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{{Technical data page}}
{{Technical data page}}
[[Temperament]]s of the '''mirkwai clan''' [[Tempering out|temper out]] the [[mirkwai comma]], {{monzo| 0 3 4 -5 }} = 16875/16807, a no-twos comma.  
The '''canopus clan''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the '''canopus''' or '''[[mirkwai 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=2| 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}}
: [[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 tuning]] ([[POTE]]): ~3 = 1\1edt, ~7/5 = 583.9584
[[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.  
The full 7-limit extensions' relation to canopus is clearer if the mapping is normalized in terms of 3.5.7.2. In fact, the strong extensions are nusecond and octoid. These temperaments are distributed into different temperament collection pages.
* ''[[Nusecond]]'' (+126/125) → [[Starling temperaments #Nusecond|Starling temperaments]]
* ''[[Octoid]]'' (+4375/4374) → [[Ragismic microtemperaments #Octoid|Ragismic microtemperaments]]


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. Miracle tempers out [[225/224]]. Pluto tempers out [[4000/3969]]. These split the generator in five. Quanharuk tempers out [[32805/32768]], splitting the generator in three with a 1/5-twelfth period. Semisept tempers out [[1728/1715]] and [[3136/3125]], splitting the generator in six. Kwai tempers out [[5120/5103]], splitting the generator in ten. 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 considered below are grendel, kwai, pluto, mirkat, eris, subsemifourth, septendesemi, gaster, subsedia, hemiseptisix, browser, and grazer. Discussed elsewhere are:  
Members of the clan discussed elsewhere are:  
* ''[[Kwai]]'' (+5120/5103) → [[Hemifamity temperaments #Kwai|Hemifamity temperaments]]
* ''[[Octokaidecal]]'' (+28/27 or 50/49) → [[Trienstonic clan #Octokaidecal|Trienstonic clan]]
* ''[[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]]
* ''[[Nusecond]]'' (+126/125) → [[Starling temperaments #Nusecond|Starling temperaments]]
* ''[[Quanharuk]]'' (+32805/32768) → [[Schismatic family #Quanharuk|Schismatic family]]
* [[Miracle]] (+225/224) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* [[Miracle]] (+225/224) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* ''[[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]]
* ''[[Quinmage]]'' (+3125/3072) → [[Magic family #Quinmage|Magic family]]
* ''[[Quinmage]]'' (+3125/3072) → [[Magic family #Quinmage|Magic family]]
* ''[[Octoid]]'' (+4375/4374) → [[Ragismic microtemperaments #Octoid|Ragismic microtemperaments]]
* ''[[Familia]]'' (+1600000/1594323) → [[Amity family #Familia|Amity family]]
* [[Sqrtphi]] (+15625/15552) → [[Kleismic family #Sqrtphi|Kleismic family]]
* [[Sqrtphi]] (+15625/15552) → [[Kleismic family #Sqrtphi|Kleismic family]]
* ''[[Quanharuk]]'' (+32805/32768) → [[Schismatic family #Quanharuk|Schismatic family]]
* ''[[Familia]]'' (+1600000/1594323) → [[Amity family #Familia|Amity family]]
* ''[[Rainwell]]'' (+2100875/2097152) → [[Semicomma family #Rainwell|Semicomma family]]
* ''[[Rainwell]]'' (+2100875/2097152) → [[Semicomma family #Rainwell|Semicomma 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-twos'' extensions, see [[No-twos subgroup temperaments #Canopus]].


== Grendel ==
Considered below are mirkat, eris, subsemifourth, septendesemi, gaster, hemiseptisix, browser, and grazer, in the order of increasing [[badness]].  
: ''For the 5-limit version of this temperament, see [[Syntonic–31 equivalence continuum #Counterwürschmidt]].''


