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Very high accuracy temperaments: Difference between revisions

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Update keys; update for no-5 nanismic
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[[Comma list]]: {{monzo| -53 10 16 }} = 9010162353515625 / 9007199254740992
[[Comma list]]: {{monzo| -53 10 16 }} = 9010162353515625 / 9007199254740992


[[Mapping]]: [{{val| 2 1 6 }}, {{val| 0 8 -5 }}]
{{Mapping|legend=1| 2 1 6 | 0 8 -5 }}


: mapping generators: ~94921875/67108864, ~1125/1024
: mapping generators: ~94921875/67108864, ~1125/1024
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[[Comma list]]: {{monzo| 91 -12 -31 }}
[[Comma list]]: {{monzo| 91 -12 -31 }}


[[Mapping]]: [{{val| 1 5 1 }}, {{val| 0 -31 12 }}]
{{Mapping|legend=1| 1 5 1 | 0 -31 12 }}


: mapping generators: ~2, ~{{monzo| -38 5 13 }}
: mapping generators: ~2, ~{{monzo| -38 5 13 }}
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[[Comma list]]: {{monzo| -70 72 -19 }}
[[Comma list]]: {{monzo| -70 72 -19 }}


[[Mapping]]: [{{val| 1 11 38 }}, {{val| 0 -19 -72 }}]
{{Mapping|legend=1| 1 11 38 | 0 -19 -72 }}


: mapping generators: ~2, ~{{monzo| 19 -19 5 }}
: mapping generators: ~2, ~{{monzo| 19 -19 5 }}
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[[Comma list]]: {{monzo| 37 25 -33 }}
[[Comma list]]: {{monzo| 37 25 -33 }}


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


: mapping generators: ~2, ~864/625
: mapping generators: ~2, ~864/625
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[[Comma list]]: {{monzo| 54 -37 2 }} = 450359962737049600 / 450283905890997363
[[Comma list]]: {{monzo| 54 -37 2 }} = 450359962737049600 / 450283905890997363


[[Mapping]]: [{{val| 1 0 -27 }}, {{val| 0 2 37 }}]
{{Mapping|legend=1| 1 0 -27 | 0 2 37 }}


: mapping generators: ~2, ~{{monzo| 27 -18 1 }}
: mapping generators: ~2, ~{{monzo| 27 -18 1 }}
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[[Comma list]]: {{monzo| -36 -52 51 }}
[[Comma list]]: {{monzo| -36 -52 51 }}


[[Mapping]]: [{{val| 1 15 16 }}, {{val| 0 -51 -52 }}]
{{Mapping|legend=1| 1 15 16 | 0 -51 -52 }}


: mapping generators: ~2, ~6/5
: mapping generators: ~2, ~6/5
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Comma list: 78125000/78121827, {{monzo| -48 0 11 8 }}
Comma list: 78125000/78121827, {{monzo| -48 0 11 8 }}


Mapping: [{{val| 1 15 16 -16 }}, {{val| 0 -102 -104 143 }}]
Mapping: {{mapping| 1 15 16 -16 | 0 -102 -104 143 }}


Optimal tuning (CTE): ~2 = 1\1, ~36756909/33554432 = 157.8240
Optimal tuning (CTE): ~2 = 1\1, ~36756909/33554432 = 157.8240
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[[Comma list]]: {{monzo| -107 47 14 }}
[[Comma list]]: {{monzo| -107 47 14 }}


[[Mapping]]: [{{val| 1 13 -36 }}, {{val| 0 -14 47 }}]
{{Mapping|legend=1| 1 13 -36 | 0 -14 47 }}


: mapping generators: ~2, 7381125/4194304
: mapping generators: ~2, 7381125/4194304
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Comma list: 78125000/78121827, {{monzo| -52 17 12 -1 }}
Comma list: 78125000/78121827, {{monzo| -52 17 12 -1 }}


Mapping: [{{val| 1 13 -36 -263 }}, {{val| 0 -14 47 326 }}]
Mapping: {{mapping| 1 13 -36 -263 | 0 -14 47 326 }}


Optimal tuning (CTE): ~2 = 1\1, ~8388608/7381125 = 221.5680
Optimal tuning (CTE): ~2 = 1\1, ~8388608/7381125 = 221.5680
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Comma list: 151263/151250, 21437500/21434787, 369140625/369098752
Comma list: 151263/151250, 21437500/21434787, 369140625/369098752


