Porwell temperaments: Difference between revisions

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This family of temperaments tempers out the ''porwell comma'', {{monzo| 11 1 -3 -2 }} = [[6144/6125]], and includes hendecatonic, hemischis, twothirdtonic, nessafof, septisuperfourth, whoops, and polypyth.  
{{Technical data page}}
This is a collection of [[regular temperament|temperaments]] that [[tempering out|temper out]] the [[porwell comma]] ({{monzo|legend=1| 11 1 -3 -2 }}, [[ratio]]: [[6144/6125]]).  


Discussed elsewhere are:  
Temperaments discussed elsewhere are:  
* ''[[Hexadecimal]]'', {36/35, 135/128} → [[Pelogic family #Armodue|Pelogic family]]
* ''[[Armodue (temperament)|Armodue]]'' (+36/35) → [[Mavila family #Armodue|Mavila family]]
* [[Porcupine]], {64/63, 250/243}, also in: [[Porcupine family #Porcupine|Porcupine family]]
* [[Mohajira]] (+81/80) → [[Meantone family #Mohajira|Meantone family]]
* [[Mohajira]], {81/80, 6144/6125}, also in: [[Meantone family #Mohajira|Meantone family]]
* ''[[Hemischis]]'' (+19683/19600) → [[Schismatic family #Hemischis|Schismatic family]]
* [[Valentine]], {126/125, 1029/1024}, also in: [[Starling temperaments #Valentine|Starling temperaments]]
* [[Porcupine]] (+64/63) → [[Porcupine family #Porcupine|Porcupine family]]
* [[Orwell]], {225/224, 1728/1715}, also in: [[Semicomma family #Orwell|Semicomma family]]
* ''[[Alphatrident]]'' (+14348907/14336000) → [[Alphatricot family #Alphatrident|Alphatricot family]]
* [[Shrutar]], {245/243, 2048/2025}, also in: [[Diaschismic family #Shrutar|Diaschismic family]]
* ''[[Shrutar]]'' (+245/243) → [[Diaschismic family #Shrutar|Diaschismic family]]
* ''[[Quinkee]]'', {1029/1000, 6144/6125} → [[Cloudy clan #Quinkee|Cloudy clan]]
* [[Amity]] (+4375/4374 or 5120/5103) → [[Amity family #Septimal amity|Amity family]]
* ''[[Hemiwürschmidt]]'', {2401/2400, 3136/3125} → [[Würschmidt family #Hemiwürschmidt|Würschmidt family]]
* [[Orwell]] (+225/224) → [[Semicomma family #Orwell|Semicomma family]]
* ''[[Hemikleismic]]'', {4000/3969, 6144/6125} → [[Kleismic family #Hemikleismic|Kleismic family]]
* ''[[Twilight]]'' (+{{monzo| 19 -22 2 4 }}) → [[Undim family #Twilight|Undim family]]
* [[Amity]], {4375/4374, 5120/5103}, also in: [[Amity family #Septimal amity|Amity family]] and [[Ragismic microtemperaments #Amity|Ragismic microtemperaments]]
* [[Valentine]] (+126/125) → [[Starling temperaments #Valentine|Starling temperaments]]
* ''[[Freivald]]'', {6144/6125, 6272/6075} → [[Passion family #Freivald|Passion family]]
* ''[[Freivald]]'' (+6272/6075) → [[Passion family #Freivald|Passion family]]
* ''[[Grendel]]'', {6144/6125, 16875/16807} → [[Mirkwai clan #Grendel|Mirkwai clan]]
* ''[[Decimaleap]]'' (+{{monzo| 15 -18 1 4 }}) → [[Quintaleap family #Decimaleap|Quintaleap family]]
* ''[[Bison]]'', {6144/6125, 78732/78125} → [[Sensipent family #Bison|Sensipent family]]
* ''[[Hemikleismic]]'' (+4000/3969) → [[Kleismic family #Hemikleismic|Kleismic family]]
* ''[[Hemimabila]]'', {6144/6125, 117649/116640} → [[Mabila family #Hemimabila|Mabila family]]
* ''[[Bison]]'' (+78732/78125) → [[Sensipent family #Bison|Sensipent family]]
* ''[[Septisuperfourth]]'', {6144/6125, 118098/117649} → [[Escapade family #Septisuperfourth|Escapade family]]
* ''[[Quinkee]]'' (+1029/1000) → [[Cloudy clan #Quinkee|Cloudy clan]]
* ''[[Trident]]'', {6144/6125, 14348907/14336000} → [[Tricot family #Trident|Tricot family]]
* ''[[Hemiwürschmidt]]'' (+2401/2400 or 3136/3125) → [[Hemimean clan #Hemiwürschmidt|Hemimean clan]]
* ''[[Hemimaquila]]'', {6144/6125, {{monzo| -5 10 5 -8 }}} → [[Maquila family #Hemimaquila|Maquila family]]
* ''[[Septisuperfourth]]'' (+118098/117649) → [[Escapade family #Septisuperfourth|Escapade family]]
* ''[[Decimaleap]]'', {6144/6125, {{monzo| 15 -18 1 4 }}} → [[Quintaleap family #Decimaleap|Quintaleap family]]
* ''[[Hemimabila]]'' (+117649/116640) → [[Mabila family #Hemimabila|Mabila family]]
* ''[[Twilight]]'', {6144/6125, {{monzo| 19 -22 2 4 }}} → [[Undim family #Twilight|Undim family]]
* ''[[Countermiracle]]'' (+823543/819200) → [[Quince clan #Countermiracle|Quince clan]]
* ''[[Hemimaquila]]'' (+{{monzo| -5 10 5 -8 }}) → [[Maquila family #Hemimaquila|Maquila family]]
 
Considered below are hendecatonic, nessafof, grendel, twothirdtonic, aufo, absurdity, polypyth, whoops, dodifo, and icositritonic, in the order of increasing [[badness]].


== Hendecatonic ==
== Hendecatonic ==
The hendecatonic temperament has a period of 1/11 octave, which represents [[16/15]] and four times of which represents [[9/7]].
: ''For the 5-limit version, see [[11th-octave temperaments #Hendecapent]].''


Subgroup: 2.3.5.7
The hendecatonic temperament has a period of 1/11 octave, which represents [[16/15]] and four times of which represent [[9/7]]. It tempers out 10976/10935, the [[hemimage comma]], and may be described as the {{nowrap| 22 & 99 }} temperament, with [[99edo]] giving an almost perfect tuning.
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 6144/6125, 10976/10935
[[Comma list]]: 6144/6125, 10976/10935


[[Mapping]]: [{{val| 11 0 43 -4 }}, {{val| 0 1 -1 2 }}]
{{Mapping|legend=1| 11 0 43 -4 | 0 1 -1 2 }}
 
: mapping generators: ~16/15, ~3
{{Multival|legend=1| 11 -11 22 -43 4 82 }}


[[POTE generator]]: ~3/2 = 703.054
[[Optimal tuning]]s:
* [[WE]]: ~16/15 = 109.0526{{c}}, ~3/2 = 702.8069{{c}}
: [[error map]]: {{val| -0.421 +0.431 +0.563 -0.265 }}
* [[CWE]]: ~16/15 = 109.0909{{c}}, ~3/2 = 702.9705{{c}}
: error map: {{val| 0.000 +1.015 +1.625 +0.751 }}


{{Val list|legend=1| 22, 55, 77, 99 }}
{{Optimal ET sequence|legend=1| 22, 55, 77, 99 }}


[[Badness]]: 0.041081
[[Badness]] (Sintel): 1.04


=== 11-limit ===
=== Hendecaton ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 121/120, 176/175, 10976/10935
Comma list: 121/120, 176/175, 10976/10935


Mapping: [{{val| 11 0 43 -4 38 }}, {{val| 0 1 -1 2 0 }}]
Mapping: {{mapping| 11 0 43 -4 38 | 0 1 -1 2 0 }}


POTE generator: ~3/2 = 702.636
Optimal tunings:  
* WE: ~16/15 = 109.0977{{c}}, ~3/2 = 702.6801{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 702.6484{{c}}


Optimal GPV sequence: {{Val list| 22, 55, 77, 99, 176e, 275e }}
{{Optimal ET sequence|legend=0| 22, 55, 77, 99 }}


Badness: 0.046088
Badness (Sintel): 1.52


==== 13-limit ====
==== 13-limit ====
Line 57: Line 68:
Comma list: 121/120, 176/175, 351/350, 4459/4455
Comma list: 121/120, 176/175, 351/350, 4459/4455


Mapping: [{{val| 11 0 43 -4 38 93 }}, {{val| 0 1 -1 2 0 -3 }}]
Mapping: {{mapping| 11 0 43 -4 38 93 | 0 1 -1 2 0 -3 }}


POTE generator: ~3/2 = 702.291
Optimal tunings:  
* WE: ~16/15 = 109.1092{{c}}, ~3/2 = 702.4093{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 702.2930{{c}}


Optimal GPV sequence: {{Val list| 22, 55, 77, 99, 176e }}
{{Optimal ET sequence|legend=0| 22, 55, 77, 99 }}


Badness: 0.040099
Badness (Sintel): 1.66


==== 17-limit ====
==== 17-limit ====
Line 70: Line 83:
Comma list: 121/120, 154/153, 176/175, 273/272, 2025/2023
Comma list: 121/120, 154/153, 176/175, 273/272, 2025/2023


Mapping: [{{val| 11 0 43 -4 38 93 45 }}, {{val| 0 1 -1 2 0 -3 0 }}]
Mapping: {{mapping| 11 0 43 -4 38 93 45 | 0 1 -1 2 0 -3 0 }}


POTE generator: ~3/2 = 702.301
Optimal tunings:  
* WE: ~16/15 = 109.0933{{c}}, ~3/2 = 702.3170{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 702.3017{{c}}


Optimal GPV sequence: {{Val list| 22, 55, 77, 99, 176eg }}
{{Optimal ET sequence|legend=0| 22, 55, 77, 99, 176eg }}


Badness: 0.029054
Badness (Sintel): 1.48


=== Cohendecatonic ===
=== Cohendecatonic ===
Line 83: Line 98:
Comma list: 540/539, 896/891, 4375/4356
Comma list: 540/539, 896/891, 4375/4356


Mapping: [{{val| 11 0 43 -4 73 }}, {{val| 0 1 -1 2 -2 }}]
Mapping: {{mapping| 11 0 43 -4 73 | 0 1 -1 2 -2 }}


POTE generator: ~3/2 = 703.686
Optimal tunings:  
* WE: ~16/15 = 109.0237{{c}}, ~3/2 = 703.2522{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.6563{{c}}


Optimal GPV sequence: {{Val list| 22, 77e, 99e, 121, 220e }}
{{Optimal ET sequence|legend=0| 22, 77e, 99e, 121, 220e }}


