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 }}
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[[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, }} …

Revision as of 10:24, 5 June 2023

Below are listed some very high accuracy temperaments (TE error < 0.005 cents/octave).

Rank-2 temperaments

Kwazy

Kwazy (118 & 612) tempers out the kwazy comma, [-53 10 16⟩. 7-limit extensions of the kwazy temperament include bisupermajor and gwazy.

Subgroup: 2.3.5

Comma list: [-53 10 16⟩ = 9010162353515625 / 9007199254740992

Mapping: [⟨2 1 6], ⟨0 8 -5]]

mapping generators: ~94921875/67108864, ~1125/1024

Optimal tuning (CTE): ~94921875/67108864 = 1\2, ~1125/1024 = 162.7433

Optimal ET sequence: 22, 74, 96, 118, 376, 494, 612, 1342, 3296, 4638, 7934

Badness: 0.014091

Astro

Astro temperament tempers out the astro comma, [91 -12 -31⟩. 7-limit extensions of the astro temperament include kastro.

Subgroup: 2.3.5

Comma list: [91 -12 -31⟩

Mapping: [⟨1 5 1], ⟨0 -31 12]]

mapping generators: ~2, ~[-38 5 13⟩

Optimal tuning (CTE): ~2 = 1\1, ~[-38 5 13⟩ = 132.1949

Optimal ET sequence: 9, 109, 118, 699, 817, 935, 1053, 1171, 13934, 15105, 16276, 17447, 18618, 19789, 20960, 22131, 23302, 24473, 25644, 26815, 27986c

Badness: 0.062778

Gaster

Main article: Gaster temperament
See also: Mirkwai clan § Gaster

Subgroup: 2.3.5

Comma list: [-70 72 -19⟩

Mapping: [⟨1 11 38], ⟨0 -19 -72]]

mapping generators: ~2, ~[19 -19 5⟩

Optimal tuning (CTE): ~2 = 1\1, ~[19 -19 5⟩ = 594.6345

Optimal ET sequence: 111, 224, 335, 559, 1342, 1901, 3243, 8387, 11630

Badness: 0.088530

Whoosh

Whoosh (152 & 289) tempers out the whoosh comma, [37 25 -33⟩. 7-limit extensions of the whoosh temperament include whoops.

Subgroup: 2.3.5

Comma list: [37 25 -33⟩

Mapping: [⟨1 17 14], ⟨0 -33 -25]]

mapping generators: ~2, ~864/625

Optimal tuning (CTE): ~2 = 1\1, ~864/625 = 560.546826

Optimal ET sequence: 15, 122, 137, 152, 289, 441, 730, 1171, 1901, 3072, 4243, 7315, 25017, 32332, 39647c

Badness: 0.025653

Monzismic

Monzismic (53 & 612) tempers out the monzisma, [54 -37 2⟩. For its 7-limit extension, see Ragismic microtemperaments #Monzismic.

Subgroup: 2.3.5

Comma list: [54 -37 2⟩ = 450359962737049600 / 450283905890997363

Mapping: [⟨1 0 -27], ⟨0 2 37]]

mapping generators: ~2, ~[27 -18 1⟩

Optimal tuning (CTE): ~2 = 1\1, ~[27 -18 1⟩ = 950.9814

Optimal ET sequence: 53, 347, 400, 453, 506, 559, 612, 1171, 1783, 4737, 6520

Badness: 0.010391

Egads

Subgroup: 2.3.5

Comma list: [-36 -52 51⟩

Mapping: [⟨1 15 16], ⟨0 -51 -52]]

mapping generators: ~2, ~6/5

Optimal tuning (CTE): ~2 = 1\1, ~6/5 = 315.6479

Optimal ET sequence: 19, 365, 384, 403, 422, 441, 901, 1342, 1783, 3125, 4908, 45955, 50863, 55771, 60679, 65587, 70495, 75403, 80311, 85219, 90127

Badness: 0.041819

Hemiegads

Subgroup: 2.3.5.7

Comma list: 78125000/78121827, [-48 0 11 8⟩

Mapping: [⟨1 15 16 -16], ⟨0 -102 -104 143]]

Optimal tuning (CTE): ~2 = 1\1, ~36756909/33554432 = 157.8240

Optimal ET sequence: 38, 403, 441, 1802, 2243, 2684, 3125, 6691, 9816, 16507, 75844, 125365c

Badness: 0.0124

Fortune

Fortune tempers out the fortune comma, [-107 47 14⟩. It can be described as 612 & 2513 temperament, which tempers out 78125000/78121827 (euzenius comma) in the 7-limit; 151263/151250, 21437500/21434787, and 369140625/369098752 in the 11-limit.

