Porwell temperaments: Difference between revisions

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Discussed elsewhere are:  
Discussed elsewhere are:  
* ''[[Hexadecimal]]'', {36/35, 135/128} → [[Pelogic family #Armodue|Pelogic family]]
* ''[[Hexadecimal]]'' (+36/35) → [[Pelogic family #Armodue|Pelogic family]]
* [[Porcupine]], {64/63, 250/243}, also in: [[Porcupine family #Porcupine|Porcupine family]]
* [[Porcupine]] (+64/63), also in: [[Porcupine family #Porcupine|Porcupine family]]
* [[Mohajira]], {81/80, 6144/6125}, also in: [[Meantone family #Mohajira|Meantone family]]
* [[Mohajira]] (+81/80), also in: [[Meantone family #Mohajira|Meantone family]]
* [[Valentine]], {126/125, 1029/1024}, also in: [[Starling temperaments #Valentine|Starling temperaments]]
* [[Valentine]] (+126/125), also in: [[Starling temperaments #Valentine|Starling temperaments]]
* [[Orwell]], {225/224, 1728/1715}, also in: [[Semicomma family #Orwell|Semicomma family]]
* [[Orwell]] (+225/224), also in: [[Semicomma family #Orwell|Semicomma family]]
* [[Shrutar]], {245/243, 2048/2025}, also in: [[Diaschismic family #Shrutar|Diaschismic family]]
* [[Shrutar]] (+245/243), also in: [[Diaschismic family #Shrutar|Diaschismic family]]
* ''[[Quinkee]]'', {1029/1000, 6144/6125} → [[Cloudy clan #Quinkee|Cloudy clan]]
* ''[[Quinkee]]'' (+1029/1000) → [[Cloudy clan #Quinkee|Cloudy clan]]
* ''[[Hemiwürschmidt]]'', {2401/2400, 3136/3125} → [[Würschmidt family #Hemiwürschmidt|Würschmidt family]]
* ''[[Hemiwürschmidt]]'' (+2401/2400 or 3136/3125) → [[Würschmidt family #Hemiwürschmidt|Würschmidt family]] and [[Hemimean clan #Hemiwürschmidt|Hemimean clan]]
* ''[[Hemikleismic]]'', {4000/3969, 6144/6125} → [[Kleismic family #Hemikleismic|Kleismic family]]
* ''[[Hemikleismic]]'' (+4000/3969) → [[Kleismic family #Hemikleismic|Kleismic family]]
* [[Amity]], {4375/4374, 5120/5103}, also in: [[Amity family #Septimal amity|Amity family]] and [[Ragismic microtemperaments #Amity|Ragismic microtemperaments]]
* [[Amity]] (+4375/4374 or 5120/5103), also in: [[Amity family #Septimal amity|Amity family]] and [[Ragismic microtemperaments #Amity|Ragismic microtemperaments]]
* ''[[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]]
* ''[[Grendel]]'' (+16875/16807) → [[Mirkwai clan #Grendel|Mirkwai clan]]
* ''[[Hemischis]]'', {6144/6125, 19683/19600} → [[Schismatic family #Hemischis|Schismatic family]]
* ''[[Hemischis]]'' (+19683/19600) → [[Schismatic family #Hemischis|Schismatic family]]
* ''[[Bison]]'', {6144/6125, 78732/78125} → [[Sensipent family #Bison|Sensipent family]]
* ''[[Bison]]'' (+78732/78125) → [[Sensipent family #Bison|Sensipent family]]
* ''[[Hemimabila]]'', {6144/6125, 117649/116640} → [[Mabila family #Hemimabila|Mabila family]]
* ''[[Hemimabila]]'' (+117649/116640) → [[Mabila family #Hemimabila|Mabila family]]
* ''[[Septisuperfourth]]'', {6144/6125, 118098/117649} → [[Escapade family #Septisuperfourth|Escapade family]]
* ''[[Septisuperfourth]]'' (+118098/117649) → [[Escapade family #Septisuperfourth|Escapade family]]
* ''[[Trident]]'', {6144/6125, 14348907/14336000} → [[Tricot family #Trident|Tricot family]]
* ''[[Trident]]'' (+14348907/14336000) → [[Tricot family #Trident|Tricot family]]
* ''[[Hemimaquila]]'', {6144/6125, {{monzo| -5 10 5 -8 }}} → [[Maquila family #Hemimaquila|Maquila family]]
* ''[[Hemimaquila]]'' (+{{monzo| -5 10 5 -8 }}) → [[Maquila family #Hemimaquila|Maquila family]]
* ''[[Decimaleap]]'', {6144/6125, {{monzo| 15 -18 1 4 }}} → [[Quintaleap family #Decimaleap|Quintaleap family]]
* ''[[Decimaleap]]'' (+{{monzo| 15 -18 1 4 }}) → [[Quintaleap family #Decimaleap|Quintaleap family]]
* ''[[Twilight]]'', {6144/6125, {{monzo| 19 -22 2 4 }}} → [[Undim family #Twilight|Undim 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]].
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
[[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 }}
{{Multival|legend=1| 11 -11 22 -43 4 82 }}


