Ragismic microtemperaments: Difference between revisions
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[[Comma list]]: 2401/2400, 4375/4374 | [[Comma list]]: 2401/2400, 4375/4374 | ||
{{Mapping|legend=1| 9 1 1 12 | 0 2 3 2 }} | |||
{{Multival|legend=1| 18 27 18 1 -22 -34 }} | {{Multival|legend=1| 18 27 18 1 -22 -34 }} | ||
: mapping generators: ~27/25, ~5/3 | |||
[[Optimal tuning]] ([[POTE]]): ~27/25 = 1\9, ~5/3 = 884.3129 (~36/35 = 49.0205) | [[Optimal tuning]] ([[POTE]]): ~27/25 = 1\9, ~5/3 = 884.3129 (~36/35 = 49.0205) | ||
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Comma list: 2401/2400, 4375/4374, 5632/5625 | Comma list: 2401/2400, 4375/4374, 5632/5625 | ||
Mapping: | Mapping: {{mapping| 9 1 1 12 -75 | 0 2 3 2 16 }} | ||
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4679 (~36/35 = 48.8654) | Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4679 (~36/35 = 48.8654) | ||
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Comma list: 1001/1000, 1716/1715, 4096/4095, 4375/4374 | Comma list: 1001/1000, 1716/1715, 4096/4095, 4375/4374 | ||
Mapping: | Mapping: {{mapping| 9 1 1 12 -75 93 | 0 2 3 2 16 -9 }} | ||
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4304 (~36/35 = 48.9030) | Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4304 (~36/35 = 48.9030) | ||
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Comma list: 715/714, 1001/1000, 1716/1715, 4096/4095, 4375/4374 | Comma list: 715/714, 1001/1000, 1716/1715, 4096/4095, 4375/4374 | ||
Mapping: | Mapping: {{mapping| 9 1 1 12 -75 93 -3 | 0 2 3 2 16 -9 6 }} | ||
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4304 (~36/35 = 48.9030) | Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4304 (~36/35 = 48.9030) | ||
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Comma list: 715/714, 1001/1000, 1216/1215, 1716/1715, 4096/4095, 4375/4374 | Comma list: 715/714, 1001/1000, 1216/1215, 1716/1715, 4096/4095, 4375/4374 | ||
Mapping: | Mapping: {{mapping| 9 1 1 12 -75 93 -3 -48 | 0 2 3 2 16 -9 6 13 }} | ||
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4304 (~36/35 = 48.9030) | Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4304 (~36/35 = 48.9030) | ||
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Comma list: 2080/2079, 2401/2400, 4375/4374, 5632/5625 | Comma list: 2080/2079, 2401/2400, 4375/4374, 5632/5625 | ||
Mapping: | Mapping: {{mapping| 9 1 1 12 -75 -106 | 0 2 3 2 16 21 }} | ||
Optimal tuning (CTE): ~27/25 = 1\9, ~5/3 = 884.4560 (~36/35 = 48.8773) | Optimal tuning (CTE): ~27/25 = 1\9, ~5/3 = 884.4560 (~36/35 = 48.8773) | ||
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Comma list: 2401/2400, 4375/4374, 131072/130977 | Comma list: 2401/2400, 4375/4374, 131072/130977 | ||
Mapping: | Mapping: {{mapping| 9 1 1 12 124 | 0 2 3 2 -14 }} | ||
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4089 (~36/35 = 48.9244) | Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4089 (~36/35 = 48.9244) | ||
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Comma list: 2080/2079, 2401/2400, 4096/4095, 4375/4374 | Comma list: 2080/2079, 2401/2400, 4096/4095, 4375/4374 | ||
Mapping: | Mapping: {{mapping| 9 1 1 12 124 93 | 0 2 3 2 -14 -9 }} | ||
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.3997 (~36/35 = 48.9336) | Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.3997 (~36/35 = 48.9336) | ||
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Comma list: 936/935, 2080/2079, 2401/2400, 4096/4095, 4375/4374 | Comma list: 936/935, 2080/2079, 2401/2400, 4096/4095, 4375/4374 | ||
Mapping: | Mapping: {{mapping| 9 1 1 12 124 93 -3 | 0 2 3 2 -14 -9 6 }} | ||
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.3997 (~36/35 = 48.9336) | Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.3997 (~36/35 = 48.9336) | ||
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Comma list: 936/935, 1216/1215, 2080/2079, 2401/2400, 4096/4095, 4375/4374 | Comma list: 936/935, 1216/1215, 2080/2079, 2401/2400, 4096/4095, 4375/4374 | ||
Mapping: | Mapping: {{mapping| 9 1 1 12 124 93 -3 -48 | 0 2 3 2 -14 -9 6 13 }} | ||
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.3997 (~36/35 = 48.9336) | Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.3997 (~36/35 = 48.9336) | ||
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Comma list: 243/242, 441/440, 4375/4356 | Comma list: 243/242, 441/440, 4375/4356 | ||
Mapping: | Mapping: {{mapping| 9 1 1 12 -2 | 0 2 3 2 5 }} | ||
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.9386 (~36/35 = 49.3948) | Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.9386 (~36/35 = 49.3948) | ||
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Comma list: 243/242, 364/363, 441/440, 625/624 | Comma list: 243/242, 364/363, 441/440, 625/624 | ||
Mapping: | Mapping: {{mapping| 9 1 1 12 -2 -33 | 0 2 3 2 5 10 }} | ||
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.9920 (~36/35 = 49.3414) | Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.9920 (~36/35 = 49.3414) | ||
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Comma list: 243/242, 364/363, 375/374, 441/440, 595/594 | Comma list: 243/242, 364/363, 375/374, 441/440, 595/594 | ||
Mapping: | Mapping: {{mapping| 9 1 1 12 -2 -33 -3 | 0 2 3 2 5 10 6 }} | ||
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.9981 (~36/35 = 49.3353) | Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.9981 (~36/35 = 49.3353) | ||
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Comma list: 243/242, 364/363, 375/374, 441/440, 513/512, 595/594 | Comma list: 243/242, 364/363, 375/374, 441/440, 513/512, 595/594 | ||
Mapping: | Mapping: {{mapping| 9 1 1 12 -2 -33 -3 78 | 0 2 3 2 5 10 6 -6 }} | ||
{{Optimal ET sequence|legend=1| 72, 171, 243 }} | {{Optimal ET sequence|legend=1| 72, 171, 243 }} | ||
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Comma list: 169/168, 243/242, 325/324, 441/440 | Comma list: 169/168, 243/242, 325/324, 441/440 | ||
Mapping: | Mapping: {{mapping| 9 1 1 12 -2 20 | 0 2 3 2 5 2 }} | ||
Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.6257 (~36/35 = 49.7076) | Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.6257 (~36/35 = 49.7076) | ||
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Comma list: 169/168, 221/220, 243/242, 325/324, 441/440 | Comma list: 169/168, 221/220, 243/242, 325/324, 441/440 | ||
Mapping: | Mapping: {{mapping| 9 1 1 12 -2 20 -3 | 0 2 3 2 5 2 6 }} | ||
Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.6257 (~36/35 = 49.7076) | Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.6257 (~36/35 = 49.7076) | ||
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Comma list: 169/168, 221/220, 243/242, 325/324, 441/440 | Comma list: 169/168, 221/220, 243/242, 325/324, 441/440 | ||
Mapping: | Mapping: {{mapping| 9 1 1 12 -2 20 -3 25 | 0 2 3 2 5 2 6 2 }} | ||
Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.6257 (~36/35 = 49.7076) | Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.6257 (~36/35 = 49.7076) | ||
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Comma list: 385/384, 1375/1372, 4375/4374 | Comma list: 385/384, 1375/1372, 4375/4374 | ||
Mapping: | Mapping: {{mapping| 9 1 1 12 51 | 0 2 3 2 -3 }} | ||
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.8298 (~36/35 = 49.5036) | Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.8298 (~36/35 = 49.5036) | ||
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Comma list: 169/168, 325/324, 385/384, 1375/1372 | Comma list: 169/168, 325/324, 385/384, 1375/1372 | ||
Mapping: | Mapping: {{mapping| 9 1 1 12 51 20 | 0 2 3 2 -3 2 }} | ||
Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.8476 (~36/35 = 49.4857) | Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.8476 (~36/35 = 49.4857) | ||
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Comma list: 169/168, 221/220, 325/324, 385/384, 1375/1372 | Comma list: 169/168, 221/220, 325/324, 385/384, 1375/1372 | ||
Mapping: | Mapping: {{mapping| 9 1 1 12 51 20 50 | 0 2 3 2 -3 2 -2 }} | ||
Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.8476 (~36/35 = 49.4857) | Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.8476 (~36/35 = 49.4857) | ||
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Comma list: 153/152, 169/168, 221/220, 325/324, 385/384, 1375/1372 | Comma list: 153/152, 169/168, 221/220, 325/324, 385/384, 1375/1372 | ||
Mapping: | Mapping: {{mapping| 9 1 1 12 51 20 50 25 | 0 2 3 2 -3 2 -2 2 }} | ||
Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.8476 (~36/35 = 49.4857) | Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.8476 (~36/35 = 49.4857) | ||
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Comma list: 2401/2400, 3025/3024, 4375/4374 | Comma list: 2401/2400, 3025/3024, 4375/4374 | ||
Mapping: | Mapping: {{mapping| 18 0 -1 22 48 | 0 2 3 2 1 }} | ||
: mapping generators: ~80/77, ~400/231 | |||
Optimal tuning (POTE): ~80/77 = 1\18, ~400/231 = 950.9553 | Optimal tuning (POTE): ~80/77 = 1\18, ~400/231 = 950.9553 | ||
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Comma list: 676/675, 1001/1000, 1716/1715, 3025/3024 | Comma list: 676/675, 1001/1000, 1716/1715, 3025/3024 | ||
Mapping: | Mapping: {{mapping| 18 0 -1 22 48 -19 | 0 2 3 2 1 6 }} | ||
Optimal tuning (POTE): ~27/26 = 1\18, ~26/15 = 951.0837 | Optimal tuning (POTE): ~27/26 = 1\18, ~26/15 = 951.0837 | ||