[[Subgroup]]: 2.3.5.7
== 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.  
[[Comma list]]: 6144/6125, 16875/16807
 
{{Mapping|legend=1| 1 9 2 7 | 0 -23 1 -13 }}
 
: mapping generators: ~2, ~5/4
 
{{Multival|legend=1| 23 -1 13 -55 -44 33 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 386.863
 
{{Optimal ET sequence|legend=1| 31, 90, 121, 152, 335d }}
 
[[Badness]]: 0.051834
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 1375/1372, 5632/5625
 
Mapping: {{mapping| 1 9 2 7 17 | 0 -23 1 -13 -42 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~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: {{mapping| 1 9 2 7 17 -5 | 0 -23 1 -13 -42 27 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~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: {{mapping| 1 9 2 7 17 -5 -3 | 0 -23 1 -13 -42 27 22 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~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
 
Mapping: {{mapping| 1 9 2 7 17 -5 -3 -8 | 0 -23 1 -13 -42 27 22 38 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 386.819
 
{{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]].''
 
Named by [[Gene Ward Smith]] in 2004 for its "bridgeability"<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10766.html Yahoo! Tuning Group | ''Kwai'']</ref>, kwai is generated by a fifth, and can be described as 41 & 70.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 5120/5103, 16875/16807
 
{{Mapping|legend=1| 1 0 -50 -40 | 0 1 33 27 }}
 
: mapping generators: ~2, ~3
 
{{Multival|legend=1| 1 33 27 50 40 -30 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~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: {{mapping| 1 0 -50 -40 32 | 0 1 33 27 -18 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~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: {{mapping| 1 0 -50 -40 32 27 | 0 1 33 27 -18 -21 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~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: {{mapping| 1 0 -50 -40 32 27 58 | 0 1 33 27 -18 -21 -34 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~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: {{mapping| 1 0 -50 -40 32 27 58 -56 | 0 1 33 27 -18 -21 -34 38 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~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: {{mapping| 1 0 -50 -40 32 -51 | 0 2 66 54 -36 69 }}
 
: mapping generators: ~2, ~26/15
 
Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 951.314
 
{{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: {{mapping| 1 0 -50 -40 32 -51 -30 | 0 2 66 54 -36 69 43 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 951.314
 
{{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: {{mapping| 1 0 -50 -40 32 -51 -30 -56 | 0 2 66 54 -36 69 43 76 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 951.313
 
{{Optimal ET sequence|legend=1| 82, 111, 193, 304dh }}
 
Badness: 0.019146
 
== Pluto ==
{{Distinguish| plutus }}
 
Pluto, named by [[Gene Ward Smith]] in 2010<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_18465.html Yahoo! Tuning Group | ''19-limit Pluto temperament'']</ref>, can be described as the 41 & 80 temperament. It is generated by a sharpened [[7/5]], and [[121edo|59\121]] is about perfect as a tuning.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4000/3969, 10976/10935
 
{{Mapping|legend=1| 1 5 15 15 | 0 -7 -26 -25 }}
 
{{Multival|legend=1| 7 26 25 25 20 -15 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~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: {{mapping| 1 5 15 15 2 | 0 -7 -26 -25 3 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~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: {{mapping| 1 5 15 15 2 -8 | 0 -7 -26 -25 3 24 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~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: {{mapping| 1 5 15 15 2 -8 -12 | 0 -7 -26 -25 3 24 33 }}
Optimal tuning (POTE): ~2 = 1\1, ~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: {{mapping| 1 5 15 15 2 -8 -12 14 | 0 -7 -26 -25 3 24 33 -20 }}
Optimal tuning (POTE): ~2 = 1\1, ~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: {{mapping| 1 5 15 15 2 12 | 0 -7 -26 -25 3 -17 }}
Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 585.111
{{Optimal ET sequence|legend=1| 39df, 41, 80f, 121ff }}
Badness: 0.033441
=== Plutino ===
Subgroup: 2.3.5.7.11
Comma list: 100/99, 245/242, 10976/10935
Mapping: {{mapping| 1 5 15 15 22 | 0 -7 -26 -25 -38 }}
Optimal tuning (POTE): ~2 = 1\1, ~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: {{mapping| 1 5 15 15 22 12 | 0 -7 -26 -25 -38 -17 }}
Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 585.232
{{Optimal ET sequence|legend=1| 39deef, 41 }}
Badness: 0.040182
== Mirkat ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 340: Line 58:


{{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


{{Multival|legend=1| 18 39 42 20 16 -12 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~63/50 = 400.0277{{c}}, ~10/9 = 183.5515{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 183.539
: [[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 356: Line 77:
Mapping: {{mapping| 3 2 1 2 9 | 0 6 13 14 3 }}
Mapping: {{mapping| 3 2 1 2 9 | 0 6 13 14 3 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 369: Line 92:
Mapping: {{mapping| 3 2 1 2 9 1 | 0 6 13 14 3 22 }}
Mapping: {{mapping| 3 2 1 2 9 1 | 0 6 13 14 3 22 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 382: Line 107:
Mapping: {{mapping| 3 2 1 2 9 1 4 | 0 6 13 14 3 22 18 }}
Mapping: {{mapping| 3 2 1 2 9 1 4 | 0 6 13 14 3 22 18 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 ==
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


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~60/49 = 348.216
[[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]]: 0.074719
[[Badness]] (Sintel): 1.89


=== 11-limit ===
=== 11-limit ===
Line 408: Line 140:
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: {{mapping| 1 -19 8 -5 -37 | 0 29 -8 11 57 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 421: Line 155:
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: {{mapping| 1 -19 8 -5 -37 47 | 0 29 -8 11 57 -61 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 ==
Line 434: Line 170:
[[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, ~144/125
: mapping generators: ~2, ~125/72


{{Multival|legend=1| 47 31 53 -60 -48 36 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1199.9182{{c}}, ~144/125 = 244.7020{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~144/125 = 244.719
: [[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 451: Line 188:
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: {{mapping| 1 -8 -4 -8 -10 | 0 47 31 53 66 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 464: Line 203:
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: {{mapping| 1 -8 -4 -8 -10 -12 | 0 0 47 31 53 66 77 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 &amp; 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 479: Line 220:
[[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, ~343/324


: mapping generators: ~2, ~648/343
[[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 }}


{{Multival|legend=1| 41 38 55 -35 -28 21 }}
{{Optimal ET sequence|legend=0| 80, 103, 183 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~343/324 = 104.916
[[Badness]] (Sintel): 3.71
 
{{Optimal ET sequence|legend=1| 80, 103, 183 }}
 
[[Badness]]: 0.146795


=== 11-limit ===
=== 11-limit ===
Line 496: Line 238:
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: {{mapping| 1 -2 -1 -2 -1 | 0 41 38 55 51 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 509: Line 253:
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: {{mapping| 1 -2 -1 -2 -1 3 | 0 41 38 55 51 8 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 522: Line 268:
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: {{mapping| 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 = 1\1, ~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 }}


The ''gaster'' temperament (111 &amp; 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.
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 540: Line 292:
[[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
{{Multival|legend=1| 19 72 69 70 56 -42 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~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 555: Line 310:
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: {{mapping| 1 -8 -34 -32 8 | 0 19 72 69 -9 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 568: Line 325:
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: {{mapping| 1 -8 -34 -32 8 -19 | 0 19 72 69 -9 45 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 581: Line 340:
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: {{mapping| 1 -8 -34 -32 8 -19 -6 | 0 19 72 69 -9 45 20 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 594: Line 355:
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: {{mapping| 1 -8 -34 -32 8 -19 -6 -24 | 0 19 72 69 -9 45 20 56 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 607: Line 370:
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: {{mapping| 1 -8 -34 -32 8 -19 -6 -24 2 | 0 19 72 69 -9 45 20 56 5 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 620: Line 385:
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: {{mapping| 1 -8 -34 -32 8 -19 -6 -24 2 21 | 0 19 72 69 -9 45 20 56 5 -32 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 633: Line 400:
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: {{mapping| 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 = 1\1, ~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 646: Line 415:
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: {{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 tuning (POTE): ~2 = 1\1, ~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 659: Line 430:
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: {{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 tuning (POTE): ~2 = 1\1, ~24/17 = 594.643
 
