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**Imported revision 319300004 - Original comment: **
m Restore the category of the corresponding temp
 
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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>2012-04-10 22:15:04 UTC</tt>.<br>
: The original revision id was <tt>319300004</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]]


This tempers out 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=3| 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 clan and collection pages.
* ''[[Nusecond]]'' (+126/125) → [[Starling temperaments #Nusecond|Starling temperaments]]
* [[Octoid]] (+4375/4374) → [[Stearnsmic clan #Octoid|Stearnsmic clan]]


[[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) → [[Aberschismic temperaments #Kwai|Aberschismic 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]]
* ''[[Browser]]'' (+78732/78125) → [[Sensipent family #Browser|Sensipent family]]
* ''[[Quintiquart]]'' (+390625000/387420489) → [[Quartonic family #Quintiquart|Quartonic family]]


==17-limit==
For ''no-2's'' 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, 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|legend=0| 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|legend=0| 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|legend=0| 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


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


POTE generator: ~60/49 = 348.219
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 10 0 6|, &lt;0 -29 8 -11|]
{{Optimal ET sequence|legend=0| 31, …, 193, 224, 703, 927d }}
EDOs: 31, 131, 162, 193, 224
Badness: 0.0747


==11-limit==
Badness (Sintel): 0.913
Commas: 540/539, 1375/1372, 65625/65536


POTE generator: ~11/9 = 348.219
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 10 0 6 20|, &lt;0 -29 8 -11 -57|]
Comma list: 540/539, 625/624, 1375/1372, 4096/4095
EDOs: 31, 193, 224, 703
Badness: 0.0276
=Mirkat=
Commas: 16875/16807, 19683/19600


POTE generator: ~10/9 = 183.539
{{Mapping|legend=0| 1 -19 8 -5 -37 47 | 0 29 -8 11 57 -61 }}


Map: [&lt;3 2 1 2|, &lt;0 6 13 14|]
Optimal tuning:  
Wedgie: &lt;&lt;18 39 42 20 16 -12||
* WE ~2 = 1199.9623{{c}}, ~18/11 = 851.7598{{c}}
EDOs: 33, 39, 72, 111, 183, 255
* CWE ~2 = 1200.0000{{c}}, ~18/11 = 851.7865{{c}}
Badness: 0.0594


==11-limit==
{{Optimal ET sequence|legend=0| 31, 193, 224 }}
Commas: 540/539, 1375/1372, 8019/8000


POTE generator: ~10/9 = 183.528
Badness (Sintel): 1.04


Map: [&lt;3 2 1 2 9|, &lt;0 6 13 14 3|]
== Subsemifourth ==
EDOs: 33, 39, 72, 183, 255
[[Subgroup]]: 2.3.5.7
Badness: 0.0221


==13-limit==
[[Comma list]]: 16875/16807, 26873856/26796875
Commas: 351/350, 540/539, 676/675, 1375/1372


POTE generator: ~10/9 = 183.577
{{Mapping|legend=1| 1 -8 -4 -8 | 0 47 31 53 }}
: mapping generators: ~2, ~144/125


Map: [&lt;3 2 1 2 9 1|, &lt;0 6 13 14 3 22|]
[[Optimal tuning]]s:  
EDOs: 33, 39, 72, 111, 183
* [[WE]]: ~2 = 1199.9182{{c}}, ~144/125 = 244.7020{{c}}
Badness: 0.0186
: [[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 }}


=Subsemifourth=
{{Optimal ET sequence|legend=1| 49, 103, 152, 255, 407 }}
Commas: 16875/16807, 26873856/26796875


POTE generator: ~144/125 = 244.719
[[Badness]] (Sintel): 3.42


Map: [&lt;1 39 27 45|, &lt;0 -47 -31 -53|]
=== 11-limit ===
Wedgie: &lt;&lt;47 31 53 -60 -48 36||
Subgroup: 2.3.5.7.11
EDOs: 49, 103, 152, 255, 407
Badness: 0.1352


