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m Restore the category of the corresponding temp
 
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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, mirkat, eris, subsemifourth, septendesemi, gaster, subsedia, hemiseptisix, browser, and grazer considered below, as well as these considered elsewhere:
{{Technical data page}}
* ''[[Octokaidecal]]'', {28/27, 50/49} → [[Trienstonic clan #Octokaidecal|Trienstonic clan]]
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.
* ''[[Meantritone]]'', {81/80, 16875/16807} → [[Meantone family #Meantritone|Meantone family]]
* ''[[Nusecond]]'', {126/125, 2430/2401} → [[Starling temperaments #Nusecond|Starling temperaments]]
* [[Miracle]], {225/224, 1029/1024} → [[Gamelismic clan #Miracle|Gamelismic clan]]
* ''[[Bohpier]]'', {245/243, 3125/3087} → [[Sensamagic clan #Bohpier|Sensamagic clan]]
* ''[[Semisept]]'', {1728/1715, 3136/3125} → [[Hemimean clan #Semisept|Hemimean clan]]
* ''[[Octoid]]'', {4375/4374, 16875/16807} → [[Ragismic microtemperaments #Octoid|Ragismic microtemperaments]]
* [[Sqrtphi]], {15625/15552, 16875/16807} → [[Kleismic family #Sqrtphi|Kleismic family]]
* ''[[Quanharuk]]'', {16875/16807, 32805/32768} → [[Schismatic family #Quanharuk|Schismatic family]]
* ''[[Familia]]'', {16875/16807, 1600000/1594323} → [[Amity family #Familia|Amity family]]
* ''[[Rainwell]]'', {16875/16807, 2100875/2097152} → [[Semicomma family #Rainwell|Semicomma family]]


== Canopus ==
== Canopus ==
{{main| Canopus }}
{{Main| Canopus }}


Subgroup: 3.5.7
[[Subgroup]]: 3.5.7


[[Comma list]]: 16875/16807
[[Comma list]]: 16875/16807


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


Sval mapping generators: ~3, ~7/5
[[Optimal tuning]]s:
 
* [[WE]]: ~3 = 1901.7826{{c}}, ~15/7 = 1317.8771{{c}}
[[POTE generator]]: ~7/5 = 583.9584
: [[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 }}


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


== Grendel ==
[[Badness]] (Sintel): 0.0996
Subgroup: 2.3.5.7


[[Comma list]]: 6144/6125, 16875/16807
=== 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 clan and collection pages.
* ''[[Nusecond]]'' (+126/125) → [[Starling temperaments #Nusecond|Starling temperaments]]
* [[Octoid]] (+4375/4374) → [[Stearnsmic clan #Octoid|Stearnsmic clan]]


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


{{Multival|legend=1| 23 -1 13 -55 -44 33 }}
Members of the clan discussed elsewhere are:
* ''[[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]]


[[POTE generator]]: ~5/4 = 386.863
For ''no-2's'' extensions, see [[No-twos subgroup temperaments #Canopus]].


{{Optimal ET sequence|legend=1| 31, 90, 121, 152, 335d }}
Considered below are mirkat, eris, subsemifourth, septendesemi, gaster, hemiseptisix, and grazer, in the order of increasing [[badness]].


[[Badness]]: 0.051834
== 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.  
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 1375/1372, 5632/5625
 
Mapping: [{{val|1 9 2 7 17}}, {{val|0 -23 1 -13 -42}}]
 
POTE generator: ~5/4 = 386.856
 
{{Optimal ET sequence|legend=1| 31, 90e, 121, 152, 335d, 487d }}
 
Badness: 0.019845
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 540/539, 625/624, 1375/1372
 
Mapping: [{{val|1 9 2 7 17 -5}}, {{val|0 -23 1 -13 -42 27}}]
 
POTE generator: ~5/4 = 386.826
 
{{Optimal ET sequence|legend=1| 31, 121, 152f, 425deff }}
 
Badness: 0.024839
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
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}}]
 
POTE generator: ~5/4 = 386.812
 
{{Optimal ET sequence|legend=1| 31, 121, 273defgg }}
 
Badness: 0.021400
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 256/255, 352/351, 375/374, 400/399, 456/455, 715/714
 
POTE generator: ~5/4 = 386.819


Mapping: [{{val|1 9 2 7 17 -5 -3 -8}}, {{val|0 -23 1 -13 -42 27 22 38}}]
[[Subgroup]]: 2.3.5.7
 
{{Optimal ET sequence|legend=1| 31, 121, 152fg, 273defgg }}
 
Badness: 0.018413
 
== Kwai ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Kwai]].''
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 5120/5103, 16875/16807
 
[[Mapping]]: [{{val|1 0 -50 -40}}, {{val|0 1 33 27}}]
 
{{Multival|legend=1|1 33 27 50 40 -30}}
 
[[POTE generator]]: ~3/2 = 702.616
 
{{Optimal ET sequence|legend=1| 41, 111, 152, 345, 497d }}
 
[[Badness]]: 0.054476
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 1375/1372, 5120/5103
 
Mapping: [{{val|1 0 -50 -40 32}}, {{val|0 1 33 27 -18}}]
 
POTE generator: ~3/2 = 702.623
 
{{Optimal ET sequence|legend=1| 29cd, 41, 111, 152 }}
 
Badness: 0.026219
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
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}}]
 
POTE generator: ~3/2 = 702.644
 
{{Optimal ET sequence|legend=1| 29cd, 41, 111, 152f }}
 
Badness: 0.024555
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 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}}]
 