{{Optimal ET sequence|legend=1| 111, 113, 224 }}
 
Badness: 0.012858
 
== Subsedia ==
The generator for ''subsedia'' (10 &amp; 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|legend=1| 1 0 5 4 | 0 16 -27 -12 }}
 
{{Multival|legend=1| 16 -27 -12 -80 -64 48 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~15/14 = 118.965


{{Optimal ET sequence|legend=1| 10, 101, 111, 121, 232d }}
Optimal tunings:
* WE: ~2 = 1199.9179{{c}}, ~17/12 = 605.3156{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3567{{c}}


[[Badness]]: 0.157658
{{Optimal ET sequence|legend=0| 111, 113, 224 }}


=== 11-limit ===
Badness (Sintel): 1.57
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 = 1\1, ~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: {{mapping| 1 0 5 4 -1 4 | 0 16 -27 -12 45 -3 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~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: {{mapping| 1 0 5 4 -1 4 3 | 0 16 -27 -12 45 -3 11 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~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: {{mapping| 1 0 5 4 -1 4 3 10 | 0 16 -27 -12 45 -3 11 -58 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 118.964
 
{{Optimal ET sequence|legend=1| 10, 101h, 111, 121, 232dg }}
 
Badness: 0.017935


== Hemiseptisix ==
== Hemiseptisix ==
The name ''hemiseptisix'' means a half of septimal major sixth ([[12/7]]). The hemiseptisix temperament (103 &amp; 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 743: Line 447:
[[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, ~98/75
: mapping generators: ~2, ~75/49
 
{{Multival|legend=1| 53 24 51 -85 -68 51 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~98/75 = 466.071
[[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]]: 0.162826
[[Badness]] (Sintel): 4.12


=== 11-limit ===
=== 11-limit ===
Line 760: Line 465:
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: {{mapping| 1 -19 -7 -17 -28 | 0 53 24 51 81 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 773: Line 480:
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: {{mapping| 1 -19 -7 -17 -28 -13 | 0 53 24 51 81 43 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 786: Line 495:
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: {{mapping| 1 -19 -7 -17 -28 -13 -13 | 0 53 24 51 81 43 44 }}


Optimal tuning (POTE): ~2 = 1\1, ~17/13 = 466.074
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=1| 103, 121, 224 }}
{{Optimal ET sequence|legend=0| 103, 121, 224 }}


Badness: 0.018611
Badness (Sintel): 0.948


== Browser ==
== Browser ==
{{See also| Sensipent family }}
{{See also| Sensipent family }}
Named by [[Xenllium]] in 2022, browser may be described as the {{nowrap| 103 & 111 }} temperament.
This can also be considered a [[non-over-1 temperament]], with considerable scope for harmony in the 2.5/3.7/3.11/3.13/3.17/3 subgroup with mos scales of 8, 15, 23, and 31 notes despite no harmonics from the root. It can be considered a detemperament of 8d-et, with a generator very slightly flat of 7\8.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 801: Line 516:
[[Comma list]]: 16875/16807, 78732/78125
[[Comma list]]: 16875/16807, 78732/78125


{{Mapping|legend=1| 1 6 8 10 | 0 -35 -45 -57 }}
{{Mapping|legend=1| 1 -29 -37 -47 | 0 35 45 57 }}
: mapping generators: ~2, ~90/49


