Canopic clan: Difference between revisions

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**Imported revision 337192502 - Original comment: **
m FloraC moved page Canopus clan to Canopic clan
 
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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 '''canopic clan''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[canopic comma]] ({{monzo|legend=1| 0 3 4 -5 }}, [[ratio]]: 16875/16807), a no-twos comma.
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-05-18 18:29:22 UTC</tt>.<br>
: The original revision id was <tt>337192502</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]]


This tempers out the //mirkwai comma//, |0 3 4 -5&gt; = 16875/16807, a no-twos comma. Members of the clan include grendel, kwai, pluto and mirkat considered below, as well as miracle, octokaidecal, nusecond, semisept and bohpier, considered elsewhere.
== Canopus ==
{{Main| Canopus }}


=Grendel=
[[Subgroup]]: 3.5.7
Commas: 6144/6125, 16875/16807


[[POTE tuning|POTE generator]]: 386.863
[[Comma list]]: 16875/16807


Map: [&lt;1 9 2 7|, &lt;0 -23 1 -13|]
{{Mapping|legend=2| 1 -2 -1 | 0 5 4 }}
Wedgie: &lt;&lt;23 -1 13 -55 -44 33||
: mapping generators: ~3, ~15/7
EDOs: 31, 90, 121, 152, 335


==11-limit==  
[[Optimal tuning]]s:
Commas: 540/539, 1375/1372, 5632/5625
* [[WE]]: ~3 = 1901.7826{{c}}, ~15/7 = 1317.8771{{c}}
: [[error map]]: {{val| +1.785 -0.771 -2.248 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~15/7 = 1317.9686{{c}}
: error map: {{val| 0.000 -0.381 +1.093 }}


[[POTE tuning|POTE generator]]: 386.856
[[Optimal ET sequence]]: [[13edt|b13]], [[62edt|b62]], [[75edt|b75]], [[88edt|b88]], [[101edt|b101]], [[114edt|b114]], [[355edt|b355]], [[469edt|b469]], [[583edt|b583]], [[697edt|b697]]


Map: [&lt;1 9 2 7 17|, &lt;0 -23 1 -13 -42|]
[[Badness]] (Sintel): 0.0996
EDOs: 31, 90, 121, 152, 335, 487


==13-limit==  
=== Overview to extensions ===
Commas: 352/351, 540/539, 625/624, 1375/1372
The full 7-limit extensions' relation to canopus is clearer if the mapping is normalized in terms of 3.5.7.2. In fact, the strong extensions are nusecond and octoid. These temperaments are distributed into different temperament collection pages.
* ''[[Nusecond]]'' (+126/125) → [[Starling temperaments #Nusecond|Starling temperaments]]
* ''[[Octoid]]'' (+4375/4374) → [[Ragismic microtemperaments #Octoid|Ragismic microtemperaments]]


[[POTE tuning|POTE generator]]: 386.826
The others are weak extensions. Mirkat tempers out [[19683/19600]], splitting the generator in two with a semitwelfth period. Sqrtphi tempers out [[15625/15552]], splitting the period in six. Semisept tempers out [[1728/1715]] and [[3136/3125]], splitting the generator in six. Miracle tempers out [[225/224]]. Pluto tempers out [[4000/3969]]. These split the generator in five. Kwai tempers out [[5120/5103]], splitting the generator in ten. Quanharuk tempers out [[32805/32768]], splitting the generator in three with a 1/5-twelfth period. Grendel tempers out [[6144/6125]], splitting the generator in eleven. Finally, eris tempers out [[65625/65536]], splitting the generator in sixteen.  


Map: [&lt;1 9 2 7 17 -5|, &lt;0 -23 1 -13 -42 27|]
Members of the clan discussed elsewhere are:  
EDOs: 31, 121, 152, 425
* ''[[Kwai]]'' (+5120/5103) → [[Hemifamity temperaments #Kwai|Hemifamity temperaments]]
* ''[[Octokaidecal]]'' (+28/27 or 50/49) → [[Trienstonic clan #Octokaidecal|Trienstonic clan]]
* ''[[Meantritone]]'' (+81/80) → [[Meantone family #Meantritone|Meantone family]]
* ''[[Quanharuk]]'' (+32805/32768) → [[Schismatic family #Quanharuk|Schismatic family]]
* [[Miracle]] (+225/224) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* ''[[Pluto]]'' (+4000/3969) → [[Octagar temperaments #Pluto|Octagar temperaments]]
* ''[[Bohpier]]'' (+245/243) → [[Sensamagic clan #Bohpier|Sensamagic clan]]
* ''[[Subsedia]]'' (+65536/64827) → [[Buzzardsmic clan #Subsedia|Buzzardsmic clan]]
* ''[[Semisept]]'' (+1728/1715 or 3136/3125) → [[Hemimean clan #Semisept|Hemimean clan]]
* ''[[Grendel]]'' (+6144/6125) → [[Porwell temperaments #Grendel|Porwell temperaments]]
* ''[[Quinmage]]'' (+3125/3072) → [[Magic family #Quinmage|Magic family]]
* ''[[Familia]]'' (+1600000/1594323) → [[Amity family #Familia|Amity family]]
* [[Sqrtphi]] (+15625/15552) → [[Kleismic family #Sqrtphi|Kleismic family]]
* ''[[Rainwell]]'' (+2100875/2097152) → [[Semicomma family #Rainwell|Semicomma family]]
* ''[[Quintiquart]]'' (+390625000/387420489) → [[Quartonic family #Quintiquart|Quartonic family]]


==17-limit==
For ''no-twos'' extensions, see [[No-twos subgroup temperaments #Canopus]].
Commas: 256/255, 352/351, 625/624, 715/714, 1275/1274


[[POTE tuning|POTE generator]]: 386.812
Considered below are mirkat, eris, subsemifourth, septendesemi, gaster, hemiseptisix, browser, and grazer, in the order of increasing [[badness]].  


Map: [&lt;1 9 2 7 17 -5 -3|, &lt;0 -23 1 -13 -42 27 22|]
== Mirkat ==
EDOs: 31, 121, 273
Mirkat tempers out 19683/19600, the [[cataharry comma]], as well as 250047/250000, the [[landscape comma]], and may be described as the {{nowrap| 72 & 111 }} temperament with a [[ploidacot]] signature of triploid alpha-hexacot.


==19-limit==
[[Subgroup]]: 2.3.5.7
Commas: 256/255, 352/351, 375/374, 400/399, 456/455, 715/714


[[POTE tuning|POTE generator]]: 386.819
[[Comma list]]: 16875/16807, 19683/19600


Map: [&lt;1 9 2 7 17 -5 -3|, &lt;0 -23 1 -13 -42 27 22|]
{{Mapping|legend=1| 3 2 1 2 | 0 6 13 14 }}
EDOs: 31, 121, 152, 273
: mapping generators: ~63/50, ~10/9


=Kwai=  
[[Optimal tuning]]s:
Commas: 5120/5103, 16875/16807
* [[WE]]: ~63/50 = 400.0277{{c}}, ~10/9 = 183.5515{{c}}
: [[error map]]: {{val| +0.083 -0.591 -0.117 +0.950 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~10/9 = 183.5470{{c}}
: error map: {{val| 0.000 -0.673 -0.203 +0.831 }}


[[POTE tuning|POTE generator]]: 702.616
{{Optimal ET sequence|legend=1| 39d, 72, 111, 183, 255 }}


Map: [&lt;1 0 -50 -40|, &lt;0 1 33 27|]
[[Badness]] (Sintel): 1.50
Wedgie: &lt;&lt;1 33 27 50 40 -30||
EDOs: 41, 111, 152, 345, 497


==11-limit==  
=== 11-limit ===
Commas: 540/539, 1375/1372, 16384/16335
Subgroup: 2.3.5.7.11


[[POTE tuning|POTE generator]]: 702.623
Comma list: 540/539, 1375/1372, 8019/8000


Map: [&lt;1 0 -50 -40 32|, &lt;0 1 33 27 -18|]
Mapping: {{mapping| 3 2 1 2 9 | 0 6 13 14 3 }}
EDOs: 29, 41, 111, 152


==13-limit==
Optimal tunings:
Commas: 352/351, 540/539, 729/728, 1375/1372
* WE: ~63/50 = 400.0463{{c}}, ~10/9 = 183.5496{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~10/9 = 183.5391{{c}}


[[POTE tuning|POTE generator]]: 702.644
{{Optimal ET sequence|legend=0| 39d, 72, 111, 183, 255 }}


Map: [&lt;1 0 -50 -40 32 27|, &lt;0 1 33 27 -18 -21|]
Badness (Sintel): 0.731
EDOs: 29, 41, 111, 152


==17-limit==  
=== 13-limit ===
Commas: 256/255, 352/351, 540/539, 715/714, 1089/1088
Subgroup: 2.3.5.7.11.13


[[POTE tuning|POTE generator]]: 702.660
Comma list: 351/350, 540/539, 676/675, 1375/1372


Map: [&lt;1 0 -50 -40 32 27 58|, &lt;0 1 33 27 -18 -21 -34|]
Mapping: {{mapping| 3 2 1 2 9 1 | 0 6 13 14 3 22 }}
EDOs: 29, 41, 111, 152, 263