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Comma list: 676/675, 715/714, 1001/1000, 1716/1715, 3025/3024 | Comma list: 676/675, 715/714, 1001/1000, 1716/1715, 3025/3024 | ||
Mapping: | Mapping: {{mapping| 18 0 -1 22 48 -19 -12 | 0 2 3 2 1 6 6 }} | ||
Optimal tuning (POTE): ~27/26 = 1\18, ~26/15 = 951.0837 | Optimal tuning (POTE): ~27/26 = 1\18, ~26/15 = 951.0837 | ||
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Comma list: 676/675, 715/714, 1001/1000, 1331/1330, 1716/1715, 3025/3024 | Comma list: 676/675, 715/714, 1001/1000, 1331/1330, 1716/1715, 3025/3024 | ||
Mapping: | Mapping: {{mapping| 18 0 -1 22 48 -19 -12 48 105 | 0 2 3 2 1 6 6 -2 }} | ||
Optimal tuning (POTE): ~27/26 = 1\18, ~26/15 = 951.0837 | Optimal tuning (POTE): ~27/26 = 1\18, ~26/15 = 951.0837 | ||
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Comma list: 2401/2400, 3025/3024, 4225/4224, 4375/4374 | Comma list: 2401/2400, 3025/3024, 4225/4224, 4375/4374 | ||
Mapping: | Mapping: {{mapping| 18 0 -1 22 48 88 | 0 4 6 4 2 -3 }} | ||
: mapping generators: ~80/77, ~1053/800 | |||
Optimal tuning (POTE): ~80/77 = 1\18, ~1053/800 = 475.4727 | Optimal tuning (POTE): ~80/77 = 1\18, ~1053/800 = 475.4727 | ||
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Comma list: 2401/2400, 2431/2430, 3025/3024, 4225/4224, 4375/4374 | Comma list: 2401/2400, 2431/2430, 3025/3024, 4225/4224, 4375/4374 | ||
Mapping: | Mapping: {{mapping| 18 0 -1 22 48 88 -119 | 0 4 6 4 2 -3 27 }} | ||
: mapping generators: ~80/77, ~1053/800 | |||
Optimal tuning (POTE): ~80/77 = 1\18, ~1053/800 = 475.4727 | Optimal tuning (POTE): ~80/77 = 1\18, ~1053/800 = 475.4727 | ||
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Comma list: 2401/2400, 2431/2430, 2926/2925, 3025/3024, 4225/4224, 4375/4374 | Comma list: 2401/2400, 2431/2430, 2926/2925, 3025/3024, 4225/4224, 4375/4374 | ||
Mapping: | Mapping: {{mapping| 18 0 -1 22 48 88 -119 -2 | 0 4 6 4 2 -3 27 11 }} | ||
: mapping generators: ~80/77, ~1053/800 | |||
Optimal tuning (POTE): ~80/77 = 1\18, ~1053/800 = 475.4727 | Optimal tuning (POTE): ~80/77 = 1\18, ~1053/800 = 475.4727 | ||
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Comma list: 2401/2400, 4000/3993, 4375/4374 | Comma list: 2401/2400, 4000/3993, 4375/4374 | ||
Mapping: | Mapping: {{mapping| 9 3 4 14 18 | 0 6 9 6 7 }} | ||
: mapping generators: ~27/25, ~140/121 | |||
Optimal tuning (POTE): ~27/25 = 1\9, ~140/121 = 250.3367 | Optimal tuning (POTE): ~27/25 = 1\9, ~140/121 = 250.3367 | ||
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Comma list: 1575/1573, 2080/2079, 2401/2400, 4375/4374 | Comma list: 1575/1573, 2080/2079, 2401/2400, 4375/4374 | ||
Mapping: | Mapping: {{mapping| 9 3 4 14 18 -8 | 0 6 9 6 7 22 }} | ||
Optimal tuning (POTE): ~27/25 = 1\9, ~140/121 = 250.3375 | Optimal tuning (POTE): ~27/25 = 1\9, ~140/121 = 250.3375 | ||
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Comma list: 2401/2400, 4375/4374, 234375/234256 | Comma list: 2401/2400, 4375/4374, 234375/234256 | ||
Mapping: | Mapping: {{mapping| 9 1 1 12 -7 | 0 8 12 8 23 }} | ||
: mapping generators: ~27/25, ~25/22 | |||
Optimal tuning (POTE): ~27/25 = 1\9, ~25/22 = 221.0717 | Optimal tuning (POTE): ~27/25 = 1\9, ~25/22 = 221.0717 | ||
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Comma list: 2401/2400, 4375/4374, 2097152/2096325 | Comma list: 2401/2400, 4375/4374, 2097152/2096325 | ||
Mapping: | Mapping: {{mapping| 27 1 0 34 177 | 0 2 3 2 -4 }} | ||
: mapping generators: ~2744/2673, ~2352/1375 | |||
Optimal tuning (POTE): ~2744/2673 = 1\27, ~2352/1375 = 928.8000 | Optimal tuning (POTE): ~2744/2673 = 1\27, ~2352/1375 = 928.8000 | ||
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=== Rhodium === | === Rhodium === | ||
{{Main|Rhodium}} | {{Main| Rhodium }} | ||
Rhodium splits the ennealimmal period in five parts and thereby features a period of 9 × 5 = 45, thus the name is given after the 45th element. | Rhodium splits the ennealimmal period in five parts and thereby features a period of 9 × 5 = 45, thus the name is given after the 45th element. | ||
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Comma list: 2401/2400, 4375/4374, 117440512/117406179 | Comma list: 2401/2400, 4375/4374, 117440512/117406179 | ||
Mapping: | Mapping: {{mapping| 45 1 -1 56 226 | 0 2 3 2 -2 }} | ||
: mapping generators: ~3072/3025, ~55/32 | |||
Optimal tuning (CTE): ~3072/3025 = 1\45, ~55/32 = 937.6658 | Optimal tuning (CTE): ~3072/3025 = 1\45, ~55/32 = 937.6658 | ||
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Comma list: 2401/2400, 4225/4224, 4375/4374, 6656/6655 | Comma list: 2401/2400, 4225/4224, 4375/4374, 6656/6655 | ||
Mapping: | Mapping: {{mapping| 45 1 -1 56 226 272 | 0 2 3 2 -2 -3 }} | ||
Optimal tuning (CTE): ~66/65 = 1\45, ~55/32 = 937.657 | Optimal tuning (CTE): ~66/65 = 1\45, ~55/32 = 937.657 | ||
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== Supermajor == | == Supermajor == | ||
The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.002 cents flat. 37 of these give (2 | The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.002 cents flat. 37 of these give (2<sup>15</sup>)/3, 46 give (2<sup>19</sup>)/5, and 75 give (2<sup>30</sup>)/7, leading to a wedgie of {{multival| 37 46 75 -13 15 45 }}. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: 1106 or 1277 can be used as tunings, leading to accuracy even greater than that of ennealimmal. The 80-note mos is presumably the place to start, and if that is not enough notes for you, there is always the 171-note mos. | ||
Subgroup: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 4375/4374, 52734375/52706752 | [[Comma list]]: 4375/4374, 52734375/52706752 | ||
{{Mapping|legend=1| 1 15 19 30 | 0 -37 -46 -75 }} | |||
{{Multival|legend=1|37 46 75 -13 15 45}} | {{Multival|legend=1| 37 46 75 -13 15 45 }} | ||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 435.082 | ||
{{Optimal ET sequence|legend=1| 11, 80, 171, 764, 1106, 1277, 3660, 4937, 6214 }} | {{Optimal ET sequence|legend=1| 11, 80, 171, 764, 1106, 1277, 3660, 4937, 6214 }} | ||
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Comma list: 3025/3024, 4375/4374, 35156250/35153041 | Comma list: 3025/3024, 4375/4374, 35156250/35153041 | ||
Mapping: | Mapping: {{mapping| 2 30 38 60 41 | 0 -37 -46 -75 -47 }} | ||
POTE | Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 435.082 | ||
{{Optimal ET sequence|legend=1| 80, 342, 764, 1106, 1448, 2554, 4002f, 6556cf }} | {{Optimal ET sequence|legend=1| 80, 342, 764, 1106, 1448, 2554, 4002f, 6556cf }} | ||
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[[Comma list]]: 4375/4374, 703125/702464 | [[Comma list]]: 4375/4374, 703125/702464 | ||
{{Mapping|legend=1| 19 0 14 -37 | 0 1 1 3 }} | |||
{{Multival|legend=1| 19 19 57 -14 37 79 }} | {{Multival|legend=1| 19 19 57 -14 37 79 }} | ||
: mapping generators: ~28/27, ~3 | |||
[[Optimal tuning]] ([[CTE]]): ~3/2 = 701.9275 (~225/224 = 7.1907) | [[Optimal tuning]] ([[CTE]]): ~28/27 = 1\19, ~3/2 = 701.9275 (~225/224 = 7.1907) | ||
{{Optimal ET sequence|legend=1| 19, …, 152, 171, 665, 836, 1007, 2185, 3192c }} | {{Optimal ET sequence|legend=1| 19, …, 152, 171, 665, 836, 1007, 2185, 3192c }} | ||
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Comma list: 540/539, 4375/4374, 16384/16335 | Comma list: 540/539, 4375/4374, 16384/16335 | ||
Mapping: | Mapping: {{mapping| 19 0 14 -37 126 | 0 1 1 3 -2 }} | ||
Optimal tuning (CTE): ~3/2 = 702.1483 (~225/224 = 7.4115) | Optimal tuning (CTE): ~28/27 = 1\19, ~3/2 = 702.1483 (~225/224 = 7.4115) | ||
{{Optimal ET sequence|legend=1| 19, 133d, 152, 323e, 475de, 627de }} | {{Optimal ET sequence|legend=1| 19, 133d, 152, 323e, 475de, 627de }} | ||
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Comma list: 540/539, 625/624, 729/728, 2205/2197 | Comma list: 540/539, 625/624, 729/728, 2205/2197 | ||
Mapping: | Mapping: {{mapping| 19 0 14 -37 126 -20 | 0 1 1 3 -2 3 }} | ||
Optimal tuning (CTE): ~3/2 = 701.9258 (~225/224 = 7.1890) | Optimal tuning (CTE): ~28/27 = 1\19, ~3/2 = 701.9258 (~225/224 = 7.1890) | ||
{{Optimal ET sequence|legend=1| 19, 133df, 152f, 323ef }} | {{Optimal ET sequence|legend=1| 19, 133df, 152f, 323ef }} | ||
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Comma list: 3025/3024, 4375/4374, 234375/234256 | Comma list: 3025/3024, 4375/4374, 234375/234256 | ||
Mapping: | Mapping: {{mapping| 38 0 28 -74 11 | 0 1 1 3 2 }} | ||
: mapping generators: ~55/54, ~3 | |||
Optimal tuning (CTE): ~3/2 = 701.9351 (~225/224 = 7.1983) | Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9351 (~225/224 = 7.1983) | ||
{{Optimal ET sequence|legend=1| 152, 342, 836, 1178, 2014, 3192ce, 5206ce }} | {{Optimal ET sequence|legend=1| 152, 342, 836, 1178, 2014, 3192ce, 5206ce }} | ||
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Comma list: 1716/1715, 2080/2079, 3025/3024, 234375/234256 | Comma list: 1716/1715, 2080/2079, 3025/3024, 234375/234256 | ||
Mapping: | Mapping: {{mapping| 38 0 28 -74 11 -281 | 0 1 1 3 2 7 }} | ||
Optimal tuning (CTE): ~3/2 = 701.9955 (~225/224 = 7.2587) | Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9955 (~225/224 = 7.2587) | ||
{{Optimal ET sequence|legend=1| 152f, 342f, 494 }} | {{Optimal ET sequence|legend=1| 152f, 342f, 494 }} | ||
