]&lt;br /&gt;
Optimal tunings:
EDOs: 94, 111, 205&lt;br /&gt;
* WE: ~99/70 = 600.0678{{c}}, ~64/63 = 25.6963{{c}}
Badness: 0.1795&lt;br /&gt;
* CWE: ~99/70 = 600.0000{{c}}, ~64/63 = 25.6956{{c}}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="Quanic-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;11-limit&lt;/h2&gt;
{{Optimal ET sequence|legend=0| 46, 94, 140 }}
Commas: 540/539, 1331/1323, 5120/5103&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.31
POTE generator: ~88/81 = 140.489&lt;br /&gt;
 
&lt;br /&gt;
=== 13-limit ===
Map: [&amp;lt;1 1 -4 0 1|, &amp;lt;0 5 54 24 21|]&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
EDOs: 94, 111, 205&lt;br /&gt;
 
Badness: 0.0587&lt;br /&gt;
Comma list: 325/324, 352/351, 385/384, 1331/1323
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc13"&gt;&lt;a name="Quanic-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;13-limit&lt;/h2&gt;
Mapping: {{mapping| 2 3 4 6 7 8 | 0 4 15 -9 -2 -14 }}
Commas: 352/351, 540/539, 729/728, 1331/1323&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
POTE generator: ~13/12 = 140.496&lt;br /&gt;
* WE: ~99/70 = 600.0612{{c}}, ~66/65 = 25.7000{{c}}
&lt;br /&gt;
* CWE: ~99/70 = 600.0000{{c}}, ~66/65 = 25.6978{{c}}
Map: [&amp;lt;1 1 -4 0 1 3|, &amp;lt;0 5 54 24 21 6|]&lt;br /&gt;
 
EDOs: 94, 111, 205&lt;br /&gt;
{{Optimal ET sequence|legend=0| 46, 94, 140 }}
Badness: 0.0325&lt;/body&gt;&lt;/html&gt;</pre></div>
 
Badness (Sintel): 1.03
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 289/288, 325/324, 352/351, 385/384, 442/441
 
Mapping: {{mapping| 2 3 4 6 7 8 8 | 0 4 15 -9 -2 -14 4 }}
 
Optimal tunings:  
* WE: ~17/12 = 600.0896{{c}}, ~66/65 = 25.7048{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.7017{{c}}
 
{{Optimal ET sequence|legend=0| 46, 94, 140 }}
 
Badness (Sintel): 0.845
 
=== 2.3.5.7.11.13.17.23 subgroup ===
Subgroup: 2.3.5.7.11.13.17.23
 
Comma list: 253/252, 289/288, 325/324, 352/351, 385/384, 391/390
 
Mapping: {{mapping| 2 3 4 6 7 8 8 9 | 0 4 15 -9 -2 -14 4 1 }}
 
Optimal tunings:  
* WE: ~17/12 = 600.1139{{c}}, ~66/65 = 25.7053{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.7013{{c}}
 
{{Optimal ET sequence|legend=0| 46, 94, 140 }}
 
Badness (Sintel): 0.772
 
== Undecental ==
Undecental adds the triwellisma to the comma list and may be described as the {{nowrap| 29 & 70 }} temperament. 5/4 is mapped to the quintuple-diminished seventh or equivalently the perfect fourth minus three [[diesis (scale theory)|dieses]]. [[99edo|58\99]] is an almost perfect generator, just as the name suggests. Another interesting tuning choice is the argent fifth, {{nowrap| 2<sup>(2 - sqrt (2))</sup> }}.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 5120/5103, 235298/234375
 
{{Mapping|legend=1| 1 0 61 71 | 0 1 -37 -43 }}
: mapping generators: ~2, ~3
 
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.6543{{c}}, ~3/2 = 702.8370{{c}}
: [[error map]]: {{val| -0.346 +0.536 +0.423 -0.494 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.0465{{c}}
: error map: {{val| 0.000 +1.092 +0.966 +0.175 }}
 
{{Optimal ET sequence|legend=1| 29, 70, 99, 722bc, 821bc, 920bc, 1019bc }}
 
[[Badness]] (Sintel): 2.39
 
== Leapday ==
{{Main| Leapday }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Leapday]].''
 
Leapday tempers out [[686/675]], the senga, in addition to the aberschisma, and may be described as the {{nowrap| 29 & 46 }} temperament. It extends [[leapfrog]], such that [[7/4]] is found by 15 generators up, as a double-augmented fifth (a major sixth and a diesis). 5/4 is found by a tritone above that, as a triple-augmented unison (a minor third and two dieses). [[46edo]] itself is an excellent tuning for this.
 