Mapping: [{{val| 1 13 -36 -263 -573 }}, {{val| 0 -14 47 326 707 }}]
Mapping: {{mapping| 1 13 -36 -263 -573 | 0 -14 47 326 707 }}


Optimal tuning (CTE): ~2 = 1\1, ~8388608/7381125 = 221.5681
Optimal tuning (CTE): ~2 = 1\1, ~8388608/7381125 = 221.5681
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[[Comma list]]: {{monzo| -17 62 -35 }}
[[Comma list]]: {{monzo| -17 62 -35 }}


[[Mapping]]: [{{val| 1 11 19 }}, {{val| 0 -35 -62 }}]
{{Mapping|legend=1| 1 11 19 | 0 -35 -62 }}


: mapping generators: ~2, ~{{monzo| -6 23 -13 }}
: mapping generators: ~2, ~{{monzo| -6 23 -13 }}
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[[Comma list]]: {{monzo| 144 -22 -47 }}
[[Comma list]]: {{monzo| 144 -22 -47 }}


[[Mapping]]: [{{val| 1 45 -18 }}, {{val| 0 -47 22 }}]
{{Mapping|legend=1| 1 45 -18 | 0 -47 22 }}


: mapping generators: ~2, ~{{monzo| -45 7 15 }}
: mapping generators: ~2, ~{{monzo| -45 7 15 }}
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== Quectismic ==
== Quectismic ==
{{See also| Quectisma }}
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| 554 -351 1 }}
[[Comma list]]: {{monzo| 554 -351 1 }}


[[Mapping]]: [{{val| 1 0 -554 }}, {{val| 0 1 351 }}]
{{Mapping|legend=1| 1 0 -554 | 0 1 351 }}


: mapping generators: ~2, ~3
: mapping generators: ~2, ~3
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Comma list: 43046721/43025920, {{monzo| 82 -19 -3 -16 }}
Comma list: 43046721/43025920, {{monzo| 82 -19 -3 -16 }}


Mapping: [{{val| 1 0 -554 109 }}, {{val| 0 1 351 -67 }}]
Mapping: {{mapping| 1 0 -554 109 | 0 1 351 -67 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.955
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.955
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Comma list: {{monzo| 33 -26 12 -7 }}, {{monzo| -32 13 17 -10 }}
Comma list: {{monzo| 33 -26 12 -7 }}, {{monzo| -32 13 17 -10 }}


Mapping: [{{val| 1 0 -554 -945 }}, {{val| 0 1 351 598 }}]
Mapping: {{mapping| 1 0 -554 -945 | 0 1 351 598 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.9548
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.9548
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[[Comma list]]: {{monzo| 71 -99 37 }}
[[Comma list]]: {{monzo| 71 -99 37 }}


[[Mapping]]: [{{val| 1 28 73 }}, {{val| 0 -37 -99 }}]
{{Mapping|legend=1| 1 28 73 | 0 -37 -99 }}


: mapping generators: ~2, ~6561/4000
: mapping generators: ~2, ~6561/4000
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[[Comma list]]: {{monzo| -90 -15 49 }}
[[Comma list]]: {{monzo| -90 -15 49 }}


[[Mapping]]: [{{val| 1 43 15 }}, {{val| 0 -49 -15 }}]
{{Mapping|legend=1| 1 43 15 | 0 -49 -15 }}


: mapping generators: ~2, ~{{monzo| -23 -4 13 }}
: mapping generators: ~2, ~{{monzo| -23 -4 13 }}
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[[Comma list]]: {{monzo| 161 -84 -12 }}
[[Comma list]]: {{monzo| 161 -84 -12 }}


[[Mapping]]: [{{val| 12 0 161 }}, {{val| 0 1 -7 }}]
{{Mapping|legend=1| 12 0 161 | 0 1 -7 }}


: mapping generators: ~{{monzo| -67 35 5 }}, ~3
: mapping generators: ~{{monzo| -67 35 5 }}, ~3
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Comma list: 250047/250000, {{monzo| -55 30 2 1 }}
Comma list: 250047/250000, {{monzo| -55 30 2 1 }}


Mapping: [{{val| 12 0 161 338 }}, {{val| 0 1 -7 -16 }}]
Mapping: {{mapping| 12 0 161 338 | 0 1 -7 -16 }}


Optimal tuning (CTE): ~30375/28672 = 1\12, ~3/2 = 701.9499 (or ~32805/32768 = 1.9499)
Optimal tuning (CTE): ~30375/28672 = 1\12, ~3/2 = 701.9499 (or ~32805/32768 = 1.9499)
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Comma list: 9801/9800, 151263/151250, 184549376/184528125
Comma list: 9801/9800, 151263/151250, 184549376/184528125


