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Temperaments of the '''Mirkwai clan''' temper out the ''mirkwai comma'' ({{Monzo| 0 3 4 -5 }} = 16875/16807), a no-twos comma. Members of the clan include grendel, kwai, pluto and mirkat considered below, as well as [[miracle]], [[Trienstonic clan #Octokaidecal|octokaidecal]], [[Starling temperaments #Nusecond|nusecond]], [[Ragismic microtemperaments #Octoid|octoid]], [[Kleismic family #Sqrtphi|sqrtphi]], [[Schismatic family #Quanharuk|quanharuk]], [[Semicomma family #Rainwell|rainwell]], [[Amity family #Familia|familia]], [[Hemimean clan #Semisept|semisept]], and [[Sensamagic clan #Bohpier|bohpier]], considered elsewhere.
{{Technical data page}}
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.  


= Grendel =
== Canopus ==
[[Comma list]]: 6144/6125, 16875/16807
{{Main| Canopus }}


[[Mapping]]: [{{val|1 9 2 7}}, {{val|0 -23 1 -13}}]
[[Subgroup]]: 3.5.7


{{Multival|legend=1|23 -1 13 -55 -44 33}}
[[Comma list]]: 16875/16807


[[POTE generator]]: ~5/4 = 386.863
{{Mapping|legend=2| 1 -2 -1 | 0 5 4 }}
: mapping generators: ~3, ~15/7


{{Val list|legend=1| 31, 90, 121, 152, 335d }}
[[Optimal tuning]]s:
* [[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 }}


[[Badness]]: 0.051834
[[Optimal ET sequence]]: [[13edt|b13]], [[62edt|b62]], [[75edt|b75]], [[88edt|b88]], [[101edt|b101]], [[114edt|b114]], [[355edt|b355]], [[469edt|b469]], [[583edt|b583]], [[697edt|b697]]


== 11-limit ==
[[Badness]] (Sintel): 0.0996
Comma list: 540/539, 1375/1372, 5632/5625


Mapping: [{{val|1 9 2 7 17}}, {{val|0 -23 1 -13 -42}}]
=== 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.
* ''[[Nusecond]]'' (+126/125) → [[Starling temperaments #Nusecond|Starling temperaments]]
* ''[[Octoid]]'' (+4375/4374) → [[Ragismic microtemperaments #Octoid|Ragismic microtemperaments]]


POTE generator: ~5/4 = 386.856
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.  


Vals: {{Val list| 31, 90e, 121, 152, 335d, 487d }}
Members of the clan discussed elsewhere are:  
* ''[[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]]


Badness: 0.019845
For ''no-twos'' extensions, see [[No-twos subgroup temperaments #Canopus]].


== 13-limit ==
Considered below are mirkat, eris, subsemifourth, septendesemi, gaster, hemiseptisix, browser, and grazer, in the order of increasing [[badness]].
Comma list: 352/351, 540/539, 625/624, 1375/1372


Map: [{{val|1 9 2 7 17 -5}}, {{val|0 -23 1 -13 -42 27}}]
== Mirkat ==
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.


POTE generator: ~5/4 = 386.826
[[Subgroup]]: 2.3.5.7


Vals: {{Val list| 31, 121, 152f, 425deff }}
[[Comma list]]: 16875/16807, 19683/19600
 
{{Mapping|legend=1| 3 2 1 2 | 0 6 13 14 }}
: mapping generators: ~63/50, ~10/9
 
[[Optimal tuning]]s:
* [[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 }}
 
{{Optimal ET sequence|legend=1| 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: {{mapping| 3 2 1 2 9 | 0 6 13 14 3 }}
 
Optimal tunings:
* WE: ~63/50 = 400.0463{{c}}, ~10/9 = 183.5496{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~10/9 = 183.5391{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 3 2 1 2 9 1 | 0 6 13 14 3 22 }}
 
Optimal tunings:
* WE: ~63/50 = 400.0245{{c}}, ~10/9 = 183.5885{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~10/9 = 183.5825{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 3 2 1 2 9 1 4 | 0 6 13 14 3 22 18 }}
 
Optimal tunings:
* WE: ~34/27 = 400.0257{{c}}, ~10/9 = 183.5906{{c}}
* CWE: ~34/27 = 400.0000{{c}}, ~10/9 = 183.5843{{c}}
 