Badness: 0.038042
Badness (Sintel): 1.26


==== 13-limit ====
==== 13-limit ====
Line 96: Line 113:
Comma list: 352/351, 364/363, 540/539, 625/624
Comma list: 352/351, 364/363, 540/539, 625/624


Mapping: [{{val| 11 0 43 -4 73 128 }}, {{val| 0 1 -1 2 -2 -5 }}]
Mapping: {{mapping| 11 0 43 -4 73 128 | 0 1 -1 2 -2 -5 }}


POTE generator: ~3/2 = 703.888
Optimal tunings:  
* WE: ~16/15 = 109.0189{{c}}, ~3/2 = 703.4228{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.9248{{c}}


Optimal GPV sequence: {{Val list| 22, 77eff, 99ef, 121, 341bdeeff }}
{{Optimal ET sequence|legend=0| 22, 99ef, 121, 341bdeeff }}


Badness: 0.036112
Badness (Sintel): 1.49


==== 17-limit ====
==== 17-limit ====
Line 109: Line 128:
Comma list: 256/255, 352/351, 364/363, 375/374, 540/539
Comma list: 256/255, 352/351, 364/363, 375/374, 540/539


Mapping: [{{val| 11 0 43 -4 73 128 45 }}, {{val| 0 1 -1 2 -2 -5 0 }}]
Mapping: {{mapping| 11 0 43 -4 73 128 45 | 0 1 -1 2 -2 -5 0 }}


POTE generator: ~3/2 = 703.877
Optimal tunings:  
* WE: ~16/15 = 109.0159{{c}}, ~3/2 = 703.3932{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.9110{{c}}


Optimal GPV sequence: {{Val list| 22, 77eff, 99ef, 121, 220efg, 341bdeeffgg }}
{{Optimal ET sequence|legend=0| 22, 99ef, 121, 220efg, 341bdeeffgg }}


Badness: 0.022590
Badness (Sintel): 1.15


=== Icosidillic ===
=== Icosidillic ===
Line 122: Line 143:
Comma list: 3388/3375, 6144/6125, 9801/9800
Comma list: 3388/3375, 6144/6125, 9801/9800


Mapping: [{{val| 22 0 86 -8 111 }}, {{val| 0 1 -1 2 -1 }}]
Mapping: {{mapping| 22 0 86 -8 111 | 0 1 -1 2 -1 }}
: mapping generators: ~33/32, ~3


POTE generator: ~3/2 = 702.914
Optimal tunings:  
* WE: ~33/32 = 54.5305{{c}}, ~3/2 = 702.7206{{c}}
* CWE: ~33/32 = 54.5455{{c}}, ~3/2 = 702.8829{{c}}


Optimal GPV sequence: {{Val list| 22, 154, 176, 198 }}
{{Optimal ET sequence|legend=0| 22, 154, 176, 198 }}


Badness: 0.057725
Badness (Sintel): 1.84


== Hemischis ==
== Nessafof ==
{{see also| Schismatic family }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Nessafof]].''


Subgroup: 2.3.5.7
Cryptically named by [[Petr Pařízek]] in 2011<ref name="petr's short post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101089.html Yahoo! Tuning Group | ''Some more unclassified temperaments'']</ref>, nessafof adds the [[landscape comma]] and has a third-octave period. The name actually refers to the fact that it has a neutral-second generator, and that a semi-augmented fourth, stacked five times, makes 5/1<ref name="petr's long post"/>.  


[[Comma list]]: 6144/6125, 19683/19600
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 0 15 -17 }}, {{val| 0 2 -16 25 }}]
[[Comma list]]: 6144/6125, 250047/250000


{{Multival|legend=1| 2 -16 25 -30 34 103 }}
{{Mapping|legend=1| 3 2 5 10 | 0 7 5 -4 }}
: mapping generators: ~63/50, ~35/32


[[POTE generator]]: ~81/70 = 249.203
[[Optimal tuning]]s:  
* [[WE]]: ~63/50 = 399.9023{{c}}, ~35/32 = 157.4418{{c}}
: [[error map]]: {{val| -0.293 -0.057 +0.407 +0.430 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~35/32 = 157.4658{{c}}
: error map: {{val| 0.000 +0.306 1.016 +1.311 }}


{{Val list|legend=1| 24, 53, 130, 183, 313 }}
{{Optimal ET sequence|legend=1| 15, 54b, 69, 84, 99, 282, 381 }}


[[Badness]]: 0.045817
[[Badness]] (Sintel): 1.14


=== 11-limit ===
=== Nessa ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 540/539, 5632/5625, 8019/8000
Comma list: 441/440, 1344/1331, 4375/4356


Mapping: [{{val| 1 0 15 -17 51 }}, {{val| 0 2 -16 25 -60 }}]
Mapping: {{mapping| 3 2 5 10 10 | 0 7 5 -4 1 }}


POTE generator: ~81/70 = 249.199
Optimal tunings:  
* WE: ~44/35 = 399.7815{{c}}, ~35/32 = 157.4527{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~35/32 = 157.5109{{c}}


Optimal GPV sequence: {{Val list| 24e, 53, 130, 183, 313 }}
{{Optimal ET sequence|legend=0| 15, 69, 84, 99e }}


Badness: 0.036289
Badness (Sintel): 1.61


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 351/350, 540/539, 676/675, 4096/4095
Comma list: 144/143, 364/363, 441/440, 625/624


Mapping: [{{val| 1 0 15 -17 51 14 }}, {{val| 0 2 -16 25 -60 -13 }}]
Mapping: {{mapping| 3 2 5 10 10 6 | 0 7 5 -4 1 13 }}


POTE generator: ~15/13 = 249.199
Optimal tunings:  
* WE: ~44/35 = 399.7595{{c}}, ~35/32 = 157.3348{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~35/32 = 157.3955{{c}}


Optimal GPV sequence: {{Val list| 24e, 53, 130, 183, 313 }}
{{Optimal ET sequence|legend=0| 15, 69, 84, 99ef, 183ef, 282eeff }}


Badness: 0.020816
Badness (Sintel): 1.55


=== 17-limit ===
=== Fof ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 176/175, 250047/250000


Comma list: 351/350, 442/441, 561/560, 676/675, 4096/4095
Mapping: {{mapping| 3 2 5 10 8 | 0 7 5 -4 6 }}


Mapping: [{{val| 1 0 15 -17 51 14 -49 }}, {{val| 0 2 -16 25 -60 -13 67 }}]
Optimal tunings:  
* WE: ~63/50 = 400.0266{{c}}, ~12/11 = 157.5301{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~12/11 = 157.5240{{c}}


POTE generator: ~15/13 = 249.190
{{Optimal ET sequence|legend=0| 15, 69e, 84e, 99 }}


Optimal GPV sequence: {{Val list| 53, 130, 183, 679df }}
Badness (Sintel): 2.26


Badness: 0.021073
== Grendel ==
: ''For the 5-limit version, see [[Syntonic–31 equivalence continuum #Counterwürschmidt]].''


== Twothirdtonic ==
Grendel tempers out 16875/16807, the [[mirkwai comma]], and may be described as the {{nowrap| 31 & 152 }} temperament. [[152edo]], [[183edo]] and especially [[335edo]] serve as good tunings.  
Subgroup: 2.3.5.7


[[Comma list]]: 686/675, 6144/6125
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 3 2 4 }}, {{val| 0 -13 3 -11 }}]
[[Comma list]]: 6144/6125, 16875/16807


{{Multival|legend=1| 13 -3 11 -35 -19 34 }}
{{Mapping|legend=1| 1 -14 3 -6 | 0 23 -1 13 }}
: mapping generators: ~2, ~8/5


[[POTE generator]]: ~15/14 = 130.401
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7348{{c}}, ~8/5 = 812.9574{{c}}
: [[error map]]: {{val| -0.265 -0.220 -0.067 +1.212 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/5 = 813.1311{{c}}
: error map: {{val| 0.000 +0.059 +0.555 +1.878 }}


{{Val list|legend=1| 9, 28b, 37, 46 }}
{{Optimal ET sequence|legend=1| 31, 90, 121, 152, 335d, 822dd }}


[[Badness]]: 0.099601
[[Badness]] (Sintel): 1.31


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 121/120, 176/175, 686/675
Comma list: 540/539, 1375/1372, 5632/5625


Mapping: [{{val| 1 3 2 4 4 }}, {{val| 0 -13 3 -11 -5 }}]
Mapping: {{mapping| 1 -14 3 -6 -25 | 0 23 -1 13 42 }}


POTE generator: ~15/14 = 130.430
Optimal tunings:  
* WE: ~2 = 1199.7355{{c}}, ~8/5 = 812.9622{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/5 = 813.1353{{c}}


Optimal GPV sequence: {{Val list| 9, 28b, 37, 46 }}
{{Optimal ET sequence|legend=0| 31, 90e, 121, 152, 335d, 487d }}


Badness: 0.040768
Badness (Sintel): 0.656


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 91/90, 121/120, 169/168, 176/175
Comma list: 352/351, 540/539, 625/624, 1375/1372
 
Mapping: {{mapping| 1 -14 3 -6 -25 22 | 0 23 -1 13 42 -27 }}
 
Optimal tunings:
* WE: ~2 = 1199.4412{{c}}, ~8/5 = 812.7956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/5 = 813.1209{{c}}
 
{{Optimal ET sequence|legend=0| 31, 90e, 121, 152f, 273def, 425deff }}
 
Badness (Sintel): 1.03
 
=== 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 -14 3 -6 -25 22 19 | 0 23 -1 13 42 -27 -22 }}


Mapping: [{{val| 1 3 2 4 4 5 }}, {{val| 0 -13 3 -11 -5 -12 }}]
Optimal tunings:  
* WE: ~2 = 1199.3029{{c}}, ~8/5 = 812.7156{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/5 = 813.1843{{c}}


POTE generator: ~13/12 = 130.409
{{Optimal ET sequence|legend=0| 31, 90e, 121, 152fg, 273defgg }}


Optimal GPV sequence: {{Val list| 9, 28b, 37, 46 }}
Badness (Sintel): 1.09


Badness: 0.025941
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


== Semaja ==
Comma list: 256/255, 352/351, 375/374, 400/399, 456/455, 715/714
Subgroup: 2.3.5.7


[[Comma list]]: 3125/3087, 6144/6125
Mapping: {{mapping| 1 -14 3 -6 -25 22 19 30 | 0 23 -1 13 42 -27 -22 -38 }}


[[Mapping]]: [{{val| 1 -2 1 3 }}, {{val| 0 19 7 -1 }}]
Optimal tunings:  
* WE: ~2 = 1199.3587{{c}}, ~8/5 = 812.7462{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/5 = 813.1796{{c}}


{{Multival|legend=1| 19 7 -1 -33 -55 -22 }}
{{Optimal ET sequence|legend=0| 31, 90e, 121, 152fg, 273defgg }}


[[POTE generator]]: ~8/7 = 226.4834
Badness (Sintel): 1.12


{{Val list|legend=1| 16, 37, 53, 196d }}
== Twothirdtonic ==
Twothirdtonic tempers out 686/675, the [[senga]], in addition to the porwell comma, and may be described as the {{nowrap| 37 & 46 }} temperament, generated by one third of a [[5/4|classical major third]] that represents [[15/14]], [[14/13]], and [[13/12]] in the [[13-limit]] interpretation. Note that in the data below, the generator is taken to be its [[octave complement]], thirteen of which [[octave reduction|octave reduced]] make the [[3/2|perfect fifth]]; it follows that the [[ploidacot]] for this temperament is 11-sheared 13-cot. [[46edo]] may be recommended as a tuning.