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


{{Optimal ET sequence|legend=1| 22, 55, 77, 99 }}
{{Optimal ET sequence|legend=1| 22, 55, 77, 99 }}
Line 45: Line 47:
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 tuning (POTE): ~16/15 = 1\11, ~3/2 = 702.636


{{Optimal ET sequence|legend=1| 22, 55, 77, 99, 176e, 275e }}
{{Optimal ET sequence|legend=1| 22, 55, 77, 99, 176e, 275e }}
Line 58: Line 60:
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 tuning (POTE): ~16/15 = 1\11, ~3/2 = 702.291


{{Optimal ET sequence|legend=1| 22, 55, 77, 99, 176e }}
{{Optimal ET sequence|legend=1| 22, 55, 77, 99, 176e }}
Line 71: Line 73:
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 tuning (POTE): ~16/15 = 1\11, ~3/2 = 702.301


{{Optimal ET sequence|legend=1| 22, 55, 77, 99, 176eg }}
{{Optimal ET sequence|legend=1| 22, 55, 77, 99, 176eg }}
Line 84: Line 86:
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 tuning (POTE): ~16/15 = 1\11, ~3/2 = 703.686


{{Optimal ET sequence|legend=1| 22, 77e, 99e, 121, 220e }}
{{Optimal ET sequence|legend=1| 22, 77e, 99e, 121, 220e }}
Line 97: Line 99:
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 tuning (POTE): ~16/15 = 1\11, ~3/2 = 703.888


{{Optimal ET sequence|legend=1| 22, 77eff, 99ef, 121, 341bdeeff }}
{{Optimal ET sequence|legend=1| 22, 77eff, 99ef, 121, 341bdeeff }}
Line 110: Line 112:
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 tuning (POTE): ~16/15 = 1\11, ~3/2 = 703.877


{{Optimal ET sequence|legend=1| 22, 77eff, 99ef, 121, 220efg, 341bdeeffgg }}
{{Optimal ET sequence|legend=1| 22, 77eff, 99ef, 121, 220efg, 341bdeeffgg }}
Line 123: Line 125:
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 }}


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


{{Optimal ET sequence|legend=1| 22, 154, 176, 198 }}
{{Optimal ET sequence|legend=1| 22, 154, 176, 198 }}
Line 132: Line 136:


== Twothirdtonic ==
== Twothirdtonic ==
Subgroup: 2.3.5.7
[[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 }}
 
: mapping generators: ~2, ~15/14


{{Multival|legend=1| 13 -3 11 -35 -19 34 }}
{{Multival|legend=1| 13 -3 11 -35 -19 34 }}


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


{{Optimal ET sequence|legend=1| 9, 28b, 37, 46 }}
{{Optimal ET sequence|legend=1| 9, 28b, 37, 46 }}
Line 151: Line 157:
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


