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== Didacus ==
== Didacus ==
Subgroup: 2.5.7
[[Subgroup]]: 2.5.7


[[Comma list]]: 3136/3125
[[Comma list]]: 3136/3125


[[Sval]] [[mapping]]: [{{val| 1 0 -3 }}, {{val| 0 2 5 }}]
{{Mapping|legend=2| 1 0 -3 | 0 2 5 }}


Sval mapping generators: ~2, ~56/25
: sval mapping generators: ~2, ~56/25


Gencom mapping: [{{val| 1 0 0 -3 }}, {{val| 0 0 2 5 }}]
{{Mapping|legend=3| 1 0 0 -3 | 0 0 2 5 }}


[[Gencom]]: [2 56/25; 3136/3125]
[[Gencom]]: [2 56/25; 3136/3125]


[[POTE generator]]: ~28/25 = 193.772
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.772


{{Optimal ET sequence|legend=1| 6, 19, 25, 31, 99, 130, 161, 353, 514c, 867c }}
{{Optimal ET sequence|legend=1| 6, 19, 25, 31, 99, 130, 161, 353, 514c, 867c }}


=== Rectified Hebrew ===
=== Rectified hebrew ===
{{Main| Rectified Hebrew }}
{{Main| Rectified hebrew }}


Rectified Hebrew (37 & 56) is derived from the [https://individual.utoronto.ca/kalendis/hebrew/rect.htm#353 calendar by the same name]. It is leap year pattern takes a stack of 18 Metonic cycle diatonic major scales and truncates the 19th one down to its generator, 11. It adds harmonic 13 through tempering out 4394/4375 and spliting the generator of didacus in three.
Rectified hebrew (37 & 56) is derived from the [https://individual.utoronto.ca/kalendis/hebrew/rect.htm#353 calendar by the same name]. It is leap year pattern takes a stack of 18 Metonic cycle diatonic major scales and truncates the 19th one down to its generator, 11. It adds harmonic 13 through tempering out [[4394/4375]] and spliting the generator of didacus in three.


Subgroup: 2.5.7.13
Subgroup: 2.5.7.13
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Comma list: 3136/3125, 4394/4375
Comma list: 3136/3125, 4394/4375


Sval mapping: [{{val| 1 2 2 3 }}, {{val| 0 6 15 13 }}]
Sval mapping: {{mapping| 1 2 2 3 | 0 6 15 13 }}


Sval mapping generators: ~2, ~26/25
: sval mapping generators: ~2, ~26/25


POTE generator: ~26/25 = 64.6086
Optimal tuning (POTE): ~2 = 1\1, ~26/25 = 64.6086


{{Optimal ET sequence|legend=1| 18, 19, 37, 93, 130 }}
{{Optimal ET sequence|legend=1| 18, 19, 37, 93, 130 }}
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'''Hemiwürschmidt''' (sometimes spelled '''hemiwuerschmidt''') tempers out [[2401/2400]], [[3136/3125]], and [[6144/6125]]. [[68edo]], [[99edo]] and [[130edo]] can all be used as tunings, but 130 is not only the most accurate, it shows how hemiwürschmidt extends to a higher limit temperament, {{multival| 16 2 5 40 -39 -49 -48 28 … }}.
'''Hemiwürschmidt''' (sometimes spelled '''hemiwuerschmidt''') tempers out [[2401/2400]], [[3136/3125]], and [[6144/6125]]. [[68edo]], [[99edo]] and [[130edo]] can all be used as tunings, but 130 is not only the most accurate, it shows how hemiwürschmidt extends to a higher limit temperament, {{multival| 16 2 5 40 -39 -49 -48 28 … }}.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 3136/3125
[[Comma list]]: 2401/2400, 3136/3125


[[Mapping]]: [{{val| 1 15 4 7 }}, {{val| 0 -16 -2 -5 }}]
{{Mapping|legend=1| 1 15 4 7 | 0 -16 -2 -5 }}


{{Multival|legend=1| 16 2 5 -34 -37 6 }}
{{Multival|legend=1| 16 2 5 -34 -37 6 }}


[[POTE generator]]: ~28/25 = 193.898
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.898


