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{{Technical data page}}
{{Technical data page}}
The '''mirkwai clan''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[mirkwai comma]] ({{monzo|legend=1| 0 3 4 -5 }}, [[ratio]]: 16875/16807), a no-twos comma.  
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.  


== Canopus ==
== Canopus ==
Line 9: Line 9:
[[Comma list]]: 16875/16807
[[Comma list]]: 16875/16807


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


[[Optimal tuning]] ([[POTE]]): ~3 = 1901.9550{{c}}, ~7/5 = 583.9584{{c}}
[[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 }}


[[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]]
[[Badness]] (Sintel): 0.0996


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


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.  
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:  
Members of the clan discussed elsewhere are:  
* ''[[Kwai]]'' (+5120/5103) → [[Hemifamity temperaments #Kwai|Hemifamity temperaments]]
* ''[[Kwai]]'' (+5120/5103) → [[Aberschismic temperaments #Kwai|Aberschismic temperaments]]
* ''[[Octokaidecal]]'' (+28/27 or 50/49) → [[Trienstonic clan #Octokaidecal|Trienstonic clan]]
* ''[[Octokaidecal]]'' (+28/27 or 50/49) → [[Trienstonic clan #Octokaidecal|Trienstonic clan]]
* ''[[Meantritone]]'' (+81/80) → [[Meantone family #Meantritone|Meantone family]]
* ''[[Meantritone]]'' (+81/80) → [[Meantone family #Meantritone|Meantone family]]
Line 31: Line 37:
* ''[[Pluto]]'' (+4000/3969) → [[Octagar temperaments #Pluto|Octagar temperaments]]
* ''[[Pluto]]'' (+4000/3969) → [[Octagar temperaments #Pluto|Octagar temperaments]]
* ''[[Bohpier]]'' (+245/243) → [[Sensamagic clan #Bohpier|Sensamagic clan]]
* ''[[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]]
* ''[[Semisept]]'' (+1728/1715 or 3136/3125) → [[Hemimean clan #Semisept|Hemimean clan]]
* ''[[Grendel]]'' (+6144/6125) → [[Porwell temperaments #Grendel|Porwell temperaments]]
* ''[[Grendel]]'' (+6144/6125) → [[Porwell temperaments #Grendel|Porwell temperaments]]
Line 37: Line 44:
* [[Sqrtphi]] (+15625/15552) → [[Kleismic family #Sqrtphi|Kleismic family]]
* [[Sqrtphi]] (+15625/15552) → [[Kleismic family #Sqrtphi|Kleismic family]]
* ''[[Rainwell]]'' (+2100875/2097152) → [[Semicomma family #Rainwell|Semicomma 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]]
* ''[[Quintiquart]]'' (+390625000/387420489) → [[Quartonic family #Quintiquart|Quartonic family]]


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


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


== Mirkat ==
== 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.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 49: Line 59:


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


[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.000{{c}}, ~10/9 = 183.539{{c}}
[[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]] (Smith): 0.059376
[[Badness]] (Sintel): 1.50


=== 11-limit ===
=== 11-limit ===
Line 61: Line 76:
Comma list: 540/539, 1375/1372, 8019/8000
Comma list: 540/539, 1375/1372, 8019/8000


Mapping: {{mapping| 3 2 1 2 9 | 0 6 13 14 3 }}
{{Mapping|legend=0| 3 2 1 2 9 | 0 6 13 14 3 }}


Optimal tuning (POTE): ~63/50 = 400.000{{c}}, ~10/9 = 183.528{{c}}
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 }}
{{Optimal ET sequence|legend=0| 39d, 72, 111, 183, 255 }}


Badness (Smith): 0.022126
Badness (Sintel): 0.731


=== 13-limit ===
=== 13-limit ===
Line 74: Line 91:
Comma list: 351/350, 540/539, 676/675, 1375/1372
Comma list: 351/350, 540/539, 676/675, 1375/1372