Subgroup: 2.3.5

Comma list: [-107 47 14⟩

Mapping: [⟨1 13 -36], ⟨0 -14 47]]

mapping generators: ~2, 7381125/4194304

Optimal tuning (CTE): ~2 = 1\1, ~8388608/7381125 = 221.5679

Optimal ET sequence: 65, 352, 417, 482, 547, 612, 1901, 2513, 5638, 8151, 18815, 45781, 64596, 110377, 174973bc

Badness: 0.027876

7-limit

Subgroup: 2.3.5.7

Comma list: 78125000/78121827, [-52 17 12 -1⟩

Mapping: [⟨1 13 -36 -263], ⟨0 -14 47 326]]

Optimal tuning (CTE): ~2 = 1\1, ~8388608/7381125 = 221.5680

Optimal ET sequence: 612, 1901, 2513, 3125, 6862, 9987, 13112, 36211, 49323

Badness: 0.023704

11-limit

Subgroup: 2.3.5.7.11

Comma list: 151263/151250, 21437500/21434787, 369140625/369098752

Mapping: [⟨1 13 -36 -263 -573], ⟨0 -14 47 326 707]]

Optimal tuning (CTE): ~2 = 1\1, ~8388608/7381125 = 221.5681

Optimal ET sequence: 612, 2513, 3125, 3737, 8086, 19909d

Badness: 0.036641

Senior

Senior (171 & 1171) tempers out the senior comma, [-17 62 -35⟩. 7-limit extensions of the senior temperament include seniority.

Subgroup: 2.3.5

Comma list: [-17 62 -35⟩

Mapping: [⟨1 11 19], ⟨0 -35 -62]]

mapping generators: ~2, ~[-6 23 -13⟩

Optimal tuning (CTE): ~2 = 1\1, ~[-6 23 -13⟩ = 322.8013

Optimal ET sequence: 171, 658, 829, 1000, 1171, 2513, 6197, 39695, 45892, 52089, 58286, 64483, 70680

Badness: 0.021220

Gross

Gross (118 & 1783) tempers out the gross comma, [144 -22 -47⟩, by which a stack of 22 3rd harmonics and 47 5th harmonics falls short of 144 (gross) octaves.

Subgroup: 2.3.5

Comma list: [144 -22 -47⟩

Mapping: [⟨1 45 -18], ⟨0 -47 22]]

mapping generators: ~2, ~[-45 7 15⟩

Optimal tuning (CTE): ~2 = 1\1, ~[46 -7 -15⟩ = 91.5310

Optimal ET sequence: 13, 105, 118, 957, 1075, 1193, 1311, 1429, 1547, 1665, 1783, 3684, 5467, 36486, 41953, 47420, 52887, 63821, 69288, 74755, 144043c, 218798c

Badness: 0.046315

Quectismic

See also: Quectisma

Subgroup: 2.3.5

Comma list: [554 -351 1⟩

Mapping: [⟨1 0 -554], ⟨0 1 351]]

mapping generators: ~2, ~3

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.9553

Optimal ET sequence: 306, 359c, 665, 2301, 2966, 3631, 4296, 46591, 50887, 55183, 59479, 63775, 68071, 72367, 76663, 80959, 85255, 89551

Badness: 0.372

Nanoquectismic

Subgroup: 2.3.5.7

Comma list: 43046721/43025920, [82 -19 -3 -16⟩

Mapping: [⟨1 0 -554 109], ⟨0 1 351 -67]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.955

Optimal ET sequence: 306, 665, 1636, 2301, 2966

Badness: 0.604

Conanoquectismic

Subgroup: 2.3.5.7

Comma list: [33 -26 12 -7⟩, [-32 13 17 -10⟩

Mapping: [⟨1 0 -554 -945], ⟨0 1 351 598]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.9548

Optimal ET sequence: 306d, 359cd, 665

Badness: 2.17

Raider

See also: Syntonic-chromatic equivalence continuum

Subgroup: 2.3.5

Comma list: [71 -99 37⟩

Mapping: [⟨1 28 73], ⟨0 -37 -99]]

mapping generators: ~2, ~6561/4000

Optimal tuning (CTE): ~2 = 1\1, ~8000/6561 = 343.2961

Optimal ET sequence: 388, 783, 1171, 1954, 3125, 4296, 5467, 7421, 9763, 14059, 18355, 22651, 26947, 31243, 35539, 66782, 102321, 137860

Badness: 0.012867

Pirate

Subgroup: 2.3.5

Comma list: [-90 -15 49⟩

Mapping: [⟨1 43 15], ⟨0 -49 -15]]

mapping generators: ~2, ~[-23 -4 13⟩

Optimal tuning (CTE): ~2 = 1\1, ~[24 4 -13⟩ = 185.7542

Optimal ET sequence: 84, 239, 323, 730, 1783, 2513, 4296, 15401, 19697, 23993, 52282, 76275, 128557, 204832 …

Badness: 0.005656

Atomic

See also: Kirnberger's atom

Atomic extensions discussed elsewhere include minutes and hafnium.