==19-limit==
Optimal tunings:
Commas: 256/255, 352/351, 400/399, 456/455, 715/714, 847/845
* WE: ~63/50 = 400.0245{{c}}, ~10/9 = 183.5885{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~10/9 = 183.5825{{c}}


[[POTE tuning|POTE generator]]: 702.657
{{Optimal ET sequence|legend=0| 39df, 72, 111, 183 }}


Map: [&lt;1 0 -50 -40 32 27 58 -56|, &lt;0 1 33 27 -18 -21 -34 38|]
Badness (Sintel): 0.770
EDOs: 29, 41, 111, 152, 263


=Pluto=  
=== 17-limit ===
Commas: 4000/3969, 10976/10935
Subgroup: 2.3.5.7.11.13.17


[[POTE tuning|POTE generator]]: 585.147
Comma list: 351/350, 442/441, 540/539, 561/560, 715/714


Map: [&lt;1 5 15 15|, &lt;0 -7 -26 -25|]
Mapping: {{mapping| 3 2 1 2 9 1 4 | 0 6 13 14 3 22 18 }}
Wedgie: &lt;&lt;7 26 25 25 20 -15||
EDOs: 39, 41, 80, 121, 404


==11-limit==
Optimal tunings:
Commas: 540/539, 896/891, 1375/1372
* WE: ~34/27 = 400.0257{{c}}, ~10/9 = 183.5906{{c}}
* CWE: ~34/27 = 400.0000{{c}}, ~10/9 = 183.5843{{c}}


[[POTE tuning|POTE generator]]: 585.114
{{Optimal ET sequence|legend=0| 39dfg, 72, 111, 183 }}


Map: [&lt;1 5 15 15 2|, &lt;0 -7 -26 -25 3|]
Badness (Sintel): 0.600
EDOs: 39, 41, 80, 121


==13-limit==  
== Eris ==
Commas: 325/324, 352/351, 364/363, 540/539
Eris tempers out 65625/65536, the [[horwell comma]], and may be described as the {{nowrap| 31 & 224 }} temperament. The [[2.5.7-subgroup|2.5.7 subgroup]] restriction of this temperament is [[exodia]].


[[POTE tuning|POTE generator]]: 585.123
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 5 15 15 2 -8|, &lt;0 -7 -26 -25 3 24|]
[[Comma list]]: 16875/16807, 65625/65536
EDOs: 39, 41, 80, 121


==17-limit==
{{Mapping|legend=1| 1 -19 8 -5 | 0 29 -8 11 }}
Commas: 256/255, 325/324, 352/351, 364/363, 540/539
: mapping generators: ~2, ~49/30


[[POTE tuning|POTE generator]]: 585.116
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0256{{c}}, ~49/30 = 851.8023{{c}}
: [[error map]]: {{val| +0.026 -0.173 -0.528 +0.872 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/30 = 851.7845{{c}}
: error map: {{val| 0.000 -0.204 -0.590 +0.804 }}


Map: [&lt;1 5 15 15 2 -8 -12|, &lt;0 -7 -26 -25 3 24 33|]
{{Optimal ET sequence|legend=1| 31, 131, 162, 193, 224 }}
EDOs: 39, 41, 80, 121


==19-limit==
[[Badness]] (Sintel): 1.89
Commas: 190/189, 256/255, 325/324, 352/351, 361/360, 595/594


[[POTE tuning|POTE generator]]: 585.109
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 5 15 15 2 -8 -12 14|, &lt;0 -7 -26 -25 3 24 33 -20|]
Comma list: 540/539, 1375/1372, 65625/65536
EDOs: 39, 41, 80, 121, 564, 685


==Plutino==
Mapping: {{mapping| 1 -19 8 -5 -37 | 0 29 -8 11 57 }}
Commas: 100/99, 245/242, 16875/16807


POTE generator: ~7/5 = 585.283
Optimal tunings:  
* WE: ~2 = 1200.0218{{c}}, ~18/11 = 851.7963{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/11 = 851.7812{{c}}


Map: [&lt;1 5 15 15 22|, &lt;0 -7 -26 -25 -38|]
{{Optimal ET sequence|legend=0| 31, …, 193, 224, 703, 927d }}
EDOs: 41, 123c
Badness: 0.0580


===13-limit===
Badness (Sintel): 0.913
Commas: 100/99, 196/195, 245/242, 729/728


POTE generator: ~7/5 = 585.232
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 5 15 15 22 12|, &lt;0 -7 -26 -25 -38 -17|]
Comma list: 540/539, 625/624, 1375/1372, 4096/4095
EDOs: 41, 121ef
Badness: 0.0402


=Eris=
Mapping: {{mapping| 1 -19 8 -5 -37 47 | 0 29 -8 11 57 -61 }}
Commas: 16875/16807, 65625/65536


POTE generator: ~60/49 = 348.216
Optimal tuning:  
* WE ~2 = 1199.9623{{c}}, ~18/11 = 851.7598{{c}}
* CWE ~2 = 1200.0000{{c}}, ~18/11 = 851.7865{{c}}


Map: [&lt;1 10 0 6|, &lt;0 -29 8 -11|]
{{Optimal ET sequence|legend=0| 31, 193, 224 }}
Wedgie: &lt;&lt;29 -8 11 -80 -64 48}}
EDOs: 31, 131, 162, 193, 224, 1823cd, 2271cd
Badness: 0.0747


==11-limit==
Badness (Sintel): 1.04
Commas: 540/539, 1375/1372, 65625/65536


POTE generator: ~11/9 = 348.219
== Subsemifourth ==
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 10 0 6 20|, &lt;0 -29 8 -11 -57|]
[[Comma list]]: 16875/16807, 26873856/26796875
EDOs: 31, 193, 224, 703, 927d, 1151cd
Badness: 0.0276


=Mirkat=
{{Mapping|legend=1| 1 -8 -4 -8 | 0 47 31 53 }}
Commas: 16875/16807, 19683/19600
: mapping generators: ~2, ~144/125


POTE generator: ~10/9 = 183.539
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9182{{c}}, ~144/125 = 244.7020{{c}}
: [[error map]]: {{val| -0.082 -0.305 -0.223 +1.037 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~144/125 = 244.7172{{c}}
: error map: {{val| 0.000 -0.248 -0.082 +1.184 }}


Map: [&lt;3 2 1 2|, &lt;0 6 13 14|]
{{Optimal ET sequence|legend=1| 49, 103, 152, 255, 407 }}
Wedgie: &lt;&lt;18 39 42 20 16 -12||
EDOs: 33, 39, 72, 111, 183, 255
Badness: 0.0594


==11-limit==
[[Badness]] (Sintel): 3.42
Commas: 540/539, 1375/1372, 8019/8000


POTE generator: ~10/9 = 183.528
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;3 2 1 2 9|, &lt;0 6 13 14 3|]
Comma list: 540/539, 1375/1372, 234375/234256
EDOs: 33, 39, 72, 183, 255
Badness: 0.0221


==13-limit==
Mapping: {{mapping| 1 -8 -4 -8 -10 | 0 47 31 53 66 }}
Commas: 351/350, 540/539, 676/675, 1375/1372


POTE generator: ~10/9 = 183.577
Optimal tunings:  
* WE: ~2 = 1199.9229{{c}}, ~121/105 = 244.7033{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~121/105 = 244.7175{{c}}


Map: [&lt;3 2 1 2 9 1|, &lt;0 6 13 14 3 22|]
{{Optimal ET sequence|legend=0| 49, 103, 152, 255, 407 }}
EDOs: 33, 39, 72, 111, 183
Badness: 0.0186


=Subsemifourth=
Badness (Sintel): 1.13
Commas: 16875/16807, 26873856/26796875


POTE generator: ~144/125 = 244.719
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 39 27 45|, &lt;0 -47 -31 -53|]
Comma list: 540/539, 847/845, 1375/1372, 1575/1573
Wedgie: &lt;&lt;47 31 53 -60 -48 36||
EDOs: 49, 103, 152, 255, 407
Badness: 0.1352


==11-limit==
Mapping: {{mapping| 1 -8 -4 -8 -10 -12 | 0 0 47 31 53 66 77 }}
Commas: 540/539, 1375/1372, 234375/234256


POTE generator: ~121/105 = 244.719
Optimal tunings:  
* WE: ~2 = 1199.9003{{c}}, ~15/13 = 244.6932{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/13 = 244.7116{{c}}


Map: [&lt;1 39 27 45 56|, &lt;0 -47 -31 -53 -66|]
{{Optimal ET sequence|legend=0| 49f, 103, 152f, 255, 407f }}
EDOs: 49, 103, 152, 255, 407, 966d
Badness: 0.0343


==13-limit==
Badness (Sintel): 1.17
Commas: 540/539, 847/845, 1375/1372, 1575/1573


POTE generator: ~15/13 = 244.714
== Septendesemi ==
Septendesemi tempers out the mirkwai comma and 1959552/1953125 (parkleiness comma) in the 7-limit, and may be described as the {{nowrap| 80 & 103 }} temperament. [[183edo]] provides an excellent tuning for 7-, 11-, 13-, and 17-limit septendesemi. Septendesemi was named by [[Xenllium]] in 2021; the name ''septendesemi'' refers to a septendecimal semitone ([[17/16]]).  


