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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]], octokaidecal, nusecond, semisept and 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


Badness: 0.024839
{{Mapping|legend=1| 3 2 1 2 | 0 6 13 14 }}
: mapping generators: ~63/50, ~10/9


== 17-limit ==
[[Optimal tuning]]s:
Comma list: 256/255, 352/351, 625/624, 715/714, 1275/1274
* [[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 }}


Mapping: [{{val|1 9 2 7 17 -5 -3}}, {{val|0 -23 1 -13 -42 27 22}}]
{{Optimal ET sequence|legend=1| 39d, 72, 111, 183, 255 }}


POTE generator: ~5/4 = 386.812
[[Badness]] (Sintel): 1.50


Vals: {{Val list| 31, 121, 273defgg }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.021400
Comma list: 540/539, 1375/1372, 8019/8000


== 19-limit ==
Mapping: {{mapping| 3 2 1 2 9 | 0 6 13 14 3 }}
Comma list: 256/255, 352/351, 375/374, 400/399, 456/455, 715/714


POTE generator: ~5/4 = 386.819
Optimal tunings:  
* WE: ~63/50 = 400.0463{{c}}, ~10/9 = 183.5496{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~10/9 = 183.5391{{c}}


Mapping: [{{val|1 9 2 7 17 -5 -3 -8}}, {{val|0 -23 1 -13 -42 27 22 38}}]
{{Optimal ET sequence|legend=0| 39d, 72, 111, 183, 255 }}


Vals: {{Val list| 31, 121, 152fg, 273defgg }}
Badness (Sintel): 0.731


Badness: 0.018413
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


= Kwai =
Comma list: 351/350, 540/539, 676/675, 1375/1372
[[Comma list]]: 5120/5103, 16875/16807


[[Mapping]]: [{{val|1 0 -50 -40}}, {{val|0 1 33 27}}]
Mapping: {{mapping| 3 2 1 2 9 1 | 0 6 13 14 3 22 }}


{{Multival|legend=1|1 33 27 50 40 -30}}
Optimal tunings:
* WE: ~63/50 = 400.0245{{c}}, ~10/9 = 183.5885{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~10/9 = 183.5825{{c}}


[[POTE generator]]: ~3/2 = 702.616
{{Optimal ET sequence|legend=0| 39df, 72, 111, 183 }}


{{Val list|legend=1| 41, 111, 152, 345, 497d }}
Badness (Sintel): 0.770


[[Badness]]: 0.054476
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


== 11-limit ==
Comma list: 351/350, 442/441, 540/539, 561/560, 715/714
Comma list: 540/539, 1375/1372, 16384/16335


Mapping: [{{val|1 0 -50 -40 32}}, {{val|0 1 33 27 -18}}]
Mapping: {{mapping| 3 2 1 2 9 1 4 | 0 6 13 14 3 22 18 }}


POTE generator: ~3/2 = 702.623
Optimal tunings:  
* WE: ~34/27 = 400.0257{{c}}, ~10/9 = 183.5906{{c}}
* CWE: ~34/27 = 400.0000{{c}}, ~10/9 = 183.5843{{c}}


Vals: {{Val list| 29cd, 41, 111, 152 }}
{{Optimal ET sequence|legend=0| 39dfg, 72, 111, 183 }}


Badness: 0.026219
Badness (Sintel): 0.600


== 13-limit ==
== Eris ==
Comma list: 352/351, 540/539, 729/728, 1375/1372
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]].


Mapping: [{{val|1 0 -50 -40 32 27}}, {{val|0 1 33 27 -18 -21}}]
[[Subgroup]]: 2.3.5.7


POTE generator: ~3/2 = 702.644
[[Comma list]]: 16875/16807, 65625/65536


Vals: {{Val list| 29cd, 41, 111, 152f }}
{{Mapping|legend=1| 1 -19 8 -5 | 0 29 -8 11 }}
: mapping generators: ~2, ~49/30


Badness: 0.024555
[[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 }}


== 17-limit ==
{{Optimal ET sequence|legend=1| 31, 131, 162, 193, 224 }}
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}}]
[[Badness]] (Sintel): 1.89


POTE generator: ~3/2 = 702.660
=== 11-limit ===
Subgroup: 2.3.5.7.11


Vals: {{Val list| 29cdg, 41, 111, 152fg, 263dfg }}
Comma list: 540/539, 1375/1372, 65625/65536