{{Optimal ET sequence|legend=0| 39dfg, 72, 111, 183 }}
 
Badness (Sintel): 0.600
 
== Eris ==
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]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 16875/16807, 65625/65536
 
{{Mapping|legend=1| 1 -19 8 -5 | 0 29 -8 11 }}
: mapping generators: ~2, ~49/30
 
[[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 }}


Badness: 0.024839
{{Optimal ET sequence|legend=1| 31, 131, 162, 193, 224 }}


== 17-limit ==
[[Badness]] (Sintel): 1.89
Comma list: 256/255, 352/351, 625/624, 715/714, 1275/1274


Mapping: [{{val|1 9 2 7 17 -5 -3}}, {{val|0 -23 1 -13 -42 27 22}}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~5/4 = 386.812
Comma list: 540/539, 1375/1372, 65625/65536


Vals: {{Val list| 31, 121, 273defgg }}
Mapping: {{mapping| 1 -19 8 -5 -37 | 0 29 -8 11 57 }}


Badness: 0.021400
Optimal tunings:  
* WE: ~2 = 1200.0218{{c}}, ~18/11 = 851.7963{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/11 = 851.7812{{c}}


== 19-limit ==
{{Optimal ET sequence|legend=0| 31, …, 193, 224, 703, 927d }}
Comma list: 256/255, 352/351, 375/374, 400/399, 456/455, 715/714


POTE generator: ~5/4 = 386.819
Badness (Sintel): 0.913


Mapping: [{{val|1 9 2 7 17 -5 -3 -8}}, {{val|0 -23 1 -13 -42 27 22 38}}]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Vals: {{Val list| 31, 121, 152fg, 273defgg }}
Comma list: 540/539, 625/624, 1375/1372, 4096/4095


Badness: 0.018413
Mapping: {{mapping| 1 -19 8 -5 -37 47 | 0 29 -8 11 57 -61 }}


= Kwai =
Optimal tuning:
[[Comma list]]: 5120/5103, 16875/16807
* WE ~2 = 1199.9623{{c}}, ~18/11 = 851.7598{{c}}
* CWE ~2 = 1200.0000{{c}}, ~18/11 = 851.7865{{c}}


[[Mapping]]: [{{val|1 0 -50 -40}}, {{val|0 1 33 27}}]
{{Optimal ET sequence|legend=0| 31, 193, 224 }}


{{Multival|legend=1|1 33 27 50 40 -30}}
Badness (Sintel): 1.04


[[POTE generator]]: ~3/2 = 702.616
== Subsemifourth ==
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 41, 111, 152, 345, 497d }}
[[Comma list]]: 16875/16807, 26873856/26796875


[[Badness]]: 0.054476
{{Mapping|legend=1| 1 -8 -4 -8 | 0 47 31 53 }}
: mapping generators: ~2, ~144/125


== 11-limit ==
[[Optimal tuning]]s:
Comma list: 540/539, 1375/1372, 16384/16335
* [[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 }}


Mapping: [{{val|1 0 -50 -40 32}}, {{val|0 1 33 27 -18}}]
{{Optimal ET sequence|legend=1| 49, 103, 152, 255, 407 }}


POTE generator: ~3/2 = 702.623
[[Badness]] (Sintel): 3.42


Vals: {{Val list| 29cd, 41, 111, 152 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 1375/1372, 234375/234256
 
Mapping: {{mapping| 1 -8 -4 -8 -10 | 0 47 31 53 66 }}
 
Optimal tunings:
* WE: ~2 = 1199.9229{{c}}, ~121/105 = 244.7033{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~121/105 = 244.7175{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 -8 -4 -8 -10 -12 | 0 0 47 31 53 66 77 }}
 
Optimal tunings:
* WE: ~2 = 1199.9003{{c}}, ~15/13 = 244.6932{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/13 = 244.7116{{c}}
 
{{Optimal ET sequence|legend=0| 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 {{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]]).
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 16875/16807, 1959552/1953125
 
{{Mapping|legend=1| 1 -2 -1 -2 | 0 41 38 55 }}
: mapping generators: ~2, ~343/324
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8649{{c}}, ~343/324 = 104.9046{{c}}
: [[error map]]: {{val| -0.135 -0.597 +0.195 +1.196 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~343/324 = 104.9134{{c}}
: error map: {{val| 0.000 -0.506 +0.395 +1.410 }}
 
{{Optimal ET sequence|legend=0| 80, 103, 183 }}
 
[[Badness]] (Sintel): 3.71
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 1375/1372, 43923/43750
 
