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Temperaments of the '''mirkwai clan''' temper out the [[mirkwai comma]], {{monzo| 0 3 4 -5 }} = 16875/16807, a no-twos comma. Members of the clan include grendel, kwai, pluto and mirkat considered below, as well as these considered elsewhere:
{{Technical data page}}
* ''[[octokaidecal]]'', {28/27, 50/49} → [[Trienstonic clan #Octokaidecal]]
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.
* ''[[meantritone]]'', {81/80, 16875/16807} → [[Meantone family #Meantritone]]
* ''[[nusecond]]'', {126/125, 2430/2401} → [[Starling temperaments #Nusecond]]
* [[miracle]], {225/224, 1029/1024} → [[Gamelismic clan #Miracle]]
* ''[[bohpier]]'', {245/243, 3125/3087} → [[Sensamagic clan #Bohpier]]
* ''[[semisept]]'', {1728/1715, 3136/3125} → [[Hemimean clan #Semisept]]
* ''[[octoid]]'', {4375/4374, 16875/16807} → [[Ragismic microtemperaments #Octoid]]
* ''[[sqrtphi]]'', {15625/15552, 16875/16807} → [[Kleismic family #Sqrtphi]]
* ''[[quanharuk]]'', {16875/16807, 32805/32768} → [[Schismatic family #Quanharuk]]
* ''[[familia]]'', {16875/16807, 1600000/1594323} → [[Amity family #Familia]]
* ''[[rainwell]]'', {16875/16807, 2100875/2097152} → [[Semicomma family #Rainwell]]


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


Subgroup: 3.5.7
[[Subgroup]]: 3.5.7


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


[[Sval]] [[mapping]]: [{{val| 1 3 3 }}, {{val| 0 -5 -4 }}]
{{Mapping|legend=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 }}


[[POTE generator]]: ~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 GPV 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


== Grendel ==
=== Overview to extensions ===
Subgroup: 2.3.5.7
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]]


[[Comma list]]: 6144/6125, 16875/16807
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.


[[Mapping]]: [{{val| 1 9 2 7 }}, {{val| 0 -23 1 -13 }}]
Members of the clan discussed elsewhere are:
* ''[[Kwai]]'' (+5120/5103) → [[Hemifamity temperaments #Kwai|Hemifamity temperaments]]
* ''[[Octokaidecal]]'' (+28/27 or 50/49) → [[Trienstonic clan #Octokaidecal|Trienstonic clan]]
* ''[[Meantritone]]'' (+81/80) → [[Meantone family #Meantritone|Meantone family]]
* ''[[Quanharuk]]'' (+32805/32768) → [[Schismatic family #Quanharuk|Schismatic family]]
* [[Miracle]] (+225/224) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* ''[[Pluto]]'' (+4000/3969) → [[Octagar temperaments #Pluto|Octagar temperaments]]
* ''[[Bohpier]]'' (+245/243) → [[Sensamagic clan #Bohpier|Sensamagic clan]]
* ''[[Subsedia]]'' (+65536/64827) → [[Buzzardsmic clan #Subsedia|Buzzardsmic clan]]
* ''[[Semisept]]'' (+1728/1715 or 3136/3125) → [[Hemimean clan #Semisept|Hemimean clan]]
* ''[[Grendel]]'' (+6144/6125) → [[Porwell temperaments #Grendel|Porwell temperaments]]
* ''[[Quinmage]]'' (+3125/3072) → [[Magic family #Quinmage|Magic family]]
* ''[[Familia]]'' (+1600000/1594323) → [[Amity family #Familia|Amity family]]
* [[Sqrtphi]] (+15625/15552) → [[Kleismic family #Sqrtphi|Kleismic family]]
* ''[[Rainwell]]'' (+2100875/2097152) → [[Semicomma family #Rainwell|Semicomma family]]
* ''[[Quintiquart]]'' (+390625000/387420489) → [[Quartonic family #Quintiquart|Quartonic family]]


{{Multival|legend=1| 23 -1 13 -55 -44 33 }}
For ''no-twos'' extensions, see [[No-twos subgroup temperaments #Canopus]].


[[POTE generator]]: ~5/4 = 386.863
Considered below are mirkat, eris, subsemifourth, septendesemi, gaster, hemiseptisix, browser, and grazer, in the order of increasing [[badness]].  