==11-limit==
Comma list: 540/539, 1375/1372, 234375/234256
Commas: 540/539, 1375/1372, 234375/234256


POTE generator: ~121/105 = 244.719
{{Mapping|legend=0| 1 -8 -4 -8 -10 | 0 47 31 53 66 }}


Map: [&lt;1 39 27 45 56|, &lt;0 -47 -31 -53 -66|]
Optimal tunings:  
EDOs: 49, 103, 152, 255, 407, 966d
* WE: ~2 = 1199.9229{{c}}, ~121/105 = 244.7033{{c}}
Badness: 0.0343
* CWE: ~2 = 1200.0000{{c}}, ~121/105 = 244.7175{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 49, 103, 152, 255, 407 }}
Commas: 540/539, 847/845, 1375/1372, 1575/1573


POTE generator: ~15/13 = 244.714
Badness (Sintel): 1.13


Map: [&lt;1 39 27 45 56 65|, &lt;0 -47 -31 -53 -66 -77|]
=== 13-limit ===
EDOs: 103, 152f, 255, 407f, 662df
Subgroup: 2.3.5.7.11.13
Badness: 0.0284


=Subneutral=
Comma list: 540/539, 847/845, 1375/1372, 1575/1573
Commas: 16875/16807, 65625/65536


POTE generator: ~60/49 = 348.216
{{Mapping|legend=0| 1 -8 -4 -8 -10 -12 | 0 0 47 31 53 66 77 }}


Map: [&lt;1 10 0 6|, &lt;0 -29 8 -11|]
Optimal tunings:  
Wedgie: &lt;&lt;29 -8 11 -80 -64 48}}
* WE: ~2 = 1199.9003{{c}}, ~15/13 = 244.6932{{c}}
EDOs: 31, 131, 162, 193, 224, 1823cd, 2271cd
* CWE: ~2 = 1200.0000{{c}}, ~15/13 = 244.7116{{c}}
Badness: 0.0747


==11-limit==
{{Optimal ET sequence|legend=0| 49f, 103, 152f, 255, 407f }}
Commas: 540/539, 1375/1372, 65625/65536


POTE generator: ~11/9 = 348.219
Badness (Sintel): 1.17


Map: [&lt;1 10 0 6 20|, &lt;0 -29 8 -11 -57|]
== Septendesemi ==
EDOs: 31, 193, 224, 703, 927d, 1151cd
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]]).
Badness: 0.0276
 