POTE generator: ~3/2 = 702.660
 
{{Optimal ET sequence|legend=1| 29cdg, 41, 111, 152fg, 263dfg }}
 
Badness: 0.021950
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
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}}]
 
POTE generator: ~3/2 = 702.657
 
{{Optimal ET sequence|legend=1| 29cdgh, 41, 111, 152fg, 263dfgh }}
 
Badness: 0.016957
 
==== Hemikwai ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 676/675, 1375/1372, 5120/5103
 
Mapping: [{{val|1 2 16 14 -4 18}}, {{val|0 -2 -66 -54 36 -69}}]
 
POTE generator: ~15/13 = 248.686
 
{{Optimal ET sequence|legend=1| 82, 111, 193, 304d }}
 
Badness: 0.044108
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 442/441, 540/539, 676/675, 715/714, 5120/5103
 
Mapping: [{{val|1 2 16 14 -4 18 13}}, {{val|0 -2 -66 -54 36 -69 -43}}]
 
POTE generator: ~15/13 = 248.686
 
{{Optimal ET sequence|legend=1| 82, 111, 193, 304d }}
 
Badness: 0.025806
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 400/399, 442/441, 540/539, 676/675, 715/714, 1445/1444
 
Mapping: [{{val|1 2 16 14 -4 18 13 20}}, {{val|0 -2 -66 -54 36 -69 -43 -76}}]
 
POTE generator: ~15/13 = 248.687
 
{{Optimal ET sequence|legend=1| 82, 111, 193, 304dh }}
 
Badness: 0.019146
 
== Pluto ==
{{distinguish|plutus}}
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 4000/3969, 10976/10935
 
[[Mapping]]: [{{val|1 5 15 15}}, {{val|0 -7 -26 -25}}]
 
{{Multival|legend=1|7 26 25 25 20 -15}}
 
[[POTE generator]]: ~7/5 = 585.147
 
{{Optimal ET sequence|legend=1| 39d, 41, 80, 121, 404bd }}
 
[[Badness]]: 0.057514
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 896/891, 1375/1372
 
Mapping: [{{val|1 5 15 15 2}}, {{val|0 -7 -26 -25 3}}]
 
POTE generator: ~7/5 = 585.114
 
{{Optimal ET sequence|legend=1| 39d, 41, 80, 121 }}
 
Badness: 0.029844
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 352/351, 364/363, 540/539
 
Mapping: [{{val|1 5 15 15 2 -8}}, {{val|0 -7 -26 -25 3 24}}]
 
POTE generator: ~7/5 = 585.123
 
{{Optimal ET sequence|legend=1| 39d, 41, 80, 121 }}
 
Badness: 0.025717
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 256/255, 325/324, 352/351, 364/363, 540/539
 
Mapping: [{{val|1 5 15 15 2 -8 -12}}, {{val|0 -7 -26 -25 3 24 33}}]
 
POTE generator: ~7/5 = 585.116
 
{{Optimal ET sequence|legend=1| 39d, 41, 80, 121 }}
 
Badness: 0.021463
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
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}}]
 
POTE generator: ~7/5 = 585.109
 
{{Optimal ET sequence|legend=1| 39d, 41, 80, 121 }}
 
Badness: 0.017650
 
==== Orcus ====
Subgroup: 2.3.5.7.11.13
 
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}}]
 
POTE generator: ~7/5 = 585.111
 
{{Optimal ET sequence|legend=1| 41, 80f, 121ff }}
 
Badness: 0.033441
 
=== Plutino ===
Subgroup: 2.3.5.7.11
 
Comma list: 100/99, 245/242, 10976/10935
 
Mapping: [{{val|1 5 15 15 22}}, {{val|0 -7 -26 -25 -38}}]
 
POTE generator: ~7/5 = 585.283
 
{{Optimal ET sequence|legend=1| 39dee, 41 }}
 
Badness: 0.057966
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 100/99, 196/195, 245/242, 729/728
 
Mapping: [{{val|1 5 15 15 22 12}}, {{val|0 -7 -26 -25 -38 -17}}]
 
POTE generator: ~7/5 = 585.232
 
{{Optimal ET sequence|legend=1| 39deef, 41 }}
 
Badness: 0.040182
 
== Mirkat ==
Subgroup: 2.3.5.7


[[Comma list]]: 16875/16807, 19683/19600
[[Comma list]]: 16875/16807, 19683/19600


[[Mapping]]: [{{val| 3 2 1 2 }}, {{val| 0 6 13 14 }}]
{{Mapping|legend=1| 3 2 1 2 | 0 6 13 14 }}
 
: mapping generators: ~63/50, ~10/9
{{Multival|legend=1|18 39 42 20 16 -12}}


[[POTE generator]]: ~10/9 = 183.539
[[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 }}
{{Optimal ET sequence|legend=1| 39d, 72, 111, 183, 255 }}


[[Badness]]: 0.059376
[[Badness]] (Sintel): 1.50


=== 11-limit ===
=== 11-limit ===
Line 330: Line 76:
Comma list: 540/539, 1375/1372, 8019/8000
Comma list: 540/539, 1375/1372, 8019/8000


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


POTE generator: ~10/9 = 183.528
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=1| 39d, 72, 111, 183, 255 }}
{{Optimal ET sequence|legend=0| 39d, 72, 111, 183, 255 }}


Badness: 0.022126
Badness (Sintel): 0.731


=== 13-limit ===
=== 13-limit ===
Line 343: Line 91:
Comma list: 351/350, 540/539, 676/675, 1375/1372
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 }}]
{{Mapping|legend=0| 3 2 1 2 9 1 | 0 6 13 14 3 22 }}