{{Val list|legend=1| 31, 90, 121, 152, 335d }}
== 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.  
[[Badness]]: 0.051834
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 1375/1372, 5632/5625
 
Mapping: [{{val|1 9 2 7 17}}, {{val|0 -23 1 -13 -42}}]
 
POTE generator: ~5/4 = 386.856
 
Optimal GPV sequence: {{Val list| 31, 90e, 121, 152, 335d, 487d }}
 
Badness: 0.019845
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 540/539, 625/624, 1375/1372
 
Mapping: [{{val|1 9 2 7 17 -5}}, {{val|0 -23 1 -13 -42 27}}]
 
POTE generator: ~5/4 = 386.826
 
Optimal GPV sequence: {{Val list| 31, 121, 152f, 425deff }}
 
Badness: 0.024839
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 256/255, 352/351, 625/624, 715/714, 1275/1274
 
Mapping: [{{val|1 9 2 7 17 -5 -3}}, {{val|0 -23 1 -13 -42 27 22}}]
 
POTE generator: ~5/4 = 386.812
 
Optimal GPV sequence: {{Val list| 31, 121, 273defgg }}
 
Badness: 0.021400
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 256/255, 352/351, 375/374, 400/399, 456/455, 715/714
 
POTE generator: ~5/4 = 386.819
 
Mapping: [{{val|1 9 2 7 17 -5 -3 -8}}, {{val|0 -23 1 -13 -42 27 22 38}}]
 
Optimal GPV sequence: {{Val list| 31, 121, 152fg, 273defgg }}
 
Badness: 0.018413
 
== Kwai ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Kwai]].''
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 5120/5103, 16875/16807
 
[[Mapping]]: [{{val|1 0 -50 -40}}, {{val|0 1 33 27}}]
 
{{Multival|legend=1|1 33 27 50 40 -30}}
 
[[POTE generator]]: ~3/2 = 702.616
 
{{Val list|legend=1| 41, 111, 152, 345, 497d }}
 
[[Badness]]: 0.054476
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 1375/1372, 16384/16335
 
Mapping: [{{val|1 0 -50 -40 32}}, {{val|0 1 33 27 -18}}]
 
POTE generator: ~3/2 = 702.623
 
Optimal GPV sequence: {{Val list| 29cd, 41, 111, 152 }}
 
Badness: 0.026219
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 540/539, 729/728, 1375/1372
 
Mapping: [{{val|1 0 -50 -40 32 27}}, {{val|0 1 33 27 -18 -21}}]
 
POTE generator: ~3/2 = 702.644
 
Optimal GPV sequence: {{Val list| 29cd, 41, 111, 152f }}
 
Badness: 0.024555
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 256/255, 352/351, 540/539, 715/714, 1089/1088
 
Mapping: [{{val|1 0 -50 -40 32 27 58}}, {{val|0 1 33 27 -18 -21 -34}}]
 
POTE generator: ~3/2 = 702.660
 
Optimal GPV sequence: {{Val list| 29cdg, 41, 111, 152fg, 263dfg }}
 
Badness: 0.021950
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 256/255, 352/351, 400/399, 456/455, 715/714, 847/845
 
Mapping: [{{val|1 0 -50 -40 32 27 58 -56}}, {{val|0 1 33 27 -18 -21 -34 38}}]
 
POTE generator: ~3/2 = 702.657
 
Optimal GPV sequence: {{Val list| 29cdgh, 41, 111, 152fg, 263dfgh }}
 
Badness: 0.016957
 
== Pluto ==
Subgroup: 2.3.5.7
 
[[Comma list]]: 4000/3969, 10976/10935
 
[[Mapping]]: [{{val|1 5 15 15}}, {{val|0 -7 -26 -25}}]
 
{{Multival|legend=1|7 26 25 25 20 -15}}
 
[[POTE generator]]: ~7/5 = 585.147
 
{{Val list|legend=1| 39d, 41, 80, 121, 404bd }}
 
[[Badness]]: 0.057514
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 896/891, 1375/1372
 
Mapping: [{{val|1 5 15 15 2}}, {{val|0 -7 -26 -25 3}}]
 