Mapping: {{mapping| 1 -2 -1 -2 -1 | 0 41 38 55 51 }}
 
Optimal tunings:
* WE: ~2 = 1199.9327{{c}}, ~35/33 = 104.9100{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 104.9144{{c}}
 
{{Optimal ET sequence|legend=0| 80, 103, 183 }}
 
Badness (Sintel): 1.37
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.026219
Comma list: 351/350, 540/539, 1375/1372, 4225/4224


== 13-limit ==
Mapping: {{mapping| 1 -2 -1 -2 -1 3 | 0 41 38 55 51 8 }}
Comma list: 352/351, 540/539, 729/728, 1375/1372


Mapping: [{{val|1 0 -50 -40 32 27}}, {{val|0 1 33 27 -18 -21}}]
Optimal tunings:  
* WE: ~2 = 1200.1082{{c}}, ~35/33 = 104.9170{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 104.9094{{c}}


POTE generator: ~3/2 = 702.644
{{Optimal ET sequence|legend=0| 80, 103, 183, 469f }}


Vals: {{Val list| 29cd, 41, 111, 152f }}
Badness (Sintel): 1.15


Badness: 0.024555
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


== 17-limit ==
Comma list: 351/350, 540/539, 561/560, 715/714, 4225/4224
Commas: 256/255, 352/351, 540/539, 715/714, 1089/1088


Mapping: [{{val|1 0 -50 -40 32 27 58}}, {{val|0 1 33 27 -18 -21 -34}}]
Mapping: {{mapping| 1 -2 -1 -2 -1 3 4 | 0 41 38 55 51 8 1 }}


POTE generator: ~3/2 = 702.660
Optimal tunings:
* WE: ~2 = 1200.0758{{c}}, ~17/16 = 104.9158{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/16 = 104.9101{{c}}


Vals: {{Val list| 29cdg, 41, 111, 152fg, 263dfg }}
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~17/16 = 104.909{{c}}


Badness: 0.021950
{{Optimal ET sequence|legend=0| 80, 103, 183, 469f }}


== 19-limit ==
Badness (Sintel): 1.03
Comma list: 256/255, 352/351, 400/399, 456/455, 715/714, 847/845


Mapping: [{{val|1 0 -50 -40 32 27 58 -56}}, {{val|0 1 33 27 -18 -21 -34 38}}]
== Gaster ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Gaster]].''
{{Main| Gaster temperament }}


POTE generator: ~3/2 = 702.657
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.  


Vals: {{Val list| 29cdgh, 41, 111, 152fg, 263dfgh }}
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.


Badness: 0.016957
[[Subgroup]]: 2.3.5.7


= Pluto =
[[Comma list]]: 16875/16807, 14348907/14336000
[[Comma list]]: 4000/3969, 10976/10935


[[Mapping]]: [{{val|1 5 15 15}}, {{val|0 -7 -26 -25}}]
{{Mapping|legend=1| 1 -8 -34 -32 | 0 19 72 69 }}
: mapping generators: ~2, ~567/400


{{Multival|legend=1|7 26 25 25 20 -15}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9920{{c}}, ~567/400 = 605.3546{{c}}
: [[error map]]: {{val| -0.008 -0.152 -0.506 +0.902 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~567/400 = 605.3586{{c}}
: error map: {{val| 0.000 -0.142 -0.497 +0.915 }}


[[POTE generator]]: ~7/5 = 585.147
{{Optimal ET sequence|legend=1| 111, 224 }}


{{Val list|legend=1| 39d, 41, 80, 121, 404bd }}
[[Badness]] (Sintel): 3.91


[[Badness]]: 0.057514
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 11-limit ==
Comma list: 540/539, 1375/1372, 14348907/14336000
Comma list: 540/539, 896/891, 1375/1372


Mapping: [{{val|1 5 15 15 2}}, {{val|0 -7 -26 -25 3}}]
Mapping: {{mapping| 1 -8 -34 -32 8 | 0 19 72 69 -9 }}


POTE generator: ~7/5 = 585.114
Optimal tunings:  
* WE: ~2 = 1199.9387{{c}}, ~363/256 = 605.3300{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~363/256 = 605.3603{{c}}


Vals: {{Val list| 39d, 41, 80, 121 }}
{{Optimal ET sequence|legend=0| 111, 224, 783d }}