{{Optimal ET sequence|legend=1| 9, 28b, 37, 46 }}
{{Optimal ET sequence|legend=1| 9, 28b, 37, 46 }}
Line 164: Line 170:
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: ~13/12 = 130.409
Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 130.409


{{Optimal ET sequence|legend=1| 9, 28b, 37, 46 }}
{{Optimal ET sequence|legend=1| 9, 28b, 37, 46 }}
Line 173: Line 179:


== Semaja ==
== Semaja ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 3125/3087, 6144/6125
[[Comma list]]: 3125/3087, 6144/6125


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


{{Multival|legend=1| 19 7 -1 -33 -55 -22 }}
{{Multival|legend=1| 19 7 -1 -33 -55 -22 }}


[[POTE generator]]: ~8/7 = 226.4834
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 226.4834


{{Optimal ET sequence|legend=1| 16, 37, 53, 196d }}
{{Optimal ET sequence|legend=1| 16, 37, 53, 196d }}
Line 192: Line 200:
Comma list: 121/120, 176/175, 3125/3087
Comma list: 121/120, 176/175, 3125/3087


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


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


{{Optimal ET sequence|legend=1| 16, 37, 53 }}
{{Optimal ET sequence|legend=1| 16, 37, 53 }}
Line 205: Line 213:
Comma list: 121/120, 169/168, 176/175, 275/273
Comma list: 121/120, 169/168, 176/175, 275/273


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


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


{{Optimal ET sequence|legend=1| 16, 37, 53 }}
{{Optimal ET sequence|legend=1| 16, 37, 53 }}
Line 272: Line 280:


== Aufo ==
== Aufo ==
{{see also| High badness temperaments #Untriton }}
:''For the 5-limit version of this temperament, see [[High badness temperaments #Untriton]].''


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 6 -7 19 | 0 -9 19 -33 }}


{{Multival|legend=1|9 -19 33 -51 27 130}}
: mapping generators: ~2, ~45/32


[[POTE generator]]: ~45/32 = 588.782
{{Multival|legend=1| 9 -19 33 -51 27 130 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~45/32 = 588.782


{{Optimal ET sequence|legend=1| 53, 161, 214 }}
{{Optimal ET sequence|legend=1| 53, 161, 214 }}
Line 293: Line 303:
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 6 -7 19 1 | 0 -9 19 -33 5 }}


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


{{Optimal ET sequence|legend=1| 53, 108e, 161e }}
{{Optimal ET sequence|legend=1| 53, 108e, 161e }}
Line 306: Line 316:
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 6 -7 19 1 -12 | 0 -9 19 -33 5 32 }}


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


{{Optimal ET sequence|legend=1| 53, 108e, 161e, 214ee }}
{{Optimal ET sequence|legend=1| 53, 108e, 161e, 214ee }}
Line 319: Line 329:
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 6 -7 19 -25 | 0 -9 19 -33 58 }}


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


{{Optimal ET sequence|legend=1| 53, 108, 161, 214, 375 }}
{{Optimal ET sequence|legend=1| 53, 108, 161, 214, 375 }}
Line 332: Line 342:
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 6 -7 19 -25 -12 | 0 -9 19 -33 58 32 }}


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


{{Optimal ET sequence|legend=1| 53, 108, 161, 214, 375, 589be }}
{{Optimal ET sequence|legend=1| 53, 108, 161, 214, 375, 589be }}
Line 341: Line 351:


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


Subgroup: 2.3.5.7
[[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 }}
 
: mapping generators: ~2, ~441/320


{{Multival|legend=1| 33 25 -21 -37 -126 -119 }}
{{Multival|legend=1| 33 25 -21 -37 -126 -119 }}


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


{{Optimal ET sequence|legend=1| 15, 122d, 137, 152, 608d, 623bd, 775bcd }}
{{Optimal ET sequence|legend=1| 15, 122d, 137, 152, 608d, 623bd, 775bcd }}
Line 362: Line 374:
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 }}


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


{{Optimal ET sequence|legend=1| 15, 122d, 137, 152, 608de, 623bde, 775bcde }}
{{Optimal ET sequence|legend=1| 15, 122d, 137, 152, 608de, 623bde, 775bcde }}
Line 371: Line 383:


== Polypyth ==
== Polypyth ==
{{see also| High badness temperaments #Leapday }}
:''For the 5-limit version of this temperament, see [[High badness temperaments #Leapday]].''