Badness: 0.021950
Mapping: {{mapping| 1 -19 8 -5 -37 | 0 29 -8 11 57 }}


== 19-limit ==
Optimal tunings:
Comma list: 256/255, 352/351, 400/399, 456/455, 715/714, 847/845
* WE: ~2 = 1200.0218{{c}}, ~18/11 = 851.7963{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/11 = 851.7812{{c}}


Mapping: [{{val|1 0 -50 -40 32 27 58 -56}}, {{val|0 1 33 27 -18 -21 -34 38}}]
{{Optimal ET sequence|legend=0| 31, …, 193, 224, 703, 927d }}


POTE generator: ~3/2 = 702.657
Badness (Sintel): 0.913


Vals: {{Val list| 29cdgh, 41, 111, 152fg, 263dfgh }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.016957
Comma list: 540/539, 625/624, 1375/1372, 4096/4095


=Pluto=
Mapping: {{mapping| 1 -19 8 -5 -37 47 | 0 29 -8 11 57 -61 }}
Commas: 4000/3969, 10976/10935


[[POTE_tuning|POTE generator]]: ~7/5 = 585.147
Optimal tuning:  
* WE ~2 = 1199.9623{{c}}, ~18/11 = 851.7598{{c}}
* CWE ~2 = 1200.0000{{c}}, ~18/11 = 851.7865{{c}}


Map: [<1 5 15 15|, <0 -7 -26 -25|]
{{Optimal ET sequence|legend=0| 31, 193, 224 }}


Wedgie: <<7 26 25 25 20 -15||
Badness (Sintel): 1.04


EDOs: 39d, 41, 80, 121, 404bd
== Subsemifourth ==
[[Subgroup]]: 2.3.5.7


Badness: 0.057514
[[Comma list]]: 16875/16807, 26873856/26796875


==11-limit==
{{Mapping|legend=1| 1 -8 -4 -8 | 0 47 31 53 }}
Commas: 540/539, 896/891, 1375/1372
: mapping generators: ~2, ~144/125


[[POTE_tuning|POTE generator]]: ~7/5 = 585.114
[[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: [<1 5 15 15 2|, <0 -7 -26 -25 3|]
{{Optimal ET sequence|legend=1| 49, 103, 152, 255, 407 }}


EDOs: 39d, 41, 80, 121
[[Badness]] (Sintel): 3.42


Badness: 0.029844
=== 11-limit ===
Subgroup: 2.3.5.7.11


===13-limit===
Comma list: 540/539, 1375/1372, 234375/234256
Commas: 325/324, 352/351, 364/363, 540/539


[[POTE_tuning|POTE generator]]: ~7/5 = 585.123
Mapping: {{mapping| 1 -8 -4 -8 -10 | 0 47 31 53 66 }}


Map: [<1 5 15 15 2 -8|, <0 -7 -26 -25 3 24|]
Optimal tunings:  
* WE: ~2 = 1199.9229{{c}}, ~121/105 = 244.7033{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~121/105 = 244.7175{{c}}


EDOs: 39d, 41, 80, 121
{{Optimal ET sequence|legend=0| 49, 103, 152, 255, 407 }}


Badness: 0.025717
Badness (Sintel): 1.13


===17-limit===
=== 13-limit ===
Commas: 256/255, 325/324, 352/351, 364/363, 540/539
Subgroup: 2.3.5.7.11.13


[[POTE_tuning|POTE generator]]: ~7/5 = 585.116
Comma list: 540/539, 847/845, 1375/1372, 1575/1573


Map: [<1 5 15 15 2 -8 -12|, <0 -7 -26 -25 3 24 33|]
Mapping: {{mapping| 1 -8 -4 -8 -10 -12 | 0 0 47 31 53 66 77 }}


EDOs: 39d, 41, 80, 121
Optimal tunings:  
* WE: ~2 = 1199.9003{{c}}, ~15/13 = 244.6932{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/13 = 244.7116{{c}}


Badness: 0.021463
{{Optimal ET sequence|legend=0| 49f, 103, 152f, 255, 407f }}


===19-limit===
Badness (Sintel): 1.17
Commas: 190/189, 256/255, 325/324, 352/351, 361/360, 595/594


[[POTE_tuning|POTE generator]]: ~7/5 = 585.109
== 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: [<1 5 15 15 2 -8 -12 14|, <0 -7 -26 -25 3 24 33 -20|]
[[Subgroup]]: 2.3.5.7


