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== 11-limit ==
== 11-limit ==
{{see also| Chords of magic }}


Tempering 100/99 allows for a tritone substitution where the extended 5-limit tuning of a dominant seventh with a 9/5 above the root shares its tritone with an 8:10:11:12:16 chord rooted on the seventh of the original chord. (The tritone of the dominant seventh is (9/5)/(5/4) = 36/25. (16/11)/(36/25) = 100/99.)
Tempering 100/99 allows for a tritone substitution where the extended 5-limit tuning of a dominant seventh with a 9/5 above the root shares its tritone with an 8:10:11:12:16 chord rooted on the seventh of the original chord. (The tritone of the dominant seventh is (9/5)/(5/4) = 36/25. (16/11)/(36/25) = 100/99.)
Line 163: Line 162:
Comma list: 55/54, 99/98, 176/175
Comma list: 55/54, 99/98, 176/175


Mapping: [<1 0 2 -1 -1|, <0 5 1 12 14|]
Mapping: [{{val| 1 0 2 -1 -1 }}, {{val| 0 5 1 12 14 }}]


POTE generator: ~5/4 = 381.019
POTE generator: ~5/4 = 381.019
Line 177: Line 176:
Comma list: 55/54, 65/64, 91/90, 99/98
Comma list: 55/54, 65/64, 91/90, 99/98


Mapping: [<1 0 2 -1 -1 4|, <0 5 1 12 14 -1|]
Mapping: [{{val| 1 0 2 -1 -1 4 }}, {{val| 0 5 1 12 14 -1 }}]


POTE generator: ~5/4 = 380.520
POTE generator: ~5/4 = 380.520
Line 191: Line 190:
Comma list: 45/44, 56/55, 245/243
Comma list: 45/44, 56/55, 245/243


Mapping: [<1 0 2 -1 0|, <0 5 1 12 11|]
Mapping: [{{val| 1 0 2 -1 0 }}, {{val| 0 5 1 12 11 }}]


POTE generator: ~5/4 = 379.642
POTE generator: ~5/4 = 379.642
Line 205: Line 204:
Comma list: 121/120, 225/224, 245/243
Comma list: 121/120, 225/224, 245/243


Mapping: [<2 0 4 -2 5|, <0 5 1 12 3|]
Mapping: [{{val| 2 0 4 -2 5 }}, {{val| 0 5 1 12 3 }}]


POTE generator: ~5/4 = 380.233
POTE generator: ~5/4 = 380.233
Line 219: Line 218:
Comma list: 105/104, 121/120, 196/195, 245/243
Comma list: 105/104, 121/120, 196/195, 245/243


Mapping: [<2 0 4 -2 5 -4|, <0 5 1 12 3 18|]
Mapping: [{{val| 2 0 4 -2 5 -4 }}, {{val| 0 5 1 12 3 18 }}]


POTE generator: ~5/4 = 379.920
POTE generator: ~5/4 = 379.920
Line 233: Line 232:
Comma list: 100/99, 225/224, 245/243, 1352/1331
Comma list: 100/99, 225/224, 245/243, 1352/1331


Mapping: [<2 0 4 -2 12 15|, <0 5 1 12 -8 -12|]
Mapping: [{{val| 2 0 4 -2 12 15 }}, {{val| 0 5 1 12 -8 -12 }}]


POTE generator: ~5/4 = 380.508
POTE generator: ~5/4 = 380.508
Line 247: Line 246:
Comma list: 225/224, 245/243, 441/440
Comma list: 225/224, 245/243, 441/440


Mapping: [<1 0 2 -1 -7|, <0 5 1 12 33|]
Mapping: [{{val| 1 0 2 -1 -7 }}, {{val| 0 5 1 12 33 }}]


POTE generator: ~5/4 = 380.232
POTE generator: ~5/4 = 380.232
Line 261: Line 260:
Comma list: 105/104, 196/195, 245/243, 275/273
Comma list: 105/104, 196/195, 245/243, 275/273


Mapping: [<1 0 2 -1 -7 -2|, <0 5 1 12 33 18|]
Mapping: [{{val| 1 0 2 -1 -7 -2 }}, {{val| 0 5 1 12 33 18 }}]


POTE generator: ~5/4 = 380.189
POTE generator: ~5/4 = 380.189
Line 268: Line 267:


Badness: 0.0235
Badness: 0.0235
== Hocus ==
Subgroup: 2.3.5.7.11
Comma list: 225/224, 243/242, 245/242
Mapping: [{{val| 1 5 3 11 12 }}, {{val| 0 -10 -2 -24 -25 }}]
POTE generator: ~14/11 = 409.910
{{Val list|legend=1| 38d, 41, 120cd, 161cd, 202cd }}
Badness: 0.0385
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Comma list: 105/104, 196/195, 243/242, 245/242
Mapping: [{{val| 1 5 3 11 12 16 }}, {{val| 0 -10 -2 -24 -25 -36 }}]
POTE generator: ~14/11 = 410.004
{{Val list|legend=1| 41, 79d, 120cd }}
Badness: 0.0303