Badness: 0.029844
Badness (Sintel): 1.79


=== 13-limit ===
=== 13-limit ===
Comma list: 325/324, 352/351, 364/363, 540/539
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 729/728, 1375/1372, 2200/2197
 
Mapping: {{mapping| 1 -8 -34 -32 8 -19 | 0 19 72 69 -9 45 }}
 
Optimal tunings:
* WE: ~2 = 1199.9154{{c}}, ~78/55 = 605.3183{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~78/55 = 605.3601{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 -8 -34 -32 8 -19 -6 | 0 19 72 69 -9 45 20 }}
 
Optimal tunings:
* WE: ~2 = 1199.8076{{c}}, ~17/12 = 605.2674{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3626{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 -8 -34 -32 8 -19 -6 -24 | 0 19 72 69 -9 45 20 56 }}
 
Optimal tunings:
* WE: ~2 = 1199.7542{{c}}, ~17/12 = 605.2674{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3613{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{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{{c}}, ~17/12 = 605.2946{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3575{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{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{{c}}, ~17/12 = 605.3263{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3541{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{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{{c}}, ~17/12 = 605.3107{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3556{{c}}
 
{{Optimal ET sequence|legend=0| 111, 113, 224 }}


Mapping: [{{val|1 5 15 15 2 -8}}, {{val|0 -7 -26 -25 3 24}}]
Badness (Sintel): 1.42


POTE generator: ~7/5 = 585.123
=== 37-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37


Vals: {{Val list| 39d, 41, 80, 121 }}
Comma list: 290/289, 324/323, 400/399, 435/434, 460/459, 495/494, 528/527, 540/539, 667/666, 715/714


Badness: 0.025717
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 }}


==== 17-limit ====
Optimal tunings:
Comma list: 256/255, 325/324, 352/351, 364/363, 540/539
* WE: ~2 = 1199.9087{{c}}, ~17/12 = 605.3101{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3559{{c}}


Mapping: [{{val|1 5 15 15 2 -8 -12}}, {{val|0 -7 -26 -25 3 24 33}}]
{{Optimal ET sequence|legend=0| 111, 113, 224 }}


POTE generator: ~7/5 = 585.116
Badness (Sintel): 1.56


Vals: {{Val list| 39d, 41, 80, 121 }}
=== 41-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41


Badness: 0.021463
Comma list: 290/289, 324/323, 400/399, 435/434, 460/459, 495/494, 528/527, 533/532, 540/539, 575/574, 667/666


==== 19-limit ====
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 }}
Comma list: 190/189, 256/255, 325/324, 352/351, 361/360, 595/594


Mapping: [{{val|1 5 15 15 2 -8 -12 14}}, {{val|0 -7 -26 -25 3 24 33 -20}}]
Optimal tunings:  
* WE: ~2 = 1199.9179{{c}}, ~17/12 = 605.3156{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3567{{c}}


POTE generator: ~7/5 = 585.109
{{Optimal ET sequence|legend=0| 111, 113, 224 }}


Vals: {{Val list| 39d, 41, 80, 121 }}
Badness (Sintel): 1.57


Badness: 0.017650
== 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]]).  


=== Orcus ===
[[Subgroup]]: 2.3.5.7
Comma list: 144/143, 196/195, 275/273, 896/891


Mapping: [{{val|1 5 15 15 2 12}}, {{val|0 -7 -26 -25 3 -17}}]
[[Comma list]]: 16875/16807, 95703125/95551488


POTE generator: ~7/5 = 585.111
{{Mapping|legend=1| 1 -19 -7 -17 | 0 53 24 51 }}
: mapping generators: ~2, ~98/75


Vals: {{Val list| 41, 80f, 121ff}}
[[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 }}


Badness: 0.033441
{{Optimal ET sequence|legend=1| 103, 121, 224 }}


== Plutino ==
[[Badness]] (Sintel): 4.12
Comma list: 100/99, 245/242, 10976/10935


Mapping: [{{val|1 5 15 15 22}}, {{val|0 -7 -26 -25 -38}}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~7/5 = 585.283
Comma list: 540/539, 1375/1372, 2734375/2725888


Vals: {{Val list| 2cde, 39dee, 41 }}
Mapping: {{mapping| 1 -19 -7 -17 -28 | 0 53 24 51 81 }}


Badness: 0.057966
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 ===
=== 13-limit ===
Comma list: 100/99, 196/195, 245/242, 729/728
Subgroup: 2.3.5.7.11.13