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


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 }}
 
: mapping generators: ~2, ~3


[[POTE generator]]: ~3/2 = 704.174
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.174


{{Optimal ET sequence|legend=1| 46, 121, 167, 288b, 455bcd, 743bcd }}
{{Optimal ET sequence|legend=1| 46, 121, 167, 288b, 455bcd, 743bcd }}
Line 392: Line 406:
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


{{Optimal ET sequence|legend=1| 46, 121, 167, 288be, 455bcde }}
{{Optimal ET sequence|legend=1| 46, 121, 167, 288be, 455bcde }}
Line 405: Line 419:
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


{{Optimal ET sequence|legend=1| 46, 121, 167, 288be }}
{{Optimal ET sequence|legend=1| 46, 121, 167, 288be }}
Line 418: Line 432:
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


{{Optimal ET sequence|legend=1| 46, 121, 167, 288beg }}
{{Optimal ET sequence|legend=1| 46, 121, 167, 288beg }}
Line 427: Line 441:


== Icositritonic ==
== Icositritonic ==
The ''icositritonic'' temperament (46&161) has a period of 1/23 octave, so six period represents [[6/5]] and nine period represents [[21/16]].
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
[[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 37 54 64 | 0 -1 -1 1 }}


{{Multival|legend=1| 23 23 -23 -17 -101 -118 }}
{{Multival|legend=1| 23 23 -23 -17 -101 -118 }}


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


{{Optimal ET sequence|legend=1| 23, 46, 115, 161, 207, 368c }}
{{Optimal ET sequence|legend=1| 23, 46, 115, 161, 207, 368c }}
Line 448: Line 462:
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 37 54 64 79 | 0 -1 -1 1 1 }}


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


{{Optimal ET sequence|legend=1| 23, 46, 115, 161, 207, 368c }}
{{Optimal ET sequence|legend=1| 23, 46, 115, 161, 207, 368c }}
Line 461: Line 475:
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 37 54 64 79 84 | 0 -1 -1 1 1 2 }}


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


{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368c }}
{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368c }}
Line 474: Line 488:
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 37 54 64 79 84 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


{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368c }}
{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368c }}
Line 487: Line 501:
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 37 54 64 79 84 94 96 | 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


{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368c }}
{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368c }}
Line 500: Line 514:
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 37 54 64 79 84 94 96 104 | 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


{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368ci }}
{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368ci }}
Line 509: Line 523:


== Countermiracle ==
== Countermiracle ==
The ''countermiracle'' temperament (31&145) tempers out the trimyna, 50421/50000 and the [[Quince clan|quince comma]], 823543/819200.
The ''countermiracle'' temperament (31 & 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 }}
{{Multival|legend=1| 25 7 2 -47 -67 -15 }}


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


{{Optimal ET sequence|legend=1| 31, 114, 145, 176, 559cc, 735cc }}
{{Optimal ET sequence|legend=1| 31, 114, 145, 176, 559cc, 735cc }}
Line 530: Line 544:
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 = 115.9158
Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.9158


{{Optimal ET sequence|legend=1| 31, 114e, 145, 176 }}
{{Optimal ET sequence|legend=1| 31, 114e, 145, 176 }}
Line 543: Line 557:
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 = 115.8803
Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.8803


{{Optimal ET sequence|legend=1| 31, 83e, 114e, 145, 321ceff }}
{{Optimal ET sequence|legend=1| 31, 83e, 114e, 145, 321ceff }}
Line 556: Line 570:
Comma list: 196/195, 256/255, 352/351, 1001/1000, 1225/1224
Comma list: 196/195, 256/255, 352/351, 1001/1000, 1225/1224