EDOs: 39d, 41, 80, 121
[[Comma list]]: 16875/16807, 1959552/1953125


Badness: 0.017650
{{Mapping|legend=1| 1 -2 -1 -2 | 0 41 38 55 }}
: mapping generators: ~2, ~343/324


===Orcus===
[[Optimal tuning]]s:
Commas: 144/143, 196/195, 275/273, 896/891
* [[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 }}


POTE generator: ~7/5 = 585.111
{{Optimal ET sequence|legend=0| 80, 103, 183 }}


Map: [<1 5 15 15 2 12|, <0 -7 -26 -25 3 -17|]
[[Badness]] (Sintel): 3.71


EDOs: 41, 80f, 121ff
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.033441
Comma list: 540/539, 1375/1372, 43923/43750


==Plutino==
Mapping: {{mapping| 1 -2 -1 -2 -1 | 0 41 38 55 51 }}
Commas: 100/99, 245/242, 10976/10935


POTE generator: ~7/5 = 585.283
Optimal tunings:  
* WE: ~2 = 1199.9327{{c}}, ~35/33 = 104.9100{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 104.9144{{c}}


Map: [<1 5 15 15 22|, <0 -7 -26 -25 -38|]
{{Optimal ET sequence|legend=0| 80, 103, 183 }}


EDOs: 2cde, 39dee, 41
Badness (Sintel): 1.37


Badness: 0.057966
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


===13-limit===
Comma list: 351/350, 540/539, 1375/1372, 4225/4224
Commas: 100/99, 196/195, 245/242, 729/728


POTE generator: ~7/5 = 585.232
Mapping: {{mapping| 1 -2 -1 -2 -1 3 | 0 41 38 55 51 8 }}


Map: [<1 5 15 15 22 12|, <0 -7 -26 -25 -38 -17|]
Optimal tunings:  
* WE: ~2 = 1200.1082{{c}}, ~35/33 = 104.9170{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 104.9094{{c}}


EDOs: 2cde, 39deef, 41
{{Optimal ET sequence|legend=0| 80, 103, 183, 469f }}


Badness: 0.040182
Badness (Sintel): 1.15


=Eris=
=== 17-limit ===
Commas: 16875/16807, 65625/65536
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~60/49 = 348.216
Comma list: 351/350, 540/539, 561/560, 715/714, 4225/4224


Map: [<1 10 0 6|, <0 -29 8 -11|]
Mapping: {{mapping| 1 -2 -1 -2 -1 3 4 | 0 41 38 55 51 8 1 }}


Wedgie: <<29 -8 11 -80 -64 48}}
Optimal tunings:  
* WE: ~2 = 1200.0758{{c}}, ~17/16 = 104.9158{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/16 = 104.9101{{c}}


EDOs: 31, 131, 162, 193, 224, 1823cd, 2271cd
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~17/16 = 104.909{{c}}


Badness: 0.0747
{{Optimal ET sequence|legend=0| 80, 103, 183, 469f }}


==11-limit==
Badness (Sintel): 1.03
Commas: 540/539, 1375/1372, 65625/65536


POTE generator: ~11/9 = 348.219
== Gaster ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Gaster]].''
{{Main| Gaster temperament }}


Map: [<1 10 0 6 20|, <0 -29 8 -11 -57|]
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.


EDOs: 31, 193, 224, 703, 927d, 1151cd
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: 0.0276
[[Subgroup]]: 2.3.5.7


=Mirkat=
[[Comma list]]: 16875/16807, 14348907/14336000
Commas: 16875/16807, 19683/19600


POTE generator: ~10/9 = 183.539
{{Mapping|legend=1| 1 -8 -34 -32 | 0 19 72 69 }}
: mapping generators: ~2, ~567/400


Map: [<3 2 1 2|, <0 6 13 14|]
[[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 }}


Wedgie: <<18 39 42 20 16 -12||
{{Optimal ET sequence|legend=1| 111, 224 }}


EDOs: 39d, 72, 111, 183, 255
[[Badness]] (Sintel): 3.91


Badness: 0.0594
=== 11-limit ===
Subgroup: 2.3.5.7.11


==11-limit==
Comma list: 540/539, 1375/1372, 14348907/14336000
Commas: 540/539, 1375/1372, 8019/8000


POTE generator: ~10/9 = 183.528
Mapping: {{mapping| 1 -8 -34 -32 8 | 0 19 72 69 -9 }}