Map: [&lt;1 39 27 45 56 65|, &lt;0 -47 -31 -53 -66 -77|]
[[Subgroup]]: 2.3.5.7
EDOs: 103, 152f, 255, 407f, 662df
 
Badness: 0.0284
[[Comma list]]: 16875/16807, 1959552/1953125
</pre></div>
 
<h4>Original HTML content:</h4>
{{Mapping|legend=1| 1 -2 -1 -2 | 0 41 38 55 }}
<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;Mirkwai clan&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:50:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:50 --&gt;&lt;!-- ws:start:WikiTextTocRule:51: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Grendel"&gt;Grendel&lt;/a&gt;&lt;/div&gt;
: mapping generators: ~2, ~343/324
&lt;!-- ws:end:WikiTextTocRule:51 --&gt;&lt;!-- ws:start:WikiTextTocRule:52: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Grendel-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:52 --&gt;&lt;!-- ws:start:WikiTextTocRule:53: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Grendel-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
[[Optimal tuning]]s:
&lt;!-- ws:end:WikiTextTocRule:53 --&gt;&lt;!-- ws:start:WikiTextTocRule:54: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Grendel-17-limit"&gt;17-limit&lt;/a&gt;&lt;/div&gt;
* [[WE]]: ~2 = 1199.8649{{c}}, ~343/324 = 104.9046{{c}}
&lt;!-- ws:end:WikiTextTocRule:54 --&gt;&lt;!-- ws:start:WikiTextTocRule:55: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Grendel-19-limit"&gt;19-limit&lt;/a&gt;&lt;/div&gt;
: [[error map]]: {{val| -0.135 -0.597 +0.195 +1.196 }}
&lt;!-- ws:end:WikiTextTocRule:55 --&gt;&lt;!-- ws:start:WikiTextTocRule:56: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Kwai"&gt;Kwai&lt;/a&gt;&lt;/div&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~343/324 = 104.9134{{c}}
&lt;!-- ws:end:WikiTextTocRule:56 --&gt;&lt;!-- ws:start:WikiTextTocRule:57: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Kwai-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
: error map: {{val| 0.000 -0.506 +0.395 +1.410 }}
&lt;!-- ws:end:WikiTextTocRule:57 --&gt;&lt;!-- ws:start:WikiTextTocRule:58: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Kwai-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:58 --&gt;&lt;!-- ws:start:WikiTextTocRule:59: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Kwai-17-limit"&gt;17-limit&lt;/a&gt;&lt;/div&gt;
{{Optimal ET sequence|legend=0| 80, 103, 183 }}
&lt;!-- ws:end:WikiTextTocRule:59 --&gt;&lt;!-- ws:start:WikiTextTocRule:60: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Kwai-19-limit"&gt;19-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:60 --&gt;&lt;!-- ws:start:WikiTextTocRule:61: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Pluto"&gt;Pluto&lt;/a&gt;&lt;/div&gt;
[[Badness]] (Sintel): 3.71
&lt;!-- ws:end:WikiTextTocRule:61 --&gt;&lt;!-- ws:start:WikiTextTocRule:62: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Pluto-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:62 --&gt;&lt;!-- ws:start:WikiTextTocRule:63: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Pluto-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
=== 11-limit ===
&lt;!-- ws:end:WikiTextTocRule:63 --&gt;&lt;!-- ws:start:WikiTextTocRule:64: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Pluto-17-limit"&gt;17-limit&lt;/a&gt;&lt;/div&gt;
Subgroup: 2.3.5.7.11
&lt;!-- ws:end:WikiTextTocRule:64 --&gt;&lt;!-- ws:start:WikiTextTocRule:65: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Pluto-19-limit"&gt;19-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:65 --&gt;&lt;!-- ws:start:WikiTextTocRule:66: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Pluto-Plutino"&gt;Plutino&lt;/a&gt;&lt;/div&gt;
Comma list: 540/539, 1375/1372, 43923/43750
&lt;!-- ws:end:WikiTextTocRule:66 --&gt;&lt;!-- ws:start:WikiTextTocRule:67: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Pluto-Plutino-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:67 --&gt;&lt;!-- ws:start:WikiTextTocRule:68: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Eris"&gt;Eris&lt;/a&gt;&lt;/div&gt;
Mapping: {{mapping| 1 -2 -1 -2 -1 | 0 41 38 55 51 }}
&lt;!-- ws:end:WikiTextTocRule:68 --&gt;&lt;!-- ws:start:WikiTextTocRule:69: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Eris-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:69 --&gt;&lt;!-- ws:start:WikiTextTocRule:70: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Mirkat"&gt;Mirkat&lt;/a&gt;&lt;/div&gt;
Optimal tunings:
&lt;!-- ws:end:WikiTextTocRule:70 --&gt;&lt;!-- ws:start:WikiTextTocRule:71: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Mirkat-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
* WE: ~2 = 1199.9327{{c}}, ~35/33 = 104.9100{{c}}
&lt;!-- ws:end:WikiTextTocRule:71 --&gt;&lt;!-- ws:start:WikiTextTocRule:72: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Mirkat-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 104.9144{{c}}
&lt;!-- ws:end:WikiTextTocRule:72 --&gt;&lt;!-- ws:start:WikiTextTocRule:73: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Subsemifourth"&gt;Subsemifourth&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:73 --&gt;&lt;!-- ws:start:WikiTextTocRule:74: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Subsemifourth-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
{{Optimal ET sequence|legend=0| 80, 103, 183 }}
&lt;!-- ws:end:WikiTextTocRule:74 --&gt;&lt;!-- ws:start:WikiTextTocRule:75: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Subsemifourth-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:75 --&gt;&lt;!-- ws:start:WikiTextTocRule:76: --&gt;&lt;/div&gt;
Badness (Sintel): 1.37
&lt;!-- ws:end:WikiTextTocRule:76 --&gt;&lt;br /&gt;
 
This tempers out the &lt;em&gt;mirkwai comma&lt;/em&gt;, |0 3 4 -5&amp;gt; = 16875/16807, a no-twos comma. Members of the clan include grendel, kwai, pluto and mirkat considered below, as well as miracle, octokaidecal, nusecond, semisept and bohpier, considered elsewhere.&lt;br /&gt;
=== 13-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Grendel"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Grendel&lt;/h1&gt;
 
Commas: 6144/6125, 16875/16807&lt;br /&gt;
Comma list: 351/350, 540/539, 1375/1372, 4225/4224
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 386.863&lt;br /&gt;
Mapping: {{mapping| 1 -2 -1 -2 -1 3 | 0 41 38 55 51 8 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 9 2 7|, &amp;lt;0 -23 1 -13|]&lt;br /&gt;
Optimal tunings:  
Wedgie: &amp;lt;&amp;lt;23 -1 13 -55 -44 33||&lt;br /&gt;
* WE: ~2 = 1200.1082{{c}}, ~35/33 = 104.9170{{c}}
EDOs: 31, 90, 121, 152, 335&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 104.9094{{c}}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Grendel-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;11-limit&lt;/h2&gt;
{{Optimal ET sequence|legend=0| 80, 103, 183, 469f }}
Commas: 540/539, 1375/1372, 5632/5625&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.15
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 386.856&lt;br /&gt;
 
&lt;br /&gt;
=== 17-limit ===
Map: [&amp;lt;1 9 2 7 17|, &amp;lt;0 -23 1 -13 -42|]&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17
EDOs: 31, 90, 121, 152, 335, 487&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 351/350, 540/539, 561/560, 715/714, 4225/4224
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Grendel-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;13-limit&lt;/h2&gt;
 
Commas: 352/351, 540/539, 625/624, 1375/1372&lt;br /&gt;
Mapping: {{mapping| 1 -2 -1 -2 -1 3 4 | 0 41 38 55 51 8 1 }}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 386.826&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1200.0758{{c}}, ~17/16 = 104.9158{{c}}
Map: [&amp;lt;1 9 2 7 17 -5|, &amp;lt;0 -23 1 -13 -42 27|]&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~17/16 = 104.9101{{c}}
EDOs: 31, 121, 152, 425&lt;br /&gt;
 
&lt;br /&gt;
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~17/16 = 104.909{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Grendel-17-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;17-limit&lt;/h2&gt;
 
Commas: 256/255, 352/351, 625/624, 715/714, 1275/1274&lt;br /&gt;
{{Optimal ET sequence|legend=0| 80, 103, 183, 469f }}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 386.812&lt;br /&gt;
Badness (Sintel): 1.03
&lt;br /&gt;
 
Map: [&amp;lt;1 9 2 7 17 -5 -3|, &amp;lt;0 -23 1 -13 -42 27 22|]&lt;br /&gt;
== Gaster ==
EDOs: 31, 121, 273&lt;br /&gt;
: ''For the 5-limit version, see [[Very high accuracy temperaments #Gaster]].''
&lt;br /&gt;
{{Main| Gaster temperament }}
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Grendel-19-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;19-limit&lt;/h2&gt;
 
Commas: 256/255, 352/351, 375/374, 400/399, 456/455, 715/714&lt;br /&gt;
Gaster tempers out {{monzo| -70 72 -19 }} in the 5-limit, mirkwai comma (16875/16807) and [[scheme comma]] (14348907/14336000) in the 7-limit, and may be described as the {{nowrap| 111 & 113 }} temperament.
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 386.819&lt;br /&gt;
It was named by [[Xenllium]] in 2022; the word "[[Wiktionary: gaster|gaster]]" means [[Wiktionary: abdomen|abdomen]] or [[Wiktionary: stomach|stomach]], but also a restructuring of the words "gassormic tritone", which is a generator of this temperament. This temperament is sufficient to obtain high prime limit harmonics like a stomach, so that patent vals [[111edo|111]], [[113edo|113]] and [[224edo|224]] support it even in the 41-limit.
&lt;br /&gt;
 