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Comma list: 3025/3024, 4096/4095, 4375/4374, 31250/31213 | Comma list: 3025/3024, 4096/4095, 4375/4374, 31250/31213 | ||
Mapping: | Mapping: {{mapping| 38 0 28 -74 11 502 | 0 1 1 3 2 -6 }} | ||
Optimal tuning (CTE): ~3/2 = 701.9812 (~225/224 = 7.2444) | Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9812 (~225/224 = 7.2444) | ||
{{Optimal ET sequence|legend=1| 152, 342, 494, 1330, 1824, 2318d }} | {{Optimal ET sequence|legend=1| 152, 342, 494, 1330, 1824, 2318d }} | ||
| Line 644: | Line 644: | ||
Comma list: 3025/3024, 4225/4224, 4375/4374, 78125/78078 | Comma list: 3025/3024, 4225/4224, 4375/4374, 78125/78078 | ||
Mapping: | Mapping: {{mapping| 38 1 29 -71 13 111 | 0 2 2 6 4 1 }} | ||
: mapping generators: ~55/54 = 1\38, ~55/54, ~429/250 | |||
Optimal tuning (CTE): ~429/250 = 935.1789 (~144/143 = 12.1895) | Optimal tuning (CTE): ~429/250 = 935.1789 (~144/143 = 12.1895) | ||
| Line 661: | Line 661: | ||
Comma list: 2500/2499, 3250/3249, 4225/4224, 4375/4374, 11016/11011, 57375/57344 | Comma list: 2500/2499, 3250/3249, 4225/4224, 4375/4374, 11016/11011, 57375/57344 | ||
Mapping: | Mapping: {{mapping| 19 3 17 -28 82 92 159 78 | 0 10 10 30 -6 -8 -30 1 }} | ||
Optimal tuning (CTE): ~28/27 = 1\19, ~6545/5928 = 171.244 | Optimal tuning (CTE): ~28/27 = 1\19, ~6545/5928 = 171.244 | ||
{{Optimal ET sequence|legend=1|855, 988, 1843}} | {{Optimal ET sequence|legend=1| 855, 988, 1843 }} | ||
== Semidimi == | == Semidimi == | ||
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Semidimi]].'' | : ''For the 5-limit version of this temperament, see [[High badness temperaments #Semidimi]].'' | ||
The generator of semidimi temperament is a semi-diminished fourth interval tuned between 162/125 and 35/27. It tempers out 5-limit {{monzo|-12 -73 55}} and 7-limit 3955078125/3954653486, as well as 4375/4374. | The generator of semidimi temperament is a semi-diminished fourth interval tuned between 162/125 and 35/27. It tempers out 5-limit {{monzo| -12 -73 55 }} and 7-limit 3955078125/3954653486, as well as 4375/4374. | ||
Subgroup: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 4375/4374, 3955078125/3954653486 | [[Comma list]]: 4375/4374, 3955078125/3954653486 | ||
{{Mapping|legend=1| 1 36 48 61 | 0 -55 -73 -93 }} | |||
{{Multival|legend=1|55 73 93 -12 -7 11}} | {{Multival|legend=1| 55 73 93 -12 -7 11 }} | ||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/27 = 449.1270 | ||
{{Optimal ET sequence|legend=1| 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }} | {{Optimal ET sequence|legend=1| 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }} | ||
| Line 687: | Line 687: | ||
== Brahmagupta == | == Brahmagupta == | ||
The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]], {{monzo|47 -7 -7 -7}} = 140737488355328 / 140710042265625. | The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]], {{monzo| 47 -7 -7 -7 }} = 140737488355328 / 140710042265625. | ||
Subgroup: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 4375/4374, 70368744177664/70338939985125 | [[Comma list]]: 4375/4374, 70368744177664/70338939985125 | ||
{{Mapping|legend=1| 7 2 -8 53 | 0 3 8 -11 }} | |||
: mapping generators: ~1157625/1048576, ~27/20 | |||
{{Multival|legend=1|21 56 -77 40 -181 -336}} | {{Multival|legend=1| 21 56 -77 40 -181 -336 }} | ||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~1157625/1048576 = 1\7, ~27/20 = 519.716 | ||
{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 1106, 1547 }} | {{Optimal ET sequence|legend=1| 7, 217, 224, 441, 1106, 1547 }} | ||
| Line 708: | Line 710: | ||
Comma list: 4000/3993, 4375/4374, 131072/130977 | Comma list: 4000/3993, 4375/4374, 131072/130977 | ||
Mapping: | Mapping: {{mapping| 7 2 -8 53 3 | 0 3 8 -11 7 }} | ||
POTE | Optimal tuning (POTE): ~243/220 = 1\7, ~27/20 = 519.704 | ||
{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 665, 1771ee }} | {{Optimal ET sequence|legend=1| 7, 217, 224, 441, 665, 1771ee }} | ||
| Line 721: | Line 723: | ||
Comma list: 1575/1573, 2080/2079, 4096/4095, 4375/4374 | Comma list: 1575/1573, 2080/2079, 4096/4095, 4375/4374 | ||
Mapping: | Mapping: {{mapping| 7 2 -8 53 3 35 | 0 3 8 -11 7 -3 }} | ||
POTE | Optimal tuning (POTE): ~243/220 = 1\7, ~27/20 = 519.706 | ||
{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 665, 1771eef }} | {{Optimal ET sequence|legend=1| 7, 217, 224, 441, 665, 1771eef }} | ||
| Line 730: | Line 732: | ||
== Abigail == | == Abigail == | ||
Abigail temperament tempers out the [[pessoalisma]] in addition to the ragisma in the 7-limit. It was named by Gene Ward Smith after the birthday of First Lady Abigail Fillmore.<ref>https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_17927.html#17930: "I propose Abigail as a name, on the grounds 313/1798 is an excellent generator, and Abigail Fillmore, wife of Millard, was born on 3-13-1798 at least as Americans recon things."</ref> | Abigail temperament tempers out the [[pessoalisma]] in addition to the ragisma in the 7-limit. It was named by Gene Ward Smith after the birthday of First Lady Abigail Fillmore.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_17927.html#17930]: "I propose Abigail as a name, on the grounds 313/1798 is an excellent generator, and Abigail Fillmore, wife of Millard, was born on 3-13-1798 at least as Americans recon things."</ref> | ||
Subgroup: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 4375/4374, 2147483648/2144153025 | [[Comma list]]: 4375/4374, 2147483648/2144153025 | ||
{{Mapping|legend=1| 2 7 13 -1 | 0 -11 -24 19 }} | |||
: mapping generators: ~46305/32768, ~27/20 | |||
{{Multival|legend=1|22 48 -38 25 -122 -223}} | {{Multival|legend=1| 22 48 -38 25 -122 -223 }} | ||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~46305/32768 = 1\2, ~6912/6125 = 208.899 | ||
{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764, 1034, 1798 }} | {{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764, 1034, 1798 }} | ||
| Line 751: | Line 755: | ||
Comma list: 3025/3024, 4375/4374, 131072/130977 | Comma list: 3025/3024, 4375/4374, 131072/130977 | ||
Mapping: | Mapping: {{mapping| 2 7 13 -1 1 | 0 -11 -24 19 17 }} | ||
POTE | Optimal tuning (POTE): ~99/70 = 1\2, ~1155/1024 = 208.901 | ||
{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764 }} | {{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764 }} | ||
| Line 764: | Line 768: | ||
Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095 | Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095 | ||
Mapping: | Mapping: {{mapping| 2 7 13 -1 1 -2 | 0 -11 -24 19 17 27 }} | ||
POTE | Optimal tuning (POTE): ~99/70 = 1\2, ~44/39 = 208.903 | ||
{{Optimal ET sequence|legend=1| 46, 178, 224, 270, 494, 764, 1258 }} | {{Optimal ET sequence|legend=1| 46, 178, 224, 270, 494, 764, 1258 }} | ||
| Line 773: | Line 777: | ||
== Gamera == | == Gamera == | ||
Subgroup: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 4375/4374, 589824/588245 | [[Comma list]]: 4375/4374, 589824/588245 | ||
{{Mapping|legend=1| 1 6 10 3 | 0 -23 -40 -1 }} | |||
: mapping generators: ~2, ~8/7 | |||
{{Multival|legend=1| 23 40 1 10 -63 -110 }} | {{Multival|legend=1| 23 40 1 10 -63 -110 }} | ||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 230.336 | ||
{{Optimal ET sequence|legend=1| 26, 73, 99, 224, 323, 422, 745d }} | {{Optimal ET sequence|legend=1| 26, 73, 99, 224, 323, 422, 745d }} | ||
| Line 794: | Line 798: | ||
Comma list: 3025/3024, 4375/4374, 589824/588245 | Comma list: 3025/3024, 4375/4374, 589824/588245 | ||
Mapping: | Mapping: {{mapping| 2 12 20 6 5 | 0 -23 -40 -1 5 }} | ||
: mapping generators: ~99/70, ~8/7 | |||
POTE | Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 230.3370 | ||
{{Optimal ET sequence|legend=1| 26, 198, 224, 422, 646, 1068d }} | {{Optimal ET sequence|legend=1| 26, 198, 224, 422, 646, 1068d }} | ||
| Line 809: | Line 813: | ||
Comma list: 1716/1715, 2080/2079, 2200/2197, 3025/3024 | Comma list: 1716/1715, 2080/2079, 2200/2197, 3025/3024 | ||
Mapping: | Mapping: {{mapping| 2 12 20 6 5 17 | 0 -23 -40 -1 5 -25 }} | ||
POTE | Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 230.3373 | ||
{{Optimal ET sequence|legend=1| 26, 198, 224, 422, 646f, 1068df }} | {{Optimal ET sequence|legend=1| 26, 198, 224, 422, 646f, 1068df }} | ||
| Line 822: | Line 826: | ||
Comma list: 4375/4374, 14641/14580, 15488/15435 | Comma list: 4375/4374, 14641/14580, 15488/15435 | ||
Mapping: | Mapping: {{mapping| 1 6 10 3 12 | 0 -46 -80 -2 -89 }} | ||
: mapping generators: ~2, ~77/72 | |||
POTE | Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.1642 | ||
{{Optimal ET sequence|legend=1| 73, 125, 198, 323, 521 }} | {{Optimal ET sequence|legend=1| 73, 125, 198, 323, 521 }} | ||
| Line 837: | Line 841: | ||
Comma list: 676/675, 1001/1000, 4375/4374, 14641/14580 | Comma list: 676/675, 1001/1000, 4375/4374, 14641/14580 | ||
Mapping: | Mapping: {{mapping| 1 6 10 3 12 18 | 0 -46 -80 -2 -89 -149 }} | ||
POTE | Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.1628 | ||
{{Optimal ET sequence|legend=1| 73f, 125f, 198, 323, 521 }} | {{Optimal ET sequence|legend=1| 73f, 125f, 198, 323, 521 }} | ||
| Line 846: | Line 850: | ||
== Orga == | == Orga == | ||
Subgroup: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 4375/4374, 54975581388800/54936068900769 | [[Comma list]]: 4375/4374, 54975581388800/54936068900769 | ||
{{Mapping|legend=1| 2 21 36 5 | 0 -29 -51 1 }} | |||