[[Badness]]: 0.107023
[[Subgroup]]: 2.3.5.7


=== 11-limit ===
[[Comma list]]: 686/675, 6144/6125
Subgroup: 2.3.5.7.11


Comma list: 121/120, 176/175, 3125/3087
{{Mapping|legend=1| 1 -10 5 -7 | 0 13 -3 11 }}
: mapping generators: ~2, ~28/15


Mapping: [{{val| 1 -2 1 3 1 }}, {{val| 0 19 7 -1 13 }}]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.3074{{c}}, ~28/15 = 1068.9820{{c}}
: [[error map]]: {{val| -0.693 +1.736 +3.278 -5.176 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/15 = 1069.5746{{c}}
: error map: {{val| 0.000 +2.515 +4.962 -3.505 }}


POTE generator: ~8/7 = 226.4856
{{Optimal ET sequence|legend=1| 9, 28b, 37, 46 }}


Optimal GPV sequence: {{Val list| 16, 37, 53 }}
[[Badness]] (Sintel): 2.52


Badness: 0.059838
=== 11-limit ===
Subgroup: 2.3.5.7.11


=== 13-limit ===
Comma list: 121/120, 176/175, 686/675
Subgroup: 2.3.5.7.11.13


Comma list: 121/120, 169/168, 176/175, 275/273
Mapping: {{mapping| 1 -10 5 -7 -1 | 0 13 -3 11 5 }}


Mapping: [{{val| 1 -2 1 3 1 2 }}, {{val| 0 19 7 -1 13 9 }}]
Optimal tunings:  
* WE: ~2 = 1199.7068{{c}}, ~28/15 = 1069.3084{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/15 = 1069.5600{{c}}


POTE generator: ~8/7 = 226.4794
{{Optimal ET sequence|legend=0| 9, 28b, 37, 46 }}


Optimal GPV sequence: {{Val list| 16, 37, 53 }}
Badness (Sintel): 1.35


Badness: 0.032564
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


== Nessafof ==
Comma list: 91/90, 121/120, 169/168, 176/175
Subgroup: 2.3.5.7


[[Comma list]]: 6144/6125, 250047/250000
Mapping: {{mapping| 1 -10 5 -7 -1 -7 | 0 13 -3 11 5 12 }}


[[Mapping]]: [{{val| 3 2 5 10 }}, {{val| 0 7 5 -4 }}]
Optimal tunings:  
* WE: ~2 = 1199.9531{{c}}, ~13/7 = 1069.5492{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/7 = 1069.5893{{c}}


{{Multival|legend=1| 21 15 -12 -25 -78 -70 }}
{{Optimal ET sequence|legend=0| 9, 28b, 37, 46 }}


[[POTE generator]]: ~35/32 = 157.480
Badness (Sintel): 1.07


{{Val list|legend=1| 15, 54b, 69, 84, 99, 282, 381 }}
== Semaja ==
{{See also| Llywelynsmic clan }}


[[Badness]]: 0.045048
Cryptically named by [[Petr Pařízek]] in 2011, semaja adds the [[gariboh comma]] to the comma list, and may be described as the {{nowrap| 37 & 53 }} temperament. Its [[ploidacot]] is gamma-19-cot (or alpha-heptaseph due to a much simpler [[2.5.7 subgroup|2.5.7-subgroup]] [[restriction]]). The name actually refers to the fact that two of its ~[[8/7]] generator steps reach a ~[[13/10]]<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>.  


=== 11-limit ===
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11


Comma list: 121/120, 176/175, 250047/250000
[[Comma list]]: 3125/3087, 6144/6125


Mapping: [{{val| 3 2 5 10 8 }}, {{val| 0 7 5 -4 6 }}]
{{Mapping|legend=1| 1 -2 1 3 | 0 19 7 -1 }}
: mapping generators: ~2, ~8/7


POTE generator: ~12/11 = 157.520
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.4860{{c}}, ~8/7 = 226.3864{{c}}
: [[error map]]: {{val| -0.514 +0.415 -2.123 +3.246 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 226.4697{{c}}
: error map: {{val| 0.000 +0.970 -1.026 +4.704 }}


Optimal GPV sequence: {{Val list| 15, 54be, 69e, 84e, 99 }}
{{Optimal ET sequence|legend=1| 16, 37, 53, 196d }}


Badness: 0.068427
[[Badness]] (Sintel): 2.71


=== Nessa ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 441/440, 1344/1331, 4375/4356
Comma list: 121/120, 176/175, 3125/3087


Mapping: [{{val| 3 2 5 10 10 }}, {{val| 0 7 5 -4 1 }}]
Mapping: {{mapping| 1 -2 1 3 1 | 0 19 7 -1 13 }}


POTE generator: ~35/32 = 157.539
Optimal tunings:  
* WE: ~2 = 1199.9818{{c}}, ~8/7 = 226.4821{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 226.4851{{c}}


Optimal GPV sequence: {{Val list| 15, 54b, 69, 84, 99e }}
{{Optimal ET sequence|legend=0| 16, 37, 53 }}


Badness: 0.048836
Badness (Sintel): 1.98


==== 13-limit ====
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 144/143, 364/363, 441/440, 625/624
Comma list: 121/120, 169/168, 176/175, 275/273


Mapping: [{{val| 3 2 5 10 10 6 }}, {{val| 0 7 5 -4 1 13 }}]
Mapping: {{mapping| 1 -2 1 3 1 2 | 0 19 7 -1 13 9 }}


POTE generator: ~35/32 = 157.429
Optimal tunings:  
* WE: ~2 = 1200.1020{{c}}, ~8/7 = 226.4987{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 226.4822{{c}}


Optimal GPV sequence: {{Val list| 15, 54bf, 69, 84, 99ef, 183ef, 282eeff }}
{{Optimal ET sequence|legend=0| 16, 37, 53 }}


Badness: 0.037409
Badness (Sintel): 1.35


== Aufo ==
== Aufo ==
{{see also| High badness temperaments #Untriton }}
:''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''
 
Also named by [[Petr Pařízek]] in 2011, ''aufo'' refers to the augmented fourth, which is a generator of this temperament<ref name="petr's long post"/>. The functional generator however is the [[64/45]] diminished fifth, and like its [[untriton]] variant, nine generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is delta-enneacot.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 6144/6125, 177147/175616
[[Comma list]]: 6144/6125, 177147/175616


[[Mapping]]: [{{Val|1 6 -7 19}}, {{Val|0 -9 19 -33}}]
{{Mapping|legend=1| 1 -3 12 -14 | 0 9 -19 33 }}
: mapping generators: ~2, ~64/45


{{Multival|legend=1|9 -19 33 -51 27 130}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9758{{c}}, ~64/45 = 611.2055{{c}}
: [[error map]]: {{val| -0.024 -1.303 +0.491 +1.295 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~64/45 = 611.2177{{c}}
: error map: {{val| 0.000 -0.996 +0.551 +1.357 }}


[[POTE generator]]: ~45/32 = 588.782
{{Optimal ET sequence|legend=1| 53, 161, 214 }}


{{Val list|legend=1| 53, 161, 214 }}
[[Badness]] (Sintel): 3.07
 
[[Badness]]: 0.121428


=== 11-limit ===
=== 11-limit ===
Line 344: Line 432:
Comma list: 121/120, 176/175, 177147/175616
Comma list: 121/120, 176/175, 177147/175616


Mapping: [{{Val|1 6 -7 19 1}}, {{Val|0 -9 19 -33 5}}]
Mapping: {{mapping| 1 -3 12 -14 6 | 0 9 -19 33 -5 }}


POTE generator: ~45/32 = 588.811
Optimal tunings:  
* WE: ~2 = 1200.4500{{c}}, ~64/45 = 611.4185{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~64/45 = 611.1918{{c}}


Optimal GPV sequence: {{Val list| 53, 108e, 161e }}
{{Optimal ET sequence|legend=0| 53, 108e, 161e }}


Badness: 0.088631
Badness (Sintel): 2.93


==== 13-limit ====
==== 13-limit ====
Line 357: Line 447:
Comma list: 121/120, 176/175, 351/350, 58806/57967
Comma list: 121/120, 176/175, 351/350, 58806/57967


Mapping: [{{Val|1 6 -7 19 1 -12}}, {{Val|0 -9 19 -33 5 32}}]
Mapping: {{mapping| 1 -3 12 -14 6 20 | 0 9 -19 33 -5 -32 }}


POTE generator: ~45/32 = 588.788
Optimal tunings:  
* WE: ~2 = 1200.3134{{c}}, ~64/45 = 611.3715{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~64/45 = 611.2118{{c}}


Optimal GPV sequence: {{Val list| 53, 108e, 161e, 214ee }}
{{Optimal ET sequence|legend=0| 53, 108e }}


Badness: 0.058507
Badness (Sintel): 2.42


=== Aufic ===
=== Aufic ===
Line 370: Line 462:
Comma list: 540/539, 5632/5625, 72171/71680
Comma list: 540/539, 5632/5625, 72171/71680


Mapping: [{{Val|1 6 -7 19 -25}}, {{Val|0 -9 19 -33 58}}]
Mapping: {{mapping| 1 -3 12 -14 33 | 0 9 -19 33 -58 }}


POTE generator: ~45/32 = 588.800
Optimal tunings:  
* WE: ~2 = 1200.0668{{c}}, ~64/45 = 611.2342{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~64/45 = 611.2000{{c}}


Optimal GPV sequence: {{Val list| 53, 108, 161, 214, 375 }}
{{Optimal ET sequence|legend=0| 53, 108, 161, 214, 375 }}


Badness: 0.075149
Badness (Sintel): 2.48


==== 13-limit ====
==== 13-limit ====
Line 383: Line 477:
Comma list: 351/350, 540/539, 847/845, 4096/4095
Comma list: 351/350, 540/539, 847/845, 4096/4095