POTE generator: ~10/9 = 183.577
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=1| 39df, 72, 111, 183 }}
{{Optimal ET sequence|legend=0| 39df, 72, 111, 183 }}


Badness: 0.018632
Badness (Sintel): 0.770


=== 17-limit ===
=== 17-limit ===
Line 356: Line 106:
Comma list: 351/350, 442/441, 540/539, 561/560, 715/714
Comma list: 351/350, 442/441, 540/539, 561/560, 715/714


Mapping: [{{val| 3 2 1 2 9 1 4 }}, {{val| 0 6 13 14 3 22 18 }}]
{{Mapping|legend=0| 3 2 1 2 9 1 4 | 0 6 13 14 3 22 18 }}


POTE generator: ~10/9 = 183.579
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=1| 39dfg, 72, 111, 183 }}
{{Optimal ET sequence|legend=0| 39dfg, 72, 111, 183 }}


Badness: 0.011775
Badness (Sintel): 0.600


== Eris ==
== Eris ==
Subgroup: 2.3.5.7
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
[[Comma list]]: 16875/16807, 65625/65536


[[Mapping]]: [{{val|1 10 0 6}}, {{val|0 -29 8 -11}}]
{{Mapping|legend=1| 1 -19 8 -5 | 0 29 -8 11 }}
: mapping generators: ~2, ~49/30


{{Multival|legend=1|29 -8 11 -80 -64 48}}
[[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 }}


[[POTE generator]]: ~60/49 = 348.216
{{Optimal ET sequence|legend=1| 31, 131, 162, 193, 224 }}


{{Optimal ET sequence|legend=1| 31, 131, 162, 193, 224, 1823cd, 2271cd }}
[[Badness]] (Sintel): 1.89
 
[[Badness]]: 0.074719


=== 11-limit ===
=== 11-limit ===
Line 384: Line 141:
Comma list: 540/539, 1375/1372, 65625/65536
Comma list: 540/539, 1375/1372, 65625/65536


Mapping: [{{val|1 10 0 6 20}}, {{val|0 -29 8 -11 -57}}]
{{Mapping|legend=0| 1 -19 8 -5 -37 | 0 29 -8 11 57 }}


POTE generator: ~11/9 = 348.219
Optimal tunings:  
* WE: ~2 = 1200.0218{{c}}, ~18/11 = 851.7963{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/11 = 851.7812{{c}}


{{Optimal ET sequence|legend=1| 31, 193, 224, 703, 927d, 1151cd }}
{{Optimal ET sequence|legend=0| 31, …, 193, 224, 703, 927d }}


Badness: 0.027621
Badness (Sintel): 0.913


=== 13-limit ===
=== 13-limit ===
Line 397: Line 156:
Comma list: 540/539, 625/624, 1375/1372, 4096/4095
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|legend=0| 1 -19 8 -5 -37 47 | 0 29 -8 11 57 -61 }}


POTE generator: ~11/9 = 348.213
Optimal tuning:  
* WE ~2 = 1199.9623{{c}}, ~18/11 = 851.7598{{c}}
* CWE ~2 = 1200.0000{{c}}, ~18/11 = 851.7865{{c}}


{{Optimal ET sequence|legend=1| 31, 193, 224 }}
{{Optimal ET sequence|legend=0| 31, 193, 224 }}


Badness: 0.025137
Badness (Sintel): 1.04


== Subsemifourth ==
== Subsemifourth ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val|1 -8 -4 -8}}, {{val|0 47 31 53}}]
{{Mapping|legend=1| 1 -8 -4 -8 | 0 47 31 53 }}
 
: mapping generators: ~2, ~144/125
{{Multival|legend=1|47 31 53 -60 -48 36}}


[[POTE generator]]: ~144/125 = 244.719
[[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 }}


{{Optimal ET sequence|legend=1| 49, 103, 152, 255, 407 }}
{{Optimal ET sequence|legend=1| 49, 103, 152, 255, 407 }}


[[Badness]]: 0.135173
[[Badness]] (Sintel): 3.42


=== 11-limit ===
=== 11-limit ===
Line 425: Line 189:
Comma list: 540/539, 1375/1372, 234375/234256
Comma list: 540/539, 1375/1372, 234375/234256


Mapping: [{{val|1 -8 -4 -8 -10}}, {{val|0 47 31 53 66}}]
{{Mapping|legend=0| 1 -8 -4 -8 -10 | 0 47 31 53 66 }}


POTE generator: ~121/105 = 244.719
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=1| 49, 103, 152, 255, 407, 966d }}
{{Optimal ET sequence|legend=0| 49, 103, 152, 255, 407 }}


Badness: 0.034276
Badness (Sintel): 1.13


=== 13-limit ===
=== 13-limit ===
Line 438: Line 204:
Comma list: 540/539, 847/845, 1375/1372, 1575/1573
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|legend=0| 1 -8 -4 -8 -10 -12 | 0 0 47 31 53 66 77 }}


POTE generator: ~15/13 = 244.714
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=1| 49f, 103, 152f, 255, 407f, 662df }}
{{Optimal ET sequence|legend=0| 49f, 103, 152f, 255, 407f }}


Badness: 0.028387
Badness (Sintel): 1.17


== Septendesemi ==
== Septendesemi ==
The name ''septendesemi'' means a septendecimal semitone ([[17/16]]). The septendesemi temperament (80&103) tempers out the mirkwai comma and 1959552/1953125 (''parkleiness'' comma, zotritrigu) in the 7-limit. [[183edo|183EDO]] provides an excellent tuning for 7, 11, 13, and 17-limit 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
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 16875/16807, 1959552/1953125
[[Comma list]]: 16875/16807, 1959552/1953125