POTE generator: ~7/5 = 585.114
 
Optimal GPV sequence: {{Val list| 39d, 41, 80, 121 }}
 
Badness: 0.029844
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 352/351, 364/363, 540/539
 
Mapping: [{{val|1 5 15 15 2 -8}}, {{val|0 -7 -26 -25 3 24}}]
 
POTE generator: ~7/5 = 585.123
 
Optimal GPV sequence: {{Val list| 39d, 41, 80, 121 }}
 
Badness: 0.025717
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 256/255, 325/324, 352/351, 364/363, 540/539
 
Mapping: [{{val|1 5 15 15 2 -8 -12}}, {{val|0 -7 -26 -25 3 24 33}}]
 
POTE generator: ~7/5 = 585.116
 
Optimal GPV sequence: {{Val list| 39d, 41, 80, 121 }}
 
Badness: 0.021463
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 190/189, 256/255, 325/324, 352/351, 361/360, 595/594
 
Mapping: [{{val|1 5 15 15 2 -8 -12 14}}, {{val|0 -7 -26 -25 3 24 33 -20}}]
 
POTE generator: ~7/5 = 585.109
 
Optimal GPV sequence: {{Val list| 39d, 41, 80, 121 }}
 
Badness: 0.017650
 
==== Orcus ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 144/143, 196/195, 275/273, 896/891
 
Mapping: [{{val|1 5 15 15 2 12}}, {{val|0 -7 -26 -25 3 -17}}]
 
POTE generator: ~7/5 = 585.111


Optimal GPV sequence: {{Val list| 41, 80f, 121ff }}
[[Subgroup]]: 2.3.5.7
 
Badness: 0.033441
 
=== Plutino ===
Subgroup: 2.3.5.7.11
 
Comma list: 100/99, 245/242, 10976/10935
 
Mapping: [{{val|1 5 15 15 22}}, {{val|0 -7 -26 -25 -38}}]
 
POTE generator: ~7/5 = 585.283
 
Optimal GPV sequence: {{Val list| 39dee, 41 }}
 
Badness: 0.057966
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 100/99, 196/195, 245/242, 729/728
 
Mapping: [{{val|1 5 15 15 22 12}}, {{val|0 -7 -26 -25 -38 -17}}]
 
POTE generator: ~7/5 = 585.232
 
Optimal GPV sequence: {{Val list| 39deef, 41 }}
 
Badness: 0.040182
 
== Mirkat ==
Subgroup: 2.3.5.7


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


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


[[POTE generator]]: ~10/9 = 183.539
[[Optimal tuning]]s:
* [[WE]]: ~63/50 = 400.0277{{c}}, ~10/9 = 183.5515{{c}}
: [[error map]]: {{val| +0.083 -0.591 -0.117 +0.950 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~10/9 = 183.5470{{c}}
: error map: {{val| 0.000 -0.673 -0.203 +0.831 }}


{{Val list|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 289: Line 75:
Comma list: 540/539, 1375/1372, 8019/8000
Comma list: 540/539, 1375/1372, 8019/8000


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


POTE generator: ~10/9 = 183.528
Optimal tunings:  
* WE: ~63/50 = 400.0463{{c}}, ~10/9 = 183.5496{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~10/9 = 183.5391{{c}}


Optimal GPV sequence: {{Val list| 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 302: Line 90:
Comma list: 351/350, 540/539, 676/675, 1375/1372
Comma list: 351/350, 540/539, 676/675, 1375/1372


Mapping: [{{val|3 2 1 2 9 1}}, {{val|0 6 13 14 3 22}}]
Mapping: {{mapping| 3 2 1 2 9 1 | 0 6 13 14 3 22 }}


POTE generator: ~10/9 = 183.577
Optimal tunings:  
* WE: ~63/50 = 400.0245{{c}}, ~10/9 = 183.5885{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~10/9 = 183.5825{{c}}


Optimal GPV sequence: {{Val list| 39df, 72, 111, 183, 255f }}
{{Optimal ET sequence|legend=0| 39df, 72, 111, 183 }}


Badness: 0.018632
Badness (Sintel): 0.770


=== 17-limit ===
=== 17-limit ===
Line 315: Line 105:
Comma list: 351/350, 442/441, 540/539, 561/560, 715/714
Comma list: 351/350, 442/441, 540/539, 561/560, 715/714