Mapping: [{{Val|1 6 -7 19 -25 -12}}, {{Val|0 -9 19 -33 58 32}}]
Mapping: {{mapping| 1 -3 12 -14 33 20 | 0 9 -19 33 -58 -32 }}


POTE generator: ~45/32 = 588.796
Optimal tunings:  
* WE: ~2 = 1200.0177{{c}}, ~64/45 = 611.2130{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~64/45 = 611.2039{{c}}


Optimal GPV sequence: {{Val list| 53, 108, 161, 214, 375, 589be }}
{{Optimal ET sequence|legend=0| 53, 108, 161, 214, 375 }}


Badness: 0.039050
Badness (Sintel): 1.61


== Whoops ==
== Absurdity ==
{{see also| Very high accuracy temperaments #Whoosh }}
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Absurdity (5-limit)]].''
{{See also| Fifth-chroma temperaments }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 6144/6125, 244140625/243045684
[[Comma list]]: 6144/6125, 177147/175000


[[Mapping]]: [{{val| 1 17 14 -7 }}, {{val| 0 -33 -25 21 }}]
{{Mapping|legend=1| 7 0 -17 64 | 0 1 3 -4 }}
: mapping generators: ~972/875, ~3


{{Multival|legend=1| 33 25 -21 -37 -126 -119 }}
[[Optimal tuning]]s:
* [[WE]]: ~972/875 = 171.4382{{c}}, ~3/2 = 700.6247{{c}}
: [[error map]]: {{val| +0.067 -1.263 +1.313 +0.450 }}
* [[CWE]]: ~972/875 = 171.4286{{c}}, ~3/2 = 700.5871{{c}}
: error map: {{val| 0.000 -1.368 +1.162 +0.254 }}


[[POTE generator]]: ~441/320 = 560.519
{{Optimal ET sequence|legend=1| 77, 84, 161 }}


{{Val list|legend=1| 15, 122d, 137, 152, 608d, 623bd, 775bcd }}
[[Badness]] (Sintel): 3.38
 
[[Badness]]: 0.175840


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 3025/3024, 4000/3993, 6144/6125
Comma list: 441/440, 6144/6125, 72171/71680
 
Mapping: {{mapping| 7 0 -17 64 124 | 0 1 3 -4 -9 }}
 
Optimal tunings:
* WE: ~495/448 = 171.4346{{c}}, ~3/2 = 700.6602{{c}}
* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 700.6339{{c}}
 
{{Optimal ET sequence|legend=0| 77, 84, 161 }}
 
Badness (Sintel): 2.70


Mapping: [{{val| 1 17 14 -7 10 }}, {{val| 0 -33 -25 21 -14 }}]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


POTE generator: ~242/175 = 560.519
Comma list: 351/350, 441/440, 1188/1183, 3584/3575


Optimal GPV sequence: {{Val list| 15, 122d, 137, 152, 608de, 623bde, 775bcde }}
Mapping: {{mapping| 7 0 -17 64 124 37 | 0 1 3 -4 -9 -1 }}


Badness: 0.043743
Optimal tunings:  
* WE: ~72/65 = 171.4223{{c}}, ~3/2 = 700.6036{{c}}
* CWE: ~72/65 = 171.4286{{c}}, ~3/2 = 700.6306{{c}}


== Polypyth ==
{{Optimal ET sequence|legend=0| 77, 84, 161 }}
{{see also| High badness temperaments #Leapday }}


Polypyth (46&amp;121) tempers out the same 5-limit comma as the [[Hemifamity temperaments #Leapday|leapday temperament]] (29&amp;46), but with the porwell (6144/6125) rather than the hemifamity (5120/5103) tempered out.
Badness (Sintel): 1.72


Subgroup: 2.3.5.7
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


[[Comma list]]: 6144/6125, 179200/177147
Comma list: 351/350, 441/440, 561/560, 1188/1183, 1632/1625


[[Mapping]]: [{{val| 1 0 -31 52 }}, {{val| 0 1 21 -31 }}]
Mapping: {{mapping| 7 0 -17 64 124 37 -49 | 0 1 3 -4 -9 -1 7 }}


[[POTE generator]]: ~3/2 = 704.174
Optimal tunings:  
* WE: ~72/65 = 171.4263{{c}}, ~3/2 = 700.6429{{c}}
* CWE: ~72/65 = 171.4286{{c}}, ~3/2 = 700.6525{{c}}


{{Val list|legend=1| 46, 121, 167, 288b, 455bcd, 743bcd }}
{{Optimal ET sequence|legend=0| 77, 161 }}


[[Badness]]: 0.137995
Badness (Sintel): 1.62


=== 11-limit ===
=== 19-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 896/891, 2200/2187, 6144/6125
Comma list: 324/323, 351/350, 441/440, 456/455, 476/475, 495/494


Mapping: [{{val| 1 0 -31 52 59 }}, {{val| 0 1 21 -31 -35 }}]
Mapping: {{mapping| 7 0 -17 64 124 37 -49 63 | 0 1 3 -4 -9 -1 7 -3 }}


POTE generator: ~3/2 = 704.177
Optimal tunings:  
* WE: ~21/19 = 171.4244{{c}}, ~3/2 = 700.6395{{c}}
* CWE: ~21/19 = 171.4286{{c}}, ~3/2 = 700.6568{{c}}


Optimal GPV sequence: {{Val list| 46, 121, 167, 288be, 455bcde }}
{{Optimal ET sequence|legend=0| 77, 161 }}


Badness: 0.051131
Badness (Sintel): 1.36


=== 13-limit ===
=== 23-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13.17.19.23


Comma list: 325/324, 352/351, 364/363, 1716/1715
Comma list: 276/275, 324/323, 351/350, 441/440, 456/455, 476/475, 495/494


Mapping: [{{val| 1 0 -31 52 59 64 }}, {{val| 0 1 21 -31 -35 -38 }}]
Mapping: {{mapping| 7 0 -17 64 124 37 -49 63 76 | 0 1 3 -4 -9 -1 7 -3 -4 }}


POTE generator: ~3/2 = 704.168
Optimal tunings:  
* WE: ~21/19 = 171.4321{{c}}, ~3/2 = 700.6475{{c}}
* CWE: ~21/19 = 171.4286{{c}}, ~3/2 = 700.6325{{c}}


Optimal GPV sequence: {{Val list| 46, 121, 167, 288be }}
{{Optimal ET sequence|legend=0| 77, 84, 161 }}


Badness: 0.030292
Badness (Sintel): 1.34


=== 17-limit ===
=== 29-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17.19.23.29


Comma list: 256/255, 325/324, 352/351, 364/363, 1716/1715
Comma list: 261/260, 276/275, 324/323, 351/350, 441/440, 456/455, 476/475, 495/494


Mapping: [{{val| 1 0 -31 52 59 64 39 }}, {{val| 0 1 21 -31 -35 -38 -22 }}]
Mapping: {{mapping| 7 0 -17 64 124 37 -49 63 76 34 | 0 1 3 -4 -9 -1 7 -3 -4 0 }}


POTE generator: ~3/2 = 704.168
Optimal tunings:  
* WE: ~21/19 = 171.4348{{c}}, ~3/2 = 700.6612{{c}}
* CWE: ~21/19 = 171.4286{{c}}, ~3/2 = 700.6351{{c}}


Optimal GPV sequence: {{Val list| 46, 121, 167, 288beg }}
{{Optimal ET sequence|legend=0| 77, 84, 161 }}


Badness: 0.019051
Badness (Sintel): 1.25


== Icositritonic ==
== Polypyth ==
The ''icositritonic'' temperament (46&amp;161) has a period of 1/23 octave, so six period represents [[6/5]] and nine period represents [[21/16]].
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Leapday]].''


Subgroup: 2.3.5.7
Polypyth tempers out the same 5-limit comma as [[leapday]], with which it shares the similarly sharp [[3/2|perfect-fifth]] generator, but the porwell comma (6144/6125) rather than the hemifamity comma (5120/5103) is tempered out here. It may be described as the {{nowrap| 46 & 121 }} temperament, and [[121edo]] and [[167edo]] make for good tunings.  


[[Comma list]]: 6144/6125, 9920232/9765625
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 23 37 54 64 }}, {{val| 0 -1 -1 1 }}]
[[Comma list]]: 6144/6125, 179200/177147


{{Multival|legend=1| 23 23 -23 -17 -101 -118 }}
{{Mapping|legend=1| 1 0 -31 52 | 0 1 21 -31 }}
: mapping generators: ~2, ~3


[[POTE generator]]: ~64/63 = 29.3586
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.3465{{c}}, ~3/2 = 703.7905{{c}}
: [[error map]]: {{val| -0.654 +1.182 -0.177 -0.056 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.1749{{c}}
: error map: {{val| 0.000 +2.220 +1.359 +1.752 }}


{{Val list|legend=1| 23, 46, 115, 161, 207, 368c }}
{{Optimal ET sequence|legend=1| 46, 121, 167, 288b, 455bcd }}


[[Badness]]: 0.196622
[[Badness]] (Sintel): 3.49


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 441/440, 6144/6125, 35937/35840
Comma list: 896/891, 2200/2187, 6144/6125


Mapping: [{{val| 23 37 54 64 79 }}, {{val| 0 -1 -1 1 1 }}]
Mapping: {{mapping| 1 0 -31 52 59 | 0 1 21 -31 -35 }}


POTE generator: ~64/63 = 29.3980
Optimal tunings:  
* WE: ~2 = 1199.3335{{c}}, ~3/2 = 703.7856{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1812{{c}}


Optimal GPV sequence: {{Val list| 23, 46, 115, 161, 207, 368c }}
{{Optimal ET sequence|legend=0| 46, 121, 167, 288be, 455bcde }}


Badness: 0.064613
Badness (Sintel): 1.69


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 351/350, 441/440, 847/845, 3584/3575
Comma list: 325/324, 352/351, 364/363, 1716/1715


Mapping: [{{val| 23 37 54 64 79 84 }}, {{val| 0 -1 -1 1 1 2 }}]
Mapping: {{mapping| 1 0 -31 52 59 64 | 0 1 21 -31 -35 -38 }}


POTE generator: ~64/63 = 29.2830
Optimal tunings:  
* WE: ~2 = 1199.3768{{c}}, ~3/2 = 703.8018{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1731{{c}}


Optimal GPV sequence: {{Val list| 46, 115, 161, 207, 368c }}
{{Optimal ET sequence|legend=0| 46, 75e, 121, 167, 288be }}


Badness: 0.040484
Badness (Sintel): 1.25


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 351/350, 441/440, 561/560, 847/845, 1089/1088
Comma list: 256/255, 325/324, 352/351, 364/363, 1716/1715


Mapping: [{{val| 23 37 54 64 79 84 94 }}, {{val| 0 -1 -1 1 1 2 0 }}]
Mapping: {{mapping| 1 0 -31 52 59 64 39 | 0 1 21 -31 -35 -38 -22 }}


POTE generator: ~64/63 = 29.2800
Optimal tunings:  
* WE: ~2 = 1199.3518{{c}}, ~3/2 = 703.7880{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1747{{c}}


Optimal GPV sequence: {{Val list| 46, 115, 161, 207, 368c }}
{{Optimal ET sequence|legend=0| 46, 75e, 121, 167, 288beg }}


Badness: 0.024676
Badness (Sintel): 0.971


=== 19-limit ===
== Whoops ==
Subgroup: 2.3.5.7.11.13.17.19
: ''For the 5-limit version, see [[Very high accuracy temperaments #Whoosh]].''


