Porwell temperaments: Difference between revisions
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Discussed elsewhere are: | Discussed elsewhere are: | ||
* ''[[Hexadecimal]]'' | * ''[[Hexadecimal]]'' (+36/35) → [[Pelogic family #Armodue|Pelogic family]] | ||
* [[Porcupine]] | * [[Porcupine]] (+64/63), also in: [[Porcupine family #Porcupine|Porcupine family]] | ||
* [[Mohajira]] | * [[Mohajira]] (+81/80), also in: [[Meantone family #Mohajira|Meantone family]] | ||
* [[Valentine]] | * [[Valentine]] (+126/125), also in: [[Starling temperaments #Valentine|Starling temperaments]] | ||
* [[Orwell]] | * [[Orwell]] (+225/224), also in: [[Semicomma family #Orwell|Semicomma family]] | ||
* [[Shrutar]] | * [[Shrutar]] (+245/243), also in: [[Diaschismic family #Shrutar|Diaschismic family]] | ||
* ''[[Quinkee]]'' | * ''[[Quinkee]]'' (+1029/1000) → [[Cloudy clan #Quinkee|Cloudy clan]] | ||
* ''[[Hemiwürschmidt]]'' | * ''[[Hemiwürschmidt]]'' (+2401/2400 or 3136/3125) → [[Würschmidt family #Hemiwürschmidt|Würschmidt family]] and [[Hemimean clan #Hemiwürschmidt|Hemimean clan]] | ||
* ''[[Hemikleismic]]'' | * ''[[Hemikleismic]]'' (+4000/3969) → [[Kleismic family #Hemikleismic|Kleismic family]] | ||
* [[Amity]] | * [[Amity]] (+4375/4374 or 5120/5103), also in: [[Amity family #Septimal amity|Amity family]] and [[Ragismic microtemperaments #Amity|Ragismic microtemperaments]] | ||
* ''[[Freivald]]'' | * ''[[Freivald]]'' (+6272/6075) → [[Passion family #Freivald|Passion family]] | ||
* ''[[Grendel]]'' | * ''[[Grendel]]'' (+16875/16807) → [[Mirkwai clan #Grendel|Mirkwai clan]] | ||
* ''[[Hemischis]]'' | * ''[[Hemischis]]'' (+19683/19600) → [[Schismatic family #Hemischis|Schismatic family]] | ||
* ''[[Bison]]'' | * ''[[Bison]]'' (+78732/78125) → [[Sensipent family #Bison|Sensipent family]] | ||
* ''[[Hemimabila]]'' | * ''[[Hemimabila]]'' (+117649/116640) → [[Mabila family #Hemimabila|Mabila family]] | ||
* ''[[Septisuperfourth]]'' | * ''[[Septisuperfourth]]'' (+118098/117649) → [[Escapade family #Septisuperfourth|Escapade family]] | ||
* ''[[Trident]]'' | * ''[[Trident]]'' (+14348907/14336000) → [[Tricot family #Trident|Tricot family]] | ||
* ''[[Hemimaquila]]'' | * ''[[Hemimaquila]]'' (+{{monzo| -5 10 5 -8 }}) → [[Maquila family #Hemimaquila|Maquila family]] | ||
* ''[[Decimaleap]]'' | * ''[[Decimaleap]]'' (+{{monzo| 15 -18 1 4 }}) → [[Quintaleap family #Decimaleap|Quintaleap family]] | ||
* ''[[Twilight]]'' | * ''[[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 | 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|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 | [[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: | Mapping: {{mapping| 11 0 43 -4 38 | 0 1 -1 2 0 }} | ||
POTE | 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: | Mapping: {{mapping| 11 0 43 -4 38 93 | 0 1 -1 2 0 -3 }} | ||
POTE | 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: | Mapping: {{mapping| 11 0 43 -4 38 93 45 | 0 1 -1 2 0 -3 0 }} | ||
POTE | 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: | Mapping: {{mapping| 11 0 43 -4 73 | 0 1 -1 2 -2 }} | ||
POTE | 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: | Mapping: {{mapping| 11 0 43 -4 73 128 | 0 1 -1 2 -2 -5 }} | ||
POTE | 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: | Mapping: {{mapping| 11 0 43 -4 73 128 45 | 0 1 -1 2 -2 -5 0 }} | ||
POTE | 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: | Mapping: {{mapping| 22 0 86 -8 111 | 0 1 -1 2 -1 }} | ||
POTE | : 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|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 | [[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: | Mapping: {{mapping| 1 3 2 4 4 | 0 -13 3 -11 -5 }} | ||
POTE | 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: | Mapping: {{mapping| 1 3 2 4 4 5 | 0 -13 3 -11 -5 -12 }} | ||
POTE | 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|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 | [[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: | Mapping: {{mapping| 1 -2 1 3 1 | 0 19 7 -1 13 }} | ||
POTE | 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: | Mapping: {{mapping| 1 -2 1 3 1 2 | 0 19 7 -1 13 9 }} | ||
POTE | 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 == | ||
:''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|legend=1| 1 6 -7 19 | 0 -9 19 -33 }} | |||
: mapping generators: ~2, ~45/32 | |||
[[POTE | {{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: | Mapping: {{mapping| 1 6 -7 19 1 | 0 -9 19 -33 5 }} | ||
POTE | 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: | Mapping: {{mapping| 1 6 -7 19 1 -12 | 0 -9 19 -33 5 32 }} | ||
POTE | 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: | Mapping: {{mapping| 1 6 -7 19 -25 | 0 -9 19 -33 58 }} | ||
POTE | 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: | Mapping: {{mapping| 1 6 -7 19 -25 -12 | 0 -9 19 -33 58 32 }} | ||
POTE | 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 == | ||
:''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|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 | [[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: | Mapping: {{mapping| 1 17 14 -7 10 | 0 -33 -25 21 -14 }} | ||
POTE | 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 == | ||
:''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|legend=1| 1 0 -31 52 | 0 1 21 -31 }} | |||
: mapping generators: ~2, ~3 | |||
[[POTE | [[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: | Mapping: {{mapping| 1 0 -31 52 59 | 0 1 21 -31 -35 }} | ||
POTE | 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: | Mapping: {{mapping| 1 0 -31 52 59 64 | 0 1 21 -31 -35 -38 }} | ||
POTE | 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: | Mapping: {{mapping| 1 0 -31 52 59 64 39 | 0 1 21 -31 -35 -38 -22 }} | ||
POTE | 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 | 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|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 | [[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: | Mapping: {{mapping| 23 37 54 64 79 | 0 -1 -1 1 1 }} | ||