{{Multival|legend=1| 35 45 57 -10 -8 6 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1199.9313{{c}}, ~90/49 = 1048.5414{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/45 = 151.399
: [[error map]]: {{val| -0.069 -1.013 +0.592 +1.264 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~90/49 = 1048.5998{{c}}
: error map: {{val| 0.000 -0.962 +0.677 +1.362 }}


{{Optimal ET sequence|legend=1| 103, 111, 214 }}
{{Optimal ET sequence|legend=1| 103, 111, 214 }}


[[Badness]]: 0.180645
[[Badness]] (Sintel): 4.57


=== 11-limit ===
=== 11-limit ===
Line 816: Line 534:
Comma list: 540/539, 1375/1372, 78732/78125
Comma list: 540/539, 1375/1372, 78732/78125


Mapping: {{mapping| 1 6 8 10 8 | 0 -35 -45 -57 -36 }}
Mapping: {{mapping| 1 -29 -37 -47 -28 | 0 35 45 57 36 }}


Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 151.405
Optimal tunings:
* WE: ~2 = 1200.1344{{c}}, ~11/6 = 1048.7124{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1048.5981{{c}}


{{Optimal ET sequence|legend=1| 103, 214 }}
{{Optimal ET sequence|legend=0| 103, 214 }}


Badness: 0.057634
Badness (Sintel): 1.91


=== 13-limit ===
=== 13-limit ===
Line 829: Line 549:
Comma list: 351/350, 540/539, 847/845, 1375/1372
Comma list: 351/350, 540/539, 847/845, 1375/1372


Mapping: {{mapping| 1 6 8 10 8 9 | 0 -35 -45 -57 -36 -42 }}
Mapping: {{mapping| 1 -29 -37 -47 -28 -33 | 0 35 45 57 36 42 }}


Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 151.403
Optimal tunings:
* WE: ~2 = 1200.1344{{c}}, ~11/6 = 1048.7124{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1048.5984{{c}}


{{Optimal ET sequence|legend=1| 103, 111, 214 }}
{{Optimal ET sequence|legend=0| 103, 111, 214 }}


Badness: 0.028822
Badness (Sintel): 1.19


=== 17-limit ===
=== 17-limit ===
Line 842: Line 564:
Comma list: 351/350, 540/539, 561/560, 715/714, 847/845
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 }}
Mapping: {{mapping| 1 -29 -37 -47 -28 -33 -23 | 0 35 45 57 36 42 31 }}


Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 151.397
Optimal tunings:
* WE: ~2 = 1199.9191{{c}}, ~11/6 = 1048.5324{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1048.6014{{c}}


{{Optimal ET sequence|legend=1| 103, 111, 214 }}
{{Optimal ET sequence|legend=0| 103, 111, 214 }}


Badness: 0.020384
Badness (Sintel): 1.04


=== 19-limit ===
=== 19-limit ===
Line 855: Line 579:
Comma list: 324/323, 351/350, 456/455, 495/494, 540/539, 715/714
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 }}
Mapping: {{mapping| 1 -29 -37 -47 -28 -33 -23 -91 | 0 35 45 57 36 42 31 109 }}


Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 151.396
Optimal tunings:
* WE: ~2 = 1199.9145{{c}}, ~11/6 = 1048.5290{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1048.6021{{c}}


{{Optimal ET sequence|legend=1| 103h, 111, 214 }}
{{Optimal ET sequence|legend=0| 103h, 111, 214 }}


Badness: 0.017570
Badness (Sintel): 1.07


== Grazer ==
== Grazer ==
Named by [[Xenllium]] in 2022, grazer may be described as the {{nowrap| 113 & 121 }} temperament.
[[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, ~49/45


: mapping generators: ~2, ~90/49
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1200.0310{{c}}, ~49/45 = 148.7229{{c}}
{{Multival|legend=1| 37 51 63 -5 -4 3 }}
: [[error map]]: {{val| +0.031 +0.700 -1.561 +0.563 }}
 