Leapday is more notable in the higher limits than the lower, as it nails the 13-limit pretty well from identifying [[14/11]] by a major third and [[13/11]] by a minor third, tempering out not only [[352/351]] and [[364/363]] but [[91/90]], [[121/120]], [[169/168]] and [[196/195]]. It can be further extended to include the [[17/1|17th]] and [[23/1|23rd]] [[harmonic]]s. Adding 17 would fix the valid diamond monotone tuning to 46edo, however.  
 
Leapday has an alternative extension called [[porwell temperaments #Polypyth|polypyth]], which tempers out the same 5-limit comma as leapday, but with the porwell comma ([[6144/6125]]) rather than the aberschisma tempered out.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 686/675, 5120/5103
 
{{Mapping|legend=1| 1 0 -31 -21 | 0 1 21 15 }}
: mapping generators: ~2, ~3
 
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.7167{{c}}, ~3/2 = 704.0971{{c}}
: [[error map]]: {{val| -0.283 +1.859 +2.559 -5.669 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.2504{{c}}
: error map: {{val| 0.000 +2.295 +2.945 -5.070 }}
 
{{Optimal ET sequence|legend=1| 17c, 29, 46 }}
 
[[Badness]] (Sintel): 2.43
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 441/440, 686/675
 
Mapping: {{mapping| 1 0 -31 -21 -14 | 0 1 21 15 11 }}
 
Optimal tunings:
* WE: ~2 = 1200.0731{{c}}, ~3/2 = 704.2933{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2538{{c}}
 
{{Optimal ET sequence|legend=0| 17c, 29, 46 }}
 
Badness (Sintel): 1.28
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 91/90, 121/120, 169/168, 352/351
 
Mapping: {{mapping| 1 0 -31 -21 -14 -9 | 0 1 21 15 11 8 }}
 
Optimal tunings:
* WE: ~2 = 1200.4758{{c}}, ~3/2 = 704.4930{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2346{{c}}
 
{{Optimal ET sequence|legend=0| 17c, 29, 46, 121def }}
 
Badness (Sintel): 1.02
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 91/90, 121/120, 136/135, 154/153, 169/168
 
Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 | 0 1 21 15 11 8 24 }}
 
Optimal tunings:
* WE: ~2 = 1200.4818{{c}}, ~3/2 = 704.5121{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2507{{c}}
 
{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 121defg }}
 
Badness (Sintel): 0.910
 
=== 2.3.5.7.11.13.17.23 subgroup ===
Subgroup: 2.3.5.7.11.13.17.23
 
Comma list: 91/90, 121/120, 136/135, 154/153, 161/160, 169/168
 
Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 -5 | 0 1 21 15 11 8 24 6 }}
 
Optimal tunings:
* WE: ~2 = 1200.5169{{c}}, ~3/2 = 704.5279{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2450{{c}}
 
{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 121defg }}
 
Badness (Sintel): 0.872
 
== Mystery ==
{{Main| Mystery }}
: ''For the 5-limit version, see [[29th-octave temperaments #Mystery]].''
 
Mystery tempers out [[50421/50000]] and may be described as the {{nowrap| 29 & 58 }} temperament. It has a 1\29 period and primes 5, 7, 11 and 13 are all reached by one generator step; its ploidacot is 29-ploid acot. [[145edo]] or [[232edo]] are good candidates for tunings.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 5120/5103, 50421/50000
 
{{Mapping|legend=1| 29 46 0 14 | 0 0 1 1 }}
: mapping generators: ~50/49, ~5
 
[[Optimal tuning]]s:  
* [[WE]]: ~50/49 = 41.3652{{c}}, ~5/4 = 388.5128{{c}}
: [[error map]]: {{val| -0.410 +0.842 +1.378 -2.022 }}
* [[CWE]]: ~50/49 = 41.3793{{c}}, ~5/4 = 388.3030{{c}}
: error map: {{val| 0.000 +1.493 +1.989 -1.213 }}
 
{{Optimal ET sequence|legend=1| 29, 58, 87, 145 }}
 
[[Badness]] (Sintel): 2.63
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 896/891, 3388/3375
 