Mapping: [{{val|1 5 15 15 22 12}}, {{val|0 -7 -26 -25 -38 -17}}]
Comma list: 540/539, 625/624, 1375/1372, 2200/2197


POTE generator: ~7/5 = 585.232
Mapping: {{mapping| 1 -19 -7 -17 -28 -13 | 0 53 24 51 81 43 }}


Vals: {{Val list| 2cde, 39deef, 41 }}
Optimal tunings:  
* WE: ~2 = 1199.9784{{c}}, ~55/42 = 466.0622{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~55/42 = 466.0703{{c}}


Badness: 0.040182
{{Optimal ET sequence|legend=0| 103, 121, 224 }}


= Eris =
Badness (Sintel): 0.873
[[Comma list]]: 16875/16807, 65625/65536
 
=== 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 }}


[[Mapping]]: [{{val|1 10 0 6}}, {{val|0 -29 8 -11}}]
Optimal tunings:  
* WE: ~2 = 1200.1344{{c}}, ~11/6 = 1048.7124{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1048.5981{{c}}


{{Multival|legend=1|29 -8 11 -80 -64 48}}
{{Optimal ET sequence|legend=0| 103, 214 }}


[[POTE generator]]: ~60/49 = 348.216
Badness (Sintel): 1.91


{{Val list|legend=1| 31, 131, 162, 193, 224, 1823cd, 2271cd }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[[Badness]]: 0.074719
Comma list: 351/350, 540/539, 847/845, 1375/1372


== 11-limit ==
Mapping: {{mapping| 1 -29 -37 -47 -28 -33 | 0 35 45 57 36 42 }}
Comma list: 540/539, 1375/1372, 65625/65536


Mapping: [{{val|1 10 0 6 20}}, {{val|0 -29 8 -11 -57}}]
Optimal tunings:  
* WE: ~2 = 1200.1344{{c}}, ~11/6 = 1048.7124{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1048.5984{{c}}


POTE generator: ~11/9 = 348.219
{{Optimal ET sequence|legend=0| 103, 111, 214 }}


Vals: {{Val list| 31, 193, 224, 703, 927d, 1151cd }}
Badness (Sintel): 1.19


Badness: 0.027621
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


== 13-limit ==
Comma list: 351/350, 540/539, 561/560, 715/714, 847/845
Comma list: 540/539, 625/624, 1375/1372, 4096/4095


Mapping: [{{val|1 10 0 6 20 -14}}, {{val|0 -29 8 -11 -57 61}}]
Mapping: {{mapping| 1 -29 -37 -47 -28 -33 -23 | 0 35 45 57 36 42 31 }}


POTE generator: ~11/9 = 348.213
Optimal tunings:  
* WE: ~2 = 1199.9191{{c}}, ~11/6 = 1048.5324{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1048.6014{{c}}


Vals: {{Val list| 31, 193, 224 }}
{{Optimal ET sequence|legend=0| 103, 111, 214 }}


Badness: 0.025137
Badness (Sintel): 1.04


= Mirkat =
=== 19-limit ===
[[Comma list]]: 16875/16807, 19683/19600
Subgroup: 2.3.5.7.11.13.17.19


[[Mapping]]: [{{val|3 2 1 2}}, {{val|0 6 13 14}}]
Comma list: 324/323, 351/350, 456/455, 495/494, 540/539, 715/714


{{Multival|legend=1|18 39 42 20 16 -12}}
Mapping: {{mapping| 1 -29 -37 -47 -28 -33 -23 -91 | 0 35 45 57 36 42 31 109 }}


[[POTE generator]]: ~10/9 = 183.539
Optimal tunings:  
* WE: ~2 = 1199.9145{{c}}, ~11/6 = 1048.5290{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1048.6021{{c}}


{{Val list|legend=1| 39d, 72, 111, 183, 255 }}
{{Optimal ET sequence|legend=0| 103h, 111, 214 }}


[[Badness]]: 0.059376
Badness (Sintel): 1.07


== 11-limit ==
== Grazer ==
Comma list: 540/539, 1375/1372, 8019/8000
Named by [[Xenllium]] in 2022, grazer may be described as the {{nowrap| 113 & 121 }} temperament.