[[Mapping]]: [{{val|1 -2 -1 -2}}, {{val|0 41 38 55}}]
{{Mapping|legend=1| 1 -2 -1 -2 | 0 41 38 55 }}
 
: mapping generators: ~2, ~343/324
{{Multival|legend=1|41 38 55 -35 -28 21}}


[[POTE generator]]: ~343/324 = 104.916
[[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=1| 80, 103, 183 }}
{{Optimal ET sequence|legend=0| 80, 103, 183 }}


[[Badness]]: 0.146795
[[Badness]] (Sintel): 3.71


=== 11-limit ===
=== 11-limit ===
Line 468: Line 239:
Comma list: 540/539, 1375/1372, 43923/43750
Comma list: 540/539, 1375/1372, 43923/43750


Mapping: [{{val|1 -2 -1 -2 -1}}, {{val|0 41 38 55 51}}]
{{Mapping|legend=0| 1 -2 -1 -2 -1 | 0 41 38 55 51 }}


POTE generator: ~35/33 = 104.916
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=1| 80, 103, 183 }}
{{Optimal ET sequence|legend=0| 80, 103, 183 }}


Badness: 0.041554
Badness (Sintel): 1.37


=== 13-limit ===
=== 13-limit ===
Line 481: Line 254:
Comma list: 351/350, 540/539, 1375/1372, 4225/4224
Comma list: 351/350, 540/539, 1375/1372, 4225/4224


Mapping: [{{val|1 -2 -1 -2 -1 3}}, {{val|0 41 38 55 51 8}}]
{{Mapping|legend=0| 1 -2 -1 -2 -1 3 | 0 41 38 55 51 8 }}


POTE generator: ~35/33 = 104.908
Optimal tunings:  
* WE: ~2 = 1200.1082{{c}}, ~35/33 = 104.9170{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 104.9094{{c}}


{{Optimal ET sequence|legend=1| 80, 103, 183, 469f, 652def }}
{{Optimal ET sequence|legend=0| 80, 103, 183, 469f }}


Badness: 0.027908
Badness (Sintel): 1.15


=== 17-limit ===
=== 17-limit ===
Line 494: Line 269:
Comma list: 351/350, 540/539, 561/560, 715/714, 4225/4224
Comma list: 351/350, 540/539, 561/560, 715/714, 4225/4224


Mapping: [{{val|1 -2 -1 -2 -1 3 4}}, {{val|0 41 38 55 51 8 1}}]
{{Mapping|legend=0| 1 -2 -1 -2 -1 3 4 | 0 41 38 55 51 8 1 }}
 
Optimal tunings:
* WE: ~2 = 1200.0758{{c}}, ~17/16 = 104.9158{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/16 = 104.9101{{c}}


POTE generator: ~17/16 = 104.909
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~17/16 = 104.909{{c}}


{{Optimal ET sequence|legend=1| 80, 103, 183, 469f, 652def }}
{{Optimal ET sequence|legend=0| 80, 103, 183, 469f }}


Badness: 0.020128
Badness (Sintel): 1.03


== Gaster ==
== Gaster ==
: ''For the 5-limit version of this temperament, see [[Very high accuracy temperaments #Gaster]].''
: ''For the 5-limit version, see [[Very high accuracy temperaments #Gaster]].''
{{main| Gaster temperament }}
{{Main| Gaster temperament }}
 
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.


The ''gaster'' temperament (111&113) tempers out {{monzo|-70 72 -19}} (quadbila-negu) in the 5-limit; mirkwai comma (16875/16807) and [[skeetsma]] (14348907/14336000) in the 7-limit. 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.
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.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 16875/16807, 14348907/14336000
[[Comma list]]: 16875/16807, 14348907/14336000


[[Mapping]]: [{{val|1 11 38 37}}, {{val|0 -19 -72 -69}}]
{{Mapping|legend=1| 1 -8 -34 -32 | 0 19 72 69 }}
 
: mapping generators: ~2, ~567/400
{{Multival|legend=1|19 72 69 70 56 -42}}


[[POTE generator]]: ~800/567 = 594.641
[[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 }}


{{Optimal ET sequence|legend=1| 111, 224 }}
{{Optimal ET sequence|legend=1| 111, 224 }}


[[Badness]]: 0.154521
[[Badness]] (Sintel): 3.91


=== 11-limit ===
=== 11-limit ===
Line 527: Line 311:
Comma list: 540/539, 1375/1372, 14348907/14336000
Comma list: 540/539, 1375/1372, 14348907/14336000


Mapping: [{{val|1 11 38 37 -1}}, {{val|0 -19 -72 -69 9}}]
{{Mapping|legend=0| 1 -8 -34 -32 8 | 0 19 72 69 -9 }}


POTE generator: ~512/363 = 594.639
Optimal tunings:  
* WE: ~2 = 1199.9387{{c}}, ~363/256 = 605.3300{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~363/256 = 605.3603{{c}}


{{Optimal ET sequence|legend=1| 111, 224, 783d, 1007d, 1231dd }}
{{Optimal ET sequence|legend=0| 111, 224, 783d }}


Badness: 0.054060
Badness (Sintel): 1.79


=== 13-limit ===
=== 13-limit ===
Line 540: Line 326:
Comma list: 540/539, 729/728, 1375/1372, 2200/2197
Comma list: 540/539, 729/728, 1375/1372, 2200/2197


Mapping: [{{val|1 11 38 37 -1 26}}, {{val|0 -19 -72 -69 9 -45}}]
{{Mapping|legend=0| 1 -8 -34 -32 8 -19 | 0 19 72 69 -9 45 }}