{{Optimal ET sequence|legend=1| 31, 68, 99, 229, 328, 557c, 885cc }}
{{Optimal ET sequence|legend=1| 31, 68, 99, 229, 328, 557c, 885cc }}
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Comma list: 243/242, 441/440, 3136/3125
Comma list: 243/242, 441/440, 3136/3125


Mapping: [{{val| 1 15 4 7 37 }}, {{val| 0 -16 -2 -5 -40 }}]
Mapping: {{mapping| 1 15 4 7 37 | 0 -16 -2 -5 -40 }}


POTE generator: ~28/25 = 193.840
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.840


{{Optimal ET sequence|legend=1| 31, 99e, 130, 650ce, 811ce }}
{{Optimal ET sequence|legend=1| 31, 99e, 130, 650ce, 811ce }}
Line 97: Line 97:
Comma list: 243/242, 351/350, 441/440, 3584/3575
Comma list: 243/242, 351/350, 441/440, 3584/3575


Mapping: [{{val| 1 15 4 7 37 -29 }}, {{val| 0 -16 -2 -5 -40 39 }}]
Mapping: {{mapping| 1 15 4 7 37 -29 | 0 -16 -2 -5 -40 39 }}


POTE generator: ~28/25 = 193.829
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.829


{{Optimal ET sequence|legend=1| 31, 99e, 130, 291, 421e, 551ce }}
{{Optimal ET sequence|legend=1| 31, 99e, 130, 291, 421e, 551ce }}
Line 110: Line 110:
Comma list: 121/120, 176/175, 196/195, 275/273
Comma list: 121/120, 176/175, 196/195, 275/273


Mapping: [{{val| 1 15 4 7 37 -3 }}, {{val| 0 -16 -2 -5 -40 8 }}]
Mapping: {{mapping| 1 15 4 7 37 -3 | 0 -16 -2 -5 -40 8 }}


POTE generator: ~28/25 = 193.918
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.918


{{Optimal ET sequence|legend=1| 31, 68e, 99ef }}
{{Optimal ET sequence|legend=1| 31, 68e, 99ef }}
Line 123: Line 123:
Comma list: 121/120, 176/175, 1375/1372
Comma list: 121/120, 176/175, 1375/1372


Mapping: [{{val| 1 15 4 7 11 }}, {{val| 0 -16 -2 -5 -9 }}]
Mapping: {{mapping| 1 15 4 7 11 | 0 -16 -2 -5 -9 }}


POTE generator: ~28/25 = 193.884
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.884


{{Optimal ET sequence|legend=1| 31, 68, 99, 130e, 229e }}
{{Optimal ET sequence|legend=1| 31, 68, 99, 130e, 229e }}
Line 136: Line 136:
Comma list: 121/120, 176/175, 196/195, 275/273
Comma list: 121/120, 176/175, 196/195, 275/273


Mapping: [{{val| 1 15 4 7 11 -3 }}, {{val| 0 -16 -2 -5 -9 8 }}]
Mapping: {{mapping| 1 15 4 7 11 -3 | 0 -16 -2 -5 -9 8 }}


POTE generator: ~28/25 = 194.004
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 194.004


{{Optimal ET sequence|legend=1| 31, 68, 99f, 167ef }}
{{Optimal ET sequence|legend=1| 31, 68, 99f, 167ef }}
Line 149: Line 149:
Comma list: 66/65, 105/104, 121/120, 1375/1372
Comma list: 66/65, 105/104, 121/120, 1375/1372


Mapping: [{{val| 1 15 4 7 11 23 }}, {{val| 0 -16 -2 -5 -9 -23 }}]
Mapping: {{mapping| 1 15 4 7 11 23 | 0 -16 -2 -5 -9 -23 }}


POTE generator: ~28/25 = 193.698
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.698


{{Optimal ET sequence|legend=1| 6f, 31 }}
{{Optimal ET sequence|legend=1| 6f, 31 }}
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Comma list: 2401/2400, 3025/3024, 3136/3125
Comma list: 2401/2400, 3025/3024, 3136/3125