Mapping: [{{val|3 2 1 2 9}}, {{val|0 6 13 14 3}}]
[[Subgroup]]: 2.3.5.7


POTE generator: ~10/9 = 183.528
[[Comma list]]: 16875/16807, 1071875/1062882


Vals: {{Val list| 39d, 72, 111, 183, 255 }}
{{Mapping|legend=1| 1 -3 -4 -5 | 0 37 51 63 }}
: mapping generators: ~2, ~49/45


Badness: 0.022126
[[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 }}


== 13-limit ==
{{Optimal ET sequence|legend=1| 113, 121, 234 }}
Comma list: 351/350, 540/539, 676/675, 1375/1372


Mapping: [{{val|3 2 1 2 9 1}}, {{val|0 6 13 14 3 22}}]
[[Badness]] (Sintel): 5.50


POTE generator: ~10/9 = 183.577
=== 11-limit ===
Subgroup: 2.3.5.7.11


Vals: {{Val list| 39df, 72, 111, 183, 255f }}
Comma list: 540/539, 1375/1372, 218750/216513


Badness: 0.018632
Mapping: {{mapping| 1 -3 -4 -5 -1 | 0 37 51 63 36 }}


== 17-limit ==
Optimal tunings:
Comma list: 351/350, 442/441, 540/539, 561/560, 715/714
* WE: ~2 = 1199.7242{{c}}, ~12/11 = 148.6946{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 148.7230{{c}}


Mapping: [{{val|3 2 1 2 9 1 4}}, {{val|0 6 13 14 3 22 18}}]
{{Optimal ET sequence|legend=0| 113, 121, 234 }}


POTE generator: ~10/9 = 183.579
Badness (Sintel): 2.51


Vals: {{Val list| 39dfg, 72, 111, 183, 255f }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.011775
Comma list: 325/324, 364/363, 540/539, 2200/2197


= Subsemifourth =
Mapping: {{mapping| 1 -3 -4 -5 -1 -2 | 0 37 51 63 36 46 }}
[[Comma list]]: 16875/16807, 26873856/26796875


[[Mapping]]: [{{val|1 -8 -4 -8}}, {{val|0 47 31 53}}]
Optimal tunings:  
* WE: ~2 = 1199.7257{{c}}, ~12/11 = 148.6947{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 148.7230{{c}}


{{Multival|legend=1|47 31 53 -60 -48 36}}
{{Optimal ET sequence|legend=0| 113, 121, 234 }}


[[POTE generator]]: ~144/125 = 244.719
Badness (Sintel): 1.50


{{Val list|legend=1| 49, 103, 152, 255, 407 }}
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


[[Badness]]: 0.135173
Comma list: 325/324, 364/363, 540/539, 595/594, 2000/1989


== 11-limit ==
Mapping: {{mapping| 1 -3 -4 -5 -1 -2 0 | 0 37 51 63 36 46 33 }}
Comma list: 540/539, 1375/1372, 234375/234256


Mapping: [{{val|1 -8 -4 -8 -10}}, {{val|0 47 31 53 66}}]
Optimal tunings:  
* WE: ~2 = 1199.5690{{c}}, ~12/11 = 148.6815{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 148.7267{{c}}


POTE generator: ~121/105 = 244.719
{{Optimal ET sequence|legend=0| 113, 121, 234g }}


Vals: {{Val list| 49, 103, 152, 255, 407, 966d }}
Badness (Sintel): 1.29


Badness: 0.034276
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


== 13-limit ==
Comma list: 325/324, 364/363, 400/399, 540/539, 595/594, 665/663
Comma list: 540/539, 847/845, 1375/1372, 1575/1573


Mapping: [{{val|1 -8 -4 -8 -10 -12}}, {{val|0 47 31 53 66 77}}]
Mapping: {{mapping| 1 -3 -4 -5 -1 -2 0 4 | 0 37 51 63 36 46 33 2 }}


POTE generator: ~15/13 = 244.714
Optimal tunings:  
* WE: ~2 = 1199.7269{{c}}, ~12/11 = 148.6928{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 148.7227{{c}}


Vals: {{Val list| 103, 152f, 255, 407f, 662df }}
{{Optimal ET sequence|legend=0| 113, 121, 234g }}


Badness: 0.028387
Badness (Sintel): 1.37


[[Category:Theory]]
[[Category:Temperament clans]]
[[Category:Temperament clan]]
[[Category:Mirkwai clan| ]] <!-- main article -->
[[Category:Mirkwai]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Canopic_clan"