Map: [&amp;lt;1 9 2 7 17 -5 -3|, &amp;lt;0 -23 1 -13 -42 27 22|]&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
EDOs: 31, 121, 152, 273&lt;br /&gt;
 
&lt;br /&gt;
[[Comma list]]: 16875/16807, 14348907/14336000
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Kwai"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Kwai&lt;/h1&gt;
 
Commas: 5120/5103, 16875/16807&lt;br /&gt;
{{Mapping|legend=1| 1 -8 -34 -32 | 0 19 72 69 }}
&lt;br /&gt;
: mapping generators: ~2, ~567/400
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 702.616&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]]s:  
Map: [&amp;lt;1 0 -50 -40|, &amp;lt;0 1 33 27|]&lt;br /&gt;
* [[WE]]: ~2 = 1199.9920{{c}}, ~567/400 = 605.3546{{c}}
Wedgie: &amp;lt;&amp;lt;1 33 27 50 40 -30||&lt;br /&gt;
: [[error map]]: {{val| -0.008 -0.152 -0.506 +0.902 }}
EDOs: 41, 111, 152, 345, 497&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~567/400 = 605.3586{{c}}
&lt;br /&gt;
: error map: {{val| 0.000 -0.142 -0.497 +0.915 }}
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Kwai-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;11-limit&lt;/h2&gt;
 
Commas: 540/539, 1375/1372, 16384/16335&lt;br /&gt;
{{Optimal ET sequence|legend=1| 111, 224 }}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 702.623&lt;br /&gt;
[[Badness]] (Sintel): 3.91
&lt;br /&gt;
 
Map: [&amp;lt;1 0 -50 -40 32|, &amp;lt;0 1 33 27 -18|]&lt;br /&gt;
=== 11-limit ===
EDOs: 29, 41, 111, 152&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Kwai-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;13-limit&lt;/h2&gt;
Comma list: 540/539, 1375/1372, 14348907/14336000
Commas: 352/351, 540/539, 729/728, 1375/1372&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 -8 -34 -32 8 | 0 19 72 69 -9 }}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 702.644&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
Map: [&amp;lt;1 0 -50 -40 32 27|, &amp;lt;0 1 33 27 -18 -21|]&lt;br /&gt;
* WE: ~2 = 1199.9387{{c}}, ~363/256 = 605.3300{{c}}
EDOs: 29, 41, 111, 152&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~363/256 = 605.3603{{c}}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Kwai-17-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;17-limit&lt;/h2&gt;
{{Optimal ET sequence|legend=0| 111, 224, 783d }}
Commas: 256/255, 352/351, 540/539, 715/714, 1089/1088&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.79
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 702.660&lt;br /&gt;
 
&lt;br /&gt;
=== 13-limit ===
Map: [&amp;lt;1 0 -50 -40 32 27 58|, &amp;lt;0 1 33 27 -18 -21 -34|]&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
EDOs: 29, 41, 111, 152, 263&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 540/539, 729/728, 1375/1372, 2200/2197
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Kwai-19-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;19-limit&lt;/h2&gt;
 
Commas: 256/255, 352/351, 400/399, 456/455, 715/714, 847/845&lt;br /&gt;
Mapping: {{mapping| 1 -8 -34 -32 8 -19 | 0 19 72 69 -9 45 }}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 702.657&lt;br /&gt;
Optimal tunings:
&lt;br /&gt;
* WE: ~2 = 1199.9154{{c}}, ~78/55 = 605.3183{{c}}
Map: [&amp;lt;1 0 -50 -40 32 27 58 -56|, &amp;lt;0 1 33 27 -18 -21 -34 38|]&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~78/55 = 605.3601{{c}}
EDOs: 29, 41, 111, 152, 263&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 111, 224, 783df }}
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc10"&gt;&lt;a name="Pluto"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Pluto&lt;/h1&gt;
 
Commas: 4000/3969, 10976/10935&lt;br /&gt;
Badness (Sintel): 1.03
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 585.147&lt;br /&gt;
=== 17-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17
Map: [&amp;lt;1 5 15 15|, &amp;lt;0 -7 -26 -25|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;7 26 25 25 20 -15||&lt;br /&gt;
Comma list: 540/539, 715/714, 729/728, 936/935, 2200/2197
EDOs: 39, 41, 80, 121, 404&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 -8 -34 -32 8 -19 -6 | 0 19 72 69 -9 45 20 }}
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Pluto-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;11-limit&lt;/h2&gt;
 
Commas: 540/539, 896/891, 1375/1372&lt;br /&gt;
Optimal tunings:
&lt;br /&gt;
* WE: ~2 = 1199.8076{{c}}, ~17/12 = 605.2674{{c}}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 585.114&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3626{{c}}
&lt;br /&gt;
 
Map: [&amp;lt;1 5 15 15 2|, &amp;lt;0 -7 -26 -25 3|]&lt;br /&gt;
{{Optimal ET sequence|legend=0| 111, 224, 559dgg }}
EDOs: 39, 41, 80, 121&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.09
&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="Pluto-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;13-limit&lt;/h2&gt;
 
Commas: 325/324, 352/351, 364/363, 540/539&lt;br /&gt;
=== 19-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17.19
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 585.123&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 324/323, 400/399, 495/494, 540/539, 715/714, 1445/1444
Map: [&amp;lt;1 5 15 15 2 -8|, &amp;lt;0 -7 -26 -25 3 24|]&lt;br /&gt;
 
EDOs: 39, 41, 80, 121&lt;br /&gt;
Mapping: {{mapping| 1 -8 -34 -32 8 -19 -6 -24 | 0 19 72 69 -9 45 20 56 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc13"&gt;&lt;a name="Pluto-17-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;17-limit&lt;/h2&gt;
Optimal tunings:  
Commas: 256/255, 325/324, 352/351, 364/363, 540/539&lt;br /&gt;
* WE: ~2 = 1199.7542{{c}}, ~17/12 = 605.2674{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3613{{c}}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 585.116&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 111, 224 }}
Map: [&amp;lt;1 5 15 15 2 -8 -12|, &amp;lt;0 -7 -26 -25 3 24 33|]&lt;br /&gt;
 
EDOs: 39, 41, 80, 121&lt;br /&gt;
Badness (Sintel): 1.12
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc14"&gt;&lt;a name="Pluto-19-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;19-limit&lt;/h2&gt;
=== 23-limit ===
Commas: 190/189, 256/255, 325/324, 352/351, 361/360, 595/594&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17.19.23
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 585.109&lt;br /&gt;
Comma list: 324/323, 400/399, 460/459, 495/494, 529/528, 540/539, 715/714
&lt;br /&gt;
 
Map: [&amp;lt;1 5 15 15 2 -8 -12 14|, &amp;lt;0 -7 -26 -25 3 24 33 -20|]&lt;br /&gt;
Mapping: {{mapping| 1 -8 -34 -32 8 -19 -6 -24 2 | 0 19 72 69 -9 45 20 56 5 }}
EDOs: 39, 41, 80, 121, 564, 685&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:
&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc15"&gt;&lt;a name="Pluto-Plutino"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;Plutino&lt;/h2&gt;
* WE: ~2 = 1199.8733{{c}}, ~17/12 = 605.2946{{c}}
Commas: 100/99, 245/242, 16875/16807&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3575{{c}}
&lt;br /&gt;
 
POTE generator: ~7/5 = 585.283&lt;br /&gt;
{{Optimal ET sequence|legend=0| 111, 224 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 5 15 15 22|, &amp;lt;0 -7 -26 -25 -38|]&lt;br /&gt;
Badness (Sintel): 1.26
EDOs: 41, 123c&lt;br /&gt;
 
Badness: 0.0580&lt;br /&gt;
=== 29-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17.19.23.29
&lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc16"&gt;&lt;a name="Pluto-Plutino-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt;13-limit&lt;/h3&gt;
 
Commas: 100/99, 196/195, 245/242, 729/728&lt;br /&gt;
Comma list: 290/289, 324/323, 400/399, 460/459, 495/494, 529/528, 540/539, 715/714
&lt;br /&gt;
 
POTE generator: ~7/5 = 585.232&lt;br /&gt;
Mapping: {{mapping| 1 -8 -34 -32 8 -19 -6 -24 2 21 | 0 19 72 69 -9 45 20 56 5 -32 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 5 15 15 22 12|, &amp;lt;0 -7 -26 -25 -38 -17|]&lt;br /&gt;
Optimal tunings:  
EDOs: 41, 121ef&lt;br /&gt;
* WE: ~2 = 1199.9442{{c}}, ~17/12 = 605.3263{{c}}
Badness: 0.0402&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3541{{c}}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:34:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc17"&gt;&lt;a name="Eris"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:34 --&gt;Eris&lt;/h1&gt;
{{Optimal ET sequence|legend=0| 111, 113, 224 }}
Commas: 16875/16807, 65625/65536&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.41
POTE generator: ~60/49 = 348.216&lt;br /&gt;
 