Mapping: [{{val| 1 15 4 7 24 }}, {{val| 0 -32 -4 -10 -49 }}]
Mapping: {{mapping| 1 15 4 7 24 | 0 -32 -4 -10 -49 }}


Mapping generators: ~2, ~147/110
: mapping generators: ~2, ~147/110


POTE generator: ~147/110 = 503.0404
Optimal tuning (POTE): ~2 = 1\1, ~147/110 = 503.0404


{{Optimal ET sequence|legend=1| 31, 105be, 136e, 167, 198, 427c }}
{{Optimal ET sequence|legend=1| 31, 105be, 136e, 167, 198, 427c }}
Line 179: Line 179:
Comma list: 2401/2400, 3136/3125, 9801/9800
Comma list: 2401/2400, 3136/3125, 9801/9800


Mapping: [{{val| 2 14 6 9 -10 }}, {{val| 0 -16 -2 -5 25 }}]
Mapping: {{mapping| 2 14 6 9 -10 | 0 -16 -2 -5 25 }}


Mapping generators: ~99/70, ~495/392
: mapping generators: ~99/70, ~495/392


POTE generator: ~28/25 = 193.9021
Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9021


{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328 }}
{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328 }}
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Comma list: 676/675, 1001/1000, 1716/1715, 3136/3125
Comma list: 676/675, 1001/1000, 1716/1715, 3136/3125


Mapping: [{{val| 2 14 6 9 -10 25 }}, {{val| 0 -16 -2 -5 25 -26 }}]
Mapping: {{mapping| 2 14 6 9 -10 25 | 0 -16 -2 -5 25 -26 }}


POTE generator: ~28/25 = 193.9035
Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9035


{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328 }}
{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328 }}
Line 207: Line 207:
Comma list: 289/288, 442/441, 561/560, 676/675, 1632/1625
Comma list: 289/288, 442/441, 561/560, 676/675, 1632/1625


Mapping: [{{val| 2 14 6 9 -10 25 19 }}, {{val| 0 -16 -2 -5 25 -26 -16 }}]
Mapping: {{mapping| 2 14 6 9 -10 25 19 | 0 -16 -2 -5 25 -26 -16 }}


POTE generator: ~28/25 = 193.9112
Optimal tuning (POTE): ~17/12 = 1\2, ~28/25 = 193.9112


{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328g, 526cfgg }}
{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328g, 526cfgg }}
Line 220: Line 220:
Comma list: 289/288, 442/441, 456/455, 476/475, 561/560, 627/625
Comma list: 289/288, 442/441, 456/455, 476/475, 561/560, 627/625


Mapping: [{{val| 2 14 6 9 -10 25 19 20 }}, {{val| 0 -16 -2 -5 25 -26 -16 -17 }}]
Mapping: {{mapping| 2 14 6 9 -10 25 19 20 | 0 -16 -2 -5 25 -26 -16 -17 }}


POTE generator: ~19/17 = 193.9145
Optimal tuning (POTE): ~17/12 = 1\2, ~19/17 = 193.9145


{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328g, 526cfgg }}
{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328g, 526cfgg }}
Line 233: Line 233:
Comma list: 256/255, 676/675, 715/714, 1001/1000, 1225/1224
Comma list: 256/255, 676/675, 715/714, 1001/1000, 1225/1224


Mapping: [{{val| 2 14 6 9 -10 25 -4 }}, {{val| 0 -16 -2 -5 25 -26 18 }}]
Mapping: {{mapping| 2 14 6 9 -10 25 -4 | 0 -16 -2 -5 25 -26 18 }}


POTE generator: ~28/25 = 193.9112
Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9112


{{Optimal ET sequence|legend=1| 62eg, 68, 130g, 198g }}
{{Optimal ET sequence|legend=1| 62eg, 68, 130g, 198g }}
Line 246: Line 246:
Comma list: 256/255, 286/285, 400/399, 476/475, 495/494, 1225/1224
Comma list: 256/255, 286/285, 400/399, 476/475, 495/494, 1225/1224


Mapping: [{{val| 2 14 6 9 -10 25 -4 -3 }}, {{val| 0 -16 -2 -5 25 -26 18 17 }}]
Mapping: {{mapping| 2 14 6 9 -10 25 -4 -3 | 0 -16 -2 -5 25 -26 18 17 }}


