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|tempers out]] the porwell comma, {{monzo| 11 1 -3 -2 }} ([[6144/6125]]), and includes hendecatonic, hemischis, twothirdtonic, nessafof, septisuperfourth, whoops, and polypyth.  


Discussed elsewhere are:  
Discussed elsewhere are:  
* ''[[Hexadecimal]]'' → [[Pelogic family #Armodue]]
* ''[[Armodue]]'' (+36/35) → [[Pelogic family #Armodue|Pelogic family]]
* [[Porcupine]], also in: [[Porcupine family #Porcupine]]
* [[Porcupine]] (+64/63) → [[Porcupine family #Porcupine|Porcupine family]]
* [[Mohajira]], also in: [[Meantone family #Mohajira]]
* [[Mohajira]] (+81/80) → [[Meantone family #Mohajira|Meantone family]]
* [[Valentine]], also in: [[Starling temperaments #Valentine]]
* [[Valentine]] (+126/125) → [[Starling temperaments #Valentine|Starling temperaments]]
* [[Orwell]], also in: [[Semicomma family #Orwell]]
* [[Orwell]] (+225/224) → [[Semicomma family #Orwell|Semicomma family]]
* [[Shrutar]], also in: [[Diaschismic family #Shrutar]]
* [[Shrutar]] (+245/243) → [[Diaschismic family #Shrutar|Diaschismic family]]
* ''[[Hemiwürschmidt]]'' → [[Würschmidt family #Hemiwürschmidt]]
* ''[[Quinkee]]'' (+1029/1000) → [[Cloudy clan #Quinkee|Cloudy clan]]
* ''[[Hemikleismic]]'' → [[Kleismic family #Hemikleismic]]
* ''[[Hemiwürschmidt]]'' (+2401/2400 or 3136/3125) → [[Hemimean clan #Hemiwürschmidt|Hemimean clan]]
* [[Amity]], also in: [[Amity family #Amity]]
* ''[[Hemikleismic]]'' (+4000/3969) → [[Kleismic family #Hemikleismic|Kleismic family]]
* ''[[Grendel]]'' → [[Mirkwai clan #Grendel]]
* [[Amity]] (+4375/4374 or 5120/5103) → [[Amity family #Septimal amity|Amity family]]
* ''[[Trident]]'' → [[Tricot family #Trident]]
* ''[[Freivald]]'' (+6272/6075) → [[Passion family #Freivald|Passion family]]
* ''[[Grendel]]'' (+16875/16807) → [[Mirkwai clan #Grendel|Mirkwai clan]]
* ''[[Hemischis]]'' (+19683/19600) → [[Schismatic family #Hemischis|Schismatic family]]
* ''[[Bison]]'' (+78732/78125) → [[Sensipent family #Bison|Sensipent family]]
* ''[[Hemimabila]]'' (+117649/116640) → [[Mabila family #Hemimabila|Mabila family]]
* ''[[Septisuperfourth]]'' (+118098/117649) → [[Escapade family #Septisuperfourth|Escapade family]]
* ''[[Alphatrident]]'' (+14348907/14336000) → [[Alphatricot family #Alphatrident|Alphatricot family]]
* ''[[Hemimaquila]]'' (+{{monzo| -5 10 5 -8 }}) → [[Maquila family #Hemimaquila|Maquila family]]
* ''[[Decimaleap]]'' (+{{monzo| 15 -18 1 4 }}) → [[Quintaleap family #Decimaleap|Quintaleap family]]
* ''[[Twilight]]'' (+{{monzo| 19 -22 2 4 }}) → [[Undim family #Twilight|Undim family]]


== Hendecatonic ==
== Hendecatonic ==
The hendecatonic temperament has a period of 1/11 octave, which represents [[16/15]] and four times of which represents [[9/7]].
{{see also|11th-octave temperaments}}


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]].
 
[[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 }}


{{Multival|legend=1| 11 -11 22 -43 4 82 }}
: Mapping generators: ~16/15, ~3


[[POTE generator]]: ~3/2 = 703.054
[[Optimal tuning]] ([[POTE]]): ~16/15 = 1\11, ~3/2 = 703.054


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


[[Badness]]: 0.041081
[[Badness]]: 0.041081


=== 11-limit ===
=== 11-limit ===
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|legend=1| 11 0 43 -4 38 | 0 1 -1 2 0 }}


POTE generator: ~3/2 = 702.636
Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 702.636


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


Badness: 0.046088
Badness: 0.046088


=== Icosidillic  ===
==== 13-limit ====
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13


Comma list: 3388/3375, 6144/6125, 9801/9800
Comma list: 121/120, 176/175, 351/350, 4459/4455


Mapping: [{{val| 22 0 86 -8 111 }}, {{val| 0 1 -1 2 -1 }}]
{{Mapping|legend=1| 11 0 43 -4 38 93 | 0 1 -1 2 0 -3 }}


POTE generator: ~3/2 = 702.914
Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 702.291


Vals: {{Val list| 22, 154, 176, 198 }}
{{Optimal ET sequence|legend=0| 22, 55, 77, 99, 176e }}


Badness: 0.057725
Badness: 0.040099


== Hemischis  ==
==== 17-limit ====
{{see also| Schismatic family }}
Subgroup: 2.3.5.7.11.13.17
 
Subgroup: 2.3.5.7


[[Comma list]]: 6144/6125, 19683/19600
Comma list: 121/120, 154/153, 176/175, 273/272, 2025/2023


[[Mapping]]: [{{val| 1 0 15 -17 }}, {{val| 0 2 -16 25 }}]
{{Mapping|legend=1| 11 0 43 -4 38 93 45 | 0 1 -1 2 0 -3 0 }}


{{Multival|legend=1| 2 -16 25 -30 34 103 }}
Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 702.301


[[POTE generator]]: ~81/70 = 249.203
{{Optimal ET sequence|legend=0| 22, 55, 77, 99, 176eg }}