</pre></div>
[[Subgroup]]: 2.3.5.7
<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;Mirkwai clan&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:50:&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:50 --&gt;&lt;!-- ws:start:WikiTextTocRule:51: --&gt;&lt;a href="#Grendel"&gt;Grendel&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:51 --&gt;&lt;!-- ws:start:WikiTextTocRule:52: --&gt;&lt;!-- ws:end:WikiTextTocRule:52 --&gt;&lt;!-- ws:start:WikiTextTocRule:53: --&gt;&lt;!-- ws:end:WikiTextTocRule:53 --&gt;&lt;!-- ws:start:WikiTextTocRule:54: --&gt;&lt;!-- ws:end:WikiTextTocRule:54 --&gt;&lt;!-- ws:start:WikiTextTocRule:55: --&gt;&lt;!-- ws:end:WikiTextTocRule:55 --&gt;&lt;!-- ws:start:WikiTextTocRule:56: --&gt; | &lt;a href="#Kwai"&gt;Kwai&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:56 --&gt;&lt;!-- ws:start:WikiTextTocRule:57: --&gt;&lt;!-- ws:end:WikiTextTocRule:57 --&gt;&lt;!-- ws:start:WikiTextTocRule:58: --&gt;&lt;!-- ws:end:WikiTextTocRule:58 --&gt;&lt;!-- ws:start:WikiTextTocRule:59: --&gt;&lt;!-- ws:end:WikiTextTocRule:59 --&gt;&lt;!-- ws:start:WikiTextTocRule:60: --&gt;&lt;!-- ws:end:WikiTextTocRule:60 --&gt;&lt;!-- ws:start:WikiTextTocRule:61: --&gt; | &lt;a href="#Pluto"&gt;Pluto&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:61 --&gt;&lt;!-- ws:start:WikiTextTocRule:62: --&gt;&lt;!-- ws:end:WikiTextTocRule:62 --&gt;&lt;!-- ws:start:WikiTextTocRule:63: --&gt;&lt;!-- ws:end:WikiTextTocRule:63 --&gt;&lt;!-- ws:start:WikiTextTocRule:64: --&gt;&lt;!-- ws:end:WikiTextTocRule:64 --&gt;&lt;!-- ws:start:WikiTextTocRule:65: --&gt;&lt;!-- ws:end:WikiTextTocRule:65 --&gt;&lt;!-- ws:start:WikiTextTocRule:66: --&gt; | &lt;a href="#Eris"&gt;Eris&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:66 --&gt;&lt;!-- ws:start:WikiTextTocRule:67: --&gt;&lt;!-- ws:end:WikiTextTocRule:67 --&gt;&lt;!-- ws:start:WikiTextTocRule:68: --&gt; | &lt;a href="#Mirkat"&gt;Mirkat&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:68 --&gt;&lt;!-- ws:start:WikiTextTocRule:69: --&gt;&lt;!-- ws:end:WikiTextTocRule:69 --&gt;&lt;!-- ws:start:WikiTextTocRule:70: --&gt;&lt;!-- ws:end:WikiTextTocRule:70 --&gt;&lt;!-- ws:start:WikiTextTocRule:71: --&gt; | &lt;a href="#Subsemifourth"&gt;Subsemifourth&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:71 --&gt;&lt;!-- ws:start:WikiTextTocRule:72: --&gt;&lt;!-- ws:end:WikiTextTocRule:72 --&gt;&lt;!-- ws:start:WikiTextTocRule:73: --&gt;&lt;!-- ws:end:WikiTextTocRule:73 --&gt;&lt;!-- ws:start:WikiTextTocRule:74: --&gt; | &lt;a href="#Subneutral"&gt;Subneutral&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:74 --&gt;&lt;!-- ws:start:WikiTextTocRule:75: --&gt;&lt;!-- ws:end:WikiTextTocRule:75 --&gt;&lt;!-- ws:start:WikiTextTocRule:76: --&gt;
[[Comma list]]: 16875/16807, 1959552/1953125
&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;
{{Mapping|legend=1| 1 -2 -1 -2 | 0 41 38 55 }}
&lt;br /&gt;
: mapping generators: ~2, ~343/324
&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;
[[Optimal tuning]]s:  
&lt;br /&gt;
* [[WE]]: ~2 = 1199.8649{{c}}, ~343/324 = 104.9046{{c}}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 386.863&lt;br /&gt;
: [[error map]]: {{val| -0.135 -0.597 +0.195 +1.196 }}
&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~343/324 = 104.9134{{c}}
Map: [&amp;lt;1 9 2 7|, &amp;lt;0 -23 1 -13|]&lt;br /&gt;
: error map: {{val| 0.000 -0.506 +0.395 +1.410 }}
Wedgie: &amp;lt;&amp;lt;23 -1 13 -55 -44 33||&lt;br /&gt;
 
EDOs: 31, 90, 121, 152, 335&lt;br /&gt;
{{Optimal ET sequence|legend=0| 80, 103, 183 }}
&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;
[[Badness]] (Sintel): 3.71
Commas: 540/539, 1375/1372, 5632/5625&lt;br /&gt;
 
&lt;br /&gt;
=== 11-limit ===
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 386.856&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;br /&gt;
 
Map: [&amp;lt;1 9 2 7 17|, &amp;lt;0 -23 1 -13 -42|]&lt;br /&gt;
Comma list: 540/539, 1375/1372, 43923/43750
EDOs: 31, 90, 121, 152, 335, 487&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=0| 1 -2 -1 -2 -1 | 0 41 38 55 51 }}
&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;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1199.9327{{c}}, ~35/33 = 104.9100{{c}}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 386.826&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 104.9144{{c}}
&lt;br /&gt;
 