Mapping: {{mapping| 3 2 1 2 9 1 | 0 6 13 14 3 22 }}
{{Mapping|legend=0| 3 2 1 2 9 1 | 0 6 13 14 3 22 }}


Optimal tuning (POTE): ~63/50 = 400.000{{c}}, ~10/9 = 183.577{{c}}
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 }}
{{Optimal ET sequence|legend=0| 39df, 72, 111, 183 }}


Badness (Smith): 0.018632
Badness (Sintel): 0.770


=== 17-limit ===
=== 17-limit ===
Line 87: 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: {{mapping| 3 2 1 2 9 1 4 | 0 6 13 14 3 22 18 }}
{{Mapping|legend=0| 3 2 1 2 9 1 4 | 0 6 13 14 3 22 18 }}


Optimal tuning (POTE): ~34/27 = 400.000{{c}}, ~10/9 = 183.579{{c}}
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 }}
{{Optimal ET sequence|legend=0| 39dfg, 72, 111, 183 }}


Badness (Smith): 0.011775
Badness (Sintel): 0.600


== Eris ==
== Eris ==
The [[2.5.7 subgroup]] restriction of this temperament is [[exodia]].
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
[[Subgroup]]: 2.3.5.7
Line 102: Line 123:
[[Comma list]]: 16875/16807, 65625/65536
[[Comma list]]: 16875/16807, 65625/65536


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~60/49 = 348.216{{c}}
[[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 }}


{{Optimal ET sequence|legend=1| 31, 131, 162, 193, 224, 1823cd, 2271cd }}
{{Optimal ET sequence|legend=1| 31, 131, 162, 193, 224 }}


[[Badness]] (Smith): 0.074719
[[Badness]] (Sintel): 1.89


=== 11-limit ===
=== 11-limit ===
Line 115: Line 141:
Comma list: 540/539, 1375/1372, 65625/65536
Comma list: 540/539, 1375/1372, 65625/65536


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


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/9 = 348.219{{c}}
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=0| 31, 193, 224, 703, 927d, 1151cd }}
{{Optimal ET sequence|legend=0| 31, …, 193, 224, 703, 927d }}


Badness (Smith): 0.027621
Badness (Sintel): 0.913


=== 13-limit ===
=== 13-limit ===
Line 128: Line 156:
Comma list: 540/539, 625/624, 1375/1372, 4096/4095
Comma list: 540/539, 625/624, 1375/1372, 4096/4095


Mapping: {{mapping| 1 10 0 6 20 -14 | 0 -29 8 -11 -57 61 }}
{{Mapping|legend=0| 1 -19 8 -5 -37 47 | 0 29 -8 11 57 -61 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/9 = 348.213{{c}}
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=0| 31, 193, 224 }}
{{Optimal ET sequence|legend=0| 31, 193, 224 }}


Badness (Smith): 0.025137
Badness (Sintel): 1.04


== Subsemifourth ==
== Subsemifourth ==
Line 141: Line 171:
[[Comma list]]: 16875/16807, 26873856/26796875
[[Comma list]]: 16875/16807, 26873856/26796875


{{Mapping|legend=1| 1 39 27 45 | 0 -47 -31 -53 }}
{{Mapping|legend=1| 1 -8 -4 -8 | 0 47 31 53 }}
: mapping generators: ~2, ~125/72
: mapping generators: ~2, ~144/125


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~144/125 = 244.719{{c}}
[[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]] (Smith): 0.135173
[[Badness]] (Sintel): 3.42


=== 11-limit ===
=== 11-limit ===
Line 155: Line 189:
Comma list: 540/539, 1375/1372, 234375/234256
Comma list: 540/539, 1375/1372, 234375/234256


Mapping: {{mapping| 1 39 27 45 56 | 0 -47 -31 -53 -66 }}
{{Mapping|legend=0| 1 -8 -4 -8 -10 | 0 47 31 53 66 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~121/105 = 244.719{{c}}
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, 966d }}
{{Optimal ET sequence|legend=0| 49, 103, 152, 255, 407 }}