POTE generator: ~19/17 = 193.9428
Optimal tuning (POTE): ~99/70 = 1\2, ~19/17 = 193.9428


{{Optimal ET sequence|legend=1| 62egh, 68, 130gh, 198gh }}
{{Optimal ET sequence|legend=1| 62egh, 68, 130gh, 198gh }}
Line 255: Line 255:


== Hemithirds ==
== Hemithirds ==
{{main| Hemithirds }}
{{Main| Hemithirds }}
{{see also| Luna family #Hemithirds }}
{{See also| Luna family #Hemithirds }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 1029/1024, 3136/3125
[[Comma list]]: 1029/1024, 3136/3125


[[Mapping]]: [{{val| 1 4 2 2 }}, {{val| 0 -15 2 5 }}]
{{Mapping|legend=1| 1 4 2 2 | 0 -15 2 5 }}


{{Multival|legend=1| 15 -2 -5 -38 -50 -6 }}
{{Multival|legend=1| 15 -2 -5 -38 -50 -6 }}


[[POTE generator]]: ~28/25 = 193.244
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.244


[[Minimax tuning]]:
[[Minimax tuning]]:
* [[7-odd-limit]]: ~28/25 = {{monzo| 1/10 -1/20 0 1/20 }}
* [[7-odd-limit]]: ~28/25 = {{monzo| 1/10 -1/20 0 1/20 }}
: [[Eigenmonzo]]s (unchanged-intervals): 2, 7/6
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.7/3
* [[9-odd-limit]]: ~28/25 = {{monzo| 6/25 -2/35 0 1/35 }}
* [[9-odd-limit]]: ~28/25 = {{monzo| 6/25 -2/35 0 1/35 }}
: [[Eigenmonzo]]s (unchanged-intervals): 2, 9/7
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.9/7


{{Optimal ET sequence|legend=1| 25, 31, 87, 118 }}
{{Optimal ET sequence|legend=1| 25, 31, 87, 118 }}
Line 283: Line 283:
Comma list: 385/384, 441/440, 3136/3125
Comma list: 385/384, 441/440, 3136/3125


Mapping: [{{val| 1 4 2 2 7 }}, {{val| 0 -15 2 5 -22 }}]
Mapping: {{mapping| 1 4 2 2 7 | 0 -15 2 5 -22 }}


POTE generator: ~28/25 = 193.227
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.227


Minimax tuning:
Minimax tuning:
* 11-odd-limit: ~28/25 = {{monzo| 5/27 0 0 1/27 -1/27 }}
* 11-odd-limit: ~28/25 = {{monzo| 5/27 0 0 1/27 -1/27 }}
: Eigenmonzos (unchanged-intervals): 2, 11/7
: Eigenmonzo (unchanged-interval) basis: 2.11/7


{{Optimal ET sequence|legend=1| 25e, 31, 87, 118 }}
{{Optimal ET sequence|legend=1| 25e, 31, 87, 118 }}
Line 300: Line 300:
Comma list: 196/195, 352/351, 385/384, 625/624
Comma list: 196/195, 352/351, 385/384, 625/624


Mapping: [{{val| 1 4 2 2 7 0 }}, {{val| 0 -15 2 5 -22 23 }}]
Mapping: {{mapping| 1 4 2 2 7 0 | 0 -15 2 5 -22 23 }}


POTE generator: ~28/25 = 193.166
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.166


{{Optimal ET sequence|legend=1| 31, 56, 87, 118, 205d }}
{{Optimal ET sequence|legend=1| 31, 56, 87, 118, 205d }}
Line 311: Line 311:
{{See also| Magic family }}
{{See also| Magic family }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 49/48, 3125/3072
[[Comma list]]: 49/48, 3125/3072


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


{{Multival|legend=1| 10 2 5 -20 -20 6 }}
{{Multival|legend=1| 10 2 5 -20 -20 6 }}


[[POTE generator]]: ~28/25 = 189.927
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 189.927


{{Optimal ET sequence|legend=1| 6, 19, 82dd }}
{{Optimal ET sequence|legend=1| 6, 19, 82dd }}
Line 330: Line 330:
Comma list: 49/48, 56/55, 125/121
Comma list: 49/48, 56/55, 125/121