: mapping generators: ~7411887/5242880, ~1310720/1058841 | |||
[[POTE | {{Multival|legend=1| 58 102 -2 27 -166 -291 }} | ||
[[Optimal tuning]] ([[POTE]]): ~7411887/5242880 = 1\2, ~8/7 = 231.104 | |||
{{Optimal ET sequence|legend=1| 26, 244, 270, 836, 1106, 1376, 2482 }} | {{Optimal ET sequence|legend=1| 26, 244, 270, 836, 1106, 1376, 2482 }} | ||
| Line 865: | Line 871: | ||
Comma list: 3025/3024, 4375/4374, 5767168/5764801 | Comma list: 3025/3024, 4375/4374, 5767168/5764801 | ||
Mapping: | Mapping: {{mapping| 2 21 36 5 2 | 0 -29 -51 1 8 }} | ||
POTE | Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 231.103 | ||
{{Optimal ET sequence|legend=1| 26, 244, 270, 566, 836, 1106 }} | {{Optimal ET sequence|legend=1| 26, 244, 270, 566, 836, 1106 }} | ||
| Line 878: | Line 884: | ||
Comma list: 1716/1715, 2080/2079, 3025/3024, 15379/15360 | Comma list: 1716/1715, 2080/2079, 3025/3024, 15379/15360 | ||
Mapping: | Mapping: {{mapping| 2 21 36 5 2 24 | 0 -29 -51 1 8 -27 }} | ||
POTE | Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 231.103 | ||
{{Optimal ET sequence|legend=1| 26, 244, 270, 566, 836f, 1106f }} | {{Optimal ET sequence|legend=1| 26, 244, 270, 566, 836f, 1106f }} | ||
| Line 889: | Line 895: | ||
The name of chlorine temperament comes from Chlorine, the 17th element. | The name of chlorine temperament comes from Chlorine, the 17th element. | ||
Chlorine temperament has a period of 1/17 octave. It tempers out the septendecima, {{monzo|-52 -17 34}}, by which 17 chromatic semitones (25/24) exceed an octave. This temperament can be described as 289&323 temperament, which tempers out {{monzo|-49 4 22 -3}} as well as the ragisma. Not only the semitwelfth, but also the ~5/4 can be used as a generator. | Chlorine temperament has a period of 1/17 octave. It tempers out the [[septendecima]], {{monzo| -52 -17 34 }}, by which 17 chromatic semitones (25/24) exceed an octave. This temperament can be described as 289 & 323 temperament, which tempers out {{monzo| -49 4 22 -3 }} as well as the ragisma. Not only the semitwelfth, but also the ~5/4 can be used as a generator. | ||
Subgroup: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
[[Comma]]: {{monzo| -52 -17 34 }} | [[Comma list]]: {{monzo| -52 -17 34 }} | ||
{{Mapping|legend=1| 17 0 26 | 0 2 1 }} | |||
: mapping generators: ~25/24, ~{{monzo| 26 9 -17 }} | |||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~{{monzo| 26 9 -17 }} = 950.9746 | ||
{{Optimal ET sequence|legend=1| 34, 153, 187, 221, 255, 289, 323, 612, 3349, 3961, 4573, 5185, 5797 }} | {{Optimal ET sequence|legend=1| 34, 153, 187, 221, 255, 289, 323, 612, 3349, 3961, 4573, 5185, 5797 }} | ||
| Line 906: | Line 912: | ||
=== 7-limit === | === 7-limit === | ||
Subgroup: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 4375/4374, | [[Comma list]]: 4375/4374, {{monzo| -49 4 22 -3 }} | ||
{{Mapping|legend=1| 17 0 26 -87 | 0 2 1 10 }} | |||
{{Multival|legend=1| 34 17 170 -52 174 347 }} | {{Multival|legend=1| 34 17 170 -52 174 347 }} | ||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~{{monzo| 24 -5 -9 2 }} = 950.9995 | ||
{{Optimal ET sequence|legend=1| 289, 323, 612, 935, 1547 }} | {{Optimal ET sequence|legend=1| 289, 323, 612, 935, 1547 }} | ||
| Line 925: | Line 931: | ||
Comma list: 4375/4374, 41503/41472, 1879453125/1879048192 | Comma list: 4375/4374, 41503/41472, 1879453125/1879048192 | ||
Mapping: | Mapping: {{mapping| 17 0 26 -87 207 | 0 2 1 10 -11 }} | ||
POTE | Optimal tuning (POTE): ~{{monzo| 24 -5 -9 2 }} = 950.9749 | ||
{{Optimal ET sequence|legend=1| 289, 323, 612 }} | {{Optimal ET sequence|legend=1| 289, 323, 612 }} | ||
| Line 934: | Line 940: | ||
== Seniority == | == Seniority == | ||
{{ | {{See also| Very high accuracy temperaments #Senior }} | ||
Aside from the ragisma, the seniority temperament (26&145) tempers out the wadisma, 201768035/201326592. It is so named because the senior comma ({{monzo|-17 62 -35}}, quadla-sepquingu) is tempered out. | Aside from the ragisma, the seniority temperament (26 & 145) tempers out the wadisma, 201768035/201326592. It is so named because the senior comma ({{monzo| -17 62 -35 }}, quadla-sepquingu) is tempered out. | ||
Subgroup: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 4375/4374, 201768035/201326592 | [[Comma list]]: 4375/4374, 201768035/201326592 | ||
{{Mapping|legend=1| 1 11 19 2 | 0 -35 -62 3 }} | |||
{{Multival|legend=1| 35 62 -3 17 -103 -181 }} | |||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3087/2560 = 322.804 | ||
{{Optimal ET sequence|legend=1| 26, 145, 171, 1513d, 1684d, 1855d, 2026d, 2197d, 2368d, 2539d, 2710d }} | {{Optimal ET sequence|legend=1| 26, 145, 171, 1513d, 1684d, 1855d, 2026d, 2197d, 2368d, 2539d, 2710d }} | ||
| Line 953: | Line 959: | ||
=== Senator === | === Senator === | ||
The senator temperament (26&145) is an 11-limit extension of the seniority, which tempers out 441/440 and 65536/65219. It can be extended to the 13- and 17-limit immediately, by adding 364/363 and 595/594 to the comma list in this order. | The senator temperament (26 & 145) is an 11-limit extension of the seniority, which tempers out 441/440 and 65536/65219. It can be extended to the 13- and 17-limit immediately, by adding 364/363 and 595/594 to the comma list in this order. | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 959: | Line 965: | ||
Comma list: 441/440, 4375/4374, 65536/65219 | Comma list: 441/440, 4375/4374, 65536/65219 | ||
Mapping: | Mapping: {{mapping| 1 11 19 2 4 | 0 -35 -62 3 -2 }} | ||
POTE | Optimal tuning (POTE): ~2 = 1\1, ~77/64 = 322.793 | ||
{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316e, 487ee }} | {{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316e, 487ee }} | ||
| Line 972: | Line 978: | ||
Comma list: 364/363, 441/440, 2200/2197, 4375/4374 | Comma list: 364/363, 441/440, 2200/2197, 4375/4374 | ||
Mapping: | Mapping: {{mapping| 1 11 19 2 4 15 | 0 -35 -62 3 -2 -42 }} | ||
POTE | Optimal tuning (POTE): ~2 = 1\1, ~77/64 = 322.793 | ||
{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316ef, 487eef }} | {{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316ef, 487eef }} | ||
| Line 985: | Line 991: | ||
Comma list: 364/363, 441/440, 595/594, 1156/1155, 2200/2197 | Comma list: 364/363, 441/440, 595/594, 1156/1155, 2200/2197 | ||
Mapping: | Mapping: {{mapping| 1 11 19 2 4 15 17 | 0 -35 -62 3 -2 -42 -48 }} | ||
POTE | Optimal tuning (POTE): ~77/64 = 322.793 | ||
{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316ef, 487eef }} | {{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316ef, 487eef }} | ||
| Line 994: | Line 1,000: | ||
== Monzismic == | == Monzismic == | ||
: ''For the 5-limit version of this temperament, see [[Very high accuracy temperaments #Monzismic]]. | |||
The | The monzismic temperament (53 & 612) tempers out the [[monzisma]], {{monzo| 54 -37 2 }}, and in the 7-limit, the [[nanisma]], {{monzo| 109 -67 0 -1 }}, as well as the ragisma, [[4375/4374]]. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 1,002: | Line 1,008: | ||
[[Comma list]]: 4375/4374, {{monzo| -55 30 2 1 }} | [[Comma list]]: 4375/4374, {{monzo| -55 30 2 1 }} | ||
{{Mapping|legend=1| 1 2 10 -25 | 0 -2 -37 134 }} | |||
{{Multival|legend=1| 2 37 -134 54 -218 -415 }} | {{Multival|legend=1| 2 37 -134 54 -218 -415 }} | ||
[[Optimal tuning]] ([[POTE]]): ~{{monzo| -27 11 3 1 }} = 249.0207 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~{{monzo| -27 11 3 1 }} = 249.0207 | ||
{{Optimal ET sequence|legend=1| 53, …, 559, 612, 1277, 1889 }} | {{Optimal ET sequence|legend=1| 53, …, 559, 612, 1277, 1889 }} | ||
| Line 1,017: | Line 1,023: | ||
Comma list: 4375/4374, 41503/41472, 184549376/184528125 | Comma list: 4375/4374, 41503/41472, 184549376/184528125 | ||
Mapping: | Mapping: {{mapping| 1 2 10 -25 46 | 0 -2 -37 134 -205 }} | ||
Optimal tuning (POTE): ~231/200 = 249.0193 | Optimal tuning (POTE): ~231/200 = 249.0193 | ||
| Line 1,030: | Line 1,036: | ||
Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625 | Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625 | ||
Mapping: | Mapping: {{mapping| 1 2 10 -25 46 23 | 0 -2 -37 134 -205 -93 }} | ||
Optimal tuning (POTE): ~231/200 = 249.0199 | Optimal tuning (POTE): ~231/200 = 249.0199 | ||
| Line 1,041: | Line 1,047: | ||
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Semidimfourth]].'' | : ''For the 5-limit version of this temperament, see [[High badness temperaments #Semidimfourth]].'' | ||
The | The semidimfourth temperament is featured by a semi-diminished fourth inverval which is [[128/125]] above the pythagorean major third [[81/64]]. In the 7-limit, this temperament tempers out the ragisma and the triwellisma, 235298/234375. | ||
Subgroup: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 4375/4374, 235298/234375 | [[Comma list]]: 4375/4374, 235298/234375 | ||
[[Mapping]]: | [[Mapping]]: {{mapping| 1 21 28 36 | 0 -31 -41 -53 }} | ||
{{Multival|legend=1| 31 41 53 -7 -3 8 }} | |||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/27 = 448.456 | ||
{{Optimal ET sequence|legend=1| 8d, 91, 99, 289, 388, 875, 1263d, 1651d }} | {{Optimal ET sequence|legend=1| 8d, 91, 99, 289, 388, 875, 1263d, 1651d }} | ||
| Line 1,062: | Line 1,068: | ||
Comma list: 3025/3024, 4375/4374, 235298/234375 | Comma list: 3025/3024, 4375/4374, 235298/234375 | ||