Map: [&amp;lt;1 9 2 7 17 -5|, &amp;lt;0 -23 1 -13 -42 27|]&lt;br /&gt;
{{Optimal ET sequence|legend=0| 80, 103, 183 }}
EDOs: 31, 121, 152, 425&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.37
&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;
=== 13-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 386.812&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 351/350, 540/539, 1375/1372, 4225/4224
Map: [&amp;lt;1 9 2 7 17 -5 -3|, &amp;lt;0 -23 1 -13 -42 27 22|]&lt;br /&gt;
 
EDOs: 31, 121, 273&lt;br /&gt;
{{Mapping|legend=0| 1 -2 -1 -2 -1 3 | 0 41 38 55 51 8 }}
&lt;br /&gt;
 
&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;
Optimal tunings:  
Commas: 256/255, 352/351, 375/374, 400/399, 456/455, 715/714&lt;br /&gt;
* WE: ~2 = 1200.1082{{c}}, ~35/33 = 104.9170{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 104.9094{{c}}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 386.819&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 80, 103, 183, 469f }}
Map: [&amp;lt;1 9 2 7 17 -5 -3|, &amp;lt;0 -23 1 -13 -42 27 22|]&lt;br /&gt;
 
EDOs: 31, 121, 152, 273&lt;br /&gt;
Badness (Sintel): 1.15
&lt;br /&gt;
 
&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;
=== 17-limit ===
Commas: 5120/5103, 16875/16807&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 702.616&lt;br /&gt;
Comma list: 351/350, 540/539, 561/560, 715/714, 4225/4224
&lt;br /&gt;
 
Map: [&amp;lt;1 0 -50 -40|, &amp;lt;0 1 33 27|]&lt;br /&gt;
{{Mapping|legend=0| 1 -2 -1 -2 -1 3 4 | 0 41 38 55 51 8 1 }}
Wedgie: &amp;lt;&amp;lt;1 33 27 50 40 -30||&lt;br /&gt;
 
EDOs: 41, 111, 152, 345, 497&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1200.0758{{c}}, ~17/16 = 104.9158{{c}}
&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;
* CWE: ~2 = 1200.0000{{c}}, ~17/16 = 104.9101{{c}}
Commas: 540/539, 1375/1372, 16384/16335&lt;br /&gt;
 
&lt;br /&gt;
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~17/16 = 104.909{{c}}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 702.623&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 80, 103, 183, 469f }}
Map: [&amp;lt;1 0 -50 -40 32|, &amp;lt;0 1 33 27 -18|]&lt;br /&gt;
 
EDOs: 29, 41, 111, 152&lt;br /&gt;
Badness (Sintel): 1.03
&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;
== Gaster ==
Commas: 352/351, 540/539, 729/728, 1375/1372&lt;br /&gt;
: ''For the 5-limit version, see [[Very high accuracy temperaments #Gaster]].''
&lt;br /&gt;
{{Main| Gaster temperament }}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 702.644&lt;br /&gt;
 
&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.
Map: [&amp;lt;1 0 -50 -40 32 27|, &amp;lt;0 1 33 27 -18 -21|]&lt;br /&gt;
 
EDOs: 29, 41, 111, 152&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;
 
&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;
[[Subgroup]]: 2.3.5.7
Commas: 256/255, 352/351, 540/539, 715/714, 1089/1088&lt;br /&gt;
 