&lt;br /&gt;
=== 31-limit ===
Map: [&amp;lt;1 10 0 6|, &amp;lt;0 -29 8 -11|]&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17.19.23.29.31
Wedgie: &amp;lt;&amp;lt;29 -8 11 -80 -64 48}}&lt;br /&gt;
 
EDOs: 31, 131, 162, 193, 224, 1823cd, 2271cd&lt;br /&gt;
Comma list: 290/289, 324/323, 400/399, 435/434, 460/459, 495/494, 528/527, 540/539, 715/714
Badness: 0.0747&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 -8 -34 -32 8 -19 -6 -24 2 21 10 | 0 19 72 69 -9 45 20 56 5 -32 -10 }}
&lt;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc18"&gt;&lt;a name="Eris-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;11-limit&lt;/h2&gt;
 
Commas: 540/539, 1375/1372, 65625/65536&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1199.9100{{c}}, ~17/12 = 605.3107{{c}}
POTE generator: ~11/9 = 348.219&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3556{{c}}
&lt;br /&gt;
 
Map: [&amp;lt;1 10 0 6 20|, &amp;lt;0 -29 8 -11 -57|]&lt;br /&gt;
{{Optimal ET sequence|legend=0| 111, 113, 224 }}
EDOs: 31, 193, 224, 703, 927d, 1151cd&lt;br /&gt;
 
Badness: 0.0276&lt;br /&gt;
Badness (Sintel): 1.42
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:38:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc19"&gt;&lt;a name="Mirkat"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:38 --&gt;Mirkat&lt;/h1&gt;
=== 37-limit ===
Commas: 16875/16807, 19683/19600&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37
&lt;br /&gt;
 
POTE generator: ~10/9 = 183.539&lt;br /&gt;
Comma list: 290/289, 324/323, 400/399, 435/434, 460/459, 495/494, 528/527, 540/539, 667/666, 715/714
&lt;br /&gt;
 
Map: [&amp;lt;3 2 1 2|, &amp;lt;0 6 13 14|]&lt;br /&gt;
Mapping: {{mapping| 1 -8 -34 -32 8 -19 -6 -24 2 21 10 38 | 0 19 72 69 -9 45 20 56 5 -32 -10 -65 }}
Wedgie: &amp;lt;&amp;lt;18 39 42 20 16 -12||&lt;br /&gt;
 
EDOs: 33, 39, 72, 111, 183, 255&lt;br /&gt;
Optimal tunings:
Badness: 0.0594&lt;br /&gt;
* WE: ~2 = 1199.9087{{c}}, ~17/12 = 605.3101{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3559{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:40:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc20"&gt;&lt;a name="Mirkat-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:40 --&gt;11-limit&lt;/h2&gt;
 
Commas: 540/539, 1375/1372, 8019/8000&lt;br /&gt;
{{Optimal ET sequence|legend=0| 111, 113, 224 }}
&lt;br /&gt;
 
POTE generator: ~10/9 = 183.528&lt;br /&gt;
Badness (Sintel): 1.56
&lt;br /&gt;
 
Map: [&amp;lt;3 2 1 2 9|, &amp;lt;0 6 13 14 3|]&lt;br /&gt;
=== 41-limit ===
EDOs: 33, 39, 72, 183, 255&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41
Badness: 0.0221&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 290/289, 324/323, 400/399, 435/434, 460/459, 495/494, 528/527, 533/532, 540/539, 575/574, 667/666
&lt;!-- ws:start:WikiTextHeadingRule:42:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc21"&gt;&lt;a name="Mirkat-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:42 --&gt;13-limit&lt;/h2&gt;
 
Commas: 351/350, 540/539, 676/675, 1375/1372&lt;br /&gt;
Mapping: {{mapping| 1 -8 -34 -32 8 -19 -6 -24 2 21 10 38 -35 | 0 19 72 69 -9 45 20 56 5 -32 -10 -65 80 }}
&lt;br /&gt;
 
POTE generator: ~10/9 = 183.577&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1199.9179{{c}}, ~17/12 = 605.3156{{c}}
Map: [&amp;lt;3 2 1 2 9 1|, &amp;lt;0 6 13 14 3 22|]&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3567{{c}}
EDOs: 33, 39, 72, 111, 183&lt;br /&gt;
 
Badness: 0.0186&lt;br /&gt;
{{Optimal ET sequence|legend=0| 111, 113, 224 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:44:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc22"&gt;&lt;a name="Subsemifourth"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:44 --&gt;Subsemifourth&lt;/h1&gt;
Badness (Sintel): 1.57
Commas: 16875/16807, 26873856/26796875&lt;br /&gt;
 
&lt;br /&gt;
== Hemiseptisix ==
POTE generator: ~144/125 = 244.719&lt;br /&gt;
Hemiseptisix tempers out the mirkwai comma and 95703125/95551488 (pontiqak comma) in the 7-limit, and may be described as the {{nowrap| 103 & 121 }} temperament. [[224edo]] provides an excellent tuning for 7-, 11-, and 13-limit hemiseptisix. Hemiseptisix was named by [[Xenllium]] in 2021; the name ''hemiseptisix'' refers to a half of septimal major sixth ([[12/7]]).
&lt;br /&gt;
 
Map: [&amp;lt;1 39 27 45|, &amp;lt;0 -47 -31 -53|]&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
Wedgie: &amp;lt;&amp;lt;47 31 53 -60 -48 36||&lt;br /&gt;
 
EDOs: 49, 103, 152, 255, 407 &lt;br /&gt;
[[Comma list]]: 16875/16807, 95703125/95551488
Badness: 0.1352&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=1| 1 -19 -7 -17 | 0 53 24 51 }}
&lt;!-- ws:start:WikiTextHeadingRule:46:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc23"&gt;&lt;a name="Subsemifourth-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:46 --&gt;11-limit&lt;/h2&gt;
: mapping generators: ~2, ~98/75
Commas: 540/539, 1375/1372, 234375/234256&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]]s:
POTE generator: ~121/105 = 244.719&lt;br /&gt;
* [[WE]]: ~2 = 1199.2693{{c}}, ~98/75 = 466.0801{{c}}
&lt;br /&gt;
: [[error map]]: {{val| +0.023 -0.149 -0.553 +0.866 }}
Map: [&amp;lt;1 39 27 45 56|, &amp;lt;0 -47 -31 -53 -66|]&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~98/75 = 466.0715{{c}}
EDOs: 49, 103, 152, 255, 407, 966d&lt;br /&gt;
: error map: {{val| 0.000 -0.167 -0.598 +0.819 }}
Badness: 0.0343&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 103, 121, 224 }}
&lt;!-- ws:start:WikiTextHeadingRule:48:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc24"&gt;&lt;a name="Subsemifourth-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:48 --&gt;13-limit&lt;/h2&gt;
 
Commas: 540/539, 847/845, 1375/1372, 1575/1573&lt;br /&gt;
[[Badness]] (Sintel): 4.12
&lt;br /&gt;
 
POTE generator: ~15/13 = 244.714&lt;br /&gt;
=== 11-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
Map: [&amp;lt;1 39 27 45 56 65|, &amp;lt;0 -47 -31 -53 -66 -77|]&lt;br /&gt;
 
EDOs: 103, 152f, 255, 407f, 662df&lt;br /&gt;
Comma list: 540/539, 1375/1372, 2734375/2725888
Badness: 0.0284&lt;/body&gt;&lt;/html&gt;</pre></div>
 
Mapping: {{mapping| 1 -19 -7 -17 -28 | 0 53 24 51 81 }}
 
Optimal tunings:  
* WE: ~2 = 1200.0183{{c}}, ~55/42 = 466.0767{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~55/42 = 466.0699{{c}}
 
{{Optimal ET sequence|legend=0| 103, 121, 224 }}
 
Badness (Sintel): 1.43
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 625/624, 1375/1372, 2200/2197
 
Mapping: {{mapping| 1 -19 -7 -17 -28 -13 | 0 53 24 51 81 43 }}
 
Optimal tunings:
* WE: ~2 = 1199.9784{{c}}, ~55/42 = 466.0622{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~55/42 = 466.0703{{c}}
 
{{Optimal ET sequence|legend=0| 103, 121, 224 }}
 
Badness (Sintel): 0.873
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 375/374, 540/539, 625/624, 715/714, 2200/2197
 
Mapping: {{mapping| 1 -19 -7 -17 -28 -13 -13 | 0 53 24 51 81 43 44 }}
 
Optimal tunings:  
* WE: ~2 = 1199.8544{{c}}, ~17/13 = 466.0174{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/13 = 466.0718{{c}}
 
{{Optimal ET sequence|legend=0| 103, 121, 224 }}
 
Badness (Sintel): 0.948
 
== Browser ==
{{See also| Sensipent family }}
 
Named by [[Xenllium]] in 2022, browser may be described as the {{nowrap| 103 & 111 }} temperament.
 