Map: [<3 2 1 2 9|, <0 6 13 14 3|]
Optimal tunings:  
* WE: ~2 = 1199.9387{{c}}, ~363/256 = 605.3300{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~363/256 = 605.3603{{c}}


EDOs: 39d, 72, 111, 183, 255
{{Optimal ET sequence|legend=0| 111, 224, 783d }}


Badness: 0.0221
Badness (Sintel): 1.79


==13-limit==
=== 13-limit ===
Commas: 351/350, 540/539, 676/675, 1375/1372
Subgroup: 2.3.5.7.11.13


POTE generator: ~10/9 = 183.577
Comma list: 540/539, 729/728, 1375/1372, 2200/2197


Map: [<3 2 1 2 9 1|, <0 6 13 14 3 22|]
Mapping: {{mapping| 1 -8 -34 -32 8 -19 | 0 19 72 69 -9 45 }}


EDOs: 39df, 72, 111, 183, 255f
Optimal tunings:  
* WE: ~2 = 1199.9154{{c}}, ~78/55 = 605.3183{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~78/55 = 605.3601{{c}}


Badness: 0.0186
{{Optimal ET sequence|legend=0| 111, 224, 783df }}


==17-limit==
Badness (Sintel): 1.03
Commas: 351/350, 442/441, 540/539, 561/560, 715/714


POTE generator: ~10/9 = 183.579
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Map: [<3 2 1 2 9 1 4|, <0 6 13 14 3 22 18|]
Comma list: 540/539, 715/714, 729/728, 936/935, 2200/2197


EDOs: 39dfg, 72, 111, 183, 255f
Mapping: {{mapping| 1 -8 -34 -32 8 -19 -6 | 0 19 72 69 -9 45 20 }}


=Subsemifourth=
Optimal tunings:
Commas: 16875/16807, 26873856/26796875
* WE: ~2 = 1199.8076{{c}}, ~17/12 = 605.2674{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3626{{c}}


POTE generator: ~144/125 = 244.719
{{Optimal ET sequence|legend=0| 111, 224, 559dgg }}


Map: [<1 39 27 45|, <0 -47 -31 -53|]
Badness (Sintel): 1.09


Wedgie: <<47 31 53 -60 -48 36||
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


EDOs: 49, 103, 152, 255, 407
Comma list: 324/323, 400/399, 495/494, 540/539, 715/714, 1445/1444


Badness: 0.1352
Mapping: {{mapping| 1 -8 -34 -32 8 -19 -6 -24 | 0 19 72 69 -9 45 20 56 }}


==11-limit==
Optimal tunings:
Commas: 540/539, 1375/1372, 234375/234256
* WE: ~2 = 1199.7542{{c}}, ~17/12 = 605.2674{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3613{{c}}


POTE generator: ~121/105 = 244.719
{{Optimal ET sequence|legend=0| 111, 224 }}


Map: [<1 39 27 45 56|, <0 -47 -31 -53 -66|]
Badness (Sintel): 1.12


EDOs: 49, 103, 152, 255, 407, 966d
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23


Badness: 0.0343
Comma list: 324/323, 400/399, 460/459, 495/494, 529/528, 540/539, 715/714


==13-limit==
Mapping: {{mapping| 1 -8 -34 -32 8 -19 -6 -24 2 | 0 19 72 69 -9 45 20 56 5 }}
Commas: 540/539, 847/845, 1375/1372, 1575/1573


POTE generator: ~15/13 = 244.714
Optimal tunings:  
* WE: ~2 = 1199.8733{{c}}, ~17/12 = 605.2946{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3575{{c}}


Map: [<1 39 27 45 56 65|, <0 -47 -31 -53 -66 -77|]
{{Optimal ET sequence|legend=0| 111, 224 }}


EDOs: 103, 152f, 255, 407f, 662df
Badness (Sintel): 1.26


Badness: 0.0284
=== 29-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29


[[Category:Theory]]
Comma list: 290/289, 324/323, 400/399, 460/459, 495/494, 529/528, 540/539, 715/714
[[Category:Temperament clan]]
 
[[Category:Mirkwai]]
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 }}
 
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: {{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{{c}}, ~17/12 = 605.3101{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3559{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{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{{c}}, ~17/12 = 605.3156{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3567{{c}}
 
{{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: {{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]]
Retrieved from "https://en.xen.wiki/w/Canopic_clan"