POTE generator: ~55/39 = 594.639
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=1| 111, 224, 783df, 1007df, 1231ddf }}
{{Optimal ET sequence|legend=0| 111, 224, 783df }}


Badness: 0.024882
Badness (Sintel): 1.03


=== 17-limit ===
=== 17-limit ===
Line 553: Line 341:
Comma list: 540/539, 715/714, 729/728, 936/935, 2200/2197
Comma list: 540/539, 715/714, 729/728, 936/935, 2200/2197


Mapping: [{{val|1 11 38 37 -1 26 14}}, {{val|0 -19 -72 -69 9 -45 -20}}]
{{Mapping|legend=0| 1 -8 -34 -32 8 -19 -6 | 0 19 72 69 -9 45 20 }}


POTE generator: ~24/17 = 594.636
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=1| 111, 224, 559dgg }}
{{Optimal ET sequence|legend=0| 111, 224, 559dgg }}


Badness: 0.021436
Badness (Sintel): 1.09


=== 19-limit ===
=== 19-limit ===
Line 566: Line 356:
Comma list: 324/323, 400/399, 495/494, 540/539, 715/714, 1445/1444
Comma list: 324/323, 400/399, 495/494, 540/539, 715/714, 1445/1444


Mapping: [{{val|1 11 38 37 -1 26 14 32}}, {{val|0 -19 -72 -69 9 -45 -20 -56}}]
{{Mapping|legend=0| 1 -8 -34 -32 8 -19 -6 -24 | 0 19 72 69 -9 45 20 56 }}


POTE generator: ~24/17 = 594.636
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=1| 111, 224 }}
{{Optimal ET sequence|legend=0| 111, 224 }}


Badness: 0.018370
Badness (Sintel): 1.12


=== 23-limit ===
=== 23-limit ===
Line 579: Line 371:
Comma list: 324/323, 400/399, 460/459, 495/494, 529/528, 540/539, 715/714
Comma list: 324/323, 400/399, 460/459, 495/494, 529/528, 540/539, 715/714


Mapping: [{{val|1 11 38 37 -1 26 14 32 7}}, {{val|0 -19 -72 -69 9 -45 -20 -56 -5}}]
{{Mapping|legend=0| 1 -8 -34 -32 8 -19 -6 -24 2 | 0 19 72 69 -9 45 20 56 5 }}


POTE generator: ~24/17 = 594.641
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=1| 111, 224 }}
{{Optimal ET sequence|legend=0| 111, 224 }}


Badness: 0.017619
Badness (Sintel): 1.26


=== 29-limit ===
=== 29-limit ===
Line 592: Line 386:
Comma list: 290/289, 324/323, 400/399, 460/459, 495/494, 529/528, 540/539, 715/714
Comma list: 290/289, 324/323, 400/399, 460/459, 495/494, 529/528, 540/539, 715/714


Mapping: [{{val|1 11 38 37 -1 26 14 32 7 -11}}, {{val|0 -19 -72 -69 9 -45 -20 -56 -5 32}}]
{{Mapping|legend=0| 1 -8 -34 -32 8 -19 -6 -24 2 21 | 0 19 72 69 -9 45 20 56 5 -32 }}


POTE generator: ~24/17 = 594.646
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=1| 111, 113, 224 }}
{{Optimal ET sequence|legend=0| 111, 113, 224 }}


Badness: 0.016815
Badness (Sintel): 1.41


=== 31-limit ===
=== 31-limit ===
Line 605: Line 401:
Comma list: 290/289, 324/323, 400/399, 435/434, 460/459, 495/494, 528/527, 540/539, 715/714
Comma list: 290/289, 324/323, 400/399, 435/434, 460/459, 495/494, 528/527, 540/539, 715/714


Mapping: [{{val|1 11 38 37 -1 26 14 32 7 -11 0}}, {{val|0 -19 -72 -69 9 -45 -20 -56 -5 32 10}}]
{{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 }}


POTE generator: ~24/17 = 594.644
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=1| 111, 113, 224 }}
{{Optimal ET sequence|legend=0| 111, 113, 224 }}


Badness: 0.014790
Badness (Sintel): 1.42


=== 37-limit ===
=== 37-limit ===
Line 618: Line 416:
Comma list: 290/289, 324/323, 400/399, 435/434, 460/459, 495/494, 528/527, 540/539, 667/666, 715/714
Comma list: 290/289, 324/323, 400/399, 435/434, 460/459, 495/494, 528/527, 540/539, 667/666, 715/714


Mapping: [{{val|1 11 38 37 -1 26 14 32 7 -11 0 -27}}, {{val|0 -19 -72 -69 9 -45 -20 -56 -5 32 10 65}}]
{{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 }}


POTE generator: ~24/17 = 594.644
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=1| 111, 113, 224 }}
{{Optimal ET sequence|legend=0| 111, 113, 224 }}


Badness: 0.014377
Badness (Sintel): 1.56


=== 41-limit ===
=== 41-limit ===
Line 631: Line 431:
Comma list: 290/289, 324/323, 400/399, 435/434, 460/459, 495/494, 528/527, 533/532, 540/539, 575/574, 667/666
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: [{{val|1 11 38 37 -1 26 14 32 7 -11 0 -27 45}}, {{val|0 -19 -72 -69 9 -45 -20 -56 -5 32 10 65 -80}}]
{{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: ~24/17 = 594.643
 