Comma list: 351/350, 441/440, 456/455, 476/475, 513/512, 847/845
Also named by [[Petr Pařízek]] in 2011, whoops is a relatively simple extension to the otherwise very accurate microtemperament known as ''whoosh''<ref name="petr's long post"/>.


Mapping: [{{val| 23 37 54 64 79 84 94 96 }}, {{val| 0 -1 -1 1 1 2 0 3 }}]
[[Subgroup]]: 2.3.5.7


POTE generator: ~64/63 = 29.3760
[[Comma list]]: 6144/6125, 244140625/243045684
 
Optimal GPV sequence: {{Val list| 46, 115, 161, 207, 368c }}
 
Badness: 0.021579
 
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 276/275, 351/350, 391/390, 441/440, 456/455, 476/475, 847/845
 
Mapping: [{{val| 23 37 54 64 79 84 94 96 104 }}, {{val| 0 -1 -1 1 1 2 0 3 0 }}]
 
POTE generator: ~64/63 = 29.3471
 
Optimal GPV sequence: {{Val list| 46, 115, 161, 207, 368ci }}
 
Badness: 0.017745
 
== Countermiracle ==
The ''countermiracle'' temperament (31&amp;145) tempers out the trimyna, 50421/50000 and the [[Quince clan|quince comma]], 823543/819200.
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 6144/6125, 50421/50000


[[Mapping]]: [{{val| 1 4 3 3 }}, {{val| 0 -25 -7 -2 }}]
{{Mapping|legend=1| 1 -16 -11 14 | 0 33 25 -21 }}
: mapping generators: ~2, ~640/441


{{Multival|legend=1| 25 7 2 -47 -67 -15 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5944{{c}}, ~640/441 = 639.2648{{c}}
: [[error map]]: {{val| -0.406 +0.272 -0.233 +0.936 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~640/441 = 639.4769{{c}}
: error map: {{val| 0.000 +0.783 +0.609 +2.159 }}


[[POTE generator]]: ~343/320 = 115.9169
{{Optimal ET sequence|legend=1| 15, 122d, 137, 152, 623bdd, 775bcdd, 927bcddd, 1079bcddd }}


{{Val list|legend=1| 31, 114, 145, 176, 559cc, 735cc }}
[[Badness]] (Sintel): 4.45
 
[[Badness]]: 0.102326


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 441/440, 3388/3375, 6144/6125
Comma list: 3025/3024, 4000/3993, 6144/6125


Mapping: [{{val| 1 4 3 3 8 }}, {{val| 0 -25 -7 -2 -47 }}]
Mapping: {{mapping| 1 -16 -11 14 -4 | 0 33 25 -21 14 }}


POTE generator: ~77/72 = 115.9158
Optimal tunings:  
* WE: ~2 = 1199.5936{{c}}, ~175/121 = 639.264{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~175/121 = 639.4770{{c}}


Optimal GPV sequence: {{Val list| 31, 114e, 145, 176 }}
{{Optimal ET sequence|legend=0| 15, 122d, 137, 152, 623bdde, 775bcdde, 927bcdddee, 1079bcdddee }}


Badness: 0.039162
Badness (Sintel): 1.45


==== Countermiraculous ====
== Dodifo ==
Subgroup: 2.3.5.7.11.13
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Dodifo]].''


Comma list: 196/195, 352/351, 1001/1000, 6144/6125
Also named by [[Petr Pařízek]] in 2011, ''dodifo'' refers to the (tetraptolemaic) double-diminished fourth, which is a generator of this temperament<ref name="petr's long post"/>. The extension here is a less accurate 7-limit interpretation.


Mapping: [{{val| 1 4 3 3 8 1 }}, {{val| 0 -25 -7 -2 -47 28 }}]
[[Subgroup]]: 2.3.5.7


POTE generator: ~77/72 = 115.8803
[[Comma list]]: 6144/6125, 2500000/2470629


Optimal GPV sequence: {{Val list| 31, 83e, 114e, 145, 321ceff }}
{{Mapping|legend=1| 1 -23 -4 0 | 0 35 9 4 }}
: mapping generators: ~2, ~80/49


Badness: 0.039271
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.6429{{c}}, ~80/49 = 842.6790{{c}}
: [[error map]]: {{val| -0.357 +0.228 -0.774 +1.890 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~80/49 = 842.9243{{c}}
: error map: {{val| 0.000 +0.396 +0.005 +2.871 }}


==== Counterbenediction ====
{{Optimal ET sequence|legend=1| 37, 84, 121, 205 }}
Subgroup: 2.3.5.7.11.13


Comma list: 351/350, 441/440, 3146/3125, 3584/3575
[[Badness]] (Sintel): 4.55


Mapping: [{{val| 1 4 3 3 8 -2 }}, {{val| 0 -25 -7 -2 -47 59 }}]
=== 11-limit ===
 
POTE generator: ~77/72 = 115.9335
 
Optimal GPV sequence: {{Val list| 31, 145f, 176, 207, 383c, 590cc }}
 
Badness: 0.045569
 
==== Countermanna ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 364/363, 441/440, 3388/3375, 6144/6125
 
Mapping: [{{val| 1 4 3 3 8 15 }}, {{val| 0 -25 -7 -2 -47 -117 }}]
 
POTE generator: ~77/72 = 115.8898
 
Optimal GPV sequence: {{Val list| 31f, 145, 176, 321ce }}
 
Badness: 0.053409
 
=== Counterrevelation ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 121/120, 176/175, 50421/50000
Comma list: 1375/1372, 2560/2541, 4375/4356


Mapping: [{{val| 1 4 3 3 5 }}, {{val| 0 -25 -7 -2 -16 }}]
Mapping: {{mapping| 1 -23 -4 0 14 | 0 35 9 4 -15 }}


POTE generator: ~343/320 = 115.9192
Optimal tunings:  
* WE: ~2 = 1199.3401{{c}}, ~80/49 = 842.4880{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~80/49 = 842.9457{{c}}


Optimal GPV sequence: {{Val list| 31, 114, 145e, 176e }}
{{Optimal ET sequence|legend=0| 37, 84, 121, 326dee }}


Badness: 0.064070
Badness (Sintel): 2.71


==== 13-limit ====
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 121/120, 176/175, 196/195, 13750/13689
Comma list: 364/363, 625/624, 640/637, 1375/1372
 
Mapping: {{mapping| 1 12 5 4 -1 4 | 0 -35 -9 -4 15 -1 }}


Mapping: [{{val| 1 4 3 3 5 1 }}, {{val| 0 -25 -7 -2 -16 28 }}]
Optimal tunings:  
* WE: ~2 = 1199.3410{{c}}, ~13/8 = 842.4885{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.9466{{c}}


POTE generator: ~273/256 = 115.8624
{{Optimal ET sequence|legend=0| 37, 84, 121, 326deef }}


Optimal GPV sequence: {{Val list| 31, 83, 114, 145e }}
Badness (Sintel): 1.63


Badness: 0.057497
== Icositritonic ==
{{See also| 23rd-octave temperaments }}


== Absurdity ==
Icositritonic has a period of 1/23 octave, so six period represents [[6/5]] and nine period represents [[21/16]]. It may be described as {{nowrap| 46 & 161 }}. It was named by [[Xenllium]] in 2019 for its number of periods per octave.  
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Absurdity]].''


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 6144/6125, 177147/175000
[[Comma list]]: 6144/6125, 9920232/9765625


[[Mapping]]: [{{val|7 11 16 20}}, {{val|0 1 3 -4}}]
{{Mapping|legend=1| 23 0 17 101 | 0 1 1 -1 }}
: mapping generators: ~1323/1280, ~3


[[POTE generator]]: ~3/2 = 700.5854 (or ~10/9 = 186.2997)
[[Optimal tuning]]s:  
* [[WE]]: ~1323/1280 = 52.1732{{c}}, ~3/2 = 701.0660{{c}}
: [[error map]]: {{val| -0.017 -0.906 +1.679 -0.386 }}
* [[CWE]]: ~1323/1280 = 52.1739{{c}}, ~3/2 = 701.0722{{c}}
: error map: {{val| 0.000 -0.883 +1.715 -0.333 }}


{{Val list|legend=1| 77, 84, 161 }}
{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368c }}


[[Badness]]: 0.133520
[[Badness]] (Sintel): 4.98


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 441/440, 6144/6125, 72171/71680
Comma list: 441/440, 6144/6125, 35937/35840


Mapping: [{{val|7 11 16 20 25}}, {{val|0 1 3 -4 -9}}]
Mapping: {{mapping| 23 0 17 101 116 | 0 1 1 -1 -1 }}


POTE generator: ~3/2 = 700.6354 (or ~10/9 = 186.3497)
Optimal tunings:  
* WE: ~33/32 = 52.1740{{c}}, ~3/2 = 701.0379{{c}}
* CWE: ~33/32 = 52.1739{{c}}, ~3/2 = 701.0370{{c}}


Optimal GPV sequence: {{Val list| 77, 84, 161 }}
{{Optimal ET sequence|legend=0| 46, 115, 161, 207, 368c }}


Badness: 0.081564
Badness (Sintel): 2.14


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 351/350, 441/440, 1188/1183, 3584/3575
Comma list: 351/350, 441/440, 847/845, 3584/3575


Mapping: [{{val|7 11 16 20 25 26}}, {{val|0 1 3 -4 -9 -1}}]
Mapping: {{mapping| 23 0 17 101 116 158 | 0 1 1 -1 -1 -2 }}


POTE generator: ~3/2 = 700.6291 (or ~10/9 = 186.3434)
Optimal tunings:  
* WE: ~33/32 = 52.1724{{c}}, ~3/2 = 701.1310{{c}}
* CWE: ~33/32 = 52.1739{{c}}, ~3/2 = 701.1524{{c}}