{{Val list|legend=1| 24, 53, 130, 183, 313 }}
Badness: 0.029054


[[Badness]]: 0.045817
=== Cohendecatonic ===
 
=== 11-limit  ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 540/539, 8019/8000, 5632/5625
Comma list: 540/539, 896/891, 4375/4356


Mapping: [{{val| 1 0 15 -17 51 }}, {{val| 0 2 -16 25 -60 }}]
{{Mapping|legend=1| 11 0 43 -4 73 | 0 1 -1 2 -2 }}


POTE generator: ~81/70 = 249.199
Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 703.686


Vals: {{Val list| 24e, 53, 130, 183, 313 }}
{{Optimal ET sequence|legend=0| 22, 77e, 99e, 121, 220e }}


Badness: 0.036289
Badness: 0.038042


=== 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: 352/351, 364/363, 540/539, 625/624


Mapping: [{{val| 1 0 15 -17 51 14 }}, {{val| 0 2 -16 25 -60 -13 }}]
{{Mapping|legend=1| 11 0 43 -4 73 128 | 0 1 -1 2 -2 -5 }}


POTE generator: ~15/13 = 249.199
Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 703.888


Vals: {{Val list| 24e, 53, 130, 183, 313 }}
{{Optimal ET sequence|legend=0| 22, 77eff, 99ef, 121, 341bdeeff }}


Badness: 0.020816
Badness: 0.036112


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


Comma list: 351/350, 442/441, 561/560, 676/675, 4096/4095
Comma list: 256/255, 352/351, 364/363, 375/374, 540/539


Map: [{{val| 1 0 15 -17 51 14 -49 }}, {{val| 0 2 -16 25 -60 -13 67 }}]
{{Mapping|legend=1| 11 0 43 -4 73 128 45 | 0 1 -1 2 -2 -5 0 }}


POTE generator: ~15/13 = 249.190
Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 703.877


Vals: {{Val list| 53, 130, 183, 679df }}
{{Optimal ET sequence|legend=0| 22, 77eff, 99ef, 121, 220efg, 341bdeeffgg }}


Badness: 0.021073
Badness: 0.022590


== Twothirdtonic ==
=== Icosidillic ===
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7.11
 
Comma list: 3388/3375, 6144/6125, 9801/9800
 
{{Mapping|legend=1| 22 0 86 -8 111 | 0 1 -1 2 -1 }}
 
: Mapping generators: ~33/32, ~3
 
Optimal tuning (POTE): ~33/32 = 1\22, ~3/2 = 702.914
 
{{Optimal ET sequence|legend=0| 22, 154, 176, 198 }}
 
Badness: 0.057725
 
== Twothirdtonic ==
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val| 1 3 2 4 }}, {{val| 0 -13 3 -11 }}]
{{Mapping|legend=1| 1 3 2 4 | 0 -13 3 -11 }}


{{Multival|legend=1| 13 -3 11 -35 -19 34 }}
: Mapping generators: ~2, ~15/14


[[POTE generator]]: ~15/14 = 130.401
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~15/14 = 130.401


{{Val list|legend=1| 9, 28b, 37, 46 }}
{{Optimal ET sequence|legend=1| 9, 28b, 37, 46 }}


[[Badness]]: 0.099601
[[Badness]]: 0.099601


=== 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: 121/120, 176/175, 686/675


Mapping: [{{val| 1 3 2 4 4 }}, {{val| 0 -13 3 -11 -5 }}]
Mapping: {{mapping| 1 3 2 4 4 | 0 -13 3 -11 -5 }}


POTE generator: ~15/14 = 130.430
Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 130.430


Vals: {{Val list| 9, 28b, 37, 46 }}
{{Optimal ET sequence|legend=1| 9, 28b, 37, 46 }}


Badness: 0.040768
Badness: 0.040768


=== 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: 91/90, 121/120, 169/168, 176/175


Mapping: [{{val| 1 3 2 4 4 5 }}, {{val| 0 -13 3 -11 -5 -12 }}]
Mapping: {{mapping| 1 3 2 4 4 5 | 0 -13 3 -11 -5 -12 }}


POTE generator: ~15/14 = 130.409
Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 130.409


Vals: {{Val list| 9, 28b, 37, 46 }}
{{Optimal ET sequence|legend=1| 9, 28b, 37, 46 }}


Badness: 0.025941
Badness: 0.025941


== Nessafof ==
== Semaja ==
Subgroup: 2.3.5.7
Cryptically named by [[Petr Pařízek]] in 2011, semaja adds the [[gariboh comma]] to the comma list. 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>.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 3125/3087, 6144/6125
 
{{Mapping|legend=1| 1 -2 1 3 | 0 19 7 -1 }}
 
: Mapping generators: ~2, ~8/7
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 226.4834
 
{{Optimal ET sequence|legend=1| 16, 37, 53, 196d }}
 
[[Badness]]: 0.107023
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 176/175, 3125/3087
 
Mapping: {{mapping| 1 -2 1 3 1 | 0 19 7 -1 13 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 226.4856
 
{{Optimal ET sequence|legend=1| 16, 37, 53 }}
 
Badness: 0.059838
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 121/120, 169/168, 176/175, 275/273
 
Mapping: {{mapping| 1 -2 1 3 1 2 | 0 19 7 -1 13 9 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 226.4794
 
{{Optimal ET sequence|legend=1| 16, 37, 53 }}
 
Badness: 0.032564
 
== Nessafof ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments#Nessafof]].''
 