Mapping: {{mapping| 29 46 0 14 33 | 0 0 1 1 1 }}
 
Optimal tunings:
* WE: ~45/44 = 41.3637{{c}}, ~5/4 = 388.3136{{c}}
* CWE: ~45/44 = 41.3793{{c}}, ~5/4 = 388.0598{{c}}
 
{{Optimal ET sequence|legend=0| 29, 58, 87, 145 }}
 
Badness (Sintel): 1.13
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 196/195, 352/351, 364/363, 676/675
 
Mapping: {{mapping| 29 46 0 14 33 40 | 0 0 1 1 1 1 }}
 
Optimal tunings:
* WE: ~45/44 = 41.3623{{c}}, ~5/4 = 388.1942{{c}}
* CWE: ~40/39 = 41.3793{{c}}, ~5/4 = 387.9017{{c}}
 
{{Optimal ET sequence|legend=0| 29, 58, 87, 145, 232 }}
 
Badness (Sintel): 0.768
 
== Hemidromeda ==
Hemidromeda may be described as the {{nowrap| 29 & 111 }} temperament. Named by [[Xenllium]] in 2023, ''hemidromeda'' comes from ''hemi-'' (Ancient Greek for "one half") and ''[[andromeda]]'', because the generator is 1/2 of andromeda's perfect twelfth (~3/1, about 1902.4 cents); the ploidacot for this temperament is alpha-dicot.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 5120/5103, 52734375/52706752
 
{{Mapping|legend=1| 1 0 38 48 | 0 2 -45 -57 }}
: mapping generator: ~2, ~12500/7203
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7236{{c}}, ~12500/7203 = 951.1864{{c}}
: [[error map]]: {{val| -0.276 +0.418 -0.205 +0.282 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~12500/7203 = 951.4098{{c}}
: error map: {{val| 0.000 +0.865 +0.243 +0.813 }}
 
{{Optimal ET sequence|legend=1| 29, 82cd, 111, 140, 251, 391, 1424bbcdd }}
 
[[Badness]] (Sintel): 2.93
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 1331/1323, 1375/1372, 5120/5103
 
Mapping: {{mapping| 1 0 38 48 32 | 0 2 -45 -57 -36 }}
 
Optimal tunings:
* WE: ~2 = 1199.8767{{c}}, ~400/231 = 951.3065{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 951.4063{{c}}
 
{{Optimal ET sequence|legend=0| 29, 82cd, 111, 140, 251, 391e }}
 
Badness (Sintel): 2.01
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 676/675, 847/845, 1331/1323
 
Mapping: {{mapping| 1 0 38 48 32 37 | 0 2 -45 -57 -36 -42 }}
 
Optimal tunings:  
* WE: ~2 = 1199.8753{{c}}, ~26/15 = 951.3054{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4064{{c}}
 
{{Optimal ET sequence|legend=0| 29, 82cdf, 111, 140, 251, 391e }}
 
Badness (Sintel): 1.18
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 352/351, 442/441, 561/560, 676/675, 715/714
 
Mapping: {{mapping| 1 0 38 48 32 37 58 | 0 2 -45 -57 -36 -42 -68 }}
 
Optimal tunings:  
* WE: ~2 = 1199.8770{{c}}, ~26/15 = 951.3039{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4035{{c}}
 
{{Optimal ET sequence|legend=0| 29g, 82cdfg, 111, 140, 251, 391e }}
 
Badness (Sintel): 0.971
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 286/285, 352/351, 363/361, 442/441, 476/475, 561/560
 
Mapping: {{mapping| 1 0 38 48 32 37 58 32 | 0 2 -45 -57 -36 -42 -68 -35 }}
 
Optimal tunings:
* WE: ~2 = 1199.7534{{c}}, ~26/15 = 951.2024{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4020{{c}}
 
{{Optimal ET sequence|legend=0| 29g, 82cdfgh, 111, 140 }}
 
Badness (Sintel): 1.01
 
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 253/252, 286/285, 352/351, 363/361, 391/390, 442/441, 460/459
 
Mapping: {{mapping| 1 0 38 48 32 37 58 32 18 | 0 2 -45 -57 -36 -42 -68 -35 -17 }}
 
Optimal tunings:
* WE: ~2 = 1199.9128{{c}}, ~26/15 = 951.3371{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4076{{c}}
 
{{Optimal ET sequence|legend=0| 29g, 82cdfgh, 111, 140 }}
 
Badness (Sintel): 1.10
 
== Countriton ==
: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic]].''
 