Optimal GPV sequence: {{Val list| 77, 84, 161 }}
{{Optimal ET sequence|legend=0| 46, 115, 161, 207, 368c }}


Badness: 0.041600
Badness (Sintel): 1.67


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 351/350, 441/440, 561/560, 1188/1183, 1632/1625
Comma list: 351/350, 441/440, 561/560, 847/845, 1089/1088


Mapping: [{{val|7 11 16 20 25 26 28}}, {{val|0 1 3 -4 -9 -1 7}}]
Mapping: {{mapping| 23 0 17 101 116 158 94 | 0 1 1 -1 -1 -2 0 }}


POTE generator: ~3/2 = 700.6524 (or ~10/9 = 186.3667)
Optimal tunings:  
* WE: ~33/32 = 52.1735{{c}}, ~3/2 = 701.1493{{c}}
* CWE: ~33/32 = 52.1739{{c}}, ~3/2 = 701.1549{{c}}


Optimal GPV sequence: {{Val list| 77, 161 }}
{{Optimal ET sequence|legend=0| 46, 115, 161, 207, 368c }}


Badness: 0.031783
Badness (Sintel): 1.26


=== 19-limit ===
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 324/323, 351/350, 441/440, 456/455, 476/475, 495/494
Comma list: 351/350, 441/440, 456/455, 476/475, 513/512, 847/845


Mapping: [{{val|7 11 16 20 25 26 28 30}}, {{val|0 1 3 -4 -9 -1 7 -3}}]
Mapping: {{mapping| 23 0 17 101 116 158 94 207 | 0 1 1 -1 -1 -2 0 -3 }}


POTE generator: ~3/2 = 700.6565 (or ~10/9 = 186.3708)
Optimal tunings:  
* WE: ~33/32 = 52.1744{{c}}, ~3/2 = 701.0649{{c}}
* CWE: ~33/32 = 52.1739{{c}}, ~3/2 = 701.0582{{c}}


Optimal GPV sequence: {{Val list| 77, 161 }}
{{Optimal ET sequence|legend=0| 46, 115, 161, 207, 368c }}


Badness: 0.022291
Badness (Sintel): 1.31


== Dodifo ==
=== 23-limit ===
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Dodifo]].''
Subgroup: 2.3.5.7.11.13.17.19.23


Subgroup: 2.3.5.7
Comma list: 276/275, 351/350, 391/390, 441/440, 456/455, 476/475, 847/845
 
[[Comma list]]: 6144/6125, 2500000/2470629
 
[[Mapping]]: [{{val|1 12 5 4}}, {{val|0 -35 -9 -4}}]
 
[[POTE generator]]: ~49/40 = 357.070
 
{{Val list|legend=1| 37, 84, 121, 205 }}
 
[[Badness]]: 0.179692
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 1375/1372, 2560/2541, 4375/4356
 
Mapping: [{{val|1 12 5 4 -1}}, {{val|0 -35 -9 -4 15}}]


POTE generator: ~49/40 = 357.048
Mapping: {{mapping| 23 0 17 101 116 158 94 207 104 | 0 1 1 -1 -1 -2 0 -3 0 }}
 
Optimal GPV sequence: {{Val list| 37, 84, 121, 326dee }}
 
Badness: 0.081923
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 364/363, 625/624, 640/637, 1375/1372


Mapping: [{{val|1 12 5 4 -1 4}}, {{val|0 -35 -9 -4 15 -1}}]
Optimal tunings:  
* WE: ~33/32 = 52.1768{{c}}, ~3/2 = 701.1259{{c}}
* CWE: ~33/32 = 52.1739{{c}}, ~3/2 = 701.0841{{c}}


POTE generator: ~16/13 = 357.049
{{Optimal ET sequence|legend=0| 46, 115, 161, 207 }}


Optimal GPV sequence: {{Val list| 37, 84, 121, 326deef }}
Badness (Sintel): 1.27


Badness: 0.039533
== References ==


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Porwell temperaments| ]] <!-- main article -->
[[Category:Porwell temperaments| ]] <!-- main article -->
[[Category:Porwell]]
[[Category:Hemischis]]
[[Category:Hendecatonic]]
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 10:17, 28 May 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.

This is a collection of temperaments that temper out the porwell comma (monzo: [11 1 -3 -2⟩, ratio: 6144/6125).

Temperaments discussed elsewhere are:

Considered below are hendecatonic, nessafof, grendel, twothirdtonic, aufo, absurdity, polypyth, whoops, dodifo, and icositritonic, in the order of increasing badness.

Hendecatonic

For the 5-limit version, see 11th-octave temperaments #Hendecapent.

The hendecatonic temperament has a period of 1/11 octave, which represents 16/15 and four times of which represent 9/7. It tempers out 10976/10935, the hemimage comma, and may be described as the 22 & 99 temperament, with 99edo giving an almost perfect tuning.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 10976/10935

Mapping: [⟨11 0 43 -4], ⟨0 1 -1 2]]

mapping generators: ~16/15, ~3

Optimal tunings:

  • WE: ~16/15 = 109.0526 ¢, ~3/2 = 702.8069 ¢
error map: ⟨-0.421 +0.431 +0.563 -0.265]
  • CWE: ~16/15 = 109.0909 ¢, ~3/2 = 702.9705 ¢
error map: ⟨0.000 +1.015 +1.625 +0.751]

Optimal ET sequence: 22, 55, 77, 99

Badness (Sintel): 1.04

Hendecaton

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 10976/10935

Mapping: [⟨11 0 43 -4 38], ⟨0 1 -1 2 0]]

Optimal tunings:

  • WE: ~16/15 = 109.0977 ¢, ~3/2 = 702.6801 ¢
  • CWE: ~16/15 = 109.0909 ¢, ~3/2 = 702.6484 ¢

Optimal ET sequence: 22, 55, 77, 99

Badness (Sintel): 1.52

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 176/175, 351/350, 4459/4455

Mapping: [⟨11 0 43 -4 38 93], ⟨0 1 -1 2 0 -3]]

Optimal tunings:

  • WE: ~16/15 = 109.1092 ¢, ~3/2 = 702.4093 ¢
  • CWE: ~16/15 = 109.0909 ¢, ~3/2 = 702.2930 ¢

Optimal ET sequence: 22, 55, 77, 99

Badness (Sintel): 1.66

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 121/120, 154/153, 176/175, 273/272, 2025/2023

Mapping: [⟨11 0 43 -4 38 93 45], ⟨0 1 -1 2 0 -3 0]]

Optimal tunings:

  • WE: ~16/15 = 109.0933 ¢, ~3/2 = 702.3170 ¢
  • CWE: ~16/15 = 109.0909 ¢, ~3/2 = 702.3017 ¢

Optimal ET sequence: 22, 55, 77, 99, 176eg

Badness (Sintel): 1.48

Cohendecatonic

Subgroup: 2.3.5.7.11

Comma list: 540/539, 896/891, 4375/4356

Mapping: [⟨11 0 43 -4 73], ⟨0 1 -1 2 -2]]

Optimal tunings:

  • WE: ~16/15 = 109.0237 ¢, ~3/2 = 703.2522 ¢
  • CWE: ~16/15 = 109.0909 ¢, ~3/2 = 703.6563 ¢

Optimal ET sequence: 22, 77e, 99e, 121, 220e

Badness (Sintel): 1.26

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 364/363, 540/539, 625/624

Mapping: [⟨11 0 43 -4 73 128], ⟨0 1 -1 2 -2 -5]]

Optimal tunings:

  • WE: ~16/15 = 109.0189 ¢, ~3/2 = 703.4228 ¢
  • CWE: ~16/15 = 109.0909 ¢, ~3/2 = 703.9248 ¢

Optimal ET sequence: 22, 99ef, 121, 341bdeeff

Badness (Sintel): 1.49

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 352/351, 364/363, 375/374, 540/539

Mapping: [⟨11 0 43 -4 73 128 45], ⟨0 1 -1 2 -2 -5 0]]

Optimal tunings:

  • WE: ~16/15 = 109.0159 ¢, ~3/2 = 703.3932 ¢
  • CWE: ~16/15 = 109.0909 ¢, ~3/2 = 703.9110 ¢

Optimal ET sequence: 22, 99ef, 121, 220efg, 341bdeeffgg

Badness (Sintel): 1.15

Icosidillic

Subgroup: 2.3.5.7.11

Comma list: 3388/3375, 6144/6125, 9801/9800

Mapping: [⟨22 0 86 -8 111], ⟨0 1 -1 2 -1]]

mapping generators: ~33/32, ~3

Optimal tunings:

  • WE: ~33/32 = 54.5305 ¢, ~3/2 = 702.7206 ¢
  • CWE: ~33/32 = 54.5455 ¢, ~3/2 = 702.8829 ¢

Optimal ET sequence: 22, 154, 176, 198

Badness (Sintel): 1.84

Nessafof

For the 5-limit version, see Miscellaneous 5-limit temperaments #Nessafof.

Cryptically named by Petr Pařízek in 2011[1], nessafof adds the landscape comma and has a third-octave period. The name actually refers to the fact that it has a neutral-second generator, and that a semi-augmented fourth, stacked five times, makes 5/1[2].

Subgroup: 2.3.5.7

Comma list: 6144/6125, 250047/250000

Mapping: [⟨3 2 5 10], ⟨0 7 5 -4]]

mapping generators: ~63/50, ~35/32

Optimal tunings:

  • WE: ~63/50 = 399.9023 ¢, ~35/32 = 157.4418 ¢
error map: ⟨-0.293 -0.057 +0.407 +0.430]
  • CWE: ~63/50 = 400.0000 ¢, ~35/32 = 157.4658 ¢
error map: ⟨0.000 +0.306 1.016 +1.311]

Optimal ET sequence: 15, 54b, 69, 84, 99, 282, 381

Badness (Sintel): 1.14

Nessa

Subgroup: 2.3.5.7.11

Comma list: 441/440, 1344/1331, 4375/4356

Mapping: [⟨3 2 5 10 10], ⟨0 7 5 -4 1]]

Optimal tunings:

  • WE: ~44/35 = 399.7815 ¢, ~35/32 = 157.4527 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~35/32 = 157.5109 ¢

Optimal ET sequence: 15, 69, 84, 99e

Badness (Sintel): 1.61

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 364/363, 441/440, 625/624

Mapping: [⟨3 2 5 10 10 6], ⟨0 7 5 -4 1 13]]

Optimal tunings:

  • WE: ~44/35 = 399.7595 ¢, ~35/32 = 157.3348 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~35/32 = 157.3955 ¢

Optimal ET sequence: 15, 69, 84, 99ef, 183ef, 282eeff

Badness (Sintel): 1.55

Fof

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 250047/250000

Mapping: [⟨3 2 5 10 8], ⟨0 7 5 -4 6]]

Optimal tunings:

  • WE: ~63/50 = 400.0266 ¢, ~12/11 = 157.5301 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~12/11 = 157.5240 ¢

Optimal ET sequence: 15, 69e, 84e, 99

Badness (Sintel): 2.26

Grendel

For the 5-limit version, see Syntonic–31 equivalence continuum #Counterwürschmidt.