&lt;br /&gt;
[[Comma list]]: 16875/16807, 14348907/14336000
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 702.660&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=1| 1 -8 -34 -32 | 0 19 72 69 }}
Map: [&amp;lt;1 0 -50 -40 32 27 58|, &amp;lt;0 1 33 27 -18 -21 -34|]&lt;br /&gt;
: mapping generators: ~2, ~567/400
EDOs: 29, 41, 111, 152, 263&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]]s:  
&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;
* [[WE]]: ~2 = 1199.9920{{c}}, ~567/400 = 605.3546{{c}}
Commas: 256/255, 352/351, 400/399, 456/455, 715/714, 847/845&lt;br /&gt;
: [[error map]]: {{val| -0.008 -0.152 -0.506 +0.902 }}
&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~567/400 = 605.3586{{c}}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 702.657&lt;br /&gt;
: error map: {{val| 0.000 -0.142 -0.497 +0.915 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 0 -50 -40 32 27 58 -56|, &amp;lt;0 1 33 27 -18 -21 -34 38|]&lt;br /&gt;
{{Optimal ET sequence|legend=1| 111, 224 }}
EDOs: 29, 41, 111, 152, 263&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]] (Sintel): 3.91
&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;
=== 11-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 585.147&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 540/539, 1375/1372, 14348907/14336000
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;
{{Mapping|legend=0| 1 -8 -34 -32 8 | 0 19 72 69 -9 }}
EDOs: 39, 41, 80, 121, 404&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
&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;
* WE: ~2 = 1199.9387{{c}}, ~363/256 = 605.3300{{c}}
Commas: 540/539, 896/891, 1375/1372&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~363/256 = 605.3603{{c}}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 585.114&lt;br /&gt;
{{Optimal ET sequence|legend=0| 111, 224, 783d }}
&lt;br /&gt;
 
Map: [&amp;lt;1 5 15 15 2|, &amp;lt;0 -7 -26 -25 3|]&lt;br /&gt;
Badness (Sintel): 1.79
EDOs: 39, 41, 80, 121&lt;br /&gt;
 
&lt;br /&gt;
=== 13-limit ===
&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;
Subgroup: 2.3.5.7.11.13
Commas: 325/324, 352/351, 364/363, 540/539&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 540/539, 729/728, 1375/1372, 2200/2197
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 585.123&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=0| 1 -8 -34 -32 8 -19 | 0 19 72 69 -9 45 }}
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;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1199.9154{{c}}, ~78/55 = 605.3183{{c}}
&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;
* CWE: ~2 = 1200.0000{{c}}, ~78/55 = 605.3601{{c}}
Commas: 256/255, 325/324, 352/351, 364/363, 540/539&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 111, 224, 783df }}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 585.116&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.03
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;
=== 17-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17
&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;
 
Commas: 190/189, 256/255, 325/324, 352/351, 361/360, 595/594&lt;br /&gt;
Comma list: 540/539, 715/714, 729/728, 936/935, 2200/2197
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 585.109&lt;br /&gt;
{{Mapping|legend=0| 1 -8 -34 -32 8 -19 -6 | 0 19 72 69 -9 45 20 }}
&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;
Optimal tunings:  
EDOs: 39, 41, 80, 121, 564, 685&lt;br /&gt;
* WE: ~2 = 1199.8076{{c}}, ~17/12 = 605.2674{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3626{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc15"&gt;&lt;a name="Eris"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;Eris&lt;/h1&gt;
 
Commas: 16875/16807, 65625/65536&lt;br /&gt;
{{Optimal ET sequence|legend=0| 111, 224, 559dgg }}
&lt;br /&gt;
 
POTE generator: ~60/49 = 348.219&lt;br /&gt;
Badness (Sintel): 1.09
&lt;br /&gt;
 
Map: [&amp;lt;1 10 0 6|, &amp;lt;0 -29 8 -11|]&lt;br /&gt;
=== 19-limit ===
EDOs: 31, 131, 162, 193, 224&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17.19
Badness: 0.0747&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 324/323, 400/399, 495/494, 540/539, 715/714, 1445/1444
&lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc16"&gt;&lt;a name="Eris-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt;11-limit&lt;/h2&gt;
 
Commas: 540/539, 1375/1372, 65625/65536&lt;br /&gt;
{{Mapping|legend=0| 1 -8 -34 -32 8 -19 -6 -24 | 0 19 72 69 -9 45 20 56 }}
&lt;br /&gt;
 