Mapping: [{{val| 1 4 3 3 8 1 1 }}, {{val| 0 -25 -7 -2 -47 28 32 }}]
Mapping: {{mapping| 1 4 3 3 8 1 1 | 0 -25 -7 -2 -47 28 32 }}


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


{{Optimal ET sequence|legend=1| 31, 83e, 114e, 145 }}
{{Optimal ET sequence|legend=1| 31, 83e, 114e, 145 }}
Line 569: Line 583:
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 = 115.9335
Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.9335


{{Optimal ET sequence|legend=1| 31, 114ef, 145f, 176, 207, 383c, 590cc }}
{{Optimal ET sequence|legend=1| 31, 114ef, 145f, 176, 207, 383c, 590cc }}
Line 582: Line 596:
Comma list: 351/350, 441/440, 561/560, 1632/1625, 3146/3125
Comma list: 351/350, 441/440, 561/560, 1632/1625, 3146/3125


Mapping: [{{val| 1 4 3 3 8 -2 -2 }}, {{val| 0 -25 -7 -2 -47 59 63 }}]
Mapping: {{mapping| 1 4 3 3 8 -2 -2 | 0 -25 -7 -2 -47 59 63 }}


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


{{Optimal ET sequence|legend=1| 31, 114efg, 145fg, 176, 207 }}
{{Optimal ET sequence|legend=1| 31, 114efg, 145fg, 176, 207 }}
Line 595: Line 609:
Comma list: 364/363, 441/440, 3388/3375, 6144/6125
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 }}]
Mapping: {{mapping| 1 4 3 3 8 15 0 -25 -7 -2 -47 -117 }}


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


{{Optimal ET sequence|legend=1| 145, 176, 321ce }}
{{Optimal ET sequence|legend=1| 145, 176, 321ce }}
Line 608: Line 622:
Comma list: 364/363, 441/440, 595/594, 1632/1625, 3388/3375
Comma list: 364/363, 441/440, 595/594, 1632/1625, 3388/3375


Mapping: [{{val| 1 4 3 3 8 15 15 }}, {{val| 0 -25 -7 -2 -47 -117 -113 }}]
Mapping: {{mapping| 1 4 3 3 8 15 15 | 0 -25 -7 -2 -47 -117 -113 }}


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


{{Optimal ET sequence|legend=1| 145, 321ce }}
{{Optimal ET sequence|legend=1| 145, 321ce }}
Line 621: Line 635:
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 = 115.9192
Optimal tuning (POTE): ~2 = 1\1, ~343/320 = 115.9192


{{Optimal ET sequence|legend=1| 31, 114, 145e, 176e }}
{{Optimal ET sequence|legend=1| 31, 114, 145e, 176e }}
Line 634: Line 648:
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 = 115.8624
Optimal tuning (POTE): ~2 = 1\1, ~273/256 = 115.8624


{{Optimal ET sequence|legend=1| 31, 83, 114, 145e }}
{{Optimal ET sequence|legend=1| 31, 83, 114, 145e }}
Line 647: Line 661:
Comma list: 121/120, 154/153, 176/175, 196/195, 10647/10625
Comma list: 121/120, 154/153, 176/175, 196/195, 10647/10625


Mapping: [{{val| 1 4 3 3 5 1 1 }}, {{val| 0 -25 -7 -2 -16 28 32 }}]
Mapping: {{mapping| 1 4 3 3 5 1 1 | 0 -25 -7 -2 -16 28 32 }}


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


{{Optimal ET sequence|legend=1| 31, 83, 114, 145e }}
{{Optimal ET sequence|legend=1| 31, 83, 114, 145e }}
Line 658: Line 672:
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Absurdity]].''
: ''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, 177147/175000


[[Mapping]]: [{{val|7 11 16 20}}, {{val|0 1 3 -4}}]
{{Mapping|legend=1| 7 11 16 20 | 0 1 3 -4 }}