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 5 times, makes 5/1<ref name="petr's long post"/>.
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 6144/6125, 250047/250000
[[Comma list]]: 6144/6125, 250047/250000


[[Mapping]]: [{{val| 3 2 5 10 }}, {{val| 0 7 5 -4 }}]
{{Mapping|legend=1| 3 2 5 10 | 0 7 5 -4 }}


{{Multival|legend=1| 21 15 -12 -25 -78 -70 }}
: Mapping generators: ~63/50, ~35/32


[[POTE generator]]: ~35/32 = 157.480
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~35/32 = 157.480


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


[[Badness]]: 0.045048
[[Badness]]: 0.045048


== Septisuperfourth  ==
=== 11-limit ===
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 176/175, 250047/250000
 
Mapping: {{mapping| 3 2 5 10 8 | 0 7 5 -4 6 }}
 
Optimal tuning (POTE): ~63/50 = 1\3, ~12/11 = 157.520
 
{{Optimal ET sequence|legend=1| 15, 54be, 69e, 84e, 99 }}
 
Badness: 0.068427
 
=== Nessa ===
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 1344/1331, 4375/4356
 
Mapping: {{mapping| 3 2 5 10 10 | 0 7 5 -4 1 }}
 
Optimal tuning (POTE): ~44/35 = 1\3, ~35/32 = 157.539
 
{{Optimal ET sequence|legend=1| 15, 54b, 69, 84, 99e }}
 
Badness: 0.048836


[[Comma list]]: 6144/6125, 118098/117649
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


[[Mapping]]: [{{val| 2 4 4 7 }}, {{val| 0 -9 7 -15 }}]
Comma list: 144/143, 364/363, 441/440, 625/624


{{Multival|legend=1| 18 -14 30 -64 -3 109 }}
Mapping: {{mapping| 3 2 5 10 10 6 | 0 7 5 -4 1 13 }}


[[POTE generator]]: ~48/35 = 544.680
Optimal tuning (POTE): ~44/35 = 1\3, ~35/32 = 157.429


{{Val list|legend=1| 22, 86, 108, 130, 152, 282 }}
{{Optimal ET sequence|legend=1| 15, 54bf, 69, 84, 99ef, 183ef, 282eeff }}


[[Badness]]: 0.059241
Badness: 0.037409


=== 11-limit ===
== Aufo ==
:''For the 5-limit version, see [[High badness 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"/>.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 6144/6125, 177147/175616
 
{{Mapping|legend=1| 1 6 -7 19 | 0 -9 19 -33 }}
 
: Mapping generators: ~2, ~45/32
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~45/32 = 588.782
 
{{Optimal ET sequence|legend=1| 53, 161, 214 }}
 
[[Badness]]: 0.121428
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 540/539, 4000/3993, 5632/5625
Comma list: 121/120, 176/175, 177147/175616


Mapping: [{{val| 2 4 4 7 6 }}, {{val| 0 -9 7 -15 10 }}]
Mapping: {{mapping| 1 6 -7 19 1 | 0 -9 19 -33 5 }}


POTE generator: ~48/35 = 544.696
Optimal tuning (POTE): ~2 = 1\1, ~45/32 = 588.811


Vals: {{Val list| 22, 86, 108, 130, 152, 282, 434de, 716de }}
{{Optimal ET sequence|legend=1| 53, 108e, 161e }}


Badness: 0.024619
Badness: 0.088631


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


Comma list: 540/539, 729/728, 4000/3993, 21168/21125
Comma list: 121/120, 176/175, 351/350, 58806/57967
 
Mapping: {{mapping| 1 6 -7 19 1 -12 | 0 -9 19 -33 5 32 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~45/32 = 588.788
 
{{Optimal ET sequence|legend=1| 53, 108e, 161e, 214ee }}
 
Badness: 0.058507
 
=== Aufic ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 5632/5625, 72171/71680


Mapping: [{{val| 2 4 4 7 6 11 }}, {{val| 0 -9 7 -15 10 -39 }}]
Mapping: {{mapping| 1 6 -7 19 -25 | 0 -9 19 -33 58 }}


POTE generator: ~48/35 = 544.675
Optimal tuning (POTE): ~2 = 1\1, ~45/32 = 588.800


Vals: {{Val list| 22f, 108f, 130, 282 }}
{{Optimal ET sequence|legend=1| 53, 108, 161, 214, 375 }}


Badness: 0.022887
Badness: 0.075149


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


Comma list: 351/350, 364/363, 540/539, 5632/5625
Comma list: 351/350, 540/539, 847/845, 4096/4095


Mapping: [{{val| 2 4 4 7 6 5 }}, {{val| 0 -9 7 -15 10 26 }}]
Mapping: {{mapping| 1 6 -7 19 -25 -12 | 0 -9 19 -33 58 32 }}


POTE generator: ~48/35 = 544.641
Optimal tuning (POTE): ~2 = 1\1, ~45/32 = 588.796


Vals: {{Val list| 22, 86f, 108, 130 }}
{{Optimal ET sequence|legend=1| 53, 108, 161, 214, 375, 589be }}


Badness: 0.033038
Badness: 0.039050


== Whoops ==
== Whoops ==
{{see also| Very high accuracy temperaments #Whoosh }}
:''For the 5-limit version, see [[Very high accuracy temperaments #Whoosh]].''


Subgroup: 2.3.5.7
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"/>.
 
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val| 1 17 14 -7 }}, {{val| 0 -33 -25 21 }}]
{{Mapping|legend=1| 1 17 14 -7 | 0 -33 -25 21 }}


{{Multival|legend=1| 33 25 -21 -37 -126 -119 }}
: Mapping generators: ~2, ~441/320


[[POTE generator]]: ~441/320 = 560.519
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~441/320 = 560.519


{{Val list|legend=1| 15, 122d, 137, 152, 608d, 623bd, 775bcd }}
{{Optimal ET sequence|legend=1| 15, 122d, 137, 152, 608d, 623bd, 775bcd }}


[[Badness]]: 0.1758
[[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: 3025/3024, 4000/3993, 6144/6125


Mapping: [{{val| 1 17 14 -7 10 }}, {{val| 0 -33 -25 21 -14 }}]
Mapping: {{mapping| 1 17 14 -7 10 | 0 -33 -25 21 -14 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~242/175 = 560.519
 
{{Optimal ET sequence|legend=1| 15, 122d, 137, 152, 608de, 623bde, 775bcde }}


POTE generator: ~242/175 = 560.519
Badness: 0.043743


Vals: {{Val list| 15, 122d, 137, 152, 608de, 623bde, 775bcde }}
== Polypyth ==
:''For the 5-limit version, see [[High badness temperaments #Leapday]].''