Mapping: [{{val| 12 0 161 338 517 }}, {{val| 0 1 -7 -16 -25 }}]
Mapping: {{mapping| 12 0 161 338 517 | 0 1 -7 -16 -25 }}


Optimal tuning (CTE): ~30375/28672 = 1\12, ~3/2 = 701.9485 (or ~32805/32768 = 1.9485)
Optimal tuning (CTE): ~30375/28672 = 1\12, ~3/2 = 701.9485 (or ~32805/32768 = 1.9485)
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[[Comma list]]: {{monzo| -875 492 41 }}
[[Comma list]]: {{monzo| -875 492 41 }}


[[Mapping]]: [{{val| 41 0 875 }}, {{val| 0 1 -12 }}]
{{Mapping|legend=1| 41 0 875 | 0 1 -12 }}


: mapping generators: ~{{monzo| -64 36 3 }}, ~3
: mapping generators: ~{{monzo| -64 36 3 }}, ~3
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Comma list: 645700815/645657712, {{monzo| 51 -13 -1 -10 }}
Comma list: 645700815/645657712, {{monzo| 51 -13 -1 -10 }}


Mapping: [{{val| 1 12 -155 5 }}, {{val| 0 -19 287 -4 }}]
Mapping: {{mapping| 1 12 -155 5 | 0 -19 287 -4 }}


Optimal tuning (CTE): ~24576/16807 = 657.792
Optimal tuning (CTE): ~24576/16807 = 657.792
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Comma list: 78125000/78121827, {{monzo| 152 -112 -41 43 }}
Comma list: 78125000/78121827, {{monzo| 152 -112 -41 43 }}


Mapping: [{{val| 87 1 -106 -406 }}, {{val| 0 4 9 19 }}]
Mapping: {{mapping| 87 1 -106 -406 | 0 4 9 19 }}


: mapping generators: ~126/125, ~{{monzo| 37 -25 -12 11 }}
: mapping generators: ~126/125, ~{{monzo| 37 -25 -12 11 }}
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Comma list: 78125000/78121827, 199297406/199290375, {{monzo| 74 -17 -5 -4 -7 }}
Comma list: 78125000/78121827, 199297406/199290375, {{monzo| 74 -17 -5 -4 -7 }}


Mapping: [{{val| 87 1 -106 -406 1225 }}, {{val| 0 4 9 19 -27 }}]
Mapping: {{mapping| 87 1 -106 -406 1225 | 0 4 9 19 -27 }}


: mapping generators: ~126/125, ~{{monzo| 35 1 -6 -3 -4 }}
: mapping generators: ~126/125, ~{{monzo| 35 1 -6 -3 -4 }}
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Badness: 0.352
Badness: 0.352


== No-five nanismic (1277&2860) ==
== Nanic ==
{{See also| Nanisma }}
 
[[Subgroup]]: 2.3.7
[[Subgroup]]: 2.3.7


[[Comma list]]: {{monzo| 109 -67 -1 }}
[[Comma list]]: {{monzo| 109 -67 -1 }}


[[Sval]] [[mapping]]: [{{val| 1 2 -25 }}, {{val| 0 -1 67 }}]
{{Mapping|legend=1| 1 0 109 | 0 1 -67 }}
 
: sval mapping generators: ~2, ~3


[[Optimal tuning]] ([[POTE]]): ~3/2 = 701.9578
[[Optimal tuning]] ([[POTE]]): ~3/2 = 701.9578


{{Optimal ET sequence|legend=1| 53, 200, 253, 306, 665, 971, 1277, 4137, 5414, 6691, 7968 }}
{{Optimal ET sequence|legend=1| 53, 200, 253, 306, 665, 971, 1277, 6691, 7968, 9245, 10522, 11799, 13076, 40505, 53581, 66657, 79733 }}
 
[[Badness]]: 0.0138


= Rank-3 temperaments =
= Rank-3 temperaments =
Line 392: Line 400:
[[Comma list]]: {{monzo| 3 -24 3 10 }} = 282475249000/282429536481
[[Comma list]]: {{monzo| 3 -24 3 10 }} = 282475249000/282429536481


[[Mapping]]: [{{val| 1 0 9 -3 }}, {{val| 0 1 8 0 }}, {{val| 0 0 10 -3 }}]
{{Mapping|legend=1| 1 0 9 -3 | 0 1 8 0 | 0 0 10 -3 }}