Badness (Smith): 0.034276
Badness (Sintel): 1.13


=== 13-limit ===
=== 13-limit ===
Line 168: Line 204:
Comma list: 540/539, 847/845, 1375/1372, 1575/1573
Comma list: 540/539, 847/845, 1375/1372, 1575/1573


Mapping: {{mapping| 1 39 27 45 56 65 | 0 -47 -31 -53 -66 -77 }}
{{Mapping|legend=0| 1 -8 -4 -8 -10 -12 | 0 0 47 31 53 66 77 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~15/13 = 244.714{{c}}
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, 662df }}
{{Optimal ET sequence|legend=0| 49f, 103, 152f, 255, 407f }}


Badness (Smith): 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]] 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
Line 183: Line 221:
[[Comma list]]: 16875/16807, 1959552/1953125
[[Comma list]]: 16875/16807, 1959552/1953125


{{Mapping|legend=1| 1 39 37 53 | 0 -41 -38 -55 }}
{{Mapping|legend=1| 1 -2 -1 -2 | 0 41 38 55 }}
: mapping generators: ~2, ~648/343
: mapping generators: ~2, ~343/324


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~343/324 = 104.916{{c}}
[[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 }}
{{Optimal ET sequence|legend=0| 80, 103, 183 }}


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


=== 11-limit ===
=== 11-limit ===
Line 197: Line 239:
Comma list: 540/539, 1375/1372, 43923/43750
Comma list: 540/539, 1375/1372, 43923/43750


Mapping: {{mapping| 1 39 37 53 50 | 0 -41 -38 -55 -51 }}
{{Mapping|legend=0| 1 -2 -1 -2 -1 | 0 41 38 55 51 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~35/33 = 104.916{{c}}
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 }}
{{Optimal ET sequence|legend=0| 80, 103, 183 }}


Badness (Smith): 0.041554
Badness (Sintel): 1.37


=== 13-limit ===
=== 13-limit ===
Line 210: Line 254:
Comma list: 351/350, 540/539, 1375/1372, 4225/4224
Comma list: 351/350, 540/539, 1375/1372, 4225/4224


Mapping: {{mapping| 1 39 37 53 50 11 | 0 -41 -38 -55 -51 -8 }}
{{Mapping|legend=0| 1 -2 -1 -2 -1 3 | 0 41 38 55 51 8 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~35/33 = 104.908{{c}}
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=0| 80, 103, 183, 469f, 652def }}
{{Optimal ET sequence|legend=0| 80, 103, 183, 469f }}


Badness (Smith): 0.027908
Badness (Sintel): 1.15


=== 17-limit ===
=== 17-limit ===
Line 223: 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: {{mapping| 1 39 37 53 50 11 5 | 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}}


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


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


Badness (Smith): 0.020128
Badness (Sintel): 1.03


== Gaster ==
== Gaster ==
Line 235: Line 285:
{{Main| Gaster temperament }}
{{Main| Gaster temperament }}


The gaster temperament (111 & 113) tempers out {{monzo| -70 72 -19 }} (quadbila-negu) in the 5-limit; mirkwai comma (16875/16807) and [[scheme comma]] (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.
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.  
 
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
Line 241: Line 293:
[[Comma list]]: 16875/16807, 14348907/14336000
[[Comma list]]: 16875/16807, 14348907/14336000


{{Mapping|legend=1| 1 11 38 37 | 0 -19 -72 -69 }}
{{Mapping|legend=1| 1 -8 -34 -32 | 0 19 72 69 }}
: mapping generators: ~2, ~567/400


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~800/567 = 594.641{{c}}
[[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]] (Smith): 0.154521
[[Badness]] (Sintel): 3.91


=== 11-limit ===
=== 11-limit ===
Line 254: Line 311:
Comma list: 540/539, 1375/1372, 14348907/14336000
Comma list: 540/539, 1375/1372, 14348907/14336000


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


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~512/363 = 594.639{{c}}
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=0| 111, 224, 783d, 1007d, 1231dd }}
{{Optimal ET sequence|legend=0| 111, 224, 783d }}