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/45 = 148.7198{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/45 = 148.719
: 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 885: Line 614:
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: {{mapping| 1 -3 -4 -5 -1 | 0 37 51 63 36 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 898: Line 629:
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: {{mapping| 1 -3 -4 -5 -1 -2 | 0 37 51 63 36 46 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 911: Line 644:
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: {{mapping| 1 -3 -4 -5 -1 -2 0 | 0 37 51 63 36 46 33 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 924: Line 659:
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: {{mapping| 1 -3 -4 -5 -1 -2 0 4 | 0 37 51 63 36 46 33 2 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 148.727


{{Optimal ET sequence|legend=1| 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: 0.022574
{{Optimal ET sequence|legend=0| 113, 121, 234g }}


== Notes ==
Badness (Sintel): 1.37


[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Mirkwai clan| ]] <!-- main article -->
[[Category:Mirkwai clan| ]] <!-- main article -->
[[Category:Mirkwai| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Revision as of 16:39, 5 June 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 canopus clan of temperaments tempers out the canopus or mirkwai 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 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-twos extensions, see No-twos subgroup temperaments #Canopus.

Considered below are mirkat, eris, subsemifourth, septendesemi, gaster, hemiseptisix, browser, 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

Browser

See also: Sensipent family

Named by Xenllium in 2022, browser may be described as the 103 & 111 temperament.

This can also be considered a non-over-1 temperament, with considerable scope for harmony in the 2.5/3.7/3.11/3.13/3.17/3 subgroup with mos scales of 8, 15, 23, and 31 notes despite no harmonics from the root. It can be considered a detemperament of 8d-et, with a generator very slightly flat of 7\8.

Subgroup: 2.3.5.7

Comma list: 16875/16807, 78732/78125

Mapping: [⟨1 -29 -37 -47], ⟨0 35 45 57]]

mapping generators: ~2, ~90/49

Optimal tunings:

  • WE: ~2 = 1199.9313 ¢, ~90/49 = 1048.5414 ¢
error map: ⟨-0.069 -1.013 +0.592 +1.264]
  • CWE: ~2 = 1200.0000 ¢, ~90/49 = 1048.5998 ¢
error map: ⟨0.000 -0.962 +0.677 +1.362]

Optimal ET sequence: 103, 111, 214

Badness (Sintel): 4.57

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 78732/78125

Mapping: [⟨1 -29 -37 -47 -28], ⟨0 35 45 57 36]]

Optimal tunings:

  • WE: ~2 = 1200.1344 ¢, ~11/6 = 1048.7124 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/6 = 1048.5981 ¢

Optimal ET sequence: 103, 214

Badness (Sintel): 1.91

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 847/845, 1375/1372

Mapping: [⟨1 -29 -37 -47 -28 -33], ⟨0 35 45 57 36 42]]

Optimal tunings:

  • WE: ~2 = 1200.1344 ¢, ~11/6 = 1048.7124 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/6 = 1048.5984 ¢

Optimal ET sequence: 103, 111, 214

Badness (Sintel): 1.19

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 540/539, 561/560, 715/714, 847/845

Mapping: [⟨1 -29 -37 -47 -28 -33 -23], ⟨0 35 45 57 36 42 31]]

Optimal tunings:

  • WE: ~2 = 1199.9191 ¢, ~11/6 = 1048.5324 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/6 = 1048.6014 ¢

Optimal ET sequence: 103, 111, 214

Badness (Sintel): 1.04

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 324/323, 351/350, 456/455, 495/494, 540/539, 715/714

Mapping: [⟨1 -29 -37 -47 -28 -33 -23 -91], ⟨0 35 45 57 36 42 31 109]]

Optimal tunings:

  • WE: ~2 = 1199.9145 ¢, ~11/6 = 1048.5290 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/6 = 1048.6021 ¢

Optimal ET sequence: 103h, 111, 214

Badness (Sintel): 1.07

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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