{{Optimal ET sequence|legend=1| 171, 764, 935, 1106, 1277, 2118, 2289, 2460, 3224, 3395, 3566, 26068, 26239, 29634, 29805, 33200, 33371, 36937, 40503, 44069 }}
{{Optimal ET sequence|legend=1| 171, 764, 935, 1106, 1277, 2118, 2289, 2460, 3224, 3395, 3566, 26068, 26239, 29634, 29805, 33200, 33371, 36937, 40503, 44069 }}
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[[Comma list]]: {{monzo| 0 -11 -7 12 }} = 13841287201/13839609375
[[Comma list]]: {{monzo| 0 -11 -7 12 }} = 13841287201/13839609375


[[Mapping]]: [{{val| 1 0 0 0 }}, {{val| 0 1 7 5 }}, {{val| 0 0 12 7 }}]
{{Mapping|legend=1| 1 0 0 0 | 0 1 7 5 | 0 0 12 7 }}


{{Optimal ET sequence|legend=1| 171, 814, 985, 1034, 1205, 1376, 1547, 1718, 1848, 2019, 3224, 3395, 3566, 8980, 9151, 12717, 47131, 50697, 59848d, 63414d, 76131d }}
{{Optimal ET sequence|legend=1| 171, 814, 985, 1034, 1205, 1376, 1547, 1718, 1848, 2019, 3224, 3395, 3566, 8980, 9151, 12717, 47131, 50697, 59848d, 63414d, 76131d }}
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[[Comma list]]: {{monzo| -7 19 -16 5 }} = 19534128475869/19531250000000
[[Comma list]]: {{monzo| -7 19 -16 5 }} = 19534128475869/19531250000000


[[Mapping]]: [{{val| 1 0 3 11 }}, {{val| 0 1 4 9 }}, {{val| 0 0 5 16 }}]
{{Mapping|legend=1| 1 0 3 11 | 0 1 4 9 | 0 0 5 16 }}


{{Optimal ET sequence|legend=1| 171, 441, 612, 2696, 2795, 2867, 2966, 3137, 3308, 3407, 3479, 3578, 3749, 3920, 4091, 4190, 4361, 4532, 4703, 4802, 4973, 5144, 10117, 10558, 15531, 15702, 52079, 62637, 67781, 78339 }}
{{Optimal ET sequence|legend=1| 171, 441, 612, 2696, 2795, 2867, 2966, 3137, 3308, 3407, 3479, 3578, 3749, 3920, 4091, 4190, 4361, 4532, 4703, 4802, 4973, 5144, 10117, 10558, 15531, 15702, 52079, 62637, 67781, 78339 }}
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[[Comma list]]: {{monzo| -1 4 11 -11 }} = 3955078125/3954653486
[[Comma list]]: {{monzo| -1 4 11 -11 }} = 3955078125/3954653486


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


{{Optimal ET sequence|legend=1| 171, 863, 894, 1034, 1065, 1205, 1236, 1376, 1407, 1547, 1578, 1718, 2612, 2783, 2954, 3125, 12329, 15454, 18579, 21704, 40283, 61987 }}
{{Optimal ET sequence|legend=1| 171, 863, 894, 1034, 1065, 1205, 1236, 1376, 1407, 1547, 1578, 1718, 2612, 2783, 2954, 3125, 12329, 15454, 18579, 21704, 40283, 61987 }}
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[[Comma list]]: 645700815/645657712
[[Comma list]]: 645700815/645657712


[[Mapping]]: [{{val| 1 0 4 0 }}, {{val| 0 1 1 2 }}, {{val| 0 0 9 1 }}]
{{Mapping|legend=1| 1 0 4 0 | 0 1 1 2 | 0 0 9 1 }}


{{Optimal ET sequence|legend=1| 171, 742, 764, 935, 1106, 1277, 1677, 1848, 2019, 2783, 2954, 3125, 8098, 11223, 25571, 33840, 36794, 45063, 48188, 59411, 70634 }}
{{Optimal ET sequence|legend=1| 171, 742, 764, 935, 1106, 1277, 1677, 1848, 2019, 2783, 2954, 3125, 8098, 11223, 25571, 33840, 36794, 45063, 48188, 59411, 70634 }}
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[[Comma list]]: {{monzo| -3 2 -17 14 }} = 6104007655641/6103515625000
[[Comma list]]: {{monzo| -3 2 -17 14 }} = 6104007655641/6103515625000