Badness (Smith): 0.054060
Badness (Sintel): 1.79


=== 13-limit ===
=== 13-limit ===
Line 267: Line 326:
Comma list: 540/539, 729/728, 1375/1372, 2200/2197
Comma list: 540/539, 729/728, 1375/1372, 2200/2197


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


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~55/39 = 594.639{{c}}
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, 1007df, 1231ddf }}
{{Optimal ET sequence|legend=0| 111, 224, 783df }}


Badness (Smith): 0.024882
Badness (Sintel): 1.03


=== 17-limit ===
=== 17-limit ===
Line 280: 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: {{mapping| 1 11 38 37 -1 26 14 | 0 -19 -72 -69 9 -45 -20 }}
{{Mapping|legend=0| 1 -8 -34 -32 8 -19 -6 | 0 19 72 69 -9 45 20 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~24/17 = 594.636{{c}}
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 }}
{{Optimal ET sequence|legend=0| 111, 224, 559dgg }}


Badness (Smith): 0.021436
Badness (Sintel): 1.09


=== 19-limit ===
=== 19-limit ===
Line 293: 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: {{mapping| 1 11 38 37 -1 26 14 32 | 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 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~24/17 = 594.636{{c}}
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 }}
{{Optimal ET sequence|legend=0| 111, 224 }}


Badness (Smith): 0.018370
Badness (Sintel): 1.12


=== 23-limit ===
=== 23-limit ===
Line 306: 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: {{mapping| 1 11 38 37 -1 26 14 32 7 | 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 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~24/17 = 594.641{{c}}
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 }}
{{Optimal ET sequence|legend=0| 111, 224 }}


Badness (Smith): 0.017619
Badness (Sintel): 1.26


=== 29-limit ===
=== 29-limit ===
Line 319: 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: {{mapping| 1 11 38 37 -1 26 14 32 7 -11 | 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 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~24/17 = 594.646{{c}}
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 }}
{{Optimal ET sequence|legend=0| 111, 113, 224 }}


Badness (Smith): 0.016815
Badness (Sintel): 1.41


=== 31-limit ===
=== 31-limit ===
Line 332: 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: {{mapping| 1 11 38 37 -1 26 14 32 7 -11 0 | 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 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~24/17 = 594.644{{c}}
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 }}
{{Optimal ET sequence|legend=0| 111, 113, 224 }}


Badness (Smith): 0.014790
Badness (Sintel): 1.42


=== 37-limit ===
=== 37-limit ===
Line 345: 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: {{mapping| 1 11 38 37 -1 26 14 32 7 -11 0 -27 | 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 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~24/17 = 594.644{{c}}
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 }}
{{Optimal ET sequence|legend=0| 111, 113, 224 }}


Badness (Smith): 0.014377
Badness (Sintel): 1.56


=== 41-limit ===
=== 41-limit ===
Line 358: 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: {{mapping| 1 11 38 37 -1 26 14 32 7 -11 0 -27 45 | 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 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~24/17 = 594.643{{c}}
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 }}
{{Optimal ET sequence|legend=0| 111, 113, 224 }}


Badness (Smith): 0.012858
Badness (Sintel): 1.57
 
== Subsedia ==
The generator for subsedia (10 & 111) is 0.5 cents flat of [[15/14]]-wide semitone and tempers out the mirkwai comma and [[buzzardsma]]. 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|legend=1| 1 0 5 4 | 0 16 -27 -12 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~15/14 = 118.965{{c}}
 
{{Optimal ET sequence|legend=1| 10, 101, 111, 121, 232d }}
 
[[Badness]] (Smith): 0.157658
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 1375/1372, 65536/64827
 
Mapping: {{mapping| 1 0 5 4 -1 | 0 16 -27 -12 45 }}
 
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~15/14 = 118.968{{c}}
 
{{Optimal ET sequence|legend=0| 10, 101, 111, 121, 232d }}
 
Badness (Smith): 0.066838
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 540/539, 676/675, 1375/1372
 