Badness: 0.0437
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.


== Polypyth  ==
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7


[[Comma list]]: 6144/6125, 179200/177147
[[Comma list]]: 6144/6125, 179200/177147


[[Mapping]]: [{{val| 1 0 -31 52 }}, {{val| 0 1 21 -31 }}]
{{Mapping|legend=1| 1 0 -31 52 | 0 1 21 -31 }}


[[POTE generator]]: ~3/2 = 704.174
: Mapping generators: ~2, ~3


{{Val list|legend=1| 46, 121, 167, 288b, 455bcd, 743bcd }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.174
 
{{Optimal ET sequence|legend=1| 46, 121, 167, 288b, 455bcd, 743bcd }}


[[Badness]]: 0.137995
[[Badness]]: 0.137995


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


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


Mapping: [{{val| 1 0 -31 52 59 }}, {{val| 0 1 21 -31 -35 }}]
Mapping: {{mapping| 1 0 -31 52 59 | 0 1 21 -31 -35 }}


POTE generator: ~3/2 = 704.177
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.177


Vals: {{Val list| 46, 121, 167, 288be, 455bcde }}
{{Optimal ET sequence|legend=1| 46, 121, 167, 288be, 455bcde }}


Badness: 0.051131
Badness: 0.051131


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


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


Mapping: [{{val| 1 0 -31 52 59 64 }}, {{val| 0 1 21 -31 -35 -38 }}]
Mapping: {{mapping| 1 0 -31 52 59 64 | 0 1 21 -31 -35 -38 }}


POTE generator: ~3/2 = 704.168
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.168


Vals: {{Val list| 46, 121, 167, 288be }}
{{Optimal ET sequence|legend=1| 46, 121, 167, 288be }}


Badness: 0.030292
Badness: 0.030292


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


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


Mapping: [{{val| 1 0 -31 52 59 64 39 }}, {{val| 0 1 21 -31 -35 -38 -22 }}]
Mapping: {{mapping| 1 0 -31 52 59 64 39 | 0 1 21 -31 -35 -38 -22 }}


POTE generator: ~3/2 = 704.168
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.168


Vals: {{Val list| 46, 121, 167, 288beg }}
{{Optimal ET sequence|legend=1| 46, 121, 167, 288beg }}


Badness: 0.019051
Badness: 0.019051


== Icositritonic ==
== Icositritonic ==
The ''icositritonic'' temperament (46&amp;161, named by [[User:Xenllium|Xenllium]]) has a period of 1/23 octave, so six period represents [[6/5]] and nine period represents [[21/16]].
{{ See also | 23rd-octave temperaments }}
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]].


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val| 23 37 54 64 }}, {{val| 0 -1 -1 1 }}]
{{Mapping|legend=1| 23 0 17 101 | 0 1 1 -1 }}


{{Multival|legend=1| 23 23 -23 -17 -101 -118 }}
: Mapping generators: ~1323/1280, ~3


[[POTE generator]]: ~64/63 = 29.3586
[[Optimal tuning]] ([[POTE]]): ~1323/1280 = 1\23, ~64/63 = 29.3586


{{Val list|legend=1| 23, 46, 115, 161, 207, 368c }}
{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368c }}


[[Badness]]: 0.196622
[[Badness]]: 0.196622


=== 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: 441/440, 6144/6125, 35937/35840


Mapping: [{{val| 23 37 54 64 79 }}, {{val| 0 -1 -1 1 1 }}]
Mapping: {{mapping| 23 0 17 101 116 | 0 1 1 -1 -1 }}


POTE generator: ~64/63 = 29.3980
Optimal tuning (POTE): ~33/32 = 1\23, ~64/63 = 29.3980


Vals: {{Val list| 23, 46, 115, 161, 207, 368c }}
{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368c }}


Badness: 0.064613
Badness: 0.064613


=== 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: 351/350, 441/440, 847/845, 3584/3575


Mapping: [{{val| 23 37 54 64 79 84 }}, {{val| 0 -1 -1 1 1 2 }}]
Mapping: {{mapping| 23 0 17 101 116 158 | 0 1 1 -1 -1 -2 }}


POTE generator: ~64/63 = 29.2830
Optimal tuning (POTE): ~33/32 = 1\23, ~64/63 = 29.2830


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


Badness: 0.040484
Badness: 0.040484


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


Mapping: [{{val| 23 37 54 64 79 84 94 }}, {{val| 0 -1 -1 1 1 2 0 }}]
Mapping: {{mapping| 23 0 17 101 116 158 94 | 0 1 1 -1 -1 -2 0 }}


POTE generator: ~64/63 = 29.2800
Optimal tuning (POTE): ~33/32 = 1\23, ~64/63 = 29.2800


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


Badness: 0.024676
Badness: 0.024676


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


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


Mapping: [{{val| 23 37 54 64 79 84 94 96 }}, {{val| 0 -1 -1 1 1 2 0 3 }}]
Mapping: {{mapping| 23 0 17 101 116 158 94 207 | 0 1 1 -1 -1 -2 0 -3 }}


POTE generator: ~64/63 = 29.3760
Optimal tuning (POTE): ~33/32 = 1\23, ~64/63 = 29.3760


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


Badness: 0.021579
Badness: 0.021579


=== 23-limit ===
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23
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
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 }}]
Mapping: {{mapping| 23 0 17 101 116 158 94 207 104 | 0 1 1 -1 -1 -2 0 -3 0 }}