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


{{Optimal ET sequence|legend=1| 77, 84, 161 }}
{{Optimal ET sequence|legend=1| 77, 84, 161 }}
Line 675: Line 689:
Comma list: 441/440, 6144/6125, 72171/71680
Comma list: 441/440, 6144/6125, 72171/71680


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


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


{{Optimal ET sequence|legend=1| 77, 84, 161 }}
{{Optimal ET sequence|legend=1| 77, 84, 161 }}
Line 688: Line 702:
Comma list: 351/350, 441/440, 1188/1183, 3584/3575
Comma list: 351/350, 441/440, 1188/1183, 3584/3575


Mapping: [{{val|7 11 16 20 25 26}}, {{val|0 1 3 -4 -9 -1}}]
Mapping: {{mapping| 7 11 16 20 25 26 | 0 1 3 -4 -9 -1 }}


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


{{Optimal ET sequence|legend=1| 77, 84, 161 }}
{{Optimal ET sequence|legend=1| 77, 84, 161 }}
Line 701: Line 715:
Comma list: 351/350, 441/440, 561/560, 1188/1183, 1632/1625
Comma list: 351/350, 441/440, 561/560, 1188/1183, 1632/1625


Mapping: [{{val|7 11 16 20 25 26 28}}, {{val|0 1 3 -4 -9 -1 7}}]
Mapping: {{mapping| 7 11 16 20 25 26 28 | 0 1 3 -4 -9 -1 7 }}


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


{{Optimal ET sequence|legend=1| 77, 161 }}
{{Optimal ET sequence|legend=1| 77, 161 }}
Line 714: Line 728:
Comma list: 324/323, 351/350, 441/440, 456/455, 476/475, 495/494
Comma list: 324/323, 351/350, 441/440, 456/455, 476/475, 495/494


Mapping: [{{val|7 11 16 20 25 26 28 30}}, {{val|0 1 3 -4 -9 -1 7 -3}}]
Mapping: {{mapping| 7 11 16 20 25 26 28 30 | 0 1 3 -4 -9 -1 7 -3 }}


POTE generator: ~3/2 = 700.6565 (or ~10/9 = 186.3708)
Optimal tuning (POTE): ~72/65 = 1\7, ~3/2 = 700.6565 (or ~10/9 = 186.3708)


{{Optimal ET sequence|legend=1| 77, 161 }}
{{Optimal ET sequence|legend=1| 77, 161 }}
Line 725: Line 739:
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Dodifo]].''
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Dodifo]].''


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 6144/6125, 2500000/2470629
[[Comma list]]: 6144/6125, 2500000/2470629


[[Mapping]]: [{{val|1 12 5 4}}, {{val|0 -35 -9 -4}}]
{{Mapping|legend=1| 1 12 5 4 | 0 -35 -9 -4 }}


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


{{Optimal ET sequence|legend=1| 37, 84, 121, 205 }}
{{Optimal ET sequence|legend=1| 37, 84, 121, 205 }}
Line 742: Line 756:
Comma list: 1375/1372, 2560/2541, 4375/4356
Comma list: 1375/1372, 2560/2541, 4375/4356


Mapping: [{{val|1 12 5 4 -1}}, {{val|0 -35 -9 -4 15}}]
Mapping: {{mapping| 1 12 5 4 -1 | 0 -35 -9 -4 15 }}


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


{{Optimal ET sequence|legend=1| 37, 84, 121, 326dee }}
{{Optimal ET sequence|legend=1| 37, 84, 121, 326dee }}
Line 755: Line 769:
Comma list: 364/363, 625/624, 640/637, 1375/1372
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}}]
Mapping: {{mapping| 1 12 5 4 -1 4 | 0 -35 -9 -4 15 -1 }}


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


{{Optimal ET sequence|legend=1| 37, 84, 121, 326deef }}
{{Optimal ET sequence|legend=1| 37, 84, 121, 326deef }}
Line 765: Line 779:
[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Porwell temperaments| ]] <!-- main article -->
[[Category:Porwell temperaments| ]] <!-- main article -->
[[Category:Porwell| ]] <!-- key article -->
[[Category:Hendecatonic]]
[[Category:Hendecatonic]]
[[Category:Rank 2]]
[[Category:Rank 2]]