{{Optimal ET sequence|legend=1| 111, 113, 224 }}
 
Badness: 0.012858
 
== Subsedia ==
The generator for ''subsedia'' (10&111) is 0.5 cents flat of [[15/14]]-wide semitone and tempers out the mirkwai comma and 65536/64827 (''buzzardisma'', saquadru comma). In this temperament, three generators makes ~[[16/13]], five of them equals ~[[24/17]], twelve of them equals ~[[16/7]], sixteen of them equals ~[[3/1]], and 45 of them equals ~22/1.
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 16875/16807, 65536/64827
 
[[Mapping]]: [{{val|1 0 5 4}}, {{val|0 16 -27 -12}}]
 
{{Multival|legend=1| 16 -27 -12 -80 -64 48 }}
 
[[POTE generator]]: ~15/14 = 118.965
 
{{Optimal ET sequence|legend=1| 10, 101, 111, 121, 232d }}
 
[[Badness]]: 0.157658
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 1375/1372, 65536/64827
 
Mapping: [{{val|1 0 5 4 -1}}, {{val|0 16 -27 -12 45}}]
 
POTE generator: ~15/14 = 118.968
 
{{Optimal ET sequence|legend=1| 10, 101, 111, 121, 232d }}
 
Badness: 0.066838
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 540/539, 676/675, 1375/1372
 
Mapping: [{{val|1 0 5 4 -1 4}}, {{val|0 16 -27 -12 45 -3}}]
 
POTE generator: ~15/14 = 118.968
 
{{Optimal ET sequence|legend=1| 10, 101, 111, 121, 232d }}
 
Badness: 0.031635
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 256/255, 352/351, 442/441, 540/539, 715/714
 
Mapping: [{{val|1 0 5 4 -1 4 3}}, {{val|0 16 -27 -12 45 -3 11}}]
 
POTE generator: ~15/14 = 118.968
 
{{Optimal ET sequence|legend=1| 10, 101, 111, 121, 232dg }}
 
Badness: 0.019707
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 256/255, 352/351, 400/399, 442/441, 456/455, 715/714
 
Mapping: [{{val|1 0 5 4 -1 4 3 10}}, {{val|0 16 -27 -12 45 -3 11 -58}}]


POTE generator: ~15/14 = 118.964
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=1| 10, 101h, 111, 121, 232dg }}
{{Optimal ET sequence|legend=0| 111, 113, 224 }}


Badness: 0.017935
Badness (Sintel): 1.57


== Hemiseptisix ==
== Hemiseptisix ==
The name ''hemiseptisix'' means a half of septimal major sixth ([[12/7]]). The hemiseptisix temperament (103&121) tempers out the mirkwai comma and 95703125/95551488 (''pontiqak'' comma, lazozotritriyo) in the 7-limit. [[224edo|224EDO]] provides an excellent tuning for 7, 11, and 13-limit 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
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 16875/16807, 95703125/95551488
[[Comma list]]: 16875/16807, 95703125/95551488


[[Mapping]]: [{{val|1 -19 -7 -17}}, {{val|0 53 24 51}}]
{{Mapping|legend=1| 1 -19 -7 -17 | 0 53 24 51 }}
: mapping generators: ~2, ~98/75


{{Multival|legend=1|53 24 51 -85 -68 51}}
[[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 }}


[[POTE generator]]: ~98/75 = 466.071
{{Optimal ET sequence|legend=1| 103, 121, 224 }}


{{Optimal ET sequence|legend=1| 18, 103, 121, 224 }}
[[Badness]] (Sintel): 4.12
 
[[Badness]]: 0.162826


=== 11-limit ===
=== 11-limit ===
Line 730: Line 466:
Comma list: 540/539, 1375/1372, 2734375/2725888
Comma list: 540/539, 1375/1372, 2734375/2725888


Mapping: [{{val|1 -19 -7 -17 -28}}, {{val|0 53 24 51 81}}]
{{Mapping|legend=0| 1 -19 -7 -17 -28 | 0 53 24 51 81 }}


POTE generator: ~55/42 = 466.070
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=1| 103, 121, 224 }}
{{Optimal ET sequence|legend=0| 103, 121, 224 }}


Badness: 0.043381
Badness (Sintel): 1.43


=== 13-limit ===
=== 13-limit ===
Line 743: Line 481:
Comma list: 540/539, 625/624, 1375/1372, 2200/2197
Comma list: 540/539, 625/624, 1375/1372, 2200/2197


Mapping: [{{val|1 -19 -7 -17 -28 -13}}, {{val|0 53 24 51 81 43}}]
{{Mapping|legend=0| 1 -19 -7 -17 -28 -13 | 0 53 24 51 81 43 }}


POTE generator: ~55/42 = 466.071
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=1| 103, 121, 224 }}
{{Optimal ET sequence|legend=0| 103, 121, 224 }}


Badness: 0.021127
Badness (Sintel): 0.873


=== 17-limit ===
=== 17-limit ===
Line 756: Line 496:
Comma list: 375/374, 540/539, 625/624, 715/714, 2200/2197
Comma list: 375/374, 540/539, 625/624, 715/714, 2200/2197


Mapping: [{{val|1 -19 -7 -17 -28 -13 -13}}, {{val|0 53 24 51 81 43 44}}]
{{Mapping|legend=0| 1 -19 -7 -17 -28 -13 -13 | 0 53 24 51 81 43 44 }}
 
POTE generator: ~17/13 = 466.074


{{Optimal ET sequence|legend=1| 103, 121, 224 }}
Optimal tunings:
* WE: ~2 = 1199.8544{{c}}, ~17/13 = 466.0174{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/13 = 466.0718{{c}}