= Muggles =
= Muggles =
Line 276: Line 303:
[[Comma list]]: 126/125, 525/512
[[Comma list]]: 126/125, 525/512


[[Mapping]]: [<1 0 2 5|, <0 5 1 -7|]
[[Mapping]]: [{{val| 1 0 2 5 }}, {{val| 0 5 1 -7 }}]


{{Multival|legend=1| 5 1 -7 -10 -25 -19 }}
{{Multival|legend=1| 5 1 -7 -10 -25 -19 }}
Line 292: Line 319:
Comma list: 45/44, 126/125, 385/384
Comma list: 45/44, 126/125, 385/384


Mapping: [<1 0 2 5 0|, <0 5 1 -7 11|]
Mapping: [{{val| 1 0 2 5 0 }}, {{val| 0 5 1 -7 11 }}]


POTE generator: ~5/4 = 377.724
POTE generator: ~5/4 = 377.724
Line 306: Line 333:
Comma list: 45/44, 65/64, 78/77, 126/125
Comma list: 45/44, 65/64, 78/77, 126/125


Mapping: [<1 0 2 5 0 4|, <0 5 1 -7 11 -1|]
Mapping: [{{val| 1 0 2 5 0 4 }}, {{val| 0 5 1 -7 11 -1 }}]


POTE generator: ~5/4 = 377.724
POTE generator: ~5/4 = 377.724
Line 320: Line 347:
[[Comma list]]: 50/49, 3125/3072
[[Comma list]]: 50/49, 3125/3072


[[Mapping]]: [<2 0 4 5|, <0 5 1 1|]
[[Mapping]]: [{{val| 2 0 4 5 }}, {{val| 0 5 1 1 }}]


{{Multival|legend=1| 10 2 2 -20 -25 -1 }}
{{Multival|legend=1| 10 2 2 -20 -25 -1 }}
Line 336: Line 363:
Comma list: 50/49, 121/120, 176/175
Comma list: 50/49, 121/120, 176/175


Mapping: [<2 0 4 5 5|, <0 5 1 1 3|]
Mapping: [{{val| 2 0 4 5 5 }}, {{val| 0 5 1 1 3 }}]


POTE generator: ~5/4 = 380.530
POTE generator: ~5/4 = 380.530
Line 350: Line 377:
Comma list: 50/49, 65/64, 78/77, 121/120
Comma list: 50/49, 65/64, 78/77, 121/120


Mapping: [<2 0 4 5 5 8|, <0 5 1 1 3 -1|]
Mapping: [{{val| 2 0 4 5 5 8 }}, {{val| 0 5 1 1 3 -1 }}]


POTE generator: ~5/4 = 379.787
POTE generator: ~5/4 = 379.787
Line 367: Line 394:
Comma list: 50/49, 66/65, 105/104, 121/120
Comma list: 50/49, 66/65, 105/104, 121/120


Mapping: [<2 0 4 5 5 3|, <0 5 1 1 3 7|]
Mapping: [{{val| 2 0 4 5 5 3 }}, {{val| 0 5 1 1 3 7 }}]


POTE generator: ~5/4 = 379.837
POTE generator: ~5/4 = 379.837
Line 381: Line 408:
[[Comma list]]: 49/48, 3125/3072
[[Comma list]]: 49/48, 3125/3072


[[Mapping]]: [<1 0 2 2|, <0 10 2 5|]
[[Mapping]]: [{{val| 1 0 2 2 }}, {{val| 0 10 2 5 }}]


{{Multival|legend=1| 10 2 5 -20 -20 6 }}
{{Multival|legend=1| 10 2 5 -20 -20 6 }}
Line 397: Line 424:
Comma list: 49/48, 56/55, 125/121
Comma list: 49/48, 56/55, 125/121


Mapping: [<1 0 2 2 3|, <0 10 2 5 3|]
Mapping: [{{val| 1 0 2 2 3 }}, {{val| 0 10 2 5 3 }}]


POTE generator: ~11/10 = 190.285
POTE generator: ~11/10 = 190.285
Line 411: Line 438:
Comma list: 49/48, 56/55, 78/77, 125/121
Comma list: 49/48, 56/55, 78/77, 125/121


Mapping: [<1 0 2 2 3 4|, <0 10 2 5 3 -2|]
Mapping: [{{val| 1 0 2 2 3 4 }}, {{val| 0 10 2 5 3 -2 }}]