Mapping: {{mapping| 1 0 5 4 -1 4 | 0 16 -27 -12 45 -3 }}
 
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~15/14 = 118.968{{c}}
 
{{Optimal ET sequence|legend=0| 10, 101, 111, 121, 232d }}
 
Badness (Smith): 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: {{mapping| 1 0 5 4 -1 4 3 | 0 16 -27 -12 45 -3 11 }}
 
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~15/14 = 118.968{{c}}
 
{{Optimal ET sequence|legend=0| 10, 101, 111, 121, 232dg }}
 
Badness (Smith): 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: {{mapping| 1 0 5 4 -1 4 3 10 | 0 16 -27 -12 45 -3 11 -58 }}
 
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~15/14 = 118.964{{c}}
 
{{Optimal ET sequence|legend=0| 10, 101h, 111, 121, 232dg }}
 
Badness (Smith): 0.017935


== 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]] 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
Line 440: Line 448:
[[Comma list]]: 16875/16807, 95703125/95551488
[[Comma list]]: 16875/16807, 95703125/95551488


{{Mapping|legend=1| 1 34 17 34 | 0 -53 -24 -51 }}
{{Mapping|legend=1| 1 -19 -7 -17 | 0 53 24 51 }}
: mapping generators: ~2, ~75/49
: mapping generators: ~2, ~98/75


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~98/75 = 466.071{{c}}
[[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 }}
{{Optimal ET sequence|legend=1| 103, 121, 224 }}


[[Badness]] (Smith): 0.162826
[[Badness]] (Sintel): 4.12


=== 11-limit ===
=== 11-limit ===
Line 454: Line 466:
Comma list: 540/539, 1375/1372, 2734375/2725888
Comma list: 540/539, 1375/1372, 2734375/2725888


Mapping: {{mapping| 1 34 17 34 53 | 0 -53 -24 -51 -81 }}
{{Mapping|legend=0| 1 -19 -7 -17 -28 | 0 53 24 51 81 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~55/42 = 466.070{{c}}
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 }}
{{Optimal ET sequence|legend=0| 103, 121, 224 }}


Badness (Smith): 0.043381
Badness (Sintel): 1.43


=== 13-limit ===
=== 13-limit ===
Line 467: Line 481:
Comma list: 540/539, 625/624, 1375/1372, 2200/2197
Comma list: 540/539, 625/624, 1375/1372, 2200/2197


Mapping: {{mapping| 1 34 17 34 53 30 | 0 -53 -24 -51 -81 -43 }}
{{Mapping|legend=0| 1 -19 -7 -17 -28 -13 | 0 53 24 51 81 43 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~55/42 = 466.071{{c}}
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 }}
{{Optimal ET sequence|legend=0| 103, 121, 224 }}


Badness (Smith): 0.021127
Badness (Sintel): 0.873


=== 17-limit ===
=== 17-limit ===
Line 480: 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: {{mapping| 1 34 17 34 53 30 31 | 0 -53 -24 -51 -81 -43 -44 }}
{{Mapping|legend=0| 1 -19 -7 -17 -28 -13 -13 | 0 53 24 51 81 43 44 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~17/13 = 466.074{{c}}
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 }}
{{Optimal ET sequence|legend=0| 103, 121, 224 }}


Badness (Smith): 0.018611
Badness (Sintel): 0.948


== Browser ==
== Grazer ==
{{See also| Sensipent family }}
Named by [[Xenllium]] in 2022, grazer may be described as the {{nowrap| 113 & 121 }} 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 MOSes of 8, 15, 23, and 31 notes despite no harmonics from the root. It can be considered a detemperament of 8et, with a generator very slightly sharp of 1\8.