Mapping: | Mapping: {{mapping| 2 11 15 19 15 | 0 -31 -41 -53 -32 }} | ||
POTE | Optimal tuning (POTE): ~99/70 = 1\2, ~12/11 = 151.547 | ||
{{Optimal ET sequence|legend=1| 8d, 190, 388 }} | {{Optimal ET sequence|legend=1| 8d, 190, 388 }} | ||
| Line 1,075: | Line 1,081: | ||
Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374 | Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374 | ||
Mapping: | Mapping: {{mapping| 2 11 15 19 15 17 | 0 -31 -41 -53 -32 -38 }} | ||
POTE | Optimal tuning (POTE): ~99/70 = 1\2, ~12/11 = 151.545 | ||
{{Optimal ET sequence|legend=1| 8d, 190, 198, 388 }} | {{Optimal ET sequence|legend=1| 8d, 190, 198, 388 }} | ||
| Line 1,084: | Line 1,090: | ||
== Acrokleismic == | == Acrokleismic == | ||
Subgroup: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 4375/4374, 2202927104/2197265625 | [[Comma list]]: 4375/4374, 2202927104/2197265625 | ||
{{Mapping|legend=1| 1 10 11 27 | 0 -32 -33 -92 }} | |||
: mapping generators: ~2, ~6/5 | |||
{{Multival|legend=1| 32 33 92 -22 56 121 }} | |||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.557 | ||
{{Optimal ET sequence|legend=1| 19, 251, 270 }} | {{Optimal ET sequence|legend=1| 19, …, 251, 270, 2449c, 2719c, 2989bc }} | ||
[[Badness]]: 0.056184 | [[Badness]]: 0.056184 | ||
| Line 1,103: | Line 1,111: | ||
Comma list: 4375/4374, 41503/41472, 172032/171875 | Comma list: 4375/4374, 41503/41472, 172032/171875 | ||
Mapping: | Mapping: {{mapping| 1 10 11 27 -16 | 0 -32 -33 -92 74 }} | ||
POTE | Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.558 | ||
{{Optimal ET sequence|legend=1| 19, 251, 270, 829, 1099, 1369, 1639 }} | {{Optimal ET sequence|legend=1| 19, 251, 270, 829, 1099, 1369, 1639 }} | ||
| Line 1,116: | Line 1,124: | ||
Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976 | Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976 | ||
Mapping: | Mapping: {{mapping| 1 10 11 27 -16 25 | 0 -32 -33 -92 74 -81 }} | ||
POTE | Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.557 | ||
{{Optimal ET sequence|legend=1| 19, 251, 270 }} | {{Optimal ET sequence|legend=1| 19, 251, 270 }} | ||
| Line 1,129: | Line 1,137: | ||
Comma list: 4375/4374, 5632/5625, 117649/117612 | Comma list: 4375/4374, 5632/5625, 117649/117612 | ||
Mapping: | Mapping: {{mapping| 1 10 11 27 55 | 0 -32 -33 -92 -196 }} | ||
POTE | Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.553 | ||
{{Optimal ET sequence|legend=1| 19e, 251e, 270, 1061e, 1331c, 1601c, 1871bc, 4012bcde }} | {{Optimal ET sequence|legend=1| 19e, 251e, 270, 1061e, 1331c, 1601c, 1871bc, 4012bcde }} | ||
| Line 1,142: | Line 1,150: | ||
Comma list: 676/675, 1716/1715, 4225/4224, 4375/4374 | Comma list: 676/675, 1716/1715, 4225/4224, 4375/4374 | ||
Mapping: | Mapping: {{mapping| 1 10 11 27 55 25 | 0 -32 -33 -92 -196 -81 }} | ||
POTE | Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.554 | ||
{{Optimal ET sequence|legend=1| 19e, 251e, 270, 1331c, 1601c, 1871bcf, 2141bcf }} | {{Optimal ET sequence|legend=1| 19e, 251e, 270, 1331c, 1601c, 1871bcf, 2141bcf }} | ||
| Line 1,151: | Line 1,159: | ||
== Quasithird == | == Quasithird == | ||
The | The quasithird temperament is featured by a major third interval which is 1600000/1594323 ([[amity comma]]) or 5120/5103 ([[5120/5103|hemifamity comma]]) below the just major third [[5/4]] as a generator, five of which give a fifth with octave reduction. This temperament has a period of a quarter octave, which allows to temper out the [[4375/4374|ragisma]] and {{monzo|-60 29 0 5}}. | ||
Subgroup: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
[[Comma]]: {{monzo| 55 -64 20 }} | [[Comma list]]: {{monzo| 55 -64 20 }} | ||
{{Mapping|legend=1| 4 0 -11 | 0 5 16 }} | |||
: mapping generators: ~51200000/43046721, ~1594323/1280000 | |||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~51200000/43046721, ~1594323/1280000 = 380.395 | ||
{{Optimal ET sequence|legend=1| 60, 104c, 164, 224, 388, 612, 1612, 2224, 2836, 6284, 9120, 15404 }} | {{Optimal ET sequence|legend=1| 60, 104c, 164, 224, 388, 612, 1612, 2224, 2836, 6284, 9120, 15404 }} | ||
| Line 1,168: | Line 1,176: | ||
=== 7-limit === | === 7-limit === | ||
Subgroup: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 4375/4374, | [[Comma list]]: 4375/4374, {{monzo| -60 29 0 5 }} | ||
{{Mapping|legend=1| 4 0 -11 48 | 0 5 16 -29 }} | |||
{{Multival|legend=1| 20 64 -116 55 -240 -449 }} | {{Multival|legend=1| 20 64 -116 55 -240 -449 }} | ||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~65536/55125 = 1\4, ~5103/4096 = 380.388 | ||
{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 1448, 2060 }} | {{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 1448, 2060 }} | ||
| Line 1,187: | Line 1,195: | ||
Comma list: 3025/3024, 4375/4374, 4296700485/4294967296 | Comma list: 3025/3024, 4375/4374, 4296700485/4294967296 | ||
Mapping: | Mapping: {{mapping| 4 0 -11 48 43 | 0 5 16 -29 -23 }} | ||
POTE | Optimal tuning (POTE): ~5103/4096 = 380.387 (or ~22/21 = 80.387) | ||
{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 836, 1448 }} | {{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 836, 1448 }} | ||
| Line 1,200: | Line 1,208: | ||
Comma list: 2200/2197, 3025/3024, 4096/4095, 4375/4374 | Comma list: 2200/2197, 3025/3024, 4096/4095, 4375/4374 | ||
Mapping: | Mapping: {{mapping| 4 0 -11 48 43 11 | 0 5 16 -29 -23 3 }} | ||
POTE | Optimal tuning (POTE): ~81/65 = 380.385 (or ~22/21 = 80.385) | ||
{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 836, 1448f, 2284f }} | {{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 836, 1448f, 2284f }} | ||
| Line 1,209: | Line 1,217: | ||
== Deca == | == Deca == | ||
: ''For 5-limit version of this temperament, see [[10th-octave temperaments #Neon]].'' | |||
Deca temperament has a period of 1/10 octave and tempers out the [[linus comma]], {{monzo| 11 -10 -10 10 }}, neon comma {{monzo| 21 60 -50 }} and {{monzo| 12 -3 -14 9 }} = 165288374272/164794921875 (satritrizo-asepbigu). | |||
Deca temperament has a period of 1/10 octave and tempers out the [[ | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 1,218: | Line 1,225: | ||
[[Comma list]]: 4375/4374, 165288374272/164794921875 | [[Comma list]]: 4375/4374, 165288374272/164794921875 | ||
{{Mapping|legend=1| 10 4 9 2 | 0 5 6 11 }} | |||
: mapping generators: ~15/14, ~6/5 | |||
{{Multival|legend=1| 50 60 110 -21 34 87 }} | {{Multival|legend=1| 50 60 110 -21 34 87 }} | ||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~15/14 = 1\10, ~6/5 = 315.577 | ||
{{Optimal ET sequence|legend=1| 80, 190, 270, 1270, 1540, 1810, 2080 }} | {{Optimal ET sequence|legend=1| 80, 190, 270, 1270, 1540, 1810, 2080 }} | ||
| Line 1,233: | Line 1,242: | ||
Comma list: 3025/3024, 4375/4374, 391314/390625 | Comma list: 3025/3024, 4375/4374, 391314/390625 | ||
Mapping: | Mapping: {{mapping| 10 4 9 2 18 | 0 5 6 11 7 }} | ||
POTE | Optimal tuning (POTE): ~15/14 = 1\10, ~6/5 = 315.582 | ||
{{Optimal ET sequence|legend=1| 80, 190, 270, 1000, 1270, 1540e, 1810e }} | {{Optimal ET sequence|legend=1| 80, 190, 270, 1000, 1270, 1540e, 1810e }} | ||
| Line 1,246: | Line 1,255: | ||
Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374 | Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374 | ||
Mapping: | Mapping: {{mapping| 10 4 9 2 18 37 | 0 5 6 11 7 0 }} | ||
POTE | Optimal tuning (POTE): ~15/14 = 1\10, ~6/5 = 315.602 | ||
{{Optimal ET sequence|legend=1| 80, 190, 270, 730, 1000 }} | {{Optimal ET sequence|legend=1| 80, 190, 270, 730, 1000 }} | ||
| Line 1,261: | Line 1,270: | ||
[[Comma list]]: 4375/4374, {{monzo| -56 1 -8 26 }} | [[Comma list]]: 4375/4374, {{monzo| -56 1 -8 26 }} | ||
{{Mapping|legend=1| 1 2 3 3 | 0 -112 -183 -52 }} | |||
: mapping generators: ~2, ~{{monzo| 21 3 1 -10 }} | |||
[[Optimal tuning]] ([[CTE]]): ~ | [[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~{{monzo| 21 3 1 -10 }} = 4.4465 | ||
{{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 2159, 4048, 18081cd }} | {{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 2159, 4048, 18081cd }} | ||
| Line 1,274: | Line 1,285: | ||
Comma list: 4375/4374, 117649/117612, 67110351/67108864 | Comma list: 4375/4374, 117649/117612, 67110351/67108864 | ||
Mapping: | Mapping: {{mapping| 1 2 3 3 3 | 0 -112 -183 -52 124 }} | ||
Optimal tuning (CTE): ~385/384 = 4.4465 | Optimal tuning (CTE): ~2 = 1\1, ~385/384 = 4.4465 | ||
{{Optimal ET sequence|legend=1| 270, 1349, 1619, 1889, 2159, 11065, 13224 }} | {{Optimal ET sequence|legend=1| 270, 1349, 1619, 1889, 2159, 11065, 13224 }} | ||
| Line 1,287: | Line 1,298: | ||
Comma list: 4225/4224, 4375/4374, 6656/6655, 117649/117612 | Comma list: 4225/4224, 4375/4374, 6656/6655, 117649/117612 | ||
Mapping: | Mapping: {{mapping| 1 2 3 3 3 3 | 0 -112 -183 -52 124 189 }} | ||
Optimal tuning (CTE): ~385/384 = 4.4466 | Optimal tuning (CTE): ~2 = 1\1, ~385/384 = 4.4466 | ||
{{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 4048 }} | {{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 4048 }} | ||
| Line 1,296: | Line 1,307: | ||
== Aluminium == | == Aluminium == | ||
Aluminium is named after the 13th element, and tempers out the {{monzo| 92 -39 -13 }} comma which sets [[135/128]] interval to be equal to 1/13th of the octave. | |||
[[Subgroup]]: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
| Line 1,302: | Line 1,313: | ||