POTE | 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: | Mapping: {{mapping| 23 37 54 64 79 84 | 0 -1 -1 1 1 2 }} | ||
POTE | 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: | Mapping: {{mapping| 23 37 54 64 79 84 94 | 0 -1 -1 1 1 2 0 }} | ||
POTE | 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: | Mapping: {{mapping| 23 37 54 64 79 84 94 96 | 0 -1 -1 1 1 2 0 3 }} | ||
POTE | 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: | Mapping: {{mapping| 23 37 54 64 79 84 94 96 104 | 0 -1 -1 1 1 2 0 3 0 }} | ||
POTE | 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 [[ | 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|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 | [[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: | Mapping: {{mapping| 1 4 3 3 8 | 0 -25 -7 -2 -47 }} | ||
POTE | 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: | Mapping: {{mapping| 1 4 3 3 8 1 | 0 -25 -7 -2 -47 28 }} | ||
POTE | 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: | Mapping: {{mapping| 1 4 3 3 8 1 1 | 0 -25 -7 -2 -47 28 32 }} | ||
POTE | 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: | Mapping: {{mapping| 1 4 3 3 8 -2 | 0 -25 -7 -2 -47 59 }} | ||
POTE | 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: | Mapping: {{mapping| 1 4 3 3 8 -2 -2 | 0 -25 -7 -2 -47 59 63 }} | ||
POTE | 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: | Mapping: {{mapping| 1 4 3 3 8 15 0 -25 -7 -2 -47 -117 }} | ||
POTE | 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: | Mapping: {{mapping| 1 4 3 3 8 15 15 | 0 -25 -7 -2 -47 -117 -113 }} | ||
POTE | 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: | Mapping: {{mapping| 1 4 3 3 5 | 0 -25 -7 -2 -16 }} | ||
POTE | 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: | Mapping: {{mapping| 1 4 3 3 5 1 | 0 -25 -7 -2 -16 28 }} | ||
POTE | 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: | Mapping: {{mapping| 1 4 3 3 5 1 1 | 0 -25 -7 -2 -16 28 32 }} | ||
POTE | 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|legend=1| 7 11 16 20 | 0 1 3 -4 }} | |||
[[POTE | [[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: | Mapping: {{mapping| 7 11 16 20 25 | 0 1 3 -4 -9 }} | ||
POTE | 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: | Mapping: {{mapping| 7 11 16 20 25 26 | 0 1 3 -4 -9 -1 }} | ||
POTE | 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: | Mapping: {{mapping| 7 11 16 20 25 26 28 | 0 1 3 -4 -9 -1 7 }} | ||
POTE | 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: | Mapping: {{mapping| 7 11 16 20 25 26 28 30 | 0 1 3 -4 -9 -1 7 -3 }} | ||
POTE | 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|legend=1| 1 12 5 4 | 0 -35 -9 -4 }} | |||
[[POTE | [[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: | Mapping: {{mapping| 1 12 5 4 -1 | 0 -35 -9 -4 15 }} | ||
POTE | 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: | Mapping: {{mapping| 1 12 5 4 -1 4 | 0 -35 -9 -4 15 -1 }} | ||
POTE | 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:
- Hexadecimal (+36/35) → Pelogic family
- Porcupine (+64/63), also in: Porcupine family
- Mohajira (+81/80), also in: Meantone family
- Valentine (+126/125), also in: Starling temperaments
- Orwell (+225/224), also in: Semicomma family
- Shrutar (+245/243), also in: Diaschismic family
- Quinkee (+1029/1000) → Cloudy clan
- Hemiwürschmidt (+2401/2400 or 3136/3125) → Würschmidt family and Hemimean clan
- Hemikleismic (+4000/3969) → Kleismic family
- Amity (+4375/4374 or 5120/5103), also in: Amity family and Ragismic microtemperaments
- Freivald (+6272/6075) → Passion family
- Grendel (+16875/16807) → Mirkwai clan
- Hemischis (+19683/19600) → Schismatic family
- Bison (+78732/78125) → Sensipent family
- Hemimabila (+117649/116640) → Mabila family
- Septisuperfourth (+118098/117649) → Escapade family
- Trident (+14348907/14336000) → Tricot family
- Hemimaquila (+[-5 10 5 -8⟩) → Maquila family
- Decimaleap (+[15 -18 1 4⟩) → Quintaleap family
- Twilight (+[19 -22 2 4⟩) → Undim family
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 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
Wedgie: ⟨⟨ 13 -3 11 -35 -19 34 ]]
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
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 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
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 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 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 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 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 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 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 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
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 sequence: 23, 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 sequence: 23, 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 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 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 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 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 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 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 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]]
Wedgie: ⟨⟨ 25 7 2 -47 -67 -15 ]]
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 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 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 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 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 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 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 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)
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)
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 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