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


POTE generator: ~11/10 = 190.285
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 190.285


{{Optimal ET sequence|legend=1| 6, 19, 44de, 63dee, 82ddee }}
{{Optimal ET sequence|legend=1| 6, 19, 44de, 63dee, 82ddee }}
Line 343: Line 343:
Comma list: 49/48, 56/55, 78/77, 125/121
Comma list: 49/48, 56/55, 78/77, 125/121


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


POTE generator: ~11/10 = 189.928
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 189.928


{{Optimal ET sequence|legend=1| 6, 19, 82ddeeff }}
{{Optimal ET sequence|legend=1| 6, 19, 82ddeeff }}
Line 356: Line 356:
Comma list: 49/48, 56/55, 91/90, 125/121
Comma list: 49/48, 56/55, 91/90, 125/121


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


POTE generator: ~11/10 = 190.360
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 190.360


{{Optimal ET sequence|legend=1| 19, 44de, 63dee, 82ddee }}
{{Optimal ET sequence|legend=1| 19, 44de, 63dee, 82ddee }}
Line 374: Line 374:


{{Mapping|legend=1| 1 12 6 12 | 0 -17 -6 -15 }}
{{Mapping|legend=1| 1 12 6 12 | 0 -17 -6 -15 }}
: mapping generators: ~2, ~75/49


{{Multival|legend=1| 17 6 15 -30 -24 18 }}
{{Multival|legend=1| 17 6 15 -30 -24 18 }}


[[Optimal tuning]] ([[POTE]]): ~75/49 = 735.155
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/49 = 735.155


{{Optimal ET sequence|legend=1| 18, 31, 80, 111 }}
{{Optimal ET sequence|legend=1| 18, 31, 80, 111 }}
Line 390: Line 392:
Mapping: {{mapping| 1 12 6 12 20 | 0 -17 -6 -15 -27 }}
Mapping: {{mapping| 1 12 6 12 20 | 0 -17 -6 -15 -27 }}


Optimal tuning (POTE): ~55/36 = 735.125
Optimal tuning (POTE): ~2 = 1\1, ~55/36 = 735.125


{{Optimal ET sequence|legend=1| 18e, 31, 80, 111, 364cd }}
{{Optimal ET sequence|legend=1| 18e, 31, 80, 111, 364cd }}
Line 403: Line 405:
Mapping: {{mapping| 1 12 6 12 20 -11 | 0 -17 -6 -15 -27 24 }}
Mapping: {{mapping| 1 12 6 12 20 -11 | 0 -17 -6 -15 -27 24 }}


Optimal tuning (POTE): ~55/36 = 735.126
Optimal tuning (POTE): ~2 = 1\1, ~55/36 = 735.126


{{Optimal ET sequence|legend=1| 31, 80, 111 }}
{{Optimal ET sequence|legend=1| 31, 80, 111 }}
Line 416: Line 418:
Mapping: {{mapping| 1 12 6 12 20 -11 -10 | 0 -17 -6 -15 -27 24 23 }}
Mapping: {{mapping| 1 12 6 12 20 -11 -10 | 0 -17 -6 -15 -27 24 23 }}


Optimal tuning (POTE): ~26/17 = 735.125
Optimal tuning (POTE): ~2 = 1\1, ~26/17 = 735.125


{{Optimal ET sequence|legend=1| 31, 80, 111 }}
{{Optimal ET sequence|legend=1| 31, 80, 111 }}
Line 429: Line 431:
Mapping: {{mapping| 1 12 6 12 20 -11 -10 -8 | 0 -17 -6 -15 -27 24 23 20 }}
Mapping: {{mapping| 1 12 6 12 20 -11 -10 -8 | 0 -17 -6 -15 -27 24 23 20 }}


Optimal tuning (POTE): ~26/17 = 735.116
Optimal tuning (POTE): ~2 = 1\1, ~26/17 = 735.116