This can also be considered a [[non-over-1 temperament]], with considerable scope for harmony in the 2.5/3.7/3.11/3.13/3.17/3 subgroup with mos scales of 8, 15, 23, and 31 notes despite no harmonics from the root. It can be considered a detemperament of 8d-et, with a generator very slightly flat of 7\8.  
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 16875/16807, 78732/78125
 
{{Mapping|legend=1| 1 -29 -37 -47 | 0 35 45 57 }}
: mapping generators: ~2, ~90/49
 
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9313{{c}}, ~90/49 = 1048.5414{{c}}
: [[error map]]: {{val| -0.069 -1.013 +0.592 +1.264 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~90/49 = 1048.5998{{c}}
: error map: {{val| 0.000 -0.962 +0.677 +1.362 }}
 
{{Optimal ET sequence|legend=1| 103, 111, 214 }}
 
[[Badness]] (Sintel): 4.57
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 1375/1372, 78732/78125
 
Mapping: {{mapping| 1 -29 -37 -47 -28 | 0 35 45 57 36 }}
 
Optimal tunings:
* WE: ~2 = 1200.1344{{c}}, ~11/6 = 1048.7124{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1048.5981{{c}}
 
{{Optimal ET sequence|legend=0| 103, 214 }}
 
Badness (Sintel): 1.91
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 540/539, 847/845, 1375/1372
 
Mapping: {{mapping| 1 -29 -37 -47 -28 -33 | 0 35 45 57 36 42 }}
 
Optimal tunings:  
* WE: ~2 = 1200.1344{{c}}, ~11/6 = 1048.7124{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1048.5984{{c}}
 
{{Optimal ET sequence|legend=0| 103, 111, 214 }}
 
Badness (Sintel): 1.19
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 351/350, 540/539, 561/560, 715/714, 847/845
 
Mapping: {{mapping| 1 -29 -37 -47 -28 -33 -23 | 0 35 45 57 36 42 31 }}
 
Optimal tunings:  
* WE: ~2 = 1199.9191{{c}}, ~11/6 = 1048.5324{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1048.6014{{c}}
 
{{Optimal ET sequence|legend=0| 103, 111, 214 }}
 
Badness (Sintel): 1.04
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 324/323, 351/350, 456/455, 495/494, 540/539, 715/714
 
Mapping: {{mapping| 1 -29 -37 -47 -28 -33 -23 -91 | 0 35 45 57 36 42 31 109 }}
 
Optimal tunings:  
* WE: ~2 = 1199.9145{{c}}, ~11/6 = 1048.5290{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1048.6021{{c}}
 
{{Optimal ET sequence|legend=0| 103h, 111, 214 }}
 
Badness (Sintel): 1.07
 
== Grazer ==
Named by [[Xenllium]] in 2022, grazer may be described as the {{nowrap| 113 & 121 }} temperament.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 16875/16807, 1071875/1062882
 
{{Mapping|legend=1| 1 -3 -4 -5 | 0 37 51 63 }}
: mapping generators: ~2, ~49/45
 
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0310{{c}}, ~49/45 = 148.7229{{c}}
: [[error map]]: {{val| +0.031 +0.700 -1.561 +0.563 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/45 = 148.7198{{c}}
: error map: {{val| 0.000 +0.676 -1.606 +0.519 }}
 
{{Optimal ET sequence|legend=1| 113, 121, 234 }}
 
[[Badness]] (Sintel): 5.50
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 1375/1372, 218750/216513
 
Mapping: {{mapping| 1 -3 -4 -5 -1 | 0 37 51 63 36 }}
 
Optimal tunings:
* WE: ~2 = 1199.7242{{c}}, ~12/11 = 148.6946{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 148.7230{{c}}
 
{{Optimal ET sequence|legend=0| 113, 121, 234 }}
 
Badness (Sintel): 2.51
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 364/363, 540/539, 2200/2197
 
Mapping: {{mapping| 1 -3 -4 -5 -1 -2 | 0 37 51 63 36 46 }}
 
Optimal tunings:
* WE: ~2 = 1199.7257{{c}}, ~12/11 = 148.6947{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 148.7230{{c}}
 
{{Optimal ET sequence|legend=0| 113, 121, 234 }}
 
Badness (Sintel): 1.50
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 325/324, 364/363, 540/539, 595/594, 2000/1989
 
Mapping: {{mapping| 1 -3 -4 -5 -1 -2 0 | 0 37 51 63 36 46 33 }}
 
Optimal tunings:  
* WE: ~2 = 1199.5690{{c}}, ~12/11 = 148.6815{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 148.7267{{c}}
 
{{Optimal ET sequence|legend=0| 113, 121, 234g }}
 
Badness (Sintel): 1.29
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 325/324, 364/363, 400/399, 540/539, 595/594, 665/663
 
Mapping: {{mapping| 1 -3 -4 -5 -1 -2 0 4 | 0 37 51 63 36 46 33 2 }}
 
Optimal tunings:  
* WE: ~2 = 1199.7269{{c}}, ~12/11 = 148.6928{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 148.7227{{c}}
 
{{Optimal ET sequence|legend=0| 113, 121, 234g }}
 
Badness (Sintel): 1.37
 
[[Category:Temperament clans]]
[[Category:Mirkwai clan| ]] <!-- main article -->
[[Category:Rank 2]]

Latest revision as of 10:18, 6 June 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 canopic clan of temperaments tempers out the canopic comma (monzo: [0 3 4 -5⟩, ratio: 16875/16807), a no-twos comma.

Canopus

Main article: Canopus

Subgroup: 3.5.7

Comma list: 16875/16807

Subgroup-val mapping: [⟨1 -2 -1], ⟨0 5 4]]

mapping generators: ~3, ~15/7

Optimal tunings:

  • WE: ~3 = 1901.7826 ¢, ~15/7 = 1317.8771 ¢
error map: ⟨+1.785 -0.771 -2.248]
  • CWE: ~3 = 1901.9550 ¢, ~15/7 = 1317.9686 ¢
error map: ⟨0.000 -0.381 +1.093]

Optimal ET sequence: b13, b62, b75, b88, b101, b114, b355, b469, b583, b697

Badness (Sintel): 0.0996

Overview to extensions

The full 7-limit extensions' relation to canopus is clearer if the mapping is normalized in terms of 3.5.7.2. In fact, the strong extensions are nusecond and octoid. These temperaments are distributed into different temperament collection pages.

The others are weak extensions. Mirkat tempers out 19683/19600, splitting the generator in two with a semitwelfth period. Sqrtphi tempers out 15625/15552, splitting the period in six. Semisept tempers out 1728/1715 and 3136/3125, splitting the generator in six. Miracle tempers out 225/224. Pluto tempers out 4000/3969. These split the generator in five. Kwai tempers out 5120/5103, splitting the generator in ten. Quanharuk tempers out 32805/32768, splitting the generator in three with a 1/5-twelfth period. Grendel tempers out 6144/6125, splitting the generator in eleven. Finally, eris tempers out 65625/65536, splitting the generator in sixteen.

Members of the clan discussed elsewhere are:

For no-twos extensions, see No-twos subgroup temperaments #Canopus.

Considered below are mirkat, eris, subsemifourth, septendesemi, gaster, hemiseptisix, browser, and grazer, in the order of increasing badness.

Mirkat

Mirkat tempers out 19683/19600, the cataharry comma, as well as 250047/250000, the landscape comma, and may be described as the 72 & 111 temperament with a ploidacot signature of triploid alpha-hexacot.

Subgroup: 2.3.5.7

Comma list: 16875/16807, 19683/19600

Mapping: [⟨3 2 1 2], ⟨0 6 13 14]]

mapping generators: ~63/50, ~10/9

Optimal tunings:

  • WE: ~63/50 = 400.0277 ¢, ~10/9 = 183.5515 ¢
error map: ⟨+0.083 -0.591 -0.117 +0.950]
  • CWE: ~63/50 = 400.0000 ¢, ~10/9 = 183.5470 ¢
error map: ⟨0.000 -0.673 -0.203 +0.831]

Optimal ET sequence: 39d, 72, 111, 183, 255

Badness (Sintel): 1.50

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 8019/8000

Mapping: [⟨3 2 1 2 9], ⟨0 6 13 14 3]]

Optimal tunings:

  • WE: ~63/50 = 400.0463 ¢, ~10/9 = 183.5496 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~10/9 = 183.5391 ¢

Optimal ET sequence: 39d, 72, 111, 183, 255

Badness (Sintel): 0.731

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 676/675, 1375/1372

Mapping: [⟨3 2 1 2 9 1], ⟨0 6 13 14 3 22]]

Optimal tunings:

  • WE: ~63/50 = 400.0245 ¢, ~10/9 = 183.5885 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~10/9 = 183.5825 ¢

Optimal ET sequence: 39df, 72, 111, 183

Badness (Sintel): 0.770

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 442/441, 540/539, 561/560, 715/714

Mapping: [⟨3 2 1 2 9 1 4], ⟨0 6 13 14 3 22 18]]

Optimal tunings:

  • WE: ~34/27 = 400.0257 ¢, ~10/9 = 183.5906 ¢
  • CWE: ~34/27 = 400.0000 ¢, ~10/9 = 183.5843 ¢

Optimal ET sequence: 39dfg, 72, 111, 183

Badness (Sintel): 0.600

Eris

Eris tempers out 65625/65536, the horwell comma, and may be described as the 31 & 224 temperament. The 2.5.7 subgroup restriction of this temperament is exodia.

