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Porwell temperaments: Difference between revisions

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


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{{Mapping|legend=1| 11 0 43 -4 | 0 1 -1 2 }}
{{Mapping|legend=1| 11 0 43 -4 | 0 1 -1 2 }}


: mapping generators: ~16/15, ~3
: Mapping generators: ~16/15, ~3


{{Multival|legend=1| 11 -11 22 -43 4 82 }}
{{Multival|legend=1| 11 -11 22 -43 4 82 }}
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Comma list: 121/120, 176/175, 10976/10935
Comma list: 121/120, 176/175, 10976/10935


Mapping: {{mapping| 11 0 43 -4 38 | 0 1 -1 2 0 }}
{{Mapping|legend=1| 11 0 43 -4 38 | 0 1 -1 2 0 }}


Optimal tuning (POTE): ~16/15 = 1\11, ~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=0| 22, 55, 77, 99, 176e, 275e }}


Badness: 0.046088
Badness: 0.046088
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Comma list: 121/120, 176/175, 351/350, 4459/4455
Comma list: 121/120, 176/175, 351/350, 4459/4455


Mapping: {{mapping| 11 0 43 -4 38 93 | 0 1 -1 2 0 -3 }}
{{Mapping|legend=1| 11 0 43 -4 38 93 | 0 1 -1 2 0 -3 }}


Optimal tuning (POTE): ~16/15 = 1\11, ~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=0| 22, 55, 77, 99, 176e }}


Badness: 0.040099
Badness: 0.040099
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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| 11 0 43 -4 38 93 45 | 0 1 -1 2 0 -3 0 }}
{{Mapping|legend=1| 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 tuning (POTE): ~16/15 = 1\11, ~3/2 = 702.301


{{Optimal ET sequence|legend=1| 22, 55, 77, 99, 176eg }}
{{Optimal ET sequence|legend=0| 22, 55, 77, 99, 176eg }}


Badness: 0.029054
Badness: 0.029054
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Comma list: 540/539, 896/891, 4375/4356
Comma list: 540/539, 896/891, 4375/4356


Mapping: {{mapping| 11 0 43 -4 73 | 0 1 -1 2 -2 }}
{{Mapping|legend=1| 11 0 43 -4 73 | 0 1 -1 2 -2 }}


Optimal tuning (POTE): ~16/15 = 1\11, ~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=0| 22, 77e, 99e, 121, 220e }}


Badness: 0.038042
Badness: 0.038042
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Comma list: 352/351, 364/363, 540/539, 625/624
Comma list: 352/351, 364/363, 540/539, 625/624


Mapping: {{mapping| 11 0 43 -4 73 128 | 0 1 -1 2 -2 -5 }}
{{Mapping|legend=1| 11 0 43 -4 73 128 | 0 1 -1 2 -2 -5 }}


Optimal tuning (POTE): ~16/15 = 1\11, ~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=0| 22, 77eff, 99ef, 121, 341bdeeff }}


Badness: 0.036112
Badness: 0.036112
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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| 11 0 43 -4 73 128 45 | 0 1 -1 2 -2 -5 0 }}
{{Mapping|legend=1| 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 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=0| 22, 77eff, 99ef, 121, 220efg, 341bdeeffgg }}


Badness: 0.022590
Badness: 0.022590
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Comma list: 3388/3375, 6144/6125, 9801/9800
Comma list: 3388/3375, 6144/6125, 9801/9800


Mapping: {{mapping| 22 0 86 -8 111 | 0 1 -1 2 -1 }}
{{Mapping|legend=1| 22 0 86 -8 111 | 0 1 -1 2 -1 }}


: mapping generators: ~33/32, ~3
: Mapping generators: ~33/32, ~3


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


{{Optimal ET sequence|legend=1| 22, 154, 176, 198 }}
{{Optimal ET sequence|legend=0| 22, 154, 176, 198 }}


Badness: 0.057725
Badness: 0.057725
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{{Mapping|legend=1| 1 3 2 4 | 0 -13 3 -11 }}
{{Mapping|legend=1| 1 3 2 4 | 0 -13 3 -11 }}


: mapping generators: ~2, ~15/14
: Mapping generators: ~2, ~15/14


{{Multival|legend=1| 13 -3 11 -35 -19 34 }}
{{Multival|legend=1| 13 -3 11 -35 -19 34 }}
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{{Mapping|legend=1| 1 -2 1 3 | 0 19 7 -1 }}
{{Mapping|legend=1| 1 -2 1 3 | 0 19 7 -1 }}