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


POTE generator: ~10/9 = 183.579
Optimal tunings:  
* WE: ~34/27 = 400.0257{{c}}, ~10/9 = 183.5906{{c}}
* CWE: ~34/27 = 400.0000{{c}}, ~10/9 = 183.5843{{c}}


Optimal GPV sequence: {{Val list| 39dfg, 72, 111, 183, 255f }}
{{Optimal ET sequence|legend=0| 39dfg, 72, 111, 183 }}


Badness: 0.011775
Badness (Sintel): 0.600


== Eris ==
== Eris ==
Subgroup: 2.3.5.7
Eris tempers out 65625/65536, the [[horwell comma]], and may be described as the {{nowrap| 31 & 224 }} temperament. The [[2.5.7-subgroup|2.5.7 subgroup]] restriction of this temperament is [[exodia]].
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 16875/16807, 65625/65536
[[Comma list]]: 16875/16807, 65625/65536


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


{{Multival|legend=1|29 -8 11 -80 -64 48}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0256{{c}}, ~49/30 = 851.8023{{c}}
: [[error map]]: {{val| +0.026 -0.173 -0.528 +0.872 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/30 = 851.7845{{c}}
: error map: {{val| 0.000 -0.204 -0.590 +0.804 }}


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


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


=== 11-limit ===
=== 11-limit ===
Line 343: Line 140:
Comma list: 540/539, 1375/1372, 65625/65536
Comma list: 540/539, 1375/1372, 65625/65536


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


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


Optimal GPV sequence: {{Val list| 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 356: Line 155:
Comma list: 540/539, 625/624, 1375/1372, 4096/4095
Comma list: 540/539, 625/624, 1375/1372, 4096/4095


Mapping: [{{val|1 10 0 6 20 -14}}, {{val|0 -29 8 -11 -57 61}}]
Mapping: {{mapping| 1 -19 8 -5 -37 47 | 0 29 -8 11 57 -61 }}


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


Optimal GPV sequence: {{Val list| 31, 193, 224 }}
{{Optimal ET sequence|legend=0| 31, 193, 224 }}


Badness: 0.025137
Badness (Sintel): 1.04


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


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


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


{{Multival|legend=1|47 31 53 -60 -48 36}}
[[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 }}


[[POTE generator]]: ~144/125 = 244.719
{{Optimal ET sequence|legend=1| 49, 103, 152, 255, 407 }}


{{Val list|legend=1| 49, 103, 152, 255, 407 }}
[[Badness]] (Sintel): 3.42
 
[[Badness]]: 0.135173


=== 11-limit ===
=== 11-limit ===
Line 384: Line 188:
Comma list: 540/539, 1375/1372, 234375/234256
Comma list: 540/539, 1375/1372, 234375/234256


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


POTE generator: ~121/105 = 244.719
Optimal tunings:  
* WE: ~2 = 1199.9229{{c}}, ~121/105 = 244.7033{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~121/105 = 244.7175{{c}}


Optimal GPV sequence: {{Val list| 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 397: Line 203:
Comma list: 540/539, 847/845, 1375/1372, 1575/1573
Comma list: 540/539, 847/845, 1375/1372, 1575/1573


Mapping: [{{val|1 -8 -4 -8 -10 -12}}, {{val|0 47 31 53 66 77}}]
Mapping: {{mapping| 1 -8 -4 -8 -10 -12 | 0 0 47 31 53 66 77 }}


POTE generator: ~15/13 = 244.714
Optimal tunings:  
* WE: ~2 = 1199.9003{{c}}, ~15/13 = 244.6932{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/13 = 244.7116{{c}}


Optimal GPV sequence: {{Val list| 49f, 103, 152f, 255, 407f, 662df }}
{{Optimal ET sequence|legend=0| 49f, 103, 152f, 255, 407f }}


Badness: 0.028387
Badness (Sintel): 1.17


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


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


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


{{Multival|legend=1|41 38 55 -35 -28 21}}
[[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 }}