POTE generator: ~64/63 = 29.3471
Optimal tuning (POTE): ~33/32 = 1\23, ~64/63 = 29.3471


Vals: {{Val list| 46, 115, 161, 207, 368ci }}
{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368ci }}


Badness: 0.017745
Badness: 0.017745


== Countermiracle ==
== Countermiracle ==
The ''countermiracle'' temperament (31&amp;145, named by [[User:Xenllium|Xenllium]]) tempers out the trimyna, 50421/50000 and the [[Quince clan|quince comma]], 823543/819200.
The ''countermiracle'' temperament (31 &amp; 145) tempers out the trimyna, 50421/50000 and the [[quince comma]], 823543/819200.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 6144/6125, 50421/50000
[[Comma list]]: 6144/6125, 50421/50000


[[Mapping]]: [{{val|1 4 3 3}}, {{val|0 -25 -7 -2}}]
{{Mapping|legend=1| 1 4 3 3 | 0 -25 -7 -2 }}


{{Multival|legend=1| 25 7 2 -47 -67 -15 }}
: Mapping generators: ~2, ~343/320


[[POTE generator]]: ~343/320 = 115.917
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~343/320 = 115.9169


{{Val list|legend=1| 31, 114, 145, 176, 559cc, 735cc }}
{{Optimal ET sequence|legend=1| 31, 114, 145, 176, 559cc, 735cc }}


[[Badness]]: 0.102326
[[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: 441/440, 3388/3375, 6144/6125


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


POTE generator: ~77/72 = 115.916
Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.9158


Vals: {{Val list| 31, 114e, 145, 176 }}
{{Optimal ET sequence|legend=1| 31, 114e, 145, 176 }}


Badness: 0.039162
Badness: 0.039162


==== Countermiraculous ====
==== Countermiraculous ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 196/195, 352/351, 1001/1000, 6144/6125
Comma list: 196/195, 352/351, 1001/1000, 6144/6125


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


POTE generator: ~77/72 = 115.880
Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.8803


Vals: {{Val list| 31, 83e, 114e, 145, 321ceff }}
{{Optimal ET sequence|legend=1| 31, 83e, 114e, 145, 321ceff }}


Badness: 0.039271
Badness: 0.039271


==== Counterbenediction ====
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 196/195, 256/255, 352/351, 1001/1000, 1225/1224
 
Mapping: {{mapping| 1 4 3 3 8 1 1 | 0 -25 -7 -2 -47 28 32 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.8756
 
{{Optimal ET sequence|legend=1| 31, 83e, 114e, 145 }}
 
Badness: 0.029496
 
==== Counterbenediction ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 351/350, 441/440, 3146/3125, 3584/3575
Comma list: 351/350, 441/440, 3146/3125, 3584/3575


Mapping: [{{val|1 4 3 3 8 -2}}, {{val|0 -25 -7 -2 -47 59}}]
Mapping: {{mapping| 1 4 3 3 8 -2 | 0 -25 -7 -2 -47 59 }}


POTE generator: ~77/72 = 115.933
Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.9335


Vals: {{Val list| 31, 145f, 176, 207, 383c, 590cc }}
{{Optimal ET sequence|legend=1| 31, 114ef, 145f, 176, 207, 383c, 590cc }}


Badness: 0.045569
Badness: 0.045569


=== Counterrevelation ===
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 351/350, 441/440, 561/560, 1632/1625, 3146/3125
 
Mapping: {{mapping| 1 4 3 3 8 -2 -2 | 0 -25 -7 -2 -47 59 63 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.9391
 
{{Optimal ET sequence|legend=1| 31, 114efg, 145fg, 176, 207 }}
 
Badness: 0.036289
 
==== Countermanna ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 364/363, 441/440, 3388/3375, 6144/6125
 
Mapping: {{mapping| 1 4 3 3 8 15  0 -25 -7 -2 -47 -117 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.8898
 
{{Optimal ET sequence|legend=1| 145, 176, 321ce }}
 
Badness: 0.053409
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 364/363, 441/440, 595/594, 1632/1625, 3388/3375
 
Mapping: {{mapping| 1 4 3 3 8 15 15 | 0 -25 -7 -2 -47 -117 -113 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.8832
 
{{Optimal ET sequence|legend=1| 145, 321ce }}
 
Badness: 0.040898
 
=== Counterrevelation ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 121/120, 176/175, 50421/50000
Comma list: 121/120, 176/175, 50421/50000


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


POTE generator: ~343/320 = 115.919
Optimal tuning (POTE): ~2 = 1\1, ~343/320 = 115.9192


Vals: {{Val list| 31, 114, 145e, 176e }}
{{Optimal ET sequence|legend=1| 31, 114, 145e, 176e }}


Badness: 0.064070
Badness: 0.064070


==== 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: 121/120, 176/175, 196/195, 13750/13689


Mapping: [{{val|1 4 3 3 5 1}}, {{val|0 -25 -7 -2 -16 28}}]
Mapping: {{mapping| 1 4 3 3 5 1 | 0 -25 -7 -2 -16 28 }}


POTE generator: ~273/256 = 115.862
Optimal tuning (POTE): ~2 = 1\1, ~273/256 = 115.8624


Vals: {{Val list| 31, 83, 114, 145e }}
{{Optimal ET sequence|legend=1| 31, 83, 114, 145e }}


Badness: 0.057497
Badness: 0.057497


[[Category:Regular temperament theory]]
==== 17-limit ====
[[Category:Temperament collection]]
Subgroup: 2.3.5.7.11.13.17
[[Category:Porwell]]
 
[[Category:Hemischis]]
Comma list: 121/120, 154/153, 176/175, 196/195, 10647/10625
 
Mapping: {{mapping| 1 4 3 3 5 1 1 | 0 -25 -7 -2 -16 28 32 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~91/85 = 115.8527
 