{{Optimal ET sequence|legend=1| 31, 80, 111 }}
{{Optimal ET sequence|legend=1| 31, 80, 111 }}
Line 442: Line 444:
Mapping: {{mapping| 1 12 6 12 20 -11 -10 -8 18 | 0 -17 -6 -15 -27 24 23 20 -22 }}
Mapping: {{mapping| 1 12 6 12 20 -11 -10 -8 18 | 0 -17 -6 -15 -27 24 23 20 -22 }}


Optimal tuning (POTE): ~26/17 = 735.106
Optimal tuning (POTE): ~2 = 1\1, ~26/17 = 735.106


{{Optimal ET sequence|legend=1| 31, 80, 111, 191cdh, 302cdgh }}
{{Optimal ET sequence|legend=1| 31, 80, 111, 191cdh, 302cdgh }}
Line 455: Line 457:
Mapping: {{mapping| 1 12 6 12 20 8 | 0 -17 -6 -15 -27 -7 }}
Mapping: {{mapping| 1 12 6 12 20 8 | 0 -17 -6 -15 -27 -7 }}


Optimal tuning (POTE): ~13/10 = 464.980
Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 464.980


{{Optimal ET sequence|legend=1| 31, 49f, 80f }}
{{Optimal ET sequence|legend=1| 31, 49f, 80f }}
Line 464: Line 466:
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Emka]].''
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Emka]].''


[[Emka]] tempers out {{monzo|-50 -8 27}} in the 5-limit. This temperament can be described as 37&50 temperament, which tempers out the hemimean and 84035/82944 (quinzo-ayo). Alternative extension [[Horwell temperaments #Emkay|emkay]] (87&224) tempers out the same 5-limit comma as the emka, but with the horwell (65625/65536) rather than the hemimean tempered out.
[[Emka]] tempers out {{monzo| -50 -8 27 }} in the 5-limit. This temperament can be described as 37 & 50 temperament, which tempers out the hemimean and 84035/82944 (quinzo-ayo). Alternative extension [[Horwell temperaments #Emkay|emkay]] (87 & 224) tempers out the same 5-limit comma as the emka, but with the horwell (65625/65536) rather than the hemimean tempered out.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 3136/3125, 84035/82944
[[Comma list]]: 3136/3125, 84035/82944


[[Mapping]]: [{{val| 1 14 6 12 }}, {{val| 0 -27 -8 -20 }}]
{{Mapping|legend=1| 1 14 6 12 | 0 -27 -8 -20 }}
 
: mapping generators: ~2, ~48/35


{{Multival|legend=1| 27 8 20 -50 -44 24 }}
{{Multival|legend=1| 27 8 20 -50 -44 24 }}


[[POTE generator]]: ~48/35 = 551.782
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~48/35 = 551.782


{{Optimal ET sequence|legend=1| 37, 50, 87, 137d, 224d }}
{{Optimal ET sequence|legend=1| 37, 50, 87, 137d, 224d }}
Line 485: Line 489:
Comma list: 385/384, 2401/2376, 3136/3125
Comma list: 385/384, 2401/2376, 3136/3125


Mapping: [{{val| 1 14 6 12 3 }}, {{val| 0 -27 -8 -20 1 }}]
Mapping: {{mapping| 1 14 6 12 3 | 0 -27 -8 -20 1 }}


POTE generator: ~11/8 = 551.765
Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 551.765


{{Optimal ET sequence|legend=1| 37, 50, 87, 224d, 311d }}
{{Optimal ET sequence|legend=1| 37, 50, 87, 224d, 311d }}
Line 498: Line 502:
Comma list: 196/195, 364/363, 385/384, 625/624
Comma list: 196/195, 364/363, 385/384, 625/624


Mapping: [{{val| 1 14 6 12 3 6 }}, {{val| 0 -27 -8 -20 1 -5 }}]
Mapping: {{mapping| 1 14 6 12 3 6 | 0 -27 -8 -20 1 -5 }}


POTE generator: ~11/8 = 551.758
Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 551.758


{{Optimal ET sequence|legend=1| 37, 50, 87, 224d, 311d, 398d }}
{{Optimal ET sequence|legend=1| 37, 50, 87, 224d, 311d, 398d }}
Retrieved from "https://en.xen.wiki/w/Hemimean_clan"