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


{{Val list|legend=1| 80, 103, 183 }}
[[Badness]] (Sintel): 3.71
 
[[Badness]]: 0.146795


=== 11-limit ===
=== 11-limit ===
Line 427: Line 238:
Comma list: 540/539, 1375/1372, 43923/43750
Comma list: 540/539, 1375/1372, 43923/43750


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


POTE generator: ~35/33 = 104.916
Optimal tunings:  
* WE: ~2 = 1199.9327{{c}}, ~35/33 = 104.9100{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 104.9144{{c}}


Optimal GPV sequence: {{Val list| 80, 103, 183 }}
{{Optimal ET sequence|legend=0| 80, 103, 183 }}


Badness: 0.041554
Badness (Sintel): 1.37


=== 13-limit ===
=== 13-limit ===
Line 440: Line 253:
Comma list: 351/350, 540/539, 1375/1372, 4225/4224
Comma list: 351/350, 540/539, 1375/1372, 4225/4224


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


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


Optimal GPV sequence: {{Val list| 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 453: 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: [{{val|1 -2 -1 -2 -1 3 4}}, {{val|0 41 38 55 51 8 1}}]
Mapping: {{mapping| 1 -2 -1 -2 -1 3 4 | 0 41 38 55 51 8 1 }}


POTE generator: ~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 GPV sequence: {{Val list| 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&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.  


Subgroup: 2.3.5.7
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


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


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


{{Multival|legend=1|19 72 69 70 56 -42}}
[[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 }}


[[POTE generator]]: ~800/567 = 594.641
{{Optimal ET sequence|legend=1| 111, 224 }}


{{Optimal GPV sequence|legend=1| 111, 224 }}
[[Badness]] (Sintel): 3.91
 
[[Badness]]: 0.154521


=== 11-limit ===
=== 11-limit ===
Line 486: Line 310:
Comma list: 540/539, 1375/1372, 14348907/14336000
Comma list: 540/539, 1375/1372, 14348907/14336000


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


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


Optimal GPV sequence: {{Val list| 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 499: Line 325:
Comma list: 540/539, 729/728, 1375/1372, 2200/2197
Comma list: 540/539, 729/728, 1375/1372, 2200/2197


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


POTE generator: ~55/39 = 594.639
Optimal tunings:  
* WE: ~2 = 1199.9154{{c}}, ~78/55 = 605.3183{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~78/55 = 605.3601{{c}}


Optimal GPV sequence: {{Val list| 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 512: 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: [{{val|1 11 38 37 -1 26 14}}, {{val|0 -19 -72 -69 9 -45 -20}}]
Mapping: {{mapping| 1 -8 -34 -32 8 -19 -6 | 0 19 72 69 -9 45 20 }}


POTE generator: ~24/17 = 594.636
Optimal tunings:  
* WE: ~2 = 1199.8076{{c}}, ~17/12 = 605.2674{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3626{{c}}


Optimal GPV sequence: {{Val list| 111, 224, 559dgg }}
{{Optimal ET sequence|legend=0| 111, 224, 559dgg }}


Badness: 0.021436
Badness (Sintel): 1.09


=== 19-limit ===
=== 19-limit ===
Line 525: 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: [{{val|1 11 38 37 -1 26 14 32}}, {{val|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 }}


POTE generator: ~24/17 = 594.636
Optimal tunings:  
* WE: ~2 = 1199.7542{{c}}, ~17/12 = 605.2674{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3613{{c}}


Optimal GPV sequence: {{Val list| 111, 224 }}
{{Optimal ET sequence|legend=0| 111, 224 }}


Badness: 0.018370
Badness (Sintel): 1.12


=== 23-limit ===
=== 23-limit ===
Line 538: 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: [{{val|1 11 38 37 -1 26 14 32 7}}, {{val|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 }}


POTE generator: ~24/17 = 594.641
Optimal tunings:  
* WE: ~2 = 1199.8733{{c}}, ~17/12 = 605.2946{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3575{{c}}


Optimal GPV sequence: {{Val list| 111, 224 }}
{{Optimal ET sequence|legend=0| 111, 224 }}


Badness: 0.017619
Badness (Sintel): 1.26


=== 29-limit ===
=== 29-limit ===
Line 551: 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: [{{val|1 11 38 37 -1 26 14 32 7 -11}}, {{val|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 }}


