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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 '''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, ~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 clan and collection pages.
* ''[[Nusecond]]'' (+126/125) → [[Starling temperaments #Nusecond|Starling temperaments]]
* [[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. 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) → [[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]]
* ''[[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]]
* ''[[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]].


== Grendel ==
Considered below are mirkat, eris, subsemifourth, septendesemi, gaster, hemiseptisix, 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
 
[[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
 
[[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 }}
 
[[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 334: 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]]): ~2 = 1\1, ~10/9 = 183.539
[[Optimal tuning]]s:
* [[WE]]: ~63/50 = 400.0277{{c}}, ~10/9 = 183.5515{{c}}
: [[error map]]: {{val| +0.083 -0.591 -0.117 +0.950 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~10/9 = 183.5470{{c}}
: error map: {{val| 0.000 -0.673 -0.203 +0.831 }}


{{Optimal ET sequence|legend=1| 39d, 72, 111, 183, 255 }}
{{Optimal ET sequence|legend=1| 39d, 72, 111, 183, 255 }}


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


=== 11-limit ===
=== 11-limit ===
Line 346: 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): ~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 359: 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): ~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 372: 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): ~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


[[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 398: 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 = 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 411: 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 = 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 424: 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, ~144/125
: mapping generators: ~2, ~125/72


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~144/125 = 244.719
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9182{{c}}, ~144/125 = 244.7020{{c}}
: [[error map]]: {{val| -0.082 -0.305 -0.223 +1.037 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~144/125 = 244.7172{{c}}
: error map: {{val| 0.000 -0.248 -0.082 +1.184 }}


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


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


=== 11-limit ===
=== 11-limit ===
Line 439: 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 = 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 452: 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 = 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 467: 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, ~343/324
: mapping generators: ~2, ~648/343


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~343/324 = 104.916
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8649{{c}}, ~343/324 = 104.9046{{c}}
: [[error map]]: {{val| -0.135 -0.597 +0.195 +1.196 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~343/324 = 104.9134{{c}}
: error map: {{val| 0.000 -0.506 +0.395 +1.410 }}


{{Optimal ET sequence|legend=1| 80, 103, 183 }}
{{Optimal ET sequence|legend=0| 80, 103, 183 }}


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


=== 11-limit ===
=== 11-limit ===
Line 482: 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 = 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 495: 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 = 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 508: 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 tuning (POTE): ~2 = 1\1, ~17/16 = 104.909
Optimal tunings:
* WE: ~2 = 1200.0758{{c}}, ~17/16 = 104.9158{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/16 = 104.9101{{c}}


{{Optimal ET sequence|legend=1| 80, 103, 183, 469f, 652def }}
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~17/16 = 104.909{{c}}


Badness: 0.020128
{{Optimal ET sequence|legend=0| 80, 103, 183, 469f }}
 
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 526: 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 = 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 539: 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 = 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 552: 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 = 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 565: 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 = 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 578: 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 = 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 591: 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 = 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 604: 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 = 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 617: 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 = 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 630: 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 = 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 643: 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 = 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 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~15/14 = 118.965
 
{{Optimal ET sequence|legend=1| 10, 101, 111, 121, 232d }}
 
[[Badness]]: 0.157658
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 1375/1372, 65536/64827
 
Mapping: {{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 tunings:  
* WE: ~2 = 1199.9179{{c}}, ~17/12 = 605.3156{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3567{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 118.968
{{Optimal ET sequence|legend=0| 111, 113, 224 }}


{{Optimal ET sequence|legend=1| 10, 101, 111, 121, 232d }}
Badness (Sintel): 1.57
 
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 725: 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, ~98/75
: mapping generators: ~2, ~75/49


[[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 740: 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 = 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 753: 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 = 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 766: 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 = 1\1, ~17/13 = 466.074


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


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


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


[[Subgroup]]: 2.3.5.7
== Grazer ==
 
Named by [[Xenllium]] in 2022, grazer may be described as the {{nowrap| 113 & 121 }} temperament.  
[[Comma list]]: 16875/16807, 78732/78125
 
{{Mapping|legend=1| 1 6 8 10 | 0 -35 -45 -57 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/45 = 151.399
 
{{Optimal ET sequence|legend=1| 103, 111, 214 }}
 
[[Badness]]: 0.180645
 
=== 11-limit ===
Subgroup: 2.3.5.7.11


Comma list: 540/539, 1375/1372, 78732/78125
Mapping: {{mapping| 1 6 8 10 8 | 0 -35 -45 -57 -36 }}
Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 151.405
{{Optimal ET sequence|legend=1| 103, 214 }}
Badness: 0.057634
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 351/350, 540/539, 847/845, 1375/1372
Mapping: {{mapping| 1 6 8 10 8 9 | 0 -35 -45 -57 -36 -42 }}
Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 151.403
{{Optimal ET sequence|legend=1| 103, 111, 214 }}
Badness: 0.028822
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Comma list: 351/350, 540/539, 561/560, 715/714, 847/845
Mapping: {{mapping| 1 6 8 10 8 9 8 | 0 -35 -45 -57 -36 -42 -31 }}
Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 151.397
{{Optimal ET sequence|legend=1| 103, 111, 214 }}
Badness: 0.020384
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 324/323, 351/350, 456/455, 495/494, 540/539, 715/714
Mapping: {{mapping| 1 6 8 10 8 9 8 18 | 0 -35 -45 -57 -36 -42 -31 -109 }}
Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 151.396
{{Optimal ET sequence|legend=1| 103h, 111, 214 }}
Badness: 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, ~49/45
: mapping generators: ~2, ~90/49


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/45 = 148.719
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0310{{c}}, ~49/45 = 148.7229{{c}}
: [[error map]]: {{val| +0.031 +0.700 -1.561 +0.563 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/45 = 148.7198{{c}}
: error map: {{val| 0.000 +0.676 -1.606 +0.519 }}


{{Optimal ET sequence|legend=1| 113, 121, 234 }}
{{Optimal ET sequence|legend=1| 113, 121, 234 }}


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


=== 11-limit ===
=== 11-limit ===
Line 861: 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 = 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 874: 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 = 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 887: 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 = 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 900: 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 = 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:Canopic clan| ]] <!-- main article -->
[[Category:Canopic| ]] <!-- key article -->
[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Pages with mostly numerical content]]
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Mirkwai clan| ]] <!-- main article -->
[[Category:Mirkwai| ]] <!-- key article -->
[[Category:Rank 2]]
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