Subgroup: 2.3.5.7

Comma list: 16875/16807, 65625/65536

Mapping: [⟨1 -19 8 -5], ⟨0 29 -8 11]]

mapping generators: ~2, ~49/30

Optimal tunings:

  • WE: ~2 = 1200.0256 ¢, ~49/30 = 851.8023 ¢
error map: ⟨+0.026 -0.173 -0.528 +0.872]
  • CWE: ~2 = 1200.0000 ¢, ~49/30 = 851.7845 ¢
error map: ⟨0.000 -0.204 -0.590 +0.804]

Optimal ET sequence: 31, 131, 162, 193, 224

Badness (Sintel): 1.89

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 65625/65536

Mapping: [⟨1 -19 8 -5 -37], ⟨0 29 -8 11 57]]

Optimal tunings:

  • WE: ~2 = 1200.0218 ¢, ~18/11 = 851.7963 ¢
  • CWE: ~2 = 1200.0000 ¢, ~18/11 = 851.7812 ¢

Optimal ET sequence: 31, …, 193, 224, 703, 927d

Badness (Sintel): 0.913

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624, 1375/1372, 4096/4095

Mapping: [⟨1 -19 8 -5 -37 47], ⟨0 29 -8 11 57 -61]]

Optimal tuning:

  • WE ~2 = 1199.9623 ¢, ~18/11 = 851.7598 ¢
  • CWE ~2 = 1200.0000 ¢, ~18/11 = 851.7865 ¢

Optimal ET sequence: 31, 193, 224

Badness (Sintel): 1.04

Subsemifourth

Subgroup: 2.3.5.7

Comma list: 16875/16807, 26873856/26796875

Mapping: [⟨1 -8 -4 -8], ⟨0 47 31 53]]

mapping generators: ~2, ~144/125

Optimal tunings:

  • WE: ~2 = 1199.9182 ¢, ~144/125 = 244.7020 ¢
error map: ⟨-0.082 -0.305 -0.223 +1.037]
  • CWE: ~2 = 1200.0000 ¢, ~144/125 = 244.7172 ¢
error map: ⟨0.000 -0.248 -0.082 +1.184]

Optimal ET sequence: 49, 103, 152, 255, 407

Badness (Sintel): 3.42

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 234375/234256

Mapping: [⟨1 -8 -4 -8 -10], ⟨0 47 31 53 66]]

Optimal tunings:

  • WE: ~2 = 1199.9229 ¢, ~121/105 = 244.7033 ¢
  • CWE: ~2 = 1200.0000 ¢, ~121/105 = 244.7175 ¢

Optimal ET sequence: 49, 103, 152, 255, 407

Badness (Sintel): 1.13

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 847/845, 1375/1372, 1575/1573

Mapping: [⟨1 -8 -4 -8 -10 -12], ⟨0 0 47 31 53 66 77]]

Optimal tunings:

  • WE: ~2 = 1199.9003 ¢, ~15/13 = 244.6932 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/13 = 244.7116 ¢

Optimal ET sequence: 49f, 103, 152f, 255, 407f

Badness (Sintel): 1.17

Septendesemi

Septendesemi tempers out the mirkwai comma and 1959552/1953125 (parkleiness comma) in the 7-limit, and may be described as the 80 & 103 temperament. 183edo provides an excellent tuning for 7-, 11-, 13-, and 17-limit septendesemi. Septendesemi was named by Xenllium in 2021; the name septendesemi refers to a septendecimal semitone (17/16).

Subgroup: 2.3.5.7

Comma list: 16875/16807, 1959552/1953125

Mapping: [⟨1 -2 -1 -2], ⟨0 41 38 55]]

mapping generators: ~2, ~343/324

Optimal tunings:

  • WE: ~2 = 1199.8649 ¢, ~343/324 = 104.9046 ¢
error map: ⟨-0.135 -0.597 +0.195 +1.196]
  • CWE: ~2 = 1200.0000 ¢, ~343/324 = 104.9134 ¢
error map: ⟨0.000 -0.506 +0.395 +1.410]

Optimal ET sequence: 80, 103, 183

Badness (Sintel): 3.71

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 43923/43750

Mapping: [⟨1 -2 -1 -2 -1], ⟨0 41 38 55 51]]

Optimal tunings:

  • WE: ~2 = 1199.9327 ¢, ~35/33 = 104.9100 ¢
  • CWE: ~2 = 1200.0000 ¢, ~35/33 = 104.9144 ¢

Optimal ET sequence: 80, 103, 183

Badness (Sintel): 1.37

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 1375/1372, 4225/4224

Mapping: [⟨1 -2 -1 -2 -1 3], ⟨0 41 38 55 51 8]]

Optimal tunings:

  • WE: ~2 = 1200.1082 ¢, ~35/33 = 104.9170 ¢
  • CWE: ~2 = 1200.0000 ¢, ~35/33 = 104.9094 ¢

Optimal ET sequence: 80, 103, 183, 469f

Badness (Sintel): 1.15

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 540/539, 561/560, 715/714, 4225/4224

Mapping: [⟨1 -2 -1 -2 -1 3 4], ⟨0 41 38 55 51 8 1]]

Optimal tunings:

  • WE: ~2 = 1200.0758 ¢, ~17/16 = 104.9158 ¢
  • CWE: ~2 = 1200.0000 ¢, ~17/16 = 104.9101 ¢

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~17/16 = 104.909 ¢

Optimal ET sequence: 80, 103, 183, 469f

Badness (Sintel): 1.03

Gaster

For the 5-limit version, see Very high accuracy temperaments #Gaster.
Main article: Gaster temperament

Gaster tempers out [-70 72 -19⟩ in the 5-limit, mirkwai comma (16875/16807) and scheme comma (14348907/14336000) in the 7-limit, and may be described as the 111 & 113 temperament.

It was named by Xenllium in 2022; the word "gaster" means abdomen or stomach, but also a restructuring of the words "gassormic tritone", which is a generator of this temperament. This temperament is sufficient to obtain high prime limit harmonics like a stomach, so that patent vals 111, 113 and 224 support it even in the 41-limit.

Subgroup: 2.3.5.7

Comma list: 16875/16807, 14348907/14336000

Mapping: [⟨1 -8 -34 -32], ⟨0 19 72 69]]

mapping generators: ~2, ~567/400

Optimal tunings:

  • WE: ~2 = 1199.9920 ¢, ~567/400 = 605.3546 ¢
error map: ⟨-0.008 -0.152 -0.506 +0.902]
  • CWE: ~2 = 1200.0000 ¢, ~567/400 = 605.3586 ¢
error map: ⟨0.000 -0.142 -0.497 +0.915]

Optimal ET sequence: 111, 224

Badness (Sintel): 3.91

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 14348907/14336000

Mapping: [⟨1 -8 -34 -32 8], ⟨0 19 72 69 -9]]

Optimal tunings:

  • WE: ~2 = 1199.9387 ¢, ~363/256 = 605.3300 ¢
  • CWE: ~2 = 1200.0000 ¢, ~363/256 = 605.3603 ¢

Optimal ET sequence: 111, 224, 783d

Badness (Sintel): 1.79

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 1375/1372, 2200/2197

Mapping: [⟨1 -8 -34 -32 8 -19], ⟨0 19 72 69 -9 45]]

Optimal tunings:

  • WE: ~2 = 1199.9154 ¢, ~78/55 = 605.3183 ¢
  • CWE: ~2 = 1200.0000 ¢, ~78/55 = 605.3601 ¢

Optimal ET sequence: 111, 224, 783df

Badness (Sintel): 1.03

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 540/539, 715/714, 729/728, 936/935, 2200/2197

Mapping: [⟨1 -8 -34 -32 8 -19 -6], ⟨0 19 72 69 -9 45 20]]

Optimal tunings:

  • WE: ~2 = 1199.8076 ¢, ~17/12 = 605.2674 ¢
  • CWE: ~2 = 1200.0000 ¢, ~17/12 = 605.3626 ¢

Optimal ET sequence: 111, 224, 559dgg

Badness (Sintel): 1.09

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 324/323, 400/399, 495/494, 540/539, 715/714, 1445/1444

Mapping: [⟨1 -8 -34 -32 8 -19 -6 -24], ⟨0 19 72 69 -9 45 20 56]]

Optimal tunings:

  • WE: ~2 = 1199.7542 ¢, ~17/12 = 605.2674 ¢
  • CWE: ~2 = 1200.0000 ¢, ~17/12 = 605.3613 ¢

Optimal ET sequence: 111, 224

Badness (Sintel): 1.12

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 324/323, 400/399, 460/459, 495/494, 529/528, 540/539, 715/714

Mapping: [⟨1 -8 -34 -32 8 -19 -6 -24 2], ⟨0 19 72 69 -9 45 20 56 5]]

Optimal tunings:

  • WE: ~2 = 1199.8733 ¢, ~17/12 = 605.2946 ¢
  • CWE: ~2 = 1200.0000 ¢, ~17/12 = 605.3575 ¢

Optimal ET sequence: 111, 224

Badness (Sintel): 1.26

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 290/289, 324/323, 400/399, 460/459, 495/494, 529/528, 540/539, 715/714

Mapping: [⟨1 -8 -34 -32 8 -19 -6 -24 2 21], ⟨0 19 72 69 -9 45 20 56 5 -32]]

Optimal tunings:

  • WE: ~2 = 1199.9442 ¢, ~17/12 = 605.3263 ¢
  • CWE: ~2 = 1200.0000 ¢, ~17/12 = 605.3541 ¢

Optimal ET sequence: 111, 113, 224

Badness (Sintel): 1.41

31-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31

Comma list: 290/289, 324/323, 400/399, 435/434, 460/459, 495/494, 528/527, 540/539, 715/714