Revision as of 10:16, 17 September 2023

This family of temperaments 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

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

Wedgie⟨⟨ 11 -11 22 -43 4 82 ]]

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

Optimal ET sequence22, 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 sequence22, 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 sequence22, 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 sequence22, 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 sequence22, 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 sequence22, 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 sequence22, 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 sequence22, 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

Wedgie⟨⟨ 13 -3 11 -35 -19 34 ]]

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

Optimal ET sequence9, 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 sequence9, 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 sequence9, 28b, 37, 46

Badness: 0.025941

Semaja

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

Wedgie⟨⟨ 19 7 -1 -33 -55 -22 ]]

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

Optimal ET sequence16, 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 sequence16, 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 sequence16, 37, 53

Badness: 0.032564

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 5 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

Wedgie⟨⟨ 21 15 -12 -25 -78 -70 ]]

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

Optimal ET sequence15, 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 sequence15, 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 sequence15, 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 sequence15, 54bf, 69, 84, 99ef, 183ef, 282eeff

Badness: 0.037409

Aufo

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

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

Wedgie⟨⟨ 9 -19 33 -51 27 130 ]]

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

Optimal ET sequence53, 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 sequence53, 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 sequence53, 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 sequence53, 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 sequence53, 108, 161, 214, 375, 589be

Badness: 0.039050

Whoops

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

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

Wedgie⟨⟨ 33 25 -21 -37 -126 -119 ]]

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

Optimal ET sequence15, 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 sequence15, 122d, 137, 152, 608de, 623bde, 775bcde

Badness: 0.043743

Polypyth

For the 5-limit version of this temperament, 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 sequence46, 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 sequence46, 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 sequence46, 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 sequence46, 121, 167, 288beg

Badness: 0.019051

Icositritonic

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 37 54 64], 0 -1 -1 1]]

Wedgie⟨⟨ 23 23 -23 -17 -101 -118 ]]

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

Optimal ET sequence23, 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 37 54 64 79], 0 -1 -1 1 1]]

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

Optimal ET sequence23, 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 37 54 64 79 84], 0 -1 -1 1 1 2]]

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

Optimal ET sequence46, 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 37 54 64 79 84 94], 0 -1 -1 1 1 2 0]]

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

Optimal ET sequence46, 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 37 54 64 79 84 94 96], 0 -1 -1 1 1 2 0 3]]

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

Optimal ET sequence46, 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 37 54 64 79 84 94 96 104], 0 -1 -1 1 1 2 0 3 0]]

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

Optimal ET sequence46, 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]]

Wedgie⟨⟨ 25 7 2 -47 -67 -15 ]]

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

Optimal ET sequence31, 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 sequence31, 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 sequence31, 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 sequence31, 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 sequence31, 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 sequence31, 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 sequence145, 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 sequence145, 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 sequence31, 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 sequence31, 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 sequence31, 83, 114, 145e

Badness: 0.044043

Absurdity

For the 5-limit version of this temperament, see High badness temperaments #Absurdity.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 177147/175000

Mapping[7 11 16 20], 0 1 3 -4]]

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

Optimal ET sequence77, 84, 161

Badness: 0.133520

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [7 11 16 20 25], 0 1 3 -4 -9]]

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

Optimal ET sequence77, 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 11 16 20 25 26], 0 1 3 -4 -9 -1]]

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

Optimal ET sequence77, 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 11 16 20 25 26 28], 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 sequence77, 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 11 16 20 25 26 28 30], 0 1 3 -4 -9 -1 7 -3]]

Optimal tuning (POTE): ~72/65 = 1\7, ~3/2 = 700.6565 (or ~10/9 = 186.3708)

Optimal ET sequence77, 161

Badness: 0.022291

Dodifo

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

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 sequence37, 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 sequence37, 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 sequence37, 84, 121, 326deef

Badness: 0.039533