POTE generator: ~24/17 = 594.646
Optimal tunings:  
* WE: ~2 = 1199.9442{{c}}, ~17/12 = 605.3263{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3541{{c}}


Optimal GPV sequence: {{Val list| 111, 113, 224 }}
{{Optimal ET sequence|legend=0| 111, 113, 224 }}


Badness: 0.016815
Badness (Sintel): 1.41


=== 31-limit ===
=== 31-limit ===
Line 564: 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: [{{val|1 11 38 37 -1 26 14 32 7 -11 0}}, {{val|0 -19 -72 -69 9 -45 -20 -56 -5 32 10}}]
Mapping: {{mapping| 1 -8 -34 -32 8 -19 -6 -24 2 21 10 | 0 19 72 69 -9 45 20 56 5 -32 -10 }}


POTE generator: ~24/17 = 594.644
Optimal tunings:  
* WE: ~2 = 1199.9100{{c}}, ~17/12 = 605.3107{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3556{{c}}


Optimal GPV sequence: {{Val list| 111, 113, 224 }}
{{Optimal ET sequence|legend=0| 111, 113, 224 }}


Badness: 0.014790
Badness (Sintel): 1.42


=== 37-limit ===
=== 37-limit ===
Line 577: 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: [{{val|1 11 38 37 -1 26 14 32 7 -11 0 -27}}, {{val|0 -19 -72 -69 9 -45 -20 -56 -5 32 10 65}}]
Mapping: {{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 }}


POTE generator: ~24/17 = 594.644
Optimal tunings:  
* WE: ~2 = 1199.9087{{c}}, ~17/12 = 605.3101{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3559{{c}}


Optimal GPV sequence: {{Val list| 111, 113, 224 }}
{{Optimal ET sequence|legend=0| 111, 113, 224 }}


Badness: 0.014377
Badness (Sintel): 1.56


=== 41-limit ===
=== 41-limit ===
Line 590: 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: [{{val|1 11 38 37 -1 26 14 32 7 -11 0 -27 45}}, {{val|0 -19 -72 -69 9 -45 -20 -56 -5 32 10 65 -80}}]
Mapping: {{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 }}


POTE generator: ~24/17 = 594.643
Optimal tunings:  
* WE: ~2 = 1199.9179{{c}}, ~17/12 = 605.3156{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3567{{c}}


Optimal GPV sequence: {{Val list| 111, 113, 224 }}
{{Optimal ET sequence|legend=0| 111, 113, 224 }}


Badness: 0.012858
Badness (Sintel): 1.57
 
== Subsedia ==
The generator for ''subsedia'' (10&111) is 0.5 cents flat of [[15/14]]-wide semitone and tempers out the mirkwai comma and 65536/64827 (''buzzardisma'', saquadru comma). In this temperament, three generators makes ~[[16/13]], five of them equals ~[[24/17]], twelve of them equals ~[[16/7]], sixteen of them equals ~[[3/1]], and 45 of them equals ~22/1.
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 16875/16807, 65536/64827
 
[[Mapping]]: [{{val|1 0 5 4}}, {{val|0 16 -27 -12}}]
 
{{Multival|legend=1| 16 -27 -12 -80 -64 48 }}
 
[[POTE generator]]: ~15/14 = 118.965
 
{{Val list|legend=1| 10, 101, 111, 121, 232d }}
 
[[Badness]]: 0.157658
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 1375/1372, 65536/64827
 
Mapping: [{{val|1 0 5 4 -1}}, {{val|0 16 -27 -12 45}}]
 
POTE generator: ~15/14 = 118.968
 
Optimal GPV sequence: {{Val list| 10, 101, 111, 121, 232d }}
 
Badness: 0.066838
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 540/539, 676/675, 1375/1372
 
Mapping: [{{val|1 0 5 4 -1 4}}, {{val|0 16 -27 -12 45 -3}}]
 
POTE generator: ~15/14 = 118.968
 
Optimal GPV sequence: {{Val list| 10, 101, 111, 121, 232d }}
 
Badness: 0.031635
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 256/255, 352/351, 442/441, 540/539, 715/714
 
Mapping: [{{val|1 0 5 4 -1 4 3}}, {{val|0 16 -27 -12 45 -3 11}}]
 