Grendel tempers out 16875/16807, the mirkwai comma, and may be described as the 31 & 152 temperament. 152edo, 183edo and especially 335edo serve as good tunings.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 16875/16807

Mapping: [⟨1 -14 3 -6], ⟨0 23 -1 13]]

mapping generators: ~2, ~8/5

Optimal tunings:

  • WE: ~2 = 1199.7348 ¢, ~8/5 = 812.9574 ¢
error map: ⟨-0.265 -0.220 -0.067 +1.212]
  • CWE: ~2 = 1200.0000 ¢, ~8/5 = 813.1311 ¢
error map: ⟨0.000 +0.059 +0.555 +1.878]

Optimal ET sequence: 31, 90, 121, 152, 335d, 822dd

Badness (Sintel): 1.31

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 5632/5625

Mapping: [⟨1 -14 3 -6 -25], ⟨0 23 -1 13 42]]

Optimal tunings:

  • WE: ~2 = 1199.7355 ¢, ~8/5 = 812.9622 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/5 = 813.1353 ¢

Optimal ET sequence: 31, 90e, 121, 152, 335d, 487d

Badness (Sintel): 0.656

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 540/539, 625/624, 1375/1372

Mapping: [⟨1 -14 3 -6 -25 22], ⟨0 23 -1 13 42 -27]]

Optimal tunings:

  • WE: ~2 = 1199.4412 ¢, ~8/5 = 812.7956 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/5 = 813.1209 ¢

Optimal ET sequence: 31, 90e, 121, 152f, 273def, 425deff

Badness (Sintel): 1.03

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 352/351, 625/624, 715/714, 1275/1274

Mapping: [⟨1 -14 3 -6 -25 22 19], ⟨0 23 -1 13 42 -27 -22]]

Optimal tunings:

  • WE: ~2 = 1199.3029 ¢, ~8/5 = 812.7156 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/5 = 813.1843 ¢

Optimal ET sequence: 31, 90e, 121, 152fg, 273defgg

Badness (Sintel): 1.09

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: [⟨1 -14 3 -6 -25 22 19 30], ⟨0 23 -1 13 42 -27 -22 -38]]

Optimal tunings:

  • WE: ~2 = 1199.3587 ¢, ~8/5 = 812.7462 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/5 = 813.1796 ¢

Optimal ET sequence: 31, 90e, 121, 152fg, 273defgg

Badness (Sintel): 1.12

Twothirdtonic

Twothirdtonic tempers out 686/675, the senga, in addition to the porwell comma, and may be described as the 37 & 46 temperament, generated by one third of a classical major third that represents 15/14, 14/13, and 13/12 in the 13-limit interpretation. Note that in the data below, the generator is taken to be its octave complement, thirteen of which octave reduced make the perfect fifth; it follows that the ploidacot for this temperament is 11-sheared 13-cot. 46edo may be recommended as a tuning.

Subgroup: 2.3.5.7

Comma list: 686/675, 6144/6125

Mapping: [⟨1 -10 5 -7], ⟨0 13 -3 11]]

mapping generators: ~2, ~28/15

Optimal tunings:

  • WE: ~2 = 1199.3074 ¢, ~28/15 = 1068.9820 ¢
error map: ⟨-0.693 +1.736 +3.278 -5.176]
  • CWE: ~2 = 1200.0000 ¢, ~28/15 = 1069.5746 ¢
error map: ⟨0.000 +2.515 +4.962 -3.505]

Optimal ET sequence: 9, 28b, 37, 46

Badness (Sintel): 2.52

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 686/675

Mapping: [⟨1 -10 5 -7 -1], ⟨0 13 -3 11 5]]

Optimal tunings:

  • WE: ~2 = 1199.7068 ¢, ~28/15 = 1069.3084 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/15 = 1069.5600 ¢

Optimal ET sequence: 9, 28b, 37, 46

Badness (Sintel): 1.35

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 121/120, 169/168, 176/175

Mapping: [⟨1 -10 5 -7 -1 -7], ⟨0 13 -3 11 5 12]]

Optimal tunings:

  • WE: ~2 = 1199.9531 ¢, ~13/7 = 1069.5492 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/7 = 1069.5893 ¢

Optimal ET sequence: 9, 28b, 37, 46

Badness (Sintel): 1.07

Semaja

See also: Llywelynsmic clan

Cryptically named by Petr Pařízek in 2011, semaja adds the gariboh comma to the comma list, and may be described as the 37 & 53 temperament. Its ploidacot is gamma-19-cot (or alpha-heptaseph due to a much simpler 2.5.7-subgroup restriction). The name actually refers to the fact that two of its ~8/7 generator steps reach a ~13/10[2].

Subgroup: 2.3.5.7

Comma list: 3125/3087, 6144/6125

Mapping: [⟨1 -2 1 3], ⟨0 19 7 -1]]

mapping generators: ~2, ~8/7

Optimal tunings:

  • WE: ~2 = 1199.4860 ¢, ~8/7 = 226.3864 ¢
error map: ⟨-0.514 +0.415 -2.123 +3.246]
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 226.4697 ¢
error map: ⟨0.000 +0.970 -1.026 +4.704]

Optimal ET sequence: 16, 37, 53, 196d

Badness (Sintel): 2.71

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 3125/3087

Mapping: [⟨1 -2 1 3 1], ⟨0 19 7 -1 13]]

Optimal tunings:

  • WE: ~2 = 1199.9818 ¢, ~8/7 = 226.4821 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 226.4851 ¢

Optimal ET sequence: 16, 37, 53

Badness (Sintel): 1.98

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 169/168, 176/175, 275/273

Mapping: [⟨1 -2 1 3 1 2], ⟨0 19 7 -1 13 9]]

Optimal tunings:

  • WE: ~2 = 1200.1020 ¢, ~8/7 = 226.4987 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 226.4822 ¢

Optimal ET sequence: 16, 37, 53

Badness (Sintel): 1.35

Aufo

For the 5-limit version, see Miscellaneous 5-limit temperaments #Untriton.

Also named by Petr Pařízek in 2011, aufo refers to the augmented fourth, which is a generator of this temperament[2]. The functional generator however is the 64/45 diminished fifth, and like its untriton variant, nine generator steps give the interval class of 3. The ploidacot for this temperament is delta-enneacot.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 177147/175616

Mapping: [⟨1 -3 12 -14], ⟨0 9 -19 33]]

mapping generators: ~2, ~64/45

Optimal tunings:

  • WE: ~2 = 1199.9758 ¢, ~64/45 = 611.2055 ¢
error map: ⟨-0.024 -1.303 +0.491 +1.295]
  • CWE: ~2 = 1200.0000 ¢, ~64/45 = 611.2177 ¢
error map: ⟨0.000 -0.996 +0.551 +1.357]

Optimal ET sequence: 53, 161, 214

Badness (Sintel): 3.07

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 177147/175616

Mapping: [⟨1 -3 12 -14 6], ⟨0 9 -19 33 -5]]

Optimal tunings:

  • WE: ~2 = 1200.4500 ¢, ~64/45 = 611.4185 ¢
  • CWE: ~2 = 1200.0000 ¢, ~64/45 = 611.1918 ¢

Optimal ET sequence: 53, 108e, 161e

Badness (Sintel): 2.93

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 176/175, 351/350, 58806/57967

Mapping: [⟨1 -3 12 -14 6 20], ⟨0 9 -19 33 -5 -32]]

Optimal tunings:

  • WE: ~2 = 1200.3134 ¢, ~64/45 = 611.3715 ¢
  • CWE: ~2 = 1200.0000 ¢, ~64/45 = 611.2118 ¢

Optimal ET sequence: 53, 108e

Badness (Sintel): 2.42

Aufic

Subgroup: 2.3.5.7.11

Comma list: 540/539, 5632/5625, 72171/71680

Mapping: [⟨1 -3 12 -14 33], ⟨0 9 -19 33 -58]]

Optimal tunings:

  • WE: ~2 = 1200.0668 ¢, ~64/45 = 611.2342 ¢
  • CWE: ~2 = 1200.0000 ¢, ~64/45 = 611.2000 ¢

Optimal ET sequence: 53, 108, 161, 214, 375

Badness (Sintel): 2.48

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 847/845, 4096/4095

Mapping: [⟨1 -3 12 -14 33 20], ⟨0 9 -19 33 -58 -32]]

Optimal tunings:

  • WE: ~2 = 1200.0177 ¢, ~64/45 = 611.2130 ¢
  • CWE: ~2 = 1200.0000 ¢, ~64/45 = 611.2039 ¢

Optimal ET sequence: 53, 108, 161, 214, 375

Badness (Sintel): 1.61

Absurdity

For the 5-limit version, see Syntonic–chromatic equivalence continuum #Absurdity (5-limit).
See also: Fifth-chroma temperaments

Subgroup: 2.3.5.7

Comma list: 6144/6125, 177147/175000

Mapping: [⟨7 0 -17 64], ⟨0 1 3 -4]]

mapping generators: ~972/875, ~3

Optimal tunings:

  • WE: ~972/875 = 171.4382 ¢, ~3/2 = 700.6247 ¢
error map: ⟨+0.067 -1.263 +1.313 +0.450]
  • CWE: ~972/875 = 171.4286 ¢, ~3/2 = 700.5871 ¢
error map: ⟨0.000 -1.368 +1.162 +0.254]

Optimal ET sequence: 77, 84, 161

Badness (Sintel): 3.38

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 6144/6125, 72171/71680

Mapping: [⟨7 0 -17 64 124], ⟨0 1 3 -4 -9]]

Optimal tunings:

  • WE: ~495/448 = 171.4346 ¢, ~3/2 = 700.6602 ¢
  • CWE: ~495/448 = 171.4286 ¢, ~3/2 = 700.6339 ¢

Optimal ET sequence: 77, 84, 161

Badness (Sintel): 2.70

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 441/440, 1188/1183, 3584/3575

Mapping: [⟨7 0 -17 64 124 37], ⟨0 1 3 -4 -9 -1]]

Optimal tunings:

  • WE: ~72/65 = 171.4223 ¢, ~3/2 = 700.6036 ¢
  • CWE: ~72/65 = 171.4286 ¢, ~3/2 = 700.6306 ¢

Optimal ET sequence: 77, 84, 161

Badness (Sintel): 1.72

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 441/440, 561/560, 1188/1183, 1632/1625