{{Optimal ET sequence|legend=1| 31, 83, 114, 145e }}
 
Badness: 0.044043
 
== Absurdity ==
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Absurdity]].''
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 6144/6125, 177147/175000
 
{{Mapping|legend=1| 7 0 -17 64 | 0 1 3 -4 }}
 
: Mapping generators: ~972/875, ~3
 
[[Optimal tuning]] ([[POTE]]): ~972/875 = 1\7, ~3/2 = 700.5854 (or ~10/9 = 186.2997)
 
{{Optimal ET sequence|legend=1| 77, 84, 161 }}
 
[[Badness]]: 0.133520
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 6144/6125, 72171/71680
 
{{Mapping|legend=1| 7 0 -17 64 124 | 0 1 3 -4 -9 }}
 
Optimal tuning (POTE): ~495/448 = 1\7, ~3/2 = 700.6354 (or ~10/9 = 186.3497)
 
{{Optimal ET sequence|legend=1| 77, 84, 161 }}
 
Badness: 0.081564
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 441/440, 1188/1183, 3584/3575
 
{{Mapping|legend=1| 7 0 -17 64 124 37 | 0 1 3 -4 -9 -1 }}
 
Optimal tuning (POTE): ~72/65 = 1\7, ~3/2 = 700.6291 (or ~10/9 = 186.3434)
 
{{Optimal ET sequence|legend=1| 77, 84, 161 }}
 
Badness: 0.041600
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 351/350, 441/440, 561/560, 1188/1183, 1632/1625
 
{{Mapping|legend=1| 7 0 -17 64 124 37 -49 | 0 1 3 -4 -9 -1 7 }}
 
Optimal tuning (POTE): ~72/65 = 1\7, ~3/2 = 700.6524 (or ~10/9 = 186.3667)
 
{{Optimal ET sequence|legend=1| 77, 161 }}
 
Badness: 0.031783
 
=== 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|legend=1| 7 0 -17 64 124 37 -49 63 | 0 1 3 -4 -9 -1 7 -3 }}
 
Optimal tuning (POTE): ~21/19 = 1\7, ~3/2 = 700.6565 (or ~10/9 = 186.3708)
 
{{Optimal ET sequence|legend=1| 77, 161 }}
 
Badness: 0.022291
 
=== 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|legend=1| 7 0 -17 64 124 37 -49 63 76 | 0 1 3 -4 -9 -1 7 -3 -4 }}
 
Optimal tuning ([[CTE]]): ~21/19 = 1\7, ~3/2 = 700.629 (or ~10/9 = 186.343)
 
{{Optimal ET sequence|legend=0| 77, 84, 161 }}
 
=== 29-limit ===
{{ See also | Fifth-chroma temperaments }}
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 261/260, 276/275, 324/323, 351/350, 441/440, 456/455, 476/475, 495/494
 
{{Mapping|legend=1| 7 0 -17 64 124 37 -49 63 76 34 | 0 1 3 -4 -9 -1 7 -3 -4 0 }}
 
Optimal tuning ([[CTE]]): ~21/19 = 1\7, ~3/2 = 700.629 (or ~10/9 = 186.343)
 
{{Optimal ET sequence|legend=0| 77, 84, 161 }}
 
== Dodifo ==
: ''For the 5-limit version, see [[High badness temperaments #Dodifo]].''
 
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 intepretation.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 6144/6125, 2500000/2470629
 
{{Mapping|legend=1| 1 12 5 4 | 0 -35 -9 -4 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 357.070
 
{{Optimal ET sequence|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: {{mapping| 1 12 5 4 -1 | 0 -35 -9 -4 15 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 357.048
 
{{Optimal ET sequence|legend=1| 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: {{mapping| 1 12 5 4 -1 4 | 0 -35 -9 -4 15 -1 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 357.049
 
{{Optimal ET sequence|legend=1| 37, 84, 121, 326deef }}
 
Badness: 0.039533
 
== Notes ==
 
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Porwell temperaments| ]] <!-- main article -->
[[Category:Porwell| ]] <!-- key article -->
[[Category:Hendecatonic]]
[[Category:Hendecatonic]]
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 00:28, 24 June 2025

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 tempers out the porwell comma, [11 1 -3 -2⟩ (6144/6125), and includes hendecatonic, hemischis, twothirdtonic, nessafof, septisuperfourth, whoops, and polypyth.

Discussed elsewhere are:

Hendecatonic

See also: 11th-octave temperaments

The hendecatonic temperament has a period of 1/11 octave, which represents 16/15 and four times of which represent 9/7.

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 tuning (POTE): ~16/15 = 1\11, ~3/2 = 703.054

Optimal ET sequence: 22, 55, 77, 99

Badness: 0.041081

11-limit

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 tuning (POTE): ~16/15 = 1\11, ~3/2 = 702.636

Optimal ET sequence: 22, 55, 77, 99, 176e, 275e

Badness: 0.046088

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 tuning (POTE): ~16/15 = 1\11, ~3/2 = 702.291

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

Badness: 0.040099

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 tuning (POTE): ~16/15 = 1\11, ~3/2 = 702.301

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

Badness: 0.029054

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 tuning (POTE): ~16/15 = 1\11, ~3/2 = 703.686

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

Badness: 0.038042

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 tuning (POTE): ~16/15 = 1\11, ~3/2 = 703.888

Optimal ET sequence: 22, 77eff, 99ef, 121, 341bdeeff

Badness: 0.036112

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 tuning (POTE): ~16/15 = 1\11, ~3/2 = 703.877

Optimal ET sequence: 22, 77eff, 99ef, 121, 220efg, 341bdeeffgg

Badness: 0.022590

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 tuning (POTE): ~33/32 = 1\22, ~3/2 = 702.914