POTE generator: ~15/14 = 118.968
 
Optimal GPV sequence: {{Val list| 10, 101, 111, 121, 232dg }}
 
Badness: 0.019707
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 256/255, 352/351, 400/399, 442/441, 456/455, 715/714
 
Mapping: [{{val|1 0 5 4 -1 4 3 10}}, {{val|0 16 -27 -12 45 -3 11 -58}}]
 
POTE generator: ~15/14 = 118.964
 
Optimal GPV sequence: {{Val list| 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&121) tempers out the mirkwai comma and 95703125/95551488 (''pontiqak'' comma, lazozotritriyo) in the 7-limit. [[224edo|224EDO]] provides an excellent tuning for 7, 11, and 13-limit hemiseptisix.
Hemiseptisix tempers out the mirkwai comma and 95703125/95551488 (pontiqak comma) in the 7-limit, and may be described as the {{nowrap| 103 & 121 }} temperament. [[224edo]] provides an excellent tuning for 7-, 11-, and 13-limit hemiseptisix. Hemiseptisix was named by [[Xenllium]] in 2021; the name ''hemiseptisix'' refers to a half of septimal major sixth ([[12/7]]).  


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


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


{{Multival|legend=1|53 24 51 -85 -68 51}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.2693{{c}}, ~98/75 = 466.0801{{c}}
: [[error map]]: {{val| +0.023 -0.149 -0.553 +0.866 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~98/75 = 466.0715{{c}}
: error map: {{val| 0.000 -0.167 -0.598 +0.819 }}


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


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


=== 11-limit ===
=== 11-limit ===
Line 689: Line 465:
Comma list: 540/539, 1375/1372, 2734375/2725888
Comma list: 540/539, 1375/1372, 2734375/2725888


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


POTE generator: ~55/42 = 466.070
Optimal tunings:  
* WE: ~2 = 1200.0183{{c}}, ~55/42 = 466.0767{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~55/42 = 466.0699{{c}}


Optimal GPV sequence: {{Val list| 103, 121, 224 }}
{{Optimal ET sequence|legend=0| 103, 121, 224 }}


Badness: 0.043381
Badness (Sintel): 1.43


=== 13-limit ===
=== 13-limit ===
Line 702: Line 480:
Comma list: 540/539, 625/624, 1375/1372, 2200/2197
Comma list: 540/539, 625/624, 1375/1372, 2200/2197


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


POTE generator: ~55/42 = 466.071
Optimal tunings:  
* WE: ~2 = 1199.9784{{c}}, ~55/42 = 466.0622{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~55/42 = 466.0703{{c}}


Optimal GPV sequence: {{Val list| 103, 121, 224 }}
{{Optimal ET sequence|legend=0| 103, 121, 224 }}


Badness: 0.021127
Badness (Sintel): 0.873


=== 17-limit ===
=== 17-limit ===
Line 715: 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: [{{val|1 -19 -7 -17 -28 -13 -13}}, {{val|0 53 24 51 81 43 44}}]
Mapping: {{mapping| 1 -19 -7 -17 -28 -13 -13 | 0 53 24 51 81 43 44 }}


POTE generator: ~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 GPV sequence: {{Val list| 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 }}


Subgroup: 2.3.5.7
Named by [[Xenllium]] in 2022, browser may be described as the {{nowrap| 103 & 111 }} temperament.
 
This can also be considered a [[non-over-1 temperament]], with considerable scope for harmony in the 2.5/3.7/3.11/3.13/3.17/3 subgroup with mos scales of 8, 15, 23, and 31 notes despite no harmonics from the root. It can be considered a detemperament of 8d-et, with a generator very slightly flat of 7\8.
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 16875/16807, 78732/78125
[[Comma list]]: 16875/16807, 78732/78125


[[Mapping]]: [{{val|1 6 8 10}}, {{val|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}}
: [[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 }}


[[POTE generator]]: ~49/45 = 151.399
{{Optimal ET sequence|legend=1| 103, 111, 214 }}


{{Val list|legend=1| 103, 111, 214 }}
[[Badness]] (Sintel): 4.57
 
[[Badness]]: 0.180645


=== 11-limit ===
=== 11-limit ===
Line 745: Line 534:
Comma list: 540/539, 1375/1372, 78732/78125
Comma list: 540/539, 1375/1372, 78732/78125