Badness: 0.018611
{{Optimal ET sequence|legend=0| 103, 121, 224 }}


== Browser ==
Badness (Sintel): 0.948
{{see also| Sensipent family }}


Subgroup: 2.3.5.7
== Grazer ==
 
Named by [[Xenllium]] in 2022, grazer may be described as the {{nowrap| 113 & 121 }} temperament.  
[[Comma list]]: 16875/16807, 78732/78125
 
[[Mapping]]: [{{val|1 6 8 10}}, {{val|0 -35 -45 -57}}]
 
{{Multival|legend=1|35 45 57 -10 -8 6}}
 
[[POTE generator]]: ~49/45 = 151.399
 
{{Optimal ET sequence|legend=1| 103, 111, 214 }}
 
[[Badness]]: 0.180645
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 1375/1372, 78732/78125
 
Mapping: [{{val|1 6 8 10 8}}, {{val|0 -35 -45 -57 -36}}]
 
POTE generator: ~12/11 = 151.405
 
{{Optimal ET sequence|legend=1| 103, 214 }}
 
Badness: 0.057634
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 540/539, 847/845, 1375/1372
 
Mapping: [{{val|1 6 8 10 8 9}}, {{val|0 -35 -45 -57 -36 -42}}]
 
POTE generator: ~12/11 = 151.403
 
{{Optimal ET sequence|legend=1| 103, 111, 214 }}
 
Badness: 0.028822
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Comma list: 351/350, 540/539, 561/560, 715/714, 847/845
[[Subgroup]]: 2.3.5.7
 
Mapping: [{{val|1 6 8 10 8 9 8}}, {{val|0 -35 -45 -57 -36 -42 -31}}]
 
POTE generator: ~12/11 = 151.397
 
{{Optimal ET sequence|legend=1| 103, 111, 214 }}
 
Badness: 0.020384
 
=== 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: [{{val|1 6 8 10 8 9 8 18}}, {{val|0 -35 -45 -57 -36 -42 -31 -109}}]
 
POTE generator: ~12/11 = 151.396
 
{{Optimal ET sequence|legend=1| 103h, 111, 214 }}
 
Badness: 0.017570
 
== Grazer ==
Subgroup: 2.3.5.7


[[Comma list]]: 16875/16807, 1071875/1062882
[[Comma list]]: 16875/16807, 1071875/1062882


[[Mapping]]: [{{val|1 -3 -4 -5}}, {{val|0 37 51 63}}]
{{Mapping|legend=1| 1 -3 -4 -5 | 0 37 51 63 }}
 
: mapping generators: ~2, ~49/45
{{Multival|legend=1|37 51 63 -5 -4 3}}


[[POTE generator]]: ~49/45 = 148.719
[[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 }}
{{Optimal ET sequence|legend=1| 113, 121, 234 }}


[[Badness]]: 0.217166
[[Badness]] (Sintel): 5.50


=== 11-limit ===
=== 11-limit ===
Line 853: Line 531:
Comma list: 540/539, 1375/1372, 218750/216513
Comma list: 540/539, 1375/1372, 218750/216513


Mapping: [{{val|1 -3 -4 -5 -1}}, {{val|0 37 51 63 36}}]
{{Mapping|legend=0| 1 -3 -4 -5 -1 | 0 37 51 63 36 }}


POTE generator: ~12/11 = 148.729
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=1| 113, 121, 234, 355e, 589cee }}
{{Optimal ET sequence|legend=0| 113, 121, 234 }}


Badness: 0.076062
Badness (Sintel): 2.51


=== 13-limit ===
=== 13-limit ===
Line 866: Line 546:
Comma list: 325/324, 364/363, 540/539, 2200/2197
Comma list: 325/324, 364/363, 540/539, 2200/2197


Mapping: [{{val|1 -3 -4 -5 -1 -2}}, {{val|0 37 51 63 36 46}}]
{{Mapping|legend=0| 1 -3 -4 -5 -1 -2 | 0 37 51 63 36 46 }}


POTE generator: ~12/11 = 148.729
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=1| 113, 121, 234, 355e, 589cee }}
{{Optimal ET sequence|legend=0| 113, 121, 234 }}


Badness: 0.036248
Badness (Sintel): 1.50


=== 17-limit ===
=== 17-limit ===
Line 879: Line 561:
Comma list: 325/324, 364/363, 540/539, 595/594, 2000/1989
Comma list: 325/324, 364/363, 540/539, 595/594, 2000/1989


Mapping: [{{val|1 -3 -4 -5 -1 -2 0}}, {{val|0 37 51 63 36 46 33}}]
{{Mapping|legend=0| 1 -3 -4 -5 -1 -2 0 | 0 37 51 63 36 46 33 }}


POTE generator: ~12/11 = 148.735
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=1| 113, 121, 234g, 355eg }}
{{Optimal ET sequence|legend=0| 113, 121, 234g }}


Badness: 0.025410
Badness (Sintel): 1.29


=== 19-limit ===
=== 19-limit ===
Line 892: Line 576:
Comma list: 325/324, 364/363, 400/399, 540/539, 595/594, 665/663
Comma list: 325/324, 364/363, 400/399, 540/539, 595/594, 665/663


Mapping: [{{val|1 -3 -4 -5 -1 -2 0 4}}, {{val|0 37 51 63 36 46 33 2}}]
{{Mapping|legend=0| 1 -3 -4 -5 -1 -2 0 4 | 0 37 51 63 36 46 33 2 }}


POTE generator: ~12/11 = 148.727
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=1| 113, 121, 234g, 355eg, 589ceegg }}
{{Optimal ET sequence|legend=0| 113, 121, 234g }}


Badness: 0.022574
Badness (Sintel): 1.37


[[Category:Canopic clan| ]] <!-- main article -->
[[Category:Canopic| ]] <!-- key article -->
[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Mirkwai clan| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Rank 2]]

Latest revision as of 18:05, 20 October 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The canopic clan of temperaments tempers out the canopic comma (monzo: [0 3 4 -5⟩, ratio: 16875/16807), a no-twos comma.