Mapping: [⟨1 -8 -34 -32 8 -19 -6 -24 2 21 10], ⟨0 19 72 69 -9 45 20 56 5 -32 -10]]

Optimal tunings:

  • WE: ~2 = 1199.9100 ¢, ~17/12 = 605.3107 ¢
  • CWE: ~2 = 1200.0000 ¢, ~17/12 = 605.3556 ¢

Optimal ET sequence: 111, 113, 224

Badness (Sintel): 1.42

37-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37

Comma list: 290/289, 324/323, 400/399, 435/434, 460/459, 495/494, 528/527, 540/539, 667/666, 715/714

Mapping: [⟨1 -8 -34 -32 8 -19 -6 -24 2 21 10 38], ⟨0 19 72 69 -9 45 20 56 5 -32 -10 -65]]

Optimal tunings:

  • WE: ~2 = 1199.9087 ¢, ~17/12 = 605.3101 ¢
  • CWE: ~2 = 1200.0000 ¢, ~17/12 = 605.3559 ¢

Optimal ET sequence: 111, 113, 224

Badness (Sintel): 1.56

41-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41

Comma list: 290/289, 324/323, 400/399, 435/434, 460/459, 495/494, 528/527, 533/532, 540/539, 575/574, 667/666

Mapping: [⟨1 -8 -34 -32 8 -19 -6 -24 2 21 10 38 -35], ⟨0 19 72 69 -9 45 20 56 5 -32 -10 -65 80]]

Optimal tunings:

  • WE: ~2 = 1199.9179 ¢, ~17/12 = 605.3156 ¢
  • CWE: ~2 = 1200.0000 ¢, ~17/12 = 605.3567 ¢

Optimal ET sequence: 111, 113, 224

Badness (Sintel): 1.57

Hemiseptisix

Hemiseptisix tempers out the mirkwai comma and 95703125/95551488 (pontiqak comma) in the 7-limit, and may be described as the 103 & 121 temperament. 224edo provides an excellent tuning for 7-, 11-, and 13-limit hemiseptisix. Hemiseptisix was named by Xenllium in 2021; the name hemiseptisix refers to a half of septimal major sixth (12/7).

Subgroup: 2.3.5.7

Comma list: 16875/16807, 95703125/95551488

Mapping: [⟨1 -19 -7 -17], ⟨0 53 24 51]]

mapping generators: ~2, ~98/75

Optimal tunings:

  • WE: ~2 = 1199.2693 ¢, ~98/75 = 466.0801 ¢
error map: ⟨+0.023 -0.149 -0.553 +0.866]
  • CWE: ~2 = 1200.0000 ¢, ~98/75 = 466.0715 ¢
error map: ⟨0.000 -0.167 -0.598 +0.819]

Optimal ET sequence: 103, 121, 224

Badness (Sintel): 4.12

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 2734375/2725888

Mapping: [⟨1 -19 -7 -17 -28], ⟨0 53 24 51 81]]

Optimal tunings:

  • WE: ~2 = 1200.0183 ¢, ~55/42 = 466.0767 ¢
  • CWE: ~2 = 1200.0000 ¢, ~55/42 = 466.0699 ¢

Optimal ET sequence: 103, 121, 224

Badness (Sintel): 1.43

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624, 1375/1372, 2200/2197

Mapping: [⟨1 -19 -7 -17 -28 -13], ⟨0 53 24 51 81 43]]

Optimal tunings:

  • WE: ~2 = 1199.9784 ¢, ~55/42 = 466.0622 ¢
  • CWE: ~2 = 1200.0000 ¢, ~55/42 = 466.0703 ¢

Optimal ET sequence: 103, 121, 224

Badness (Sintel): 0.873

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 375/374, 540/539, 625/624, 715/714, 2200/2197

Mapping: [⟨1 -19 -7 -17 -28 -13 -13], ⟨0 53 24 51 81 43 44]]

Optimal tunings:

  • WE: ~2 = 1199.8544 ¢, ~17/13 = 466.0174 ¢
  • CWE: ~2 = 1200.0000 ¢, ~17/13 = 466.0718 ¢

Optimal ET sequence: 103, 121, 224

Badness (Sintel): 0.948

Browser

See also: Sensipent family

Named by Xenllium in 2022, browser may be described as the 103 & 111 temperament.

This can also be considered a non-over-1 temperament, with considerable scope for harmony in the 2.5/3.7/3.11/3.13/3.17/3 subgroup with mos scales of 8, 15, 23, and 31 notes despite no harmonics from the root. It can be considered a detemperament of 8d-et, with a generator very slightly flat of 7\8.

Subgroup: 2.3.5.7

Comma list: 16875/16807, 78732/78125

Mapping: [⟨1 -29 -37 -47], ⟨0 35 45 57]]

mapping generators: ~2, ~90/49

Optimal tunings:

  • WE: ~2 = 1199.9313 ¢, ~90/49 = 1048.5414 ¢
error map: ⟨-0.069 -1.013 +0.592 +1.264]
  • CWE: ~2 = 1200.0000 ¢, ~90/49 = 1048.5998 ¢
error map: ⟨0.000 -0.962 +0.677 +1.362]

Optimal ET sequence: 103, 111, 214

Badness (Sintel): 4.57

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 78732/78125

Mapping: [⟨1 -29 -37 -47 -28], ⟨0 35 45 57 36]]

Optimal tunings:

  • WE: ~2 = 1200.1344 ¢, ~11/6 = 1048.7124 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/6 = 1048.5981 ¢

Optimal ET sequence: 103, 214

Badness (Sintel): 1.91

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 847/845, 1375/1372

Mapping: [⟨1 -29 -37 -47 -28 -33], ⟨0 35 45 57 36 42]]

Optimal tunings:

  • WE: ~2 = 1200.1344 ¢, ~11/6 = 1048.7124 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/6 = 1048.5984 ¢

Optimal ET sequence: 103, 111, 214

Badness (Sintel): 1.19

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 540/539, 561/560, 715/714, 847/845

Mapping: [⟨1 -29 -37 -47 -28 -33 -23], ⟨0 35 45 57 36 42 31]]

Optimal tunings:

  • WE: ~2 = 1199.9191 ¢, ~11/6 = 1048.5324 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/6 = 1048.6014 ¢

Optimal ET sequence: 103, 111, 214

Badness (Sintel): 1.04

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 324/323, 351/350, 456/455, 495/494, 540/539, 715/714

Mapping: [⟨1 -29 -37 -47 -28 -33 -23 -91], ⟨0 35 45 57 36 42 31 109]]

Optimal tunings:

  • WE: ~2 = 1199.9145 ¢, ~11/6 = 1048.5290 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/6 = 1048.6021 ¢

Optimal ET sequence: 103h, 111, 214

Badness (Sintel): 1.07

Grazer

Named by Xenllium in 2022, grazer may be described as the 113 & 121 temperament.

Subgroup: 2.3.5.7

Comma list: 16875/16807, 1071875/1062882

Mapping: [⟨1 -3 -4 -5], ⟨0 37 51 63]]

mapping generators: ~2, ~49/45

Optimal tunings:

  • WE: ~2 = 1200.0310 ¢, ~49/45 = 148.7229 ¢
error map: ⟨+0.031 +0.700 -1.561 +0.563]
  • CWE: ~2 = 1200.0000 ¢, ~49/45 = 148.7198 ¢
error map: ⟨0.000 +0.676 -1.606 +0.519]

Optimal ET sequence: 113, 121, 234

Badness (Sintel): 5.50

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 218750/216513

Mapping: [⟨1 -3 -4 -5 -1], ⟨0 37 51 63 36]]

Optimal tunings:

  • WE: ~2 = 1199.7242 ¢, ~12/11 = 148.6946 ¢
  • CWE: ~2 = 1200.0000 ¢, ~12/11 = 148.7230 ¢

Optimal ET sequence: 113, 121, 234

Badness (Sintel): 2.51

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 540/539, 2200/2197

Mapping: [⟨1 -3 -4 -5 -1 -2], ⟨0 37 51 63 36 46]]

Optimal tunings:

  • WE: ~2 = 1199.7257 ¢, ~12/11 = 148.6947 ¢
  • CWE: ~2 = 1200.0000 ¢, ~12/11 = 148.7230 ¢

Optimal ET sequence: 113, 121, 234

Badness (Sintel): 1.50

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 325/324, 364/363, 540/539, 595/594, 2000/1989

Mapping: [⟨1 -3 -4 -5 -1 -2 0], ⟨0 37 51 63 36 46 33]]

Optimal tunings:

  • WE: ~2 = 1199.5690 ¢, ~12/11 = 148.6815 ¢
  • CWE: ~2 = 1200.0000 ¢, ~12/11 = 148.7267 ¢

Optimal ET sequence: 113, 121, 234g

Badness (Sintel): 1.29

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 325/324, 364/363, 400/399, 540/539, 595/594, 665/663

Mapping: [⟨1 -3 -4 -5 -1 -2 0 4], ⟨0 37 51 63 36 46 33 2]]

Optimal tunings:

  • WE: ~2 = 1199.7269 ¢, ~12/11 = 148.6928 ¢
  • CWE: ~2 = 1200.0000 ¢, ~12/11 = 148.7227 ¢

Optimal ET sequence: 113, 121, 234g

Badness (Sintel): 1.37

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