Countriton may be described as the {{nowrap| 51c & 53 }} temperament. It splits the [[24/1|24th harmonic]] into nine tritone generators; its ploidacot is thus delta-enneacot. Among the possible tunings are [[157edo]] and [[210edo]], as well as [[104edo]] in the 104c val.
 
Countriton was named by [[Xenllium]] in 2022 as a counterpart of [[untriton]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 5120/5103, 7558272/7503125
 
{{Mapping|legend=1| 1 -3 -15 13 | 0 9 34 -20 }}
: mapping generators: ~2, ~1225/864
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.4179{{c}}, ~1225/864 = 611.1213{{c}}
: [[error map]]: {{val| -0.582 -0.117 +0.541 +1.181 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1225/864 = 611.4120{{c}}
: error map: {{val| 0.000 +0.753 +1.695 +2.934 }}
 
{{Optimal ET sequence|legend=1| 51c, 53, 157, 210, 473cdd }}
 
[[Badness]] (Sintel): 3.32
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 5120/5103, 41503/41472
 
Mapping: {{mapping| 1 -3 -15 13 -21 | 0 9 34 -20 48 }}
 
Optimal tunings:
* WE: ~2 = 1199.5178{{c}}, ~77/54 = 611.2097{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/54 = 611.4495{{c}}
 
{{Optimal ET sequence|legend=0| 51ce, 53, 104c, 157 }}
 
Badness (Sintel): 2.80
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 176/175, 351/350, 847/845, 2197/2187
 
Mapping: {{mapping| 1 -3 -15 13 -21 -7 | 0 9 34 -20 48 21 }}
 
Optimal tunings:
* WE: ~2 = 1199.5944{{c}}, ~77/54 = 611.2491{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/54 = 611.4506{{c}}
 
{{Optimal ET sequence|legend=0| 51ce, 53, 104c, 157 }}
 
Badness (Sintel): 1.75
 
== Artoneutral ==
Artoneutral can be described as the {{nowrap| 87 & 94 }} temperament. It is generated by an artoneutral third of ~11/9 (or a tendoneutral sixth of ~18/11), nine of which make the [[12/1|12th harmonic]]; its ploidacot is thus beta-enneacot. [[181edo]] may be recommended as a tuning.
 
Artoneutral was named by [[Flora Canou]] in 2023 for its generator's quality.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 5120/5103, 3828125/3779136
 
{{Mapping|legend=1| 1 -1 -4 12 | 0 9 22 -32 }}
: mapping generators: ~2, ~128/105
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1400{{c}}, ~128/105 = 344.7929{{c}}
: [[error map]]: {{val| +0.140 +1.041 -1.430 -0.518 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~128/105 = 344.7531{{c}}
: error map: {{val| 0.000 +0.823 -1.746 -0.925 }}
 
{{Optimal ET sequence|legend=1| 87, 94, 181 }}
 
[[Badness]] (Sintel): 3.98
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 2200/2187, 4000/3993
 
Mapping: {{mapping| 1 -1 -4 12 -2 | 0 9 22 -32 19 }}
 
Optimal tunings:
* WE: ~2 = 1200.1668{{c}}, ~11/9 = 344.8027{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7557{{c}}
 
{{Optimal ET sequence|legend=0| 87, 181 }}
 
Badness (Sintel): 1.52
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 352/351, 385/384, 1575/1573
 
Mapping: {{mapping| 1 -1 -4 12 -2 6 | 0 9 22 -32 19 -8 }}
 
Optimal tunings:
* WE: ~2 = 1200.0662{{c}}, ~11/9 = 344.7804{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7617{{c}}
 
{{Optimal ET sequence|legend=0| 87, 181 }}
 
Badness (Sintel): 1.08
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 325/324, 352/351, 375/374, 385/384, 595/594
 
Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 | 0 9 22 -32 19 -8 56 }}
 
Optimal tunings:
* WE: ~2 = 1200.0346{{c}}, ~11/9 = 344.7589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7492{{c}}
 
{{Optimal ET sequence|legend=0| 87, 94, 181 }}
 
Badness (Sintel): 1.16
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 325/324, 352/351, 375/374, 385/384, 400/399, 595/594
 
Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 -15 | 0 9 22 -32 19 -8 56 67 }}
 
Optimal tunings:
* WE: ~2 = 1200.0282{{c}}, ~11/9 = 344.7532{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7453{{c}}
 
{{Optimal ET sequence|legend=0| 87, 94, 181 }}
 
Badness (Sintel): 1.19
 
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 300/299, 325/324, 352/351, 375/374, 385/384, 400/399, 484/483
 
Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 -15 -13 | 0 9 22 -32 19 -8 56 67 61 }}
 
Optimal tunings:
* WE: ~2 = 1200.0163{{c}}, ~11/9 = 344.7461{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7416{{c}}
 
{{Optimal ET sequence|legend=0| 87, 94, 181 }}
 
Badness (Sintel): 1.17
 
== Quanic ==
Quanic may be described as the {{nowrap| 94 & 111 }} temperament. It splits the perfect fifth into five generators which in the 13-limit extension may be taken as ~13/12; its ploidacot is thus pentacot. [[205edo]] may be recommended as a tuning.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 5120/5103, 5832000/5764801
 
{{Mapping|legend=1| 1 1 -4 0 | 0 5 54 24 }}
: mapping generators: ~2, ~160/147
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6159{{c}}, ~160/147 = 140.4483{{c}}
: [[error map]]: {{val| -0.384 -0.098 -0.570 +1.933 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~160/147 = 140.4862{{c}}
: error map: {{val| 0.000 +0.476 -0.061 +2.842 }}
 
{{Optimal ET sequence|legend=1| 94, 111, 205 }}
 
[[Badness]] (Sintel): 4.54
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 1331/1323, 5120/5103
 
Mapping: {{mapping| 1 1 -4 0 1 | 0 5 54 24 21 }}
 
Optimal tunings:
* WE: ~2 = 1199.7834{{c}}, ~88/81 = 140.4635{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~88/81 = 140.4850{{c}}
 
{{Optimal ET sequence|legend=0| 94, 111, 205 }}
 
Badness (Sintel): 1.94
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 540/539, 729/728, 1331/1323
 
Mapping: {{mapping| 1 1 -4 0 1 3 | 0 5 54 24 21 6 }}
 
Optimal tunings:
* WE: ~2 = 1199.6639{{c}}, ~13/12 = 140.4562{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4904{{c}}
 
{{Optimal ET sequence|legend=0| 94, 111, 205 }}
 
Badness (Sintel): 1.34
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 352/351, 442/441, 540/539, 715/714, 847/845
 
Mapping: {{mapping| 1 1 -4 0 1 3 -2 | 0 5 54 24 21 6 52 }}
 
Optimal tunings:
* WE: ~2 = 1199.6699{{c}}, ~13/12 = 140.4586{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4920{{c}}
 
{{Optimal ET sequence|legend=0| 94, 111, 205 }}
 
Badness (Sintel): 1.08
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 352/351, 400/399, 442/441, 456/455, 495/494, 715/714
 
Mapping: {{mapping| 1 1 -4 0 1 3 -2 -5 | 0 5 54 24 21 6 52 79 }}
 
Optimal tunings:
* WE: ~2 = 1199.6745{{c}}, ~13/12 = 140.4574{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4908{{c}}
 
{{Optimal ET sequence|legend=0| 94, 111, 205 }}
 
Badness (Sintel): 1.05
 
== Jorgensen ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Jorgensen]].''
 
Jorgensen tempers out the [[linus comma]] in addition to the aberschisma, and may be described as the {{nowrap| 70 & 140 }} temperament, with a 70th-octave period. Its ploidacot is 70-ploid acot.
 
It is the natural 7-limit extension of the 5-limit temperament tempering out the 70-comma, named by [[Mike Battaglia]] in 2012 for historical interests<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_103982.html Yahoo! Tuning Group | ''Jorgensen Temperament'']</ref>.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 5120/5103, 578509309952/576650390625
 
{{Mapping|legend=1| 70 111 0 34 | 0 0 1 1 }}
: mapping generators: ~50421/50000, ~5
 
[[Optimal tuning]]s:
* [[WE]]: ~50421/50000 = 17.1387{{c}}, ~5/4 = 386.8071{{c}}
: [[error map]]: {{val| -0.288 +0.445 -0.084 +0.121 }}
* [[CWE]]: ~50421/50000 = 17.1429{{c}}, ~5/4 = 386.6593{{c}}
: error map: {{val| 0.000 +0.902 +0.346 +0.690 }}
 
{{Optimal ET sequence|legend=1| 70, 140, 350, 490 }}
 
[[Badness]] (Sintel): 5.40
 
== References ==
 
[[Category:Temperament collections]]
[[Category:Aberschismic temperaments| ]] <!-- main article -->
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Aberschismic_temperaments"