Canopus

Main article: Canopus

Subgroup: 3.5.7

Comma list: 16875/16807

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

mapping generators: ~3, ~15/7

Optimal tunings:

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

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

Badness (Sintel): 0.0996

Overview to extensions

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

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

Members of the clan discussed elsewhere are:

For no-2's extensions, see No-twos subgroup temperaments #Canopus.

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

Mirkat

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

Subgroup: 2.3.5.7

Comma list: 16875/16807, 19683/19600

Mapping: [⟨3 2 1 2], ⟨0 6 13 14]]

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

Optimal tunings:

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

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

Badness (Sintel): 1.50

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨3 2 1 2 9], ⟨0 6 13 14 3]]

Optimal tunings:

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

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

Badness (Sintel): 0.731

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

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

Badness (Sintel): 0.770

17-limit

Subgroup: 2.3.5.7.11.13.17

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

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

Optimal tunings:

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

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

Badness (Sintel): 0.600

Eris

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

Subgroup: 2.3.5.7

Comma list: 16875/16807, 65625/65536

Mapping: [⟨1 -19 8 -5], ⟨0 29 -8 11]]

mapping generators: ~2, ~49/30

Optimal tunings:

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

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

Badness (Sintel): 1.89

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

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

Badness (Sintel): 0.913

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning:

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

Optimal ET sequence: 31, 193, 224

Badness (Sintel): 1.04

Subsemifourth

Subgroup: 2.3.5.7

Comma list: 16875/16807, 26873856/26796875

Mapping: [⟨1 -8 -4 -8], ⟨0 47 31 53]]

mapping generators: ~2, ~144/125

Optimal tunings:

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

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

Badness (Sintel): 3.42

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

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

Badness (Sintel): 1.13

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

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

Badness (Sintel): 1.17

Septendesemi

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

Subgroup: 2.3.5.7

Comma list: 16875/16807, 1959552/1953125

Mapping: [⟨1 -2 -1 -2], ⟨0 41 38 55]]

mapping generators: ~2, ~343/324

Optimal tunings:

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

Optimal ET sequence: 80, 103, 183

Badness (Sintel): 3.71

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

Optimal ET sequence: 80, 103, 183

Badness (Sintel): 1.37

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

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

Badness (Sintel): 1.15

17-limit

Subgroup: 2.3.5.7.11.13.17

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

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

Optimal tunings:

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

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

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

Badness (Sintel): 1.03

Gaster

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

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

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

Subgroup: 2.3.5.7

Comma list: 16875/16807, 14348907/14336000

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

mapping generators: ~2, ~567/400

Optimal tunings:

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

Optimal ET sequence: 111, 224

Badness (Sintel): 3.91

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

Optimal ET sequence: 111, 224, 783d

Badness (Sintel): 1.79

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

Optimal ET sequence: 111, 224, 783df

Badness (Sintel): 1.03

17-limit

Subgroup: 2.3.5.7.11.13.17

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

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

Optimal tunings:

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

Optimal ET sequence: 111, 224, 559dgg

Badness (Sintel): 1.09

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

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

Optimal tunings:

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

Optimal ET sequence: 111, 224

Badness (Sintel): 1.12

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

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

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

Optimal tunings:

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

Optimal ET sequence: 111, 224

Badness (Sintel): 1.26

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

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

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

Optimal tunings:

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

Optimal ET sequence: 111, 113, 224

Badness (Sintel): 1.41

31-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31

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

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

Optimal tunings:

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

Optimal ET sequence: 111, 113, 224

Badness (Sintel): 1.42

37-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37

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

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

Optimal tunings:

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

Optimal ET sequence: 111, 113, 224

Badness (Sintel): 1.56

41-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41

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

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

Optimal tunings:

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

Optimal ET sequence: 111, 113, 224

Badness (Sintel): 1.57

Hemiseptisix

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

Subgroup: 2.3.5.7

Comma list: 16875/16807, 95703125/95551488

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

mapping generators: ~2, ~98/75

Optimal tunings:

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

Optimal ET sequence: 103, 121, 224

Badness (Sintel): 4.12

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

Optimal ET sequence: 103, 121, 224

Badness (Sintel): 1.43

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

Optimal ET sequence: 103, 121, 224

Badness (Sintel): 0.873

17-limit

Subgroup: 2.3.5.7.11.13.17

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

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

Optimal tunings:

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

Optimal ET sequence: 103, 121, 224

Badness (Sintel): 0.948

Grazer

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

Subgroup: 2.3.5.7

Comma list: 16875/16807, 1071875/1062882

Mapping: [⟨1 -3 -4 -5], ⟨0 37 51 63]]

mapping generators: ~2, ~49/45

Optimal tunings:

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

Optimal ET sequence: 113, 121, 234

Badness (Sintel): 5.50

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

Optimal ET sequence: 113, 121, 234

Badness (Sintel): 2.51

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

Optimal ET sequence: 113, 121, 234

Badness (Sintel): 1.50

17-limit

Subgroup: 2.3.5.7.11.13.17

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

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

Optimal tunings:

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

Optimal ET sequence: 113, 121, 234g

Badness (Sintel): 1.29

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

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

Optimal tunings:

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

Optimal ET sequence: 113, 121, 234g

Badness (Sintel): 1.37

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