[[Mapping]]: [{{val| 1 0 13 16 }}, {{val| 0 1 10 12 }}, {{val| 0 0 14 17 }}]
{{Mapping|legend=1| 1 0 13 16 | 0 1 10 12 | 0 0 14 17 }}


{{Optimal ET sequence|legend=1| 171, 894, 1065, 1236, 1407, 1578, 3053, 3224, 3395, 3566, 5144, 8368, 8539, 8710, 17249, 20815, 29525, 38064, 67589, 105653 }}
{{Optimal ET sequence|legend=1| 171, 894, 1065, 1236, 1407, 1578, 3053, 3224, 3395, 3566, 5144, 8368, 8539, 8710, 17249, 20815, 29525, 38064, 67589, 105653 }}
Line 460: Line 468:
[[Comma list]]: 78125000/78121827
[[Comma list]]: 78125000/78121827


[[Mapping]]: [{{val| 1 0 0 -5 }}, {{val| 0 2 0 -13 }}, {{val| 0 0 1 5 }}]
{{Mapping|legend=1| 1 0 0 -5 | 0 2 0 -13 | 0 0 1 5 }}


{{Optimal ET sequence|legend=1| 171, 441, 612, 1730, 1901, 2072, 2171, 2342, 2513, 2684, 2783, 2954, 3125, 6520, 6691, 9816, 16336, 16507, 23027, 26152, 32843, 42659, 58995, 68811, 101654 }}
{{Optimal ET sequence|legend=1| 171, 441, 612, 1730, 1901, 2072, 2171, 2342, 2513, 2684, 2783, 2954, 3125, 6520, 6691, 9816, 16336, 16507, 23027, 26152, 32843, 42659, 58995, 68811, 101654 }}
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[[Comma list]]: {{monzo| -55 30 2 1 }}
[[Comma list]]: {{monzo| -55 30 2 1 }}


[[Mapping]]: [{{val| 1 0 0 55 }}, {{val| 0 1 0 -30 }}, {{val| 0 0 1 -2 }}]
{{Mapping|legend=1| 1 0 0 55 | 0 1 0 -30 | 0 0 1 -2 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9516, ~5/4 = 386.3132
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9516, ~5/4 = 386.3132
Line 486: Line 494:
[[Comma list]]: {{monzo| 1 -15 -18 23 }}
[[Comma list]]: {{monzo| 1 -15 -18 23 }}


[[Mapping]]: [{{val| 1 0 9 7 }}, {{val| 0 1 3 3 }}, {{val| 0 0 23 18 }}]
{{Mapping|legend=1| 1 0 9 7 | 0 1 3 3 | 0 0 23 18 }}


{{Optimal ET sequence|legend=1| 171, 863, 1034, 1205, 1376, 1547, 1718, 3255, 3426, 3597, 4460, 4631, 4802, 4973, 5144, 6520, 6691, 11664, 18355, 48374, 48545, 66900, 145464, }} …
{{Optimal ET sequence|legend=1| 171, 863, 1034, 1205, 1376, 1547, 1718, 3255, 3426, 3597, 4460, 4631, 4802, 4973, 5144, 6520, 6691, 11664, 18355, 48374, 48545, 66900, 145464, }} …
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[[Comma list]]: {{monzo| 51 -13 -1 -10 }}
[[Comma list]]: {{monzo| 51 -13 -1 -10 }}


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


{{Optimal ET sequence|legend=1| 441, 966, 1277, 1407, 1718, 1848, 2684, 3125, 6691, 9816, 11664, 14789, 18355, 33144, 51499, 124478, 175977, }} …
{{Optimal ET sequence|legend=1| 441, 966, 1277, 1407, 1718, 1848, 2684, 3125, 6691, 9816, 11664, 14789, 18355, 33144, 51499, 124478, 175977, }} …
Line 508: Line 516:
[[Comma list]]: {{monzo| 110 -71 -11 10 }}
[[Comma list]]: {{monzo| 110 -71 -11 10 }}


[[Mapping]]: [{{val| 1 0 0 -11 }}, {{val| 0 1 -1 6 }}, {{val| 0 0 10 11 }}]
{{Mapping|legend=1| 1 0 0 -11 | 0 1 -1 6 | 0 0 10 11 }}


{{Optimal ET sequence|legend=1| 1848, 3737, 5585, 6079, 7927, 9816, 11664, 21480, 33144, 54624, 70025, 81689, 103169, }} …
{{Optimal ET sequence|legend=1| 1848, 3737, 5585, 6079, 7927, 9816, 11664, 21480, 33144, 54624, 70025, 81689, 103169, }} …
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