: mapping generators: ~2, ~8/7
: Mapping generators: ~2, ~8/7


{{Multival|legend=1| 19 7 -1 -33 -55 -22 }}
{{Multival|legend=1| 19 7 -1 -33 -55 -22 }}
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{{Mapping|legend=1| 3 2 5 10 | 0 7 5 -4 }}
{{Mapping|legend=1| 3 2 5 10 | 0 7 5 -4 }}


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


{{Multival|legend=1| 21 15 -12 -25 -78 -70 }}
{{Multival|legend=1| 21 15 -12 -25 -78 -70 }}
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{{Mapping|legend=1| 1 6 -7 19 | 0 -9 19 -33 }}
{{Mapping|legend=1| 1 6 -7 19 | 0 -9 19 -33 }}


: mapping generators: ~2, ~45/32
: Mapping generators: ~2, ~45/32


{{Multival|legend=1| 9 -19 33 -51 27 130 }}
{{Multival|legend=1| 9 -19 33 -51 27 130 }}
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{{Mapping|legend=1| 1 17 14 -7 | 0 -33 -25 21 }}
{{Mapping|legend=1| 1 17 14 -7 | 0 -33 -25 21 }}


: mapping generators: ~2, ~441/320
: Mapping generators: ~2, ~441/320


{{Multival|legend=1| 33 25 -21 -37 -126 -119 }}
{{Multival|legend=1| 33 25 -21 -37 -126 -119 }}
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{{Mapping|legend=1| 1 0 -31 52 | 0 1 21 -31 }}
{{Mapping|legend=1| 1 0 -31 52 | 0 1 21 -31 }}


: mapping generators: ~2, ~3
: Mapping generators: ~2, ~3


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.174
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.174
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{{Mapping|legend=1| 23 0 17 101 | 0 1 1 -1 }}
{{Mapping|legend=1| 23 0 17 101 | 0 1 1 -1 }}


: mapping generators: ~1323/1280, ~3
: Mapping generators: ~1323/1280, ~3


{{Multival|legend=1| 23 23 -23 -17 -101 -118 }}
{{Multival|legend=1| 23 23 -23 -17 -101 -118 }}
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{{Mapping|legend=1| 1 4 3 3 | 0 -25 -7 -2 }}
{{Mapping|legend=1| 1 4 3 3 | 0 -25 -7 -2 }}


: mapping generators: ~2, ~343/320
: Mapping generators: ~2, ~343/320


{{Multival|legend=1| 25 7 2 -47 -67 -15 }}
{{Multival|legend=1| 25 7 2 -47 -67 -15 }}
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{{Mapping|legend=1| 7 0 -17 64 | 0 1 3 -4 }}
{{Mapping|legend=1| 7 0 -17 64 | 0 1 3 -4 }}


: mapping generators: ~972/875, ~3
: Mapping generators: ~972/875, ~3


[[Optimal tuning]] ([[POTE]]): ~972/875 = 1\7, ~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)
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Subgroup: 2.3.5.7.11.13.17.19.23
Subgroup: 2.3.5.7.11.13.17.19.23


Comma list: 324/323, 351/350, 441/440, 456/455, 476/475, 495/494, 276/275
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 }}
{{Mapping|legend=1| 7 0 -17 64 124 37 -49 63 76 | 0 1 3 -4 -9 -1 7 -3 -4 }}
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Optimal tuning ([[CTE]]): ~21/19 = 1\7, ~3/2 = 700.629 (or ~10/9 = 186.343)
Optimal tuning ([[CTE]]): ~21/19 = 1\7, ~3/2 = 700.629 (or ~10/9 = 186.343)


{{Optimal ET sequence|legend=1| 77, 84, 161 }}
{{Optimal ET sequence|legend=0| 77, 84, 161 }}


=== 29-limit ===
=== 29-limit ===
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Subgroup: 2.3.5.7.11.13.17.19.23
Subgroup: 2.3.5.7.11.13.17.19.23


Comma list: 324/323, 351/350, 441/440, 456/455, 476/475, 495/494, 276/275, 261/260
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 }}
{{Mapping|legend=1| 7 0 -17 64 124 37 -49 63 76 34 | 0 1 3 -4 -9 -1 7 -3 -4 0 }}
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Optimal tuning ([[CTE]]): ~21/19 = 1\7, ~3/2 = 700.629 (or ~10/9 = 186.343)
Optimal tuning ([[CTE]]): ~21/19 = 1\7, ~3/2 = 700.629 (or ~10/9 = 186.343)


{{Optimal ET sequence|legend=1| 77, 84, 161 }}
{{Optimal ET sequence|legend=0| 77, 84, 161 }}


== Dodifo ==
== Dodifo ==
Retrieved from "https://en.xen.wiki/w/Porwell_temperaments"