[[Comma list]]: {{monzo| 92 -39 -13 }} | [[Comma list]]: {{monzo| 92 -39 -13 }} | ||
[[Mapping]]: | [[Mapping]]: {{mapping| 13 0 92 | 0 1 -3 }} | ||
: mapping generators: ~135/128, ~3 | |||
[[Optimal tuning]] ([[CTE]]): ~135/128 = 1\13, ~3/2 = 701.9897 | [[Optimal tuning]] ([[CTE]]): ~135/128 = 1\13, ~3/2 = 701.9897 | ||
| Line 1,317: | Line 1,328: | ||
[[Comma list]]: 4375/4374, {{monzo| 92 -39 -13 }} | [[Comma list]]: 4375/4374, {{monzo| 92 -39 -13 }} | ||
[[Mapping]]: | [[Mapping]]: {{mapping| 13 0 92 -355 | 0 1 -3 19 }} | ||
[[Optimal tuning]] ([[CTE]]): ~135/128 = 1\13, ~3/2 = 702.0024 | [[Optimal tuning]] ([[CTE]]): ~135/128 = 1\13, ~3/2 = 702.0024 | ||
| Line 1,330: | Line 1,341: | ||
Comma list: 4375/4374, 234375/234256, 2097152/2096325 | Comma list: 4375/4374, 234375/234256, 2097152/2096325 | ||
Mapping: | Mapping: {{mapping| 13 0 92 -355 148 | 0 1 -3 19 -5 }} | ||
Optimal tuning (CTE): ~135/128 = 1\13, ~3/2 = 702.0042 | Optimal tuning (CTE): ~135/128 = 1\13, ~3/2 = 702.0042 | ||
| Line 1,343: | Line 1,354: | ||
Comma list: 4096/4095, 4375/4374, 6656/6655, 78125/78078 | Comma list: 4096/4095, 4375/4374, 6656/6655, 78125/78078 | ||
Mapping: | Mapping: {{mapping| 13 0 92 -355 148 419 | 0 1 -3 19 -5 -18 }} | ||
Optimal tuning (CTE): ~135/128 = 1\13, ~3/2 = 702.0099 | Optimal tuning (CTE): ~135/128 = 1\13, ~3/2 = 702.0099 | ||
| Line 1,352: | Line 1,363: | ||
== Countritonic == | == Countritonic == | ||
:''For the 5-limit version of this temperament, see [[Schismic-Mercator equivalence continuum #Countritonic]] and [[High badness temperaments #Countritonic]] | : ''For the 5-limit version of this temperament, see [[Schismic-Mercator equivalence continuum #Countritonic]] and [[High badness temperaments #Countritonic]] | ||
Countritonic (''co-un-tritonic'') can be described as the 53 & 422 temperament, generated by an octave-reduced 91st harmonic or subharmonic in the 13-limit. | Countritonic (''co-un-tritonic'') can be described as the 53 & 422 temperament, generated by an octave-reduced 91st harmonic or subharmonic in the 13-limit. | ||
| Line 1,399: | Line 1,410: | ||
{{See also| Stratosphere }} | {{See also| Stratosphere }} | ||
Subgroup: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 4375/4374, {{monzo| -32 5 14 -3 }} | |||
{{Mapping|legend=1| 2 7 7 23 | 0 -13 -8 -59 }} | |||
: mapping generators: ~2278125/1605632, ~448/405 | |||
{{Multival|legend=1| 26 16 118 -35 114 229 }} | {{Multival|legend=1| 26 16 118 -35 114 229 }} | ||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~2278125/1605632 = 1\2, ~448/405 = 176.805 | ||
{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1052c, 1690bcc }} | {{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1052c, 1690bcc }} | ||
| Line 1,418: | Line 1,431: | ||
Comma list: 3025/3024, 4375/4374, 1265625/1261568 | Comma list: 3025/3024, 4375/4374, 1265625/1261568 | ||
Mapping: | Mapping: {{mapping| 2 7 7 23 19 | 0 -13 -8 -59 -41 }} | ||
POTE | Optimal tuning (POTE): ~99/70 = 1\2, ~448/405 = 176.806 | ||
{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1052c }} | {{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1052c }} | ||
| Line 1,431: | Line 1,444: | ||
Comma list: 625/624, 729/728, 1575/1573, 2200/2197 | Comma list: 625/624, 729/728, 1575/1573, 2200/2197 | ||
Mapping: | Mapping: {{mapping| 2 7 7 23 19 13 | 0 -13 -8 -59 -41 -19 }} | ||
POTE | Optimal tuning (POTE): ~99/70 = 1\2, ~195/176 = 176.804 | ||
{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1690bcc, 2328bccde }} | {{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1690bcc, 2328bccde }} | ||
| Line 1,446: | Line 1,459: | ||
[[Comma list]]: 4375/4374, {{monzo| -88 2 45 -7 }} | [[Comma list]]: 4375/4374, {{monzo| -88 2 45 -7 }} | ||
{{Mapping|legend=1| 1 57 38 248 | 0 -73 -47 -323 }} | |||
: mapping generators: ~2, ~6422528/3796875 | |||
[[Optimal tuning]] ([[CTE]]): ~ | [[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6422528/3796875 = 910.9323 | ||
{{Optimal ET sequence|legend=1| 494, 1125, 1619 }} | {{Optimal ET sequence|legend=1| 494, 1125, 1619 }} | ||
| Line 1,459: | Line 1,474: | ||
Comma list: 4375/4374, 759375/758912, 100663296/100656875 | Comma list: 4375/4374, 759375/758912, 100663296/100656875 | ||
Mapping: | Mapping: {{mapping| 1 57 38 248 -14 | 0 -73 -47 -323 23 }} | ||
Optimal tuning (CTE): ~ | Optimal tuning (CTE): ~2 = 1\1, ~1024/605 = 910.9323 | ||
{{Optimal ET sequence|legend=1| 494, 1125, 1619, 2113 }} | {{Optimal ET sequence|legend=1| 494, 1125, 1619, 2113 }} | ||
| Line 1,474: | Line 1,489: | ||
Comma list: 4225/4224, 4375/4374, 6656/6655, 78125/78078 | Comma list: 4225/4224, 4375/4374, 6656/6655, 78125/78078 | ||
Mapping: | Mapping: {{mapping| 1 57 38 248 -14 -13 | 0 -73 -47 -323 23 22 }} | ||
Optimal tuning (CTE): ~22/13 = 910.9323 | Optimal tuning (CTE): ~2 = 1\1, ~22/13 = 910.9323 | ||
{{Optimal ET sequence|legend=1| 494, 1125, 1619, 2113 }} | {{Optimal ET sequence|legend=1| 494, 1125, 1619, 2113 }} | ||
| Line 1,484: | Line 1,499: | ||
== Palladium == | == Palladium == | ||
: ''For the 5-limit version of this temperament, see [[46th-octave temperaments]]''. | : ''For the 5-limit version of this temperament, see [[46th-octave temperaments]]''. | ||
The name of the ''palladium'' temperament comes from palladium, the 46th element. Palladium has a period of 1/46 octave. It tempers out the 46-9/5-comma, {{monzo| -39 92 -46 }}, by which 46 minortones (10/9) fall short of seven octaves. This temperament can be described as 46&414 temperament, which tempers out {{monzo| -51 8 2 12 }} as well as the ragisma. | |||
The name of the ''palladium'' temperament comes from palladium, the 46th element. Palladium has a period of 1/46 octave. It tempers out the 46-9/5-comma, {{monzo| -39 92 -46 }}, by which 46 minortones (10/9) fall short of seven octaves. This temperament can be described as 46 & 414 temperament, which tempers out {{monzo| -51 8 2 12 }} as well as the ragisma. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 4375/4374, | [[Comma list]]: 4375/4374, {{monzo| -51 8 2 12 }} | ||
{{Mapping|legend=1| 46 73 107 129 | 0 -1 -2 1 }} | |||
: mapping generators: ~83349/81920, ~3 | |||
{{Multival|legend=1| 46 92 -46 39 -202 -365 }} | {{Multival|legend=1| 46 92 -46 39 -202 -365 }} | ||
[[Optimal tuning]] ([[POTE]]): ~3/2 = 701.6074 | [[Optimal tuning]] ([[POTE]]): ~83349/81920 = 1\46, ~3/2 = 701.6074 | ||
{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874d }} | {{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874d }} | ||
| Line 1,505: | Line 1,523: | ||
Comma list: 3025/3024, 4375/4374, 134775333/134217728 | Comma list: 3025/3024, 4375/4374, 134775333/134217728 | ||
Mapping: | Mapping: {{mapping| 46 73 107 129 159 | 0 -1 -2 1 1 }} | ||
Optimal tuning (POTE): ~3/2 = 701.5951 | Optimal tuning (POTE): ~8192/8085 = 1\46, ~3/2 = 701.5951 | ||
{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874de }} | {{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874de }} | ||
| Line 1,518: | Line 1,536: | ||
Comma list: 3025/3024, 4225/4224, 4375/4374, 26411/26364 | Comma list: 3025/3024, 4225/4224, 4375/4374, 26411/26364 | ||
Mapping: | Mapping: {{mapping| 46 73 107 129 159 170 | 0 -1 -2 1 1 2 }} | ||
Optimal tuning (POTE): ~3/2 = 701.6419 | Optimal tuning (POTE): ~65/64 = 1\46, ~3/2 = 701.6419 | ||
{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874de, 1334de }} | {{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874de, 1334de }} | ||
| Line 1,531: | Line 1,549: | ||
Comma list: 833/832, 1089/1088, 1225/1224, 1701/1700, 4225/4224 | Comma list: 833/832, 1089/1088, 1225/1224, 1701/1700, 4225/4224 | ||
Mapping: | Mapping: {{mapping| 46 73 107 129 159 170 188 | 0 -1 -2 1 1 2 0 }} | ||
Optimal tuning (POTE): ~3/2 = 701.6425 | Optimal tuning (POTE): ~65/64 = 1\46, ~3/2 = 701.6425 | ||
{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874de, 1334deg }} | {{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874de, 1334deg }} | ||
| Line 1,548: | Line 1,566: | ||
[[Comma list]]: 4375/4374, {{monzo| -100 53 48 -34 }} | [[Comma list]]: 4375/4374, {{monzo| -100 53 48 -34 }} | ||
{{Mapping|legend=1| 1 50 51 147 | 0 -184 -185 -548 }} | |||
: mapping generators: ~2, ~6/5 | |||
[[Optimal tuning]] ([[CTE]]): ~6/5 = 315.7501 | [[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6/5 = 315.7501 | ||
{{Optimal ET sequence|legend=1| 19, …, 1600, 1619, 4838, 6457c }} | {{Optimal ET sequence|legend=1| 19, …, 1600, 1619, 4838, 6457c }} | ||
| Line 1,557: | Line 1,577: | ||
== Octoid == | == Octoid == | ||
The | The octoid temperament has a period of 1/8 octave and tempers out 4375/4374 ([[4375/4374|ragisma]]) and 16875/16807 ([[16875/16807|mirkwai]]). In the 11-limit, it tempers out 540/539, 1375/1372, and 6250/6237. In this temperament, one period gives both 12/11 and 49/45, two gives 25/21, three gives 35/27, and four gives both 99/70 and 140/99. | ||
Subgroup: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 4375/4374, 16875/16807 | [[Comma list]]: 4375/4374, 16875/16807 | ||
{{Mapping|legend=1| 8 1 3 3 | 0 3 4 5 }} | |||
{{Multival|legend=1| 24 32 40 -5 -4 3 }} | |||
: mapping generators: ~49/45, ~7/5 | |||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~49/45 = 1\8, ~7/5 = 583.940 | ||
[[Tuning ranges]]: | [[Tuning ranges]]: | ||
| Line 1,583: | Line 1,603: | ||
[[Badness]]: 0.042670 | [[Badness]]: 0.042670 | ||