POTE generator: ~11/9 = 348.219&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1199.7542{{c}}, ~17/12 = 605.2674{{c}}
Map: [&amp;lt;1 10 0 6 20|, &amp;lt;0 -29 8 -11 -57|]&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3613{{c}}
EDOs: 31, 193, 224, 703&lt;br /&gt;
 
Badness: 0.0276&lt;br /&gt;
{{Optimal ET sequence|legend=0| 111, 224 }}
&lt;!-- ws:start:WikiTextHeadingRule:34:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc17"&gt;&lt;a name="Mirkat"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:34 --&gt;Mirkat&lt;/h1&gt;
 
Commas: 16875/16807, 19683/19600&lt;br /&gt;
Badness (Sintel): 1.12
&lt;br /&gt;
 
POTE generator: ~10/9 = 183.539&lt;br /&gt;
=== 23-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17.19.23
Map: [&amp;lt;3 2 1 2|, &amp;lt;0 6 13 14|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;18 39 42 20 16 -12||&lt;br /&gt;
Comma list: 324/323, 400/399, 460/459, 495/494, 529/528, 540/539, 715/714
EDOs: 33, 39, 72, 111, 183, 255&lt;br /&gt;
 
Badness: 0.0594&lt;br /&gt;
{{Mapping|legend=0| 1 -8 -34 -32 8 -19 -6 -24 2 | 0 19 72 69 -9 45 20 56 5 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc18"&gt;&lt;a name="Mirkat-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;11-limit&lt;/h2&gt;
Optimal tunings:  
Commas: 540/539, 1375/1372, 8019/8000&lt;br /&gt;
* WE: ~2 = 1199.8733{{c}}, ~17/12 = 605.2946{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3575{{c}}
POTE generator: ~10/9 = 183.528&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 111, 224 }}
Map: [&amp;lt;3 2 1 2 9|, &amp;lt;0 6 13 14 3|]&lt;br /&gt;
 
EDOs: 33, 39, 72, 183, 255&lt;br /&gt;
Badness (Sintel): 1.26
Badness: 0.0221&lt;br /&gt;
 
&lt;br /&gt;
=== 29-limit ===
&lt;!-- ws:start:WikiTextHeadingRule:38:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc19"&gt;&lt;a name="Mirkat-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:38 --&gt;13-limit&lt;/h2&gt;
Subgroup: 2.3.5.7.11.13.17.19.23.29
Commas: 351/350, 540/539, 676/675, 1375/1372&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 290/289, 324/323, 400/399, 460/459, 495/494, 529/528, 540/539, 715/714
POTE generator: ~10/9 = 183.577&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=0| 1 -8 -34 -32 8 -19 -6 -24 2 21 | 0 19 72 69 -9 45 20 56 5 -32 }}
Map: [&amp;lt;3 2 1 2 9 1|, &amp;lt;0 6 13 14 3 22|]&lt;br /&gt;
 
EDOs: 33, 39, 72, 111, 183&lt;br /&gt;
Optimal tunings:  
Badness: 0.0186&lt;br /&gt;
* WE: ~2 = 1199.9442{{c}}, ~17/12 = 605.3263{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3541{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:40:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc20"&gt;&lt;a name="Subsemifourth"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:40 --&gt;Subsemifourth&lt;/h1&gt;
 
Commas: 16875/16807, 26873856/26796875&lt;br /&gt;
{{Optimal ET sequence|legend=0| 111, 113, 224 }}
&lt;br /&gt;
 
POTE generator: ~144/125 = 244.719&lt;br /&gt;
Badness (Sintel): 1.41
&lt;br /&gt;
 
Map: [&amp;lt;1 39 27 45|, &amp;lt;0 -47 -31 -53|]&lt;br /&gt;
=== 31-limit ===
Wedgie: &amp;lt;&amp;lt;47 31 53 -60 -48 36||&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17.19.23.29.31
EDOs: 49, 103, 152, 255, 407 &lt;br /&gt;
 