Subgroup: 2.3.5

Comma list: [161 -84 -12⟩

Mapping: [⟨12 0 161], ⟨0 1 -7]]

mapping generators: ~[-67 35 5⟩, ~3

Optimal tuning (CTE): ~[-67 35 5⟩ = 1\12, ~3/2 = 701.9552 (or ~32805/32768 = 1.9552)

Optimal ET sequence: 12, …, 576, 588, 600, 612, 2460, 3072, 3684, 4296, 12276, 16572, 20868, 25164, 46032

Badness: 0.003803

7-limit

Subgroup: 2.3.5.7

Comma list: 250047/250000, [-55 30 2 1⟩

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 ET sequence: 12, …, 600, 612, 1236, 1848, 4308, 10464, 14772, 25236c, 40008ccd

Badness: 0.045818

11-limit

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 151263/151250, 184549376/184528125

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 ET sequence: 12, …, 600e, 612, 1236, 1848

Badness: 0.016029

Niobium

Niobium can be viewed as the 41st-octave counterpart of atomic.

Subgroup: 2.3.5

Comma list: [-875 492 41⟩

Mapping: [⟨41 0 875], ⟨0 1 -12]]

mapping generators: ~[-64 36 3⟩, ~3

Optimal tuning (CTE): ~[-64 36 3⟩ = 1\41, ~3/2 = 701.9536

Optimal ET sequence: 41, …, 2378, 2419, 2460, 4961, 7421, 17302, 24723, 66748, 91471, 116194

Badness: 5.57

Revopent

Described as the 1848 & 3125 temperament, revopent is named because the year 1848 is known for revolutions while 3125 is 5 to the 5.

Subgroup: 2.3.5.7

Comma list: 645700815/645657712, [51 -13 -1 -10⟩

Mapping: [⟨1 12 -155 5], ⟨0 -19 287 -4]]

Optimal tuning (CTE): ~24576/16807 = 657.792

Optimal ET sequence: 135, 436c, 571, 1277, 1848, 3125, 8098, 11223, 25571, 36794, 121605d

Badness: 0.263

Francium

Francium tempers out the euzenius comma and the 87-126/125 comma in the 7-limit, hereby dividing the octave into 87 parts.

Subgroup: 2.3.5.7

Comma list: 78125000/78121827, [152 -112 -41 43⟩

Mapping: [⟨87 1 -106 -406], ⟨0 4 9 19]]

mapping generators: ~126/125, ~[37 -25 -12 11⟩

Optimal tuning (CTE): ~126/125 = 1\87, ~[37 -25 -12 11⟩ = 472.043

Optimal ET sequence: 783, …, 8178, 8961, 9744, 10527, 20271, 30798

Badness: 0.762

11-limit

Subgroup: 2.3.5.7.11

Comma list: 78125000/78121827, 199297406/199290375, [74 -17 -5 -4 -7⟩

Mapping: [⟨87 1 -106 -406 1225], ⟨0 4 9 19 -27]]

mapping generators: ~126/125, ~[35 1 -6 -3 -4⟩

Optimal tuning (CTE): ~126/125 = 1\87, ~[35 1 -6 -3 -4⟩ = 472.043

Optimal ET sequence: 783, 8178, 8961, 9744, 28449de, 38193bcde, 47937bcdee

Badness: 0.352

Nanic

See also: Nanisma

Subgroup: 2.3.7

Comma list: [109 -67 -1⟩

Mapping: [⟨1 0 109], ⟨0 1 -67]]

sval mapping generators: ~2, ~3

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

Optimal ET sequence: 53, 200, 253, 306, 665, 971, 1277, 6691, 7968, 9245, 10522, 11799, 13076, 40505, 53581, 66657, 79733

Badness: 0.0138

Rank-3 temperaments

Sasaquinbizo-atriyo (171 & 1106 & 3566)

Subgroup: 2.3.5.7

Comma list: [3 -24 3 10⟩ = 282475249000/282429536481

Mapping: [⟨1 0 9 -3], ⟨0 1 8 0], ⟨0 0 10 -3]]

Optimal ET sequence: 171, 764, 935, 1106, 1277, 2118, 2289, 2460, 3224, 3395, 3566, 26068, 26239, 29634, 29805, 33200, 33371, 36937, 40503, 44069