Optimal ET sequence: 22, 154, 176, 198

Badness: 0.057725

Twothirdtonic

Subgroup: 2.3.5.7

Comma list: 686/675, 6144/6125

Mapping: [⟨1 3 2 4], ⟨0 -13 3 -11]]

Mapping generators: ~2, ~15/14

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 130.401

Optimal ET sequence: 9, 28b, 37, 46

Badness: 0.099601

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 3 2 4 4], ⟨0 -13 3 -11 -5]]

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 130.430

Optimal ET sequence: 9, 28b, 37, 46

Badness: 0.040768

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 3 2 4 4 5], ⟨0 -13 3 -11 -5 -12]]

Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 130.409

Optimal ET sequence: 9, 28b, 37, 46

Badness: 0.025941

Semaja

Cryptically named by Petr Pařízek in 2011, semaja adds the gariboh comma to the comma list. The name actually refers to the fact that two of its ~8/7 generator steps reach a 13/10[1].

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 tuning (POTE): ~2 = 1\1, ~8/7 = 226.4834

Optimal ET sequence: 16, 37, 53, 196d

Badness: 0.107023

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 tuning (POTE): ~2 = 1\1, ~8/7 = 226.4856

Optimal ET sequence: 16, 37, 53

Badness: 0.059838

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 tuning (POTE): ~2 = 1\1, ~8/7 = 226.4794

Optimal ET sequence: 16, 37, 53

Badness: 0.032564

Nessafof

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

Cryptically named by Petr Pařízek in 2011[2], 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 5 times, makes 5/1[1].

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 tuning (POTE): ~63/50 = 1\3, ~35/32 = 157.480

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

Badness: 0.045048

11-limit

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 tuning (POTE): ~63/50 = 1\3, ~12/11 = 157.520

Optimal ET sequence: 15, 54be, 69e, 84e, 99

Badness: 0.068427

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 tuning (POTE): ~44/35 = 1\3, ~35/32 = 157.539

Optimal ET sequence: 15, 54b, 69, 84, 99e

Badness: 0.048836

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 tuning (POTE): ~44/35 = 1\3, ~35/32 = 157.429

Optimal ET sequence: 15, 54bf, 69, 84, 99ef, 183ef, 282eeff

Badness: 0.037409

Aufo

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

Also named by Petr Pařízek in 2011, aufo refers to the augmented fourth, which is a generator of this temperament[1].

Subgroup: 2.3.5.7

Comma list: 6144/6125, 177147/175616

Mapping: [⟨1 6 -7 19], ⟨0 -9 19 -33]]

Mapping generators: ~2, ~45/32

Optimal tuning (POTE): ~2 = 1\1, ~45/32 = 588.782

Optimal ET sequence: 53, 161, 214

Badness: 0.121428

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 6 -7 19 1], ⟨0 -9 19 -33 5]]

Optimal tuning (POTE): ~2 = 1\1, ~45/32 = 588.811

Optimal ET sequence: 53, 108e, 161e

Badness: 0.088631

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 6 -7 19 1 -12], ⟨0 -9 19 -33 5 32]]

Optimal tuning (POTE): ~2 = 1\1, ~45/32 = 588.788

Optimal ET sequence: 53, 108e, 161e, 214ee

Badness: 0.058507

Aufic

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 6 -7 19 -25], ⟨0 -9 19 -33 58]]

Optimal tuning (POTE): ~2 = 1\1, ~45/32 = 588.800

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

Badness: 0.075149

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 6 -7 19 -25 -12], ⟨0 -9 19 -33 58 32]]

Optimal tuning (POTE): ~2 = 1\1, ~45/32 = 588.796

Optimal ET sequence: 53, 108, 161, 214, 375, 589be

Badness: 0.039050

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[1].

Subgroup: 2.3.5.7

Comma list: 6144/6125, 244140625/243045684

Mapping: [⟨1 17 14 -7], ⟨0 -33 -25 21]]

Mapping generators: ~2, ~441/320

Optimal tuning (POTE): ~2 = 1\1, ~441/320 = 560.519

Optimal ET sequence: 15, 122d, 137, 152, 608d, 623bd, 775bcd

Badness: 0.175840

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 17 14 -7 10], ⟨0 -33 -25 21 -14]]

Optimal tuning (POTE): ~2 = 1\1, ~242/175 = 560.519

Optimal ET sequence: 15, 122d, 137, 152, 608de, 623bde, 775bcde

Badness: 0.043743

Polypyth

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

Polypyth (46 & 121) tempers out the same 5-limit comma as the leapday temperament (29 & 46), but with the porwell (6144/6125) rather than the hemifamity (5120/5103) tempered out.

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 tuning (POTE): ~2 = 1\1, ~3/2 = 704.174

Optimal ET sequence: 46, 121, 167, 288b, 455bcd, 743bcd

Badness: 0.137995

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 tuning (POTE): ~2 = 1\1, ~3/2 = 704.177

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

Badness: 0.051131

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 tuning (POTE): ~2 = 1\1, ~3/2 = 704.168

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

Badness: 0.030292

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 tuning (POTE): ~2 = 1\1, ~3/2 = 704.168

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

Badness: 0.019051

Icositritonic

See also: 23rd-octave temperaments

The icositritonic temperament (46 & 161) has a period of 1/23 octave, so six period represents 6/5 and nine period represents 21/16.