Badness: 0.1352&lt;br /&gt;
Comma list: 290/289, 324/323, 400/399, 435/434, 460/459, 495/494, 528/527, 540/539, 715/714
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:42:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc21"&gt;&lt;a name="Subsemifourth-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:42 --&gt;11-limit&lt;/h2&gt;
{{Mapping|legend=0| 1 -8 -34 -32 8 -19 -6 -24 2 21 10 | 0 19 72 69 -9 45 20 56 5 -32 -10 }}
Commas: 540/539, 1375/1372, 234375/234256&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
POTE generator: ~121/105 = 244.719&lt;br /&gt;
* WE: ~2 = 1199.9100{{c}}, ~17/12 = 605.3107{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3556{{c}}
Map: [&amp;lt;1 39 27 45 56|, &amp;lt;0 -47 -31 -53 -66|]&lt;br /&gt;
 
EDOs: 49, 103, 152, 255, 407, 966d&lt;br /&gt;
{{Optimal ET sequence|legend=0| 111, 113, 224 }}
Badness: 0.0343&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.42
&lt;!-- ws:start:WikiTextHeadingRule:44:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc22"&gt;&lt;a name="Subsemifourth-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:44 --&gt;13-limit&lt;/h2&gt;
 
Commas: 540/539, 847/845, 1375/1372, 1575/1573&lt;br /&gt;
=== 37-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37
POTE generator: ~15/13 = 244.714&lt;br /&gt;
 
&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
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;
{{Mapping|legend=0| 1 -8 -34 -32 8 -19 -6 -24 2 21 10 38 | 0 19 72 69 -9 45 20 56 5 -32 -10 -65 }}
Badness: 0.0284&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:
&lt;!-- ws:start:WikiTextHeadingRule:46:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc23"&gt;&lt;a name="Subneutral"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:46 --&gt;Subneutral&lt;/h1&gt;
* WE: ~2 = 1199.9087{{c}}, ~17/12 = 605.3101{{c}}
Commas: 16875/16807, 65625/65536&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3559{{c}}
&lt;br /&gt;
 
POTE generator: ~60/49 = 348.216&lt;br /&gt;
{{Optimal ET sequence|legend=0| 111, 113, 224 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 10 0 6|, &amp;lt;0 -29 8 -11|]&lt;br /&gt;
Badness (Sintel): 1.56
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;
=== 41-limit ===
Badness: 0.0747&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:48:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc24"&gt;&lt;a name="Subneutral-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:48 --&gt;11-limit&lt;/h2&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
Commas: 540/539, 1375/1372, 65625/65536&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=0| 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 }}
POTE generator: ~11/9 = 348.219&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:
Map: [&amp;lt;1 10 0 6 20|, &amp;lt;0 -29 8 -11 -57|]&lt;br /&gt;
* WE: ~2 = 1199.9179{{c}}, ~17/12 = 605.3156{{c}}
EDOs: 31, 193, 224, 703, 927d, 1151cd&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3567{{c}}
Badness: 0.0276&lt;/body&gt;&lt;/html&gt;</pre></div>
 
{{Optimal ET sequence|legend=0| 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 {{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]]).
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 16875/16807, 95703125/95551488
 
{{Mapping|legend=1| 1 -19 -7 -17 | 0 53 24 51 }}
: mapping generators: ~2, ~98/75
 
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.2693{{c}}, ~98/75 = 466.0801{{c}}
: [[error map]]: {{val| +0.023 -0.149 -0.553 +0.866 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~98/75 = 466.0715{{c}}
: error map: {{val| 0.000 -0.167 -0.598 +0.819 }}
 
{{Optimal ET sequence|legend=1| 103, 121, 224 }}
 
[[Badness]] (Sintel): 4.12
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 1375/1372, 2734375/2725888
 
{{Mapping|legend=0| 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|legend=0| 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|legend=0| 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
 
== 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|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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:Canopic clan| ]] <!-- main article -->
[[Category:Canopic| ]] <!-- key article -->
[[Category:Temperament clans]]
[[Category:Catalogs of rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/Canopic_clan"