Badness: 0.281 × 10-3

Saquadtrizo-asepgu (171 & 2019 & 3566)

Subgroup: 2.3.5.7

Comma list: [0 -11 -7 12⟩ = 13841287201/13839609375

Mapping: [⟨1 0 0 0], ⟨0 1 7 5], ⟨0 0 12 7]]

Optimal ET sequence: 171, 814, 985, 1034, 1205, 1376, 1547, 1718, 1848, 2019, 3224, 3395, 3566, 8980, 9151, 12717, 47131, 50697, 59848d, 63414d, 76131d

Badness: 0.135 × 10-3

Laquinzo-aquadquadgu (171 & 441 & 4973)

Subgroup: 2.3.5.7

Comma list: [-7 19 -16 5⟩ = 19534128475869/19531250000000

Mapping: [⟨1 0 3 11], ⟨0 1 4 9], ⟨0 0 5 16]]

Optimal ET sequence: 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

Badness: 0.297 × 10-3

Laleruyo (171 & 1547 & 3125)

Subgroup: 2.3.5.7

Comma list: [-1 4 11 -11⟩ = 3955078125/3954653486

Mapping: [⟨1 3 0 1], ⟨0 11 0 4], ⟨0 0 1 1]]

Optimal ET sequence: 171, 863, 894, 1034, 1065, 1205, 1236, 1376, 1407, 1547, 1578, 1718, 2612, 2783, 2954, 3125, 12329, 15454, 18579, 21704, 40283, 61987

Badness: 0.114 × 10-3

Starscape or Latritriru-ayo

Subgroup: 2.3.5.7

Comma list: 645700815/645657712

Mapping: [⟨1 0 4 0], ⟨0 1 1 2], ⟨0 0 9 1]]

Optimal ET sequence: 171, 742, 764, 935, 1106, 1277, 1677, 1848, 2019, 2783, 2954, 3125, 8098, 11223, 25571, 33840, 36794, 45063, 48188, 59411, 70634

Badness: 0.545 × 10-3

Sepbizo-asegu (171 & 3566 & 4973)

Subgroup: 2.3.5.7

Comma list: [-3 2 -17 14⟩ = 6104007655641/6103515625000

Mapping: [⟨1 0 13 16], ⟨0 1 10 12], ⟨0 0 14 17]]

Optimal ET sequence: 171, 894, 1065, 1236, 1407, 1578, 3053, 3224, 3395, 3566, 5144, 8368, 8539, 8710, 17249, 20815, 29525, 38064, 67589, 105653

Badness: 0.223 × 10-3

Euzenius or Sabiruquinyo

See also: Euzenius comma

Subgroup: 2.3.5.7

Comma list: 78125000/78121827

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

Optimal ET sequence: 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

Badness: 0.205 × 10-3

Nommismic or Quinla-zoyoyo

See also: Nommisma

Subgroup: 2.3.5.7

Comma list: [-55 30 2 1⟩

Mapping: [⟨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 ET sequence: 612, 1236, 1277, 1848, 2513, 3125, 6862, 8098, 8710, 11835, 14960, 26795, 195663b, …

Badness: 0.366 × 10-3

Satwethezo-atritribigu (171 & 1547 & 4973)

Subgroup: 2.3.5.7

Comma list: [1 -15 -18 23⟩

Mapping: [⟨1 0 9 7], ⟨0 1 3 3], ⟨0 0 23 18]]

Optimal ET sequence: 171, 863, 1034, 1205, 1376, 1547, 1718, 3255, 3426, 3597, 4460, 4631, 4802, 4973, 5144, 6520, 6691, 11664, 18355, 48374, 48545, 66900, 145464, …

Badness: 0.114 × 10-3

Trisa-quinbiru-agu (441 & 1848 & 3125)

Subgroup: 2.3.5.7

Comma list: [51 -13 -1 -10⟩

Mapping: [⟨1 0 1 5], ⟨0 1 7 -2], ⟨0 0 10 -1]]

Optimal ET sequence: 441, 966, 1277, 1407, 1718, 1848, 2684, 3125, 6691, 9816, 11664, 14789, 18355, 33144, 51499, 124478, 175977, …

Badness: 0.263 × 10-3

Quinbisa-quinbizo-alegu (1848 & 11664 & 21480)

Subgroup: 2.3.5.7

Comma list: [110 -71 -11 10⟩

Mapping: [⟨1 0 0 -11], ⟨0 1 -1 6], ⟨0 0 10 11]]

Optimal ET sequence: 1848, 3737, 5585, 6079, 7927, 9816, 11664, 21480, 33144, 54624, 70025, 81689, 103169, …

Badness: 0.122 × 10-3

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