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


POTE generator: ~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 GPV sequence: {{Val list| 103, 214 }}
{{Optimal ET sequence|legend=0| 103, 214 }}


Badness: 0.057634
Badness (Sintel): 1.91


=== 13-limit ===
=== 13-limit ===
Line 758: Line 549:
Comma list: 351/350, 540/539, 847/845, 1375/1372
Comma list: 351/350, 540/539, 847/845, 1375/1372


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


POTE generator: ~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 GPV sequence: {{Val list| 103, 111, 214 }}
{{Optimal ET sequence|legend=0| 103, 111, 214 }}


Badness: 0.028822
Badness (Sintel): 1.19


=== 17-limit ===
=== 17-limit ===
Line 771: 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: [{{val|1 6 8 10 8 9 8}}, {{val|0 -35 -45 -57 -36 -42 -31}}]
Mapping: {{mapping| 1 -29 -37 -47 -28 -33 -23 | 0 35 45 57 36 42 31 }}


POTE generator: ~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 GPV sequence: {{Val list| 103, 111, 214 }}
{{Optimal ET sequence|legend=0| 103, 111, 214 }}


Badness: 0.020384
Badness (Sintel): 1.04


=== 19-limit ===
=== 19-limit ===
Line 784: 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: [{{val|1 6 8 10 8 9 8 18}}, {{val|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 }}


POTE generator: ~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 GPV sequence: {{Val list| 103h, 111, 214 }}
{{Optimal ET sequence|legend=0| 103h, 111, 214 }}


Badness: 0.017570
Badness (Sintel): 1.07


== Grazer ==
== Grazer ==
Subgroup: 2.3.5.7
Named by [[Xenllium]] in 2022, grazer may be described as the {{nowrap| 113 & 121 }} temperament.
 
[[Subgroup]]: 2.3.5.7


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


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


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


{{Val list|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 812: Line 614:
Comma list: 540/539, 1375/1372, 218750/216513
Comma list: 540/539, 1375/1372, 218750/216513


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


POTE generator: ~12/11 = 148.729
Optimal tunings:  
* WE: ~2 = 1199.7242{{c}}, ~12/11 = 148.6946{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 148.7230{{c}}


Optimal GPV sequence: {{Val list| 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 825: Line 629:
Comma list: 325/324, 364/363, 540/539, 2200/2197
Comma list: 325/324, 364/363, 540/539, 2200/2197


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


POTE generator: ~12/11 = 148.729
Optimal tunings:  
* WE: ~2 = 1199.7257{{c}}, ~12/11 = 148.6947{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 148.7230{{c}}


Optimal GPV sequence: {{Val list| 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 838: 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: [{{val|1 -3 -4 -5 -1 -2 0}}, {{val|0 37 51 63 36 46 33}}]
Mapping: {{mapping| 1 -3 -4 -5 -1 -2 0 | 0 37 51 63 36 46 33 }}


POTE generator: ~12/11 = 148.735
Optimal tunings:  
* WE: ~2 = 1199.5690{{c}}, ~12/11 = 148.6815{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 148.7267{{c}}


Optimal GPV sequence: {{Val list| 113, 121, 234g, 355eg }}
{{Optimal ET sequence|legend=0| 113, 121, 234g }}


Badness: 0.025410
Badness (Sintel): 1.29


=== 19-limit ===
=== 19-limit ===
Line 851: 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: [{{val|1 -3 -4 -5 -1 -2 0 4}}, {{val|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 }}


POTE generator: ~12/11 = 148.727
Optimal tunings:  
* WE: ~2 = 1199.7269{{c}}, ~12/11 = 148.6928{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 148.7227{{c}}


Optimal GPV sequence: {{Val list| 113, 121, 234g, 355eg, 589ceegg }}
{{Optimal ET sequence|legend=0| 113, 121, 234g }}


Badness: 0.022574
Badness (Sintel): 1.37


[[Category:Regular temperament theory]]
[[Category:Temperament clans]]
[[Category:Temperament clan]]
[[Category:Mirkwai clan| ]] <!-- main article -->
[[Category:Mirkwai clan| ]] <!-- main 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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