Mapping: [⟨7 0 -17 64 124 37 -49], ⟨0 1 3 -4 -9 -1 7]]

Optimal tunings:

  • WE: ~72/65 = 171.4263 ¢, ~3/2 = 700.6429 ¢
  • CWE: ~72/65 = 171.4286 ¢, ~3/2 = 700.6525 ¢

Optimal ET sequence: 77, 161

Badness (Sintel): 1.62

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 324/323, 351/350, 441/440, 456/455, 476/475, 495/494

Mapping: [⟨7 0 -17 64 124 37 -49 63], ⟨0 1 3 -4 -9 -1 7 -3]]

Optimal tunings:

  • WE: ~21/19 = 171.4244 ¢, ~3/2 = 700.6395 ¢
  • CWE: ~21/19 = 171.4286 ¢, ~3/2 = 700.6568 ¢

Optimal ET sequence: 77, 161

Badness (Sintel): 1.36

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 276/275, 324/323, 351/350, 441/440, 456/455, 476/475, 495/494

Mapping: [⟨7 0 -17 64 124 37 -49 63 76], ⟨0 1 3 -4 -9 -1 7 -3 -4]]

Optimal tunings:

  • WE: ~21/19 = 171.4321 ¢, ~3/2 = 700.6475 ¢
  • CWE: ~21/19 = 171.4286 ¢, ~3/2 = 700.6325 ¢

Optimal ET sequence: 77, 84, 161

Badness (Sintel): 1.34

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 261/260, 276/275, 324/323, 351/350, 441/440, 456/455, 476/475, 495/494

Mapping: [⟨7 0 -17 64 124 37 -49 63 76 34], ⟨0 1 3 -4 -9 -1 7 -3 -4 0]]

Optimal tunings:

  • WE: ~21/19 = 171.4348 ¢, ~3/2 = 700.6612 ¢
  • CWE: ~21/19 = 171.4286 ¢, ~3/2 = 700.6351 ¢

Optimal ET sequence: 77, 84, 161

Badness (Sintel): 1.25

Polypyth

For the 5-limit version, see Miscellaneous 5-limit temperaments #Leapday.

Polypyth tempers out the same 5-limit comma as leapday, with which it shares the similarly sharp perfect-fifth generator, but the porwell comma (6144/6125) rather than the hemifamity comma (5120/5103) is tempered out here. It may be described as the 46 & 121 temperament, and 121edo and 167edo make for good tunings.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 179200/177147

Mapping: [⟨1 0 -31 52], ⟨0 1 21 -31]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1199.3465 ¢, ~3/2 = 703.7905 ¢
error map: ⟨-0.654 +1.182 -0.177 -0.056]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.1749 ¢
error map: ⟨0.000 +2.220 +1.359 +1.752]

Optimal ET sequence: 46, 121, 167, 288b, 455bcd

Badness (Sintel): 3.49

11-limit

Subgroup: 2.3.5.7.11

Comma list: 896/891, 2200/2187, 6144/6125

Mapping: [⟨1 0 -31 52 59], ⟨0 1 21 -31 -35]]

Optimal tunings:

  • WE: ~2 = 1199.3335 ¢, ~3/2 = 703.7856 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.1812 ¢

Optimal ET sequence: 46, 121, 167, 288be, 455bcde

Badness (Sintel): 1.69

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 364/363, 1716/1715

Mapping: [⟨1 0 -31 52 59 64], ⟨0 1 21 -31 -35 -38]]

Optimal tunings:

  • WE: ~2 = 1199.3768 ¢, ~3/2 = 703.8018 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.1731 ¢

Optimal ET sequence: 46, 75e, 121, 167, 288be

Badness (Sintel): 1.25

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 325/324, 352/351, 364/363, 1716/1715

Mapping: [⟨1 0 -31 52 59 64 39], ⟨0 1 21 -31 -35 -38 -22]]

Optimal tunings:

  • WE: ~2 = 1199.3518 ¢, ~3/2 = 703.7880 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.1747 ¢

Optimal ET sequence: 46, 75e, 121, 167, 288beg

Badness (Sintel): 0.971

Whoops

For the 5-limit version, see Very high accuracy temperaments #Whoosh.

Also named by Petr Pařízek in 2011, whoops is a relatively simple extension to the otherwise very accurate microtemperament known as whoosh[2].

Subgroup: 2.3.5.7

Comma list: 6144/6125, 244140625/243045684

Mapping: [⟨1 -16 -11 14], ⟨0 33 25 -21]]

mapping generators: ~2, ~640/441

Optimal tunings:

  • WE: ~2 = 1199.5944 ¢, ~640/441 = 639.2648 ¢
error map: ⟨-0.406 +0.272 -0.233 +0.936]
  • CWE: ~2 = 1200.0000 ¢, ~640/441 = 639.4769 ¢
error map: ⟨0.000 +0.783 +0.609 +2.159]

Optimal ET sequence: 15, 122d, 137, 152, 623bdd, 775bcdd, 927bcddd, 1079bcddd

Badness (Sintel): 4.45

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4000/3993, 6144/6125

Mapping: [⟨1 -16 -11 14 -4], ⟨0 33 25 -21 14]]

Optimal tunings:

  • WE: ~2 = 1199.5936 ¢, ~175/121 = 639.264 ¢
  • CWE: ~2 = 1200.0000 ¢, ~175/121 = 639.4770 ¢

Optimal ET sequence: 15, 122d, 137, 152, 623bdde, 775bcdde, 927bcdddee, 1079bcdddee

Badness (Sintel): 1.45

Dodifo

For the 5-limit version, see Miscellaneous 5-limit temperaments #Dodifo.

Also named by Petr Pařízek in 2011, dodifo refers to the (tetraptolemaic) double-diminished fourth, which is a generator of this temperament[2]. The extension here is a less accurate 7-limit interpretation.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 2500000/2470629

Mapping: [⟨1 -23 -4 0], ⟨0 35 9 4]]

mapping generators: ~2, ~80/49

Optimal tunings:

  • WE: ~2 = 1199.6429 ¢, ~80/49 = 842.6790 ¢
error map: ⟨-0.357 +0.228 -0.774 +1.890]
  • CWE: ~2 = 1200.0000 ¢, ~80/49 = 842.9243 ¢
error map: ⟨0.000 +0.396 +0.005 +2.871]

Optimal ET sequence: 37, 84, 121, 205

Badness (Sintel): 4.55

11-limit

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 2560/2541, 4375/4356

Mapping: [⟨1 -23 -4 0 14], ⟨0 35 9 4 -15]]

Optimal tunings:

  • WE: ~2 = 1199.3401 ¢, ~80/49 = 842.4880 ¢
  • CWE: ~2 = 1200.0000 ¢, ~80/49 = 842.9457 ¢

Optimal ET sequence: 37, 84, 121, 326dee

Badness (Sintel): 2.71

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 625/624, 640/637, 1375/1372

Mapping: [⟨1 12 5 4 -1 4], ⟨0 -35 -9 -4 15 -1]]

Optimal tunings:

  • WE: ~2 = 1199.3410 ¢, ~13/8 = 842.4885 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/8 = 842.9466 ¢

Optimal ET sequence: 37, 84, 121, 326deef

Badness (Sintel): 1.63

Icositritonic

See also: 23rd-octave temperaments

Icositritonic has a period of 1/23 octave, so six period represents 6/5 and nine period represents 21/16. It may be described as 46 & 161. It was named by Xenllium in 2019 for its number of periods per octave.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 9920232/9765625

Mapping: [⟨23 0 17 101], ⟨0 1 1 -1]]

mapping generators: ~1323/1280, ~3

Optimal tunings:

  • WE: ~1323/1280 = 52.1732 ¢, ~3/2 = 701.0660 ¢
error map: ⟨-0.017 -0.906 +1.679 -0.386]
  • CWE: ~1323/1280 = 52.1739 ¢, ~3/2 = 701.0722 ¢
error map: ⟨0.000 -0.883 +1.715 -0.333]

Optimal ET sequence: 46, 115, 161, 207, 368c

Badness (Sintel): 4.98

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 6144/6125, 35937/35840

Mapping: [⟨23 0 17 101 116], ⟨0 1 1 -1 -1]]

Optimal tunings:

  • WE: ~33/32 = 52.1740 ¢, ~3/2 = 701.0379 ¢
  • CWE: ~33/32 = 52.1739 ¢, ~3/2 = 701.0370 ¢

Optimal ET sequence: 46, 115, 161, 207, 368c

Badness (Sintel): 2.14

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 441/440, 847/845, 3584/3575

Mapping: [⟨23 0 17 101 116 158], ⟨0 1 1 -1 -1 -2]]

Optimal tunings:

  • WE: ~33/32 = 52.1724 ¢, ~3/2 = 701.1310 ¢
  • CWE: ~33/32 = 52.1739 ¢, ~3/2 = 701.1524 ¢

Optimal ET sequence: 46, 115, 161, 207, 368c

Badness (Sintel): 1.67

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 441/440, 561/560, 847/845, 1089/1088

Mapping: [⟨23 0 17 101 116 158 94], ⟨0 1 1 -1 -1 -2 0]]

Optimal tunings:

  • WE: ~33/32 = 52.1735 ¢, ~3/2 = 701.1493 ¢
  • CWE: ~33/32 = 52.1739 ¢, ~3/2 = 701.1549 ¢

Optimal ET sequence: 46, 115, 161, 207, 368c

Badness (Sintel): 1.26

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 351/350, 441/440, 456/455, 476/475, 513/512, 847/845

Mapping: [⟨23 0 17 101 116 158 94 207], ⟨0 1 1 -1 -1 -2 0 -3]]

Optimal tunings:

  • WE: ~33/32 = 52.1744 ¢, ~3/2 = 701.0649 ¢
  • CWE: ~33/32 = 52.1739 ¢, ~3/2 = 701.0582 ¢

Optimal ET sequence: 46, 115, 161, 207, 368c

Badness (Sintel): 1.31

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 276/275, 351/350, 391/390, 441/440, 456/455, 476/475, 847/845

Mapping: [⟨23 0 17 101 116 158 94 207 104], ⟨0 1 1 -1 -1 -2 0 -3 0]]

Optimal tunings:

  • WE: ~33/32 = 52.1768 ¢, ~3/2 = 701.1259 ¢
  • CWE: ~33/32 = 52.1739 ¢, ~3/2 = 701.0841 ¢

Optimal ET sequence: 46, 115, 161, 207

Badness (Sintel): 1.27

References

  1. ↑ Yahoo! Tuning Group | Some more unclassified temperaments
  2. ↑ 2.0 2.1 2.2 2.3 2.4 Yahoo! Tuning Group | Suggested names for the unclasified temperaments
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