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 tuning (POTE): ~1323/1280 = 1\23, ~64/63 = 29.3586

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

Badness: 0.196622

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 tuning (POTE): ~33/32 = 1\23, ~64/63 = 29.3980

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

Badness: 0.064613

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 tuning (POTE): ~33/32 = 1\23, ~64/63 = 29.2830

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

Badness: 0.040484

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 tuning (POTE): ~33/32 = 1\23, ~64/63 = 29.2800

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

Badness: 0.024676

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 tuning (POTE): ~33/32 = 1\23, ~64/63 = 29.3760

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

Optimal tuning (POTE): ~33/32 = 1\23, ~64/63 = 29.3471

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

Badness: 0.017745

Countermiracle

The countermiracle temperament (31 & 145) tempers out the trimyna, 50421/50000 and the quince comma, 823543/819200.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 50421/50000

Mapping: [⟨1 4 3 3], ⟨0 -25 -7 -2]]

Mapping generators: ~2, ~343/320

Optimal tuning (POTE): ~2 = 1\1, ~343/320 = 115.9169

Optimal ET sequence: 31, 114, 145, 176, 559cc, 735cc

Badness: 0.102326

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3388/3375, 6144/6125

Mapping: [⟨1 4 3 3 8], ⟨0 -25 -7 -2 -47]]

Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.9158

Optimal ET sequence: 31, 114e, 145, 176

Badness: 0.039162

Countermiraculous

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 1001/1000, 6144/6125

Mapping: [⟨1 4 3 3 8 1], ⟨0 -25 -7 -2 -47 28]]

Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.8803

Optimal ET sequence: 31, 83e, 114e, 145, 321ceff

Badness: 0.039271

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 196/195, 256/255, 352/351, 1001/1000, 1225/1224

Mapping: [⟨1 4 3 3 8 1 1], ⟨0 -25 -7 -2 -47 28 32]]

Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.8756

Optimal ET sequence: 31, 83e, 114e, 145

Badness: 0.029496

Counterbenediction

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 441/440, 3146/3125, 3584/3575

Mapping: [⟨1 4 3 3 8 -2], ⟨0 -25 -7 -2 -47 59]]

Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.9335

Optimal ET sequence: 31, 114ef, 145f, 176, 207, 383c, 590cc

Badness: 0.045569

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 441/440, 561/560, 1632/1625, 3146/3125

Mapping: [⟨1 4 3 3 8 -2 -2], ⟨0 -25 -7 -2 -47 59 63]]

Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.9391

Optimal ET sequence: 31, 114efg, 145fg, 176, 207

Badness: 0.036289

Countermanna

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 3388/3375, 6144/6125

Mapping: [⟨1 4 3 3 8 15 0 -25 -7 -2 -47 -117]]

Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.8898

Optimal ET sequence: 145, 176, 321ce

Badness: 0.053409

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 441/440, 595/594, 1632/1625, 3388/3375

Mapping: [⟨1 4 3 3 8 15 15], ⟨0 -25 -7 -2 -47 -117 -113]]

Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.8832

Optimal ET sequence: 145, 321ce

Badness: 0.040898

Counterrevelation

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 50421/50000

Mapping: [⟨1 4 3 3 5], ⟨0 -25 -7 -2 -16]]

Optimal tuning (POTE): ~2 = 1\1, ~343/320 = 115.9192

Optimal ET sequence: 31, 114, 145e, 176e

Badness: 0.064070

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 176/175, 196/195, 13750/13689

Mapping: [⟨1 4 3 3 5 1], ⟨0 -25 -7 -2 -16 28]]

Optimal tuning (POTE): ~2 = 1\1, ~273/256 = 115.8624

Optimal ET sequence: 31, 83, 114, 145e

Badness: 0.057497

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 121/120, 154/153, 176/175, 196/195, 10647/10625

Mapping: [⟨1 4 3 3 5 1 1], ⟨0 -25 -7 -2 -16 28 32]]

Optimal tuning (POTE): ~2 = 1\1, ~91/85 = 115.8527

Optimal ET sequence: 31, 83, 114, 145e

Badness: 0.044043

Absurdity

For the 5-limit version, see Syntonic–chromatic equivalence continuum #Absurdity.

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 tuning (POTE): ~972/875 = 1\7, ~3/2 = 700.5854 (or ~10/9 = 186.2997)

Optimal ET sequence: 77, 84, 161

Badness: 0.133520

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 tuning (POTE): ~495/448 = 1\7, ~3/2 = 700.6354 (or ~10/9 = 186.3497)

Optimal ET sequence: 77, 84, 161

Badness: 0.081564

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 tuning (POTE): ~72/65 = 1\7, ~3/2 = 700.6291 (or ~10/9 = 186.3434)

Optimal ET sequence: 77, 84, 161

Badness: 0.041600

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 tuning (POTE): ~72/65 = 1\7, ~3/2 = 700.6524 (or ~10/9 = 186.3667)

Optimal ET sequence: 77, 161

Badness: 0.031783

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 tuning (POTE): ~21/19 = 1\7, ~3/2 = 700.6565 (or ~10/9 = 186.3708)

Optimal ET sequence: 77, 161

Badness: 0.022291

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 tuning (CTE): ~21/19 = 1\7, ~3/2 = 700.629 (or ~10/9 = 186.343)

Optimal ET sequence: 77, 84, 161

29-limit

See also: Fifth-chroma temperaments

Subgroup: 2.3.5.7.11.13.17.19.23

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 tuning (CTE): ~21/19 = 1\7, ~3/2 = 700.629 (or ~10/9 = 186.343)

Optimal ET sequence: 77, 84, 161

Dodifo

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

Also named by Petr Pařízek in 2011, dodifo refers to the (tetraptolemaic) double-diminished fourth, which is a generator of this temperament[1]. The extension here is a less accurate 7-limit intepretation.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 2500000/2470629

Mapping: [⟨1 12 5 4], ⟨0 -35 -9 -4]]

Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 357.070

Optimal ET sequence: 37, 84, 121, 205

Badness: 0.179692

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 357.048

Optimal ET sequence: 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: [⟨1 12 5 4 -1 4], ⟨0 -35 -9 -4 15 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 357.049

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

Badness: 0.039533

Notes

  1. ↑ 1.0 1.1 1.2 1.3 1.4 Yahoo! Tuning Group | Suggested names for the unclasified temperaments
  2. ↑ Yahoo! Tuning Group | Some more unclassified temperaments
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