[[Subgroup]]: 2.3.5.7
[[Comma list]]: 16875/16807, 78732/78125
{{Mapping|legend=1| 1 6 8 10 | 0 -35 -45 -57 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~49/45 = 151.399{{c}}
{{Optimal ET sequence|legend=1| 103, 111, 214 }}
[[Badness]] (Smith): 0.180645
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 540/539, 1375/1372, 78732/78125
Mapping: {{mapping| 1 6 8 10 8 | 0 -35 -45 -57 -36 }}
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~12/11 = 151.405{{c}}
{{Optimal ET sequence|legend=0| 103, 214 }}
Badness (Smith): 0.057634
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 351/350, 540/539, 847/845, 1375/1372
Mapping: {{mapping| 1 6 8 10 8 9 | 0 -35 -45 -57 -36 -42 }}
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~12/11 = 151.403{{c}}
{{Optimal ET sequence|legend=0| 103, 111, 214 }}
Badness (Smith): 0.028822
=== 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 6 8 10 8 9 8 | 0 -35 -45 -57 -36 -42 -31 }}
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~12/11 = 151.397{{c}}
{{Optimal ET sequence|legend=0| 103, 111, 214 }}
Badness (Smith): 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: {{mapping| 1 6 8 10 8 9 8 18 | 0 -35 -45 -57 -36 -42 -31 -109 }}
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~12/11 = 151.396{{c}}
{{Optimal ET sequence|legend=0| 103h, 111, 214 }}
Badness (Smith): 0.017570
== Grazer ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


{{Mapping|legend=1| 1 34 47 58 | 0 -37 -51 -63 }}
{{Mapping|legend=1| 1 -3 -4 -5 | 0 37 51 63 }}
: mapping generators: ~2, ~90/49
: mapping generators: ~2, ~49/45


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~49/45 = 148.719{{c}}
[[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]] (Smith): 0.217166
[[Badness]] (Sintel): 5.50


=== 11-limit ===
=== 11-limit ===
Line 576: Line 531:
Comma list: 540/539, 1375/1372, 218750/216513
Comma list: 540/539, 1375/1372, 218750/216513


Mapping: {{mapping| 1 34 47 58 35 | 0 -37 -51 -63 -36 }}
{{Mapping|legend=0| 1 -3 -4 -5 -1 | 0 37 51 63 36 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~12/11 = 148.729{{c}}
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, 355e, 589cee }}
{{Optimal ET sequence|legend=0| 113, 121, 234 }}


Badness (Smith): 0.076062
Badness (Sintel): 2.51


=== 13-limit ===
=== 13-limit ===
Line 589: Line 546:
Comma list: 325/324, 364/363, 540/539, 2200/2197
Comma list: 325/324, 364/363, 540/539, 2200/2197


Mapping: {{mapping| 1 34 47 58 35 44 | 0 -37 -51 -63 -36 -46 }}
{{Mapping|legend=0| 1 -3 -4 -5 -1 -2 | 0 37 51 63 36 46 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~12/11 = 148.729{{c}}
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, 355e, 589cee }}
{{Optimal ET sequence|legend=0| 113, 121, 234 }}


Badness (Smith): 0.036248
Badness (Sintel): 1.50


=== 17-limit ===
=== 17-limit ===
Line 602: 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: {{mapping| 1 34 47 58 35 44 33 | 0 -37 -51 -63 -36 -46 -33 }}
{{Mapping|legend=0| 1 -3 -4 -5 -1 -2 0 | 0 37 51 63 36 46 33 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~12/11 = 148.735{{c}}
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, 355eg }}
{{Optimal ET sequence|legend=0| 113, 121, 234g }}


Badness (Smith): 0.025410
Badness (Sintel): 1.29


=== 19-limit ===
=== 19-limit ===
Line 615: 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: {{mapping| 1 34 47 58 35 44 33 6 | 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 }}
 
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~12/11 = 148.727{{c}}


{{Optimal ET sequence|legend=0| 113, 121, 234g, 355eg, 589ceegg }}
Optimal tunings:
* WE: ~2 = 1199.7269{{c}}, ~12/11 = 148.6928{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 148.7227{{c}}


Badness (Smith): 0.022574
{{Optimal ET sequence|legend=0| 113, 121, 234g }}


== References ==
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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