Scales: [[ | Scales: [[octoid72]], [[octoid80]] | ||
=== 11-limit === | === 11-limit === | ||
| Line 1,590: | Line 1,610: | ||
Comma list: 540/539, 1375/1372, 4000/3993 | Comma list: 540/539, 1375/1372, 4000/3993 | ||
Mapping: | Mapping: {{mapping| 8 1 3 3 16 | 0 3 4 5 3 }} | ||
POTE | Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.962 | ||
Tuning ranges: | Tuning ranges: | ||
| Line 1,603: | Line 1,623: | ||
Badness: 0.014097 | Badness: 0.014097 | ||
Scales: [[ | Scales: [[octoid72]], [[octoid80]] | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
| Line 1,610: | Line 1,630: | ||
Comma list: 540/539, 625/624, 729/728, 1375/1372 | Comma list: 540/539, 625/624, 729/728, 1375/1372 | ||
Mapping: | Mapping: {{mapping| 8 1 3 3 16 -21 | 0 3 4 5 3 13 }} | ||
POTE | Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.905 | ||
{{Optimal ET sequence|legend=1| 72, 152f, 224 }} | {{Optimal ET sequence|legend=1| 72, 152f, 224 }} | ||
| Line 1,618: | Line 1,638: | ||
Badness: 0.015274 | Badness: 0.015274 | ||
Scales: [[ | Scales: [[octoid72]], [[octoid80]] | ||
; Music | ; Music | ||
* [https://www.archive.org/details/Dreyfus | * [https://www.archive.org/details/Dreyfus ''Dreyfus''] [https://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play] by [[Gene Ward Smith]] | ||
===== 17-limit ===== | ===== 17-limit ===== | ||
| Line 1,628: | Line 1,648: | ||
Comma list: 375/374, 540/539, 625/624, 715/714, 729/728 | Comma list: 375/374, 540/539, 625/624, 715/714, 729/728 | ||
Mapping: | Mapping: {{mapping| 8 1 3 3 16 -21 -14 | 0 3 4 5 3 13 12 }} | ||
POTE | Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.842 | ||
{{Optimal ET sequence|legend=1| 72, 152fg, 224, 296, 520g }} | {{Optimal ET sequence|legend=1| 72, 152fg, 224, 296, 520g }} | ||
| Line 1,636: | Line 1,656: | ||
Badness: 0.014304 | Badness: 0.014304 | ||
Scales: [[ | Scales: [[octoid72]], [[octoid80]] | ||
===== 19-limit ===== | ===== 19-limit ===== | ||
| Line 1,643: | Line 1,663: | ||
Comma list: 324/323, 375/374, 400/399, 495/494, 540/539, 715/714 | Comma list: 324/323, 375/374, 400/399, 495/494, 540/539, 715/714 | ||
Mapping: | Mapping: {{mapping| 8 1 3 3 16 -21 -14 34 | 0 3 4 5 3 13 12 0 }} | ||
POTE | Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.932 | ||
{{Optimal ET sequence|legend=1| 72, 152fg, 224 }} | {{Optimal ET sequence|legend=1| 72, 152fg, 224 }} | ||
| Line 1,651: | Line 1,671: | ||
Badness: 0.016036 | Badness: 0.016036 | ||
Scales: [[ | Scales: [[octoid72]], [[octoid80]] | ||
==== Octopus ==== | ==== Octopus ==== | ||
| Line 1,658: | Line 1,678: | ||
Comma list: 169/168, 325/324, 364/363, 540/539 | Comma list: 169/168, 325/324, 364/363, 540/539 | ||
Mapping: | Mapping: {{mapping| 8 1 3 3 16 14 | 0 3 4 5 3 4 }} | ||
POTE | Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.892 | ||
{{Optimal ET sequence|legend=1| 72, 152, 224f }} | {{Optimal ET sequence|legend=1| 72, 152, 224f }} | ||
| Line 1,666: | Line 1,686: | ||
Badness: 0.021679 | Badness: 0.021679 | ||
Scales: [[ | Scales: [[octoid72]], [[octoid80]] | ||
===== 17-limit ===== | ===== 17-limit ===== | ||
| Line 1,673: | Line 1,693: | ||
Comma list: 169/168, 221/220, 289/288, 325/324, 540/539 | Comma list: 169/168, 221/220, 289/288, 325/324, 540/539 | ||
Mapping: | Mapping: {{mapping| 8 1 3 3 16 14 21 | 0 3 4 5 3 4 3 }} | ||
POTE | Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.811 | ||
{{Optimal ET sequence|legend=1| 72, 152, 224fg, 296ffg }} | {{Optimal ET sequence|legend=1| 72, 152, 224fg, 296ffg }} | ||
| Line 1,688: | Line 1,708: | ||
Comma list: 169/168, 221/220, 286/285, 289/288, 325/324, 400/399 | Comma list: 169/168, 221/220, 286/285, 289/288, 325/324, 400/399 | ||
Mapping: | Mapping: {{mapping| 8 1 3 3 16 14 21 34 | 0 3 4 5 3 4 3 0 }} | ||
POTE | Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 584.064 | ||
{{Optimal ET sequence|legend=1| 72, 152, 224fg, 376ffgh }} | {{Optimal ET sequence|legend=1| 72, 152, 224fg, 376ffgh }} | ||
| Line 1,699: | Line 1,719: | ||
==== Hexadecoid ==== | ==== Hexadecoid ==== | ||
Hexadecoid (80&144) has a period of 1/16 octave and tempers out 4225/4224. | Hexadecoid (80 & 144) has a period of 1/16 octave and tempers out 4225/4224. | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 1,705: | Line 1,725: | ||
Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224 | Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224 | ||
Mapping: | Mapping: {{mapping| 16 26 38 46 56 59 | 0 -3 -4 -5 -3 1 }} | ||
: mapping generators: ~448/429, ~7/5 | |||
POTE | Optimal tuning (POTE): ~448/429 = 1\16, ~13/8 = 841.015 | ||
{{Optimal ET sequence|legend=1| 80, 144, 224 }} | {{Optimal ET sequence|legend=1| 80, 144, 224 }} | ||
| Line 1,718: | Line 1,740: | ||
Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224 | Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224 | ||
Mapping: | Mapping: {{mapping| 16 26 38 46 56 59 65 | 0 -3 -4 -5 -3 1 2 }} | ||
POTE | Optimal tuning (POTE): ~117/112 = 1\16, ~13/8 = 840.932 | ||
{{Optimal ET sequence|legend=1| 80, 144, 224, 528dg }} | {{Optimal ET sequence|legend=1| 80, 144, 224, 528dg }} | ||
| Line 1,731: | Line 1,753: | ||
Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444 | Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444 | ||
Mapping: | Mapping: {{mapping| 16 26 38 46 56 59 65 68 | 0 -3 -4 -5 -3 1 2 0 }} | ||
POTE | Optimal tuning (POTE): ~117/112 = 1\16, ~13/8 = 840.896 | ||
{{Optimal ET sequence|legend=1| 80, 144, 224, 304dh, 528dghh }} | {{Optimal ET sequence|legend=1| 80, 144, 224, 304dh, 528dghh }} | ||
| Line 1,742: | Line 1,764: | ||
{{Main| Parakleismic }} | {{Main| Parakleismic }} | ||
In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, {{monzo|8 14 -13}}, with the [[118edo | In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, {{monzo| 8 14 -13 }}, with the [[118edo]] tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat 6/5, 13 of which give 32/3, and 14 give 64/5. However while 118 no longer has better than a cent of accuracy in the 7- or 11-limit, it is a decent temperament there nonetheless, and this allows an extension, with the 7-limit wedgie being {{multival| 13 14 35 -8 19 42 }} and adding 3136/3125 and 4375/4374, and the 11-limit wedgie {{multival| 13 14 35 -36 -8 19 -102 42 -132 -222 }} adding 385/384. For the 7-limit [[99edo]] may be preferred, but in the 11-limit it is best to stick with 118. | ||
Subgroup: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
[[Comma list]]: 1224440064/1220703125 | [[Comma list]]: 1224440064/1220703125 | ||
{{Mapping|legend=1| 1 5 6 | 0 -13 -14 }} | |||
: mapping generators: ~2, ~6/5 | |||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.240 | ||
{{Optimal ET sequence|legend=1| 19, 61, 80, 99, 118, 453, 571, 689, 1496 }} | {{Optimal ET sequence|legend=1| 19, 61, 80, 99, 118, 453, 571, 689, 1496 }} | ||
| Line 1,761: | Line 1,785: | ||
[[Comma list]]: 3136/3125, 4375/4374 | [[Comma list]]: 3136/3125, 4375/4374 | ||
{{Mapping|legend=1| 1 5 6 12 | 0 -13 -14 -35 }} | |||
{{Multival|legend=1| 13 14 35 -8 19 42 }} | |||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.181 | ||
{{Optimal ET sequence|legend=1| 19, 80, 99, 217, 316, 415 }} | {{Optimal ET sequence|legend=1| 19, 80, 99, 217, 316, 415 }} | ||
| Line 1,776: | Line 1,800: | ||
Comma list: 385/384, 3136/3125, 4375/4374 | Comma list: 385/384, 3136/3125, 4375/4374 | ||
Mapping: | Mapping: {{mapping| 1 5 6 12 -6 | 0 -13 -14 -35 36 }} | ||
POTE | Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.251 | ||
{{Optimal ET sequence|legend=1| 19, 99, 118 }} | {{Optimal ET sequence|legend=1| 19, 99, 118 }} | ||
| Line 1,785: | Line 1,809: | ||
=== Paralytic === | === Paralytic === | ||
The ''paralytic'' temperament (118&217) tempers out 441/440, 5632/5625, and 19712/19683. In 13-limit, 118&217 tempers out 1001/1000, 1575/1573, and 3584/3575. | The ''paralytic'' temperament (118&217) tempers out 441/440, 5632/5625, and 19712/19683. In 13-limit, 118 & 217 tempers out 1001/1000, 1575/1573, and 3584/3575. | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 1,791: | Line 1,815: | ||
Comma list: 441/440, 3136/3125, 4375/4374 | Comma list: 441/440, 3136/3125, 4375/4374 | ||
Mapping: | Mapping: {{mapping| 1 5 6 12 25 | 0 -13 -14 -35 -82 }} | ||
POTE | Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.220 | ||
{{Optimal ET sequence|legend=1| 19e, 99e, 118, 217, 335, 552d, 887dd }} | {{Optimal ET sequence|legend=1| 19e, 99e, 118, 217, 335, 552d, 887dd }} | ||
| Line 1,804: | Line 1,828: | ||
Comma list: 441/440, 1001/1000, 3136/3125, 4375/4374 | Comma list: 441/440, 1001/1000, 3136/3125, 4375/4374 | ||
Mapping: | Mapping: {{mapping| 1 5 6 12 25 -16 | 0 -13 -14 -35 -82 75 }} | ||
POTE | Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.214 | ||
{{Optimal ET sequence|legend=1| 99e, 118, 217, 552d, 769de }} | {{Optimal ET sequence|legend=1| 99e, 118, 217, 552d, 769de }} | ||
| Line 1,813: | Line 1,837: | ||
==== Paraklein ==== | ==== Paraklein ==== | ||
The ''paraklein'' temperament (19e&118) is another 13-limit extension of paralytic, which equates [[13/11]] with [[32/27]], [[14/13]] with [[15/14]], [[25/24]] with [[26/25]], and [[27/26]] with [[28/27]]. | The ''paraklein'' temperament (19e & 118) is another 13-limit extension of paralytic, which equates [[13/11]] with [[32/27]], [[14/13]] with [[15/14]], [[25/24]] with [[26/25]], and [[27/26]] with [[28/27]]. | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 1,819: | Line 1,843: | ||