POTE generator: ~11/10 = 189.928
POTE generator: ~11/10 = 189.928
Line 425: Line 452:
Comma list: 49/48, 56/55, 91/90, 125/121
Comma list: 49/48, 56/55, 91/90, 125/121


Mapping: [<1 0 2 2 3 1|, <0 10 2 5 3 17|]
Mapping: [{{val| 1 0 2 2 3 1 }}, {{val| 0 10 2 5 3 17 }}]


POTE generator: ~11/10 = 190.360
POTE generator: ~11/10 = 190.360
Line 439: Line 466:
[[Comma list]]: 3125/3072, 4000/3969
[[Comma list]]: 3125/3072, 4000/3969


[[Mapping]]: [<1 5 3 -3|, <0 -10 -2 17|]
[[Mapping]]: [{{val| 1 5 3 -3 }}, {{val| 0 -10 -2 17 }}]


{{Multival|legend=1| 10 2 -17 -20 -55 -45 }}
{{Multival|legend=1| 10 2 -17 -20 -55 -45 }}
Line 448: Line 475:


[[Badness]]: 0.1071
[[Badness]]: 0.1071
== Hocus ==
Subgroup: 2.3.5.7.11
Comma list: 225/224, 243/242, 245/242
Mapping: [<1 5 3 11 12|, <0 -10 -2 -24 -25|]
POTE generator: ~14/11 = 409.910
{{Val list|legend=1| 38d, 41, 120cd, 161cd, 202cd }}
Badness: 0.0385
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Comma list: 105/104, 196/195, 243/242, 245/242
Mapping: [<1 5 3 11 12 16|, <0 -10 -2 -24 -25 -36|]
POTE generator: ~14/11 = 410.004
{{Val list|legend=1| 41, 79d, 120cd }}
Badness: 0.0303


= Trismegistus =
= Trismegistus =
Line 481: Line 480:
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


[[Comma list]]: 3125/3072, 1029/1024
[[Comma list]]: 1029/1024, 3125/3072


[[Mapping]]: [<1 10 4 0|, <0 -15 -3 5|]
[[Mapping]]: [{{val| 1 10 4 0 }}, {{val| 0 -15 -3 5 }}]


{{Multival|legend=1| 15 3 -5 -30 -50 -20 }}
{{Multival|legend=1| 15 3 -5 -30 -50 -20 }}
Line 501: Line 500:
POTE generator: ~22/15 = 673.340
POTE generator: ~22/15 = 673.340


Mapping: [<1 10 4 0 13|, <0 -15 -3 5 -17|]
Mapping: [{{val| 1 10 4 0 13 }}, {{val| 0 -15 -3 5 -17 }}]


{{Val list|legend=1| 16, 25e, 41, 98c, 139c, 180c }}
{{Val list|legend=1| 16, 25e, 41, 98c, 139c, 180c }}
Line 513: Line 512:
Comma list: 105/104, 144/143, 275/273, 625/616
Comma list: 105/104, 144/143, 275/273, 625/616


Mapping: [<1 10 4 0 13 11|, <0 -15 -3 5 -17 -13|]
Mapping: [{{val| 1 10 4 0 13 11 }}, {{val| 0 -15 -3 5 -17 -13 }}]


POTE generator: ~22/15 = 673.359
POTE generator: ~22/15 = 673.359
Line 527: Line 526:
[[Comma list]]: 2401/2400, 3125/3072
[[Comma list]]: 2401/2400, 3125/3072


[[Mapping]]: [<1 5 3 4|, <0 -20 -4 -7|]
[[Mapping]]: [{{val| 1 5 3 4 }}, {{val| 0 -20 -4 -7 }}]


{{Multival|legend=1| 20 4 7 -40 -45 5 }}
{{Multival|legend=1| 20 4 7 -40 -45 5 }}
Line 543: Line 542:
Comma list: 245/242, 385/384, 625/616
Comma list: 245/242, 385/384, 625/616


Mapping: [<1 5 3 4 5|, <0 -20 -4 -7 -9|]
Mapping: [{{val| 1 5 3 4 5 }}, {{val| 0 -20 -4 -7 -9 }}]


POTE generator: ~28/25 = 204.956
POTE generator: ~28/25 = 204.956
Line 557: Line 556:
Comma list: 105/104, 144/143, 245/242, 625/616
Comma list: 105/104, 144/143, 245/242, 625/616


Mapping: [<1 5 3 4 5 9|, <0 -20 -4 -7 -9 -31|]
Mapping: [{{val| 1 5 3 4 5 9 }}, {{val| 0 -20 -4 -7 -9 -31 }}]


POTE generator: ~28/25 = 205.028
POTE generator: ~28/25 = 205.028
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