Comma list: 196/195, 352/351, 625/624, 729/728 | Comma list: 196/195, 352/351, 625/624, 729/728 | ||
Mapping: | Mapping: {{mapping| 1 5 6 12 25 15 | 0 -13 -14 -35 -82 -43 }} | ||
POTE | Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.225 | ||
{{Optimal ET sequence|legend=1| 19e, 99ef, 118, 217ff, 335ff }} | {{Optimal ET sequence|legend=1| 19e, 99ef, 118, 217ff, 335ff }} | ||
| Line 1,832: | Line 1,856: | ||
Comma list: 176/175, 1375/1372, 2200/2187 | Comma list: 176/175, 1375/1372, 2200/2187 | ||
Mapping: | Mapping: {{mapping| 1 5 6 12 20 | 0 -13 -14 -35 -63 }} | ||
POTE | Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.060 | ||
{{Optimal ET sequence|legend=1| 19e, 80, 179, 259cd }} | {{Optimal ET sequence|legend=1| 19e, 80, 179, 259cd }} | ||
| Line 1,845: | Line 1,869: | ||
Comma list: 169/168, 176/175, 325/324, 1375/1372 | Comma list: 169/168, 176/175, 325/324, 1375/1372 | ||
Mapping: | Mapping: {{mapping| 1 5 6 12 20 10 | 0 -13 -14 -35 -63 -24 }} | ||
POTE | Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.075 | ||
{{Optimal ET sequence|legend=1| 19e, 80, 179 }} | {{Optimal ET sequence|legend=1| 19e, 80, 179 }} | ||
| Line 1,858: | Line 1,882: | ||
Comma list: 540/539, 896/891, 3136/3125 | Comma list: 540/539, 896/891, 3136/3125 | ||
Mapping: | Mapping: {{mapping| 1 5 6 12 -1 | 0 -13 -14 -35 17 }} | ||
POTE | Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.096 | ||
{{Optimal ET sequence|legend=1| 19, 61d, 80, 99e, 179e }} | {{Optimal ET sequence|legend=1| 19, 61d, 80, 99e, 179e }} | ||
| Line 1,871: | Line 1,895: | ||
Comma list: 169/168, 325/324, 540/539, 832/825 | Comma list: 169/168, 325/324, 540/539, 832/825 | ||
Mapping: | Mapping: {{mapping| 1 5 6 12 -1 10 | 0 -13 -14 -35 17 -24 }} | ||
POTE | Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.080 | ||
{{Optimal ET sequence|legend=1| 19, 61d, 80, 99e, 179e }} | {{Optimal ET sequence|legend=1| 19, 61d, 80, 99e, 179e }} | ||
| Line 1,884: | Line 1,908: | ||
Comma list: 3025/3024, 3136/3125, 4375/4374 | Comma list: 3025/3024, 3136/3125, 4375/4374 | ||
Mapping: | Mapping: {{mapping| 2 10 12 24 19 | 0 -13 -14 -35 -23 }} | ||
POTE | Optimal tuning (POTE): ~99/70 = 1\2, ~6/5 = 315.181 | ||
{{Optimal ET sequence|legend=1| 80, 118, 198, 316, 514c, 830c }} | {{Optimal ET sequence|legend=1| 80, 118, 198, 316, 514c, 830c }} | ||
| Line 1,899: | Line 1,923: | ||
Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374 | Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374 | ||
Mapping: | Mapping: {{mapping| 2 10 12 24 19 -1 | 0 -13 -14 -35 -23 16 }} | ||
POTE | Optimal tuning (POTE): ~99/70 = 1\2, ~6/5 = 315.156 | ||
{{Optimal ET sequence|legend=1| 80, 118, 198 }} | {{Optimal ET sequence|legend=1| 80, 118, 198 }} | ||
| Line 1,914: | Line 1,938: | ||
Comma list: 169/168, 325/324, 364/363, 3136/3125 | Comma list: 169/168, 325/324, 364/363, 3136/3125 | ||
Mapping: | Mapping: {{mapping| 2 10 12 24 19 20 | 0 -13 -14 -35 -23 -24 }} | ||
POTE | Optimal tuning (POTE): ~55/39 = 1\2, ~6/5 = 315.184 | ||
{{Optimal ET sequence|legend=1| 80, 118f, 198f }} | {{Optimal ET sequence|legend=1| 80, 118f, 198f }} | ||
| Line 1,923: | Line 1,947: | ||
== Counterkleismic == | == Counterkleismic == | ||
{{ | {{See also| High badness temperaments #Counterhanson}} | ||
In the 5-limit, the counterhanson temperament tempers out the counterhanson (quinquinyo) comma, {{monzo|-20 -24 25}}, the amount by which six [[648/625|major dieses (648/625)]] fall short of the [[5/4|classic major third (5/4)]]. It can be described as 19&224 temperament (''counterkleismic'', named by analogy to [[catakleismic]] and parakleismic), tempering out the ragisma and 158203125/157351936 (laquadru-atritriyo comma). | In the 5-limit, the counterhanson temperament tempers out the counterhanson (quinquinyo) comma, {{monzo| -20 -24 25 }}, the amount by which six [[648/625|major dieses (648/625)]] fall short of the [[5/4|classic major third (5/4)]]. It can be described as 19 & 224 temperament (''counterkleismic'', named by analogy to [[catakleismic]] and parakleismic), tempering out the ragisma and 158203125/157351936 (laquadru-atritriyo comma). | ||
Subgroup: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 4375/4374, 158203125/157351936 | [[Comma list]]: 4375/4374, 158203125/157351936 | ||
{{Mapping|legend=1| 1 20 20 61 | 0 -25 -24 -79 }} | |||
: mapping generators: ~2, ~5/3 | |||
[[POTE | {{Multival|legend=1| 25 24 79 -20 55 116 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 316.060 | |||
{{Optimal ET sequence|legend=1| 19, 205, 224, 243, 467 }} | {{Optimal ET sequence|legend=1| 19, 205, 224, 243, 467 }} | ||
| Line 1,946: | Line 1,972: | ||
Comma list: 540/539, 4375/4374, 2097152/2096325 | Comma list: 540/539, 4375/4374, 2097152/2096325 | ||
Mapping: | Mapping: {{mapping| 1 20 20 61 -40 | 0 -25 -24 -79 59 }} | ||
POTE | Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.071 | ||
{{Optimal ET sequence|legend=1| 19, 205, 224 }} | {{Optimal ET sequence|legend=1| 19, 205, 224 }} | ||
| Line 1,959: | Line 1,985: | ||
Comma list: 540/539, 625/624, 729/728, 10985/10976 | Comma list: 540/539, 625/624, 729/728, 10985/10976 | ||
Mapping: | Mapping: {{mapping| 1 20 20 61 -40 56 | 0 -25 -24 -79 59 -71 }} | ||
POTE | Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.070 | ||
{{Optimal ET sequence|legend=1| 19, 205, 224, 1587cde, 1811ccdef, 2035ccddeef, 2259ccddeef, 2483ccddeef, 2707ccddeef }} | {{Optimal ET sequence|legend=1| 19, 205, 224, 1587cde, 1811ccdef, 2035ccddeef, 2259ccddeef, 2483ccddeef, 2707ccddeef }} | ||
| Line 1,972: | Line 1,998: | ||
Comma list: 1375/1372, 4375/4374, 496125/495616 | Comma list: 1375/1372, 4375/4374, 496125/495616 | ||
Mapping: | Mapping: {{mapping| 1 20 20 61 125 | 0 -25 -24 -79 -165 }} | ||
POTE | Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.065 | ||
{{Optimal ET sequence|legend=1| 19e, 205e, 224 }} | {{Optimal ET sequence|legend=1| 19e, 205e, 224 }} | ||
| Line 1,985: | Line 2,011: | ||
Comma list: 625/624, 729/728, 1375/1372, 10985/10976 | Comma list: 625/624, 729/728, 1375/1372, 10985/10976 | ||
Mapping: | Mapping: {{mapping| 1 20 20 61 125 56 | 0 -25 -24 -79 -165 -71 }} | ||
POTE | Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.065 | ||
{{Optimal ET sequence|legend=1| 19e, 205e, 224 }} | {{Optimal ET sequence|legend=1| 19e, 205e, 224 }} | ||
| Line 1,994: | Line 2,020: | ||
== Quincy == | == Quincy == | ||
Subgroup: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 4375/4374, 823543/819200 | [[Comma list]]: 4375/4374, 823543/819200 | ||
{{Mapping|legend=1| 1 2 3 3 | 0 -30 -49 -14 }} | |||
{{Multival|legend=1| 30 49 14 8 -62 -105 }} | |||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1728/1715 = 16.613 | ||
{{Optimal ET sequence|legend=1| 72, 217, 289 }} | {{Optimal ET sequence|legend=1| 72, 217, 289 }} | ||
| Line 2,013: | Line 2,039: | ||
Comma list: 441/440, 4000/3993, 4375/4374 | Comma list: 441/440, 4000/3993, 4375/4374 | ||
Mapping: | Mapping: {{mapping| 1 2 3 3 4 | 0 -30 -49 -14 -39 }} | ||
POTE | Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.613 | ||
{{Optimal ET sequence|legend=1| 72, 217, 289 }} | {{Optimal ET sequence|legend=1| 72, 217, 289 }} | ||
| Line 2,026: | Line 2,052: | ||
Comma list: 364/363, 441/440, 676/675, 4375/4374 | Comma list: 364/363, 441/440, 676/675, 4375/4374 | ||
Mapping: | Mapping: {{mapping| 1 2 3 3 4 5 | 0 -30 -49 -14 -39 -94 }} | ||
POTE | Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.602 | ||
{{Optimal ET sequence|legend=1| 72, 145, 217, 289 }} | {{Optimal ET sequence|legend=1| 72, 145, 217, 289 }} | ||
| Line 2,039: | Line 2,065: | ||
Comma list: 364/363, 441/440, 595/594, 676/675, 1156/1155 | Comma list: 364/363, 441/440, 595/594, 676/675, 1156/1155 | ||
Mapping: | Mapping: {{mapping| 1 2 3 3 4 5 5 | 0 -30 -49 -14 -39 -94 -66 }} | ||
POTE | Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.602 | ||
{{Optimal ET sequence|legend=1| 72, 145, 217, 289 }} | {{Optimal ET sequence|legend=1| 72, 145, 217, 289 }} | ||
| Line 2,052: | Line 2,078: | ||
Comma list: 343/342, 364/363, 441/440, 476/475, 595/594, 676/675 | Comma list: 343/342, 364/363, 441/440, 476/475, 595/594, 676/675 | ||
Mapping: | Mapping: {{mapping| 1 2 3 3 4 5 5 4 | 0 -30 -49 -14 -39 -94 -66 18 }} | ||
POTE | Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.594 | ||
{{Optimal ET sequence|legend=1| 72, 145, 217 }} | {{Optimal ET sequence|legend=1| 72, 145, 217 }} | ||
| Line 2,063: | Line 2,089: | ||
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Sfourth]].'' | : ''For the 5-limit version of this temperament, see [[High badness temperaments #Sfourth]].'' | ||
Subgroup: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 4375/4374, 64827/64000 | [[Comma list]]: 4375/4374, 64827/64000 | ||
{{Mapping|legend=1| 1 2 3 3 | 0 -19 -31 -9 }} | |||
{{Multival|legend=1|19 31 9 5 -39 -66}} | {{Multival|legend=1| 19 31 9 5 -39 -66 }} | ||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/48 = 26.287 | ||
{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }} | {{Optimal ET sequence|legend=1| 45, 46, 91, 137d }} | ||
| Line 2,082: | Line 2,108: | ||
Comma list: 121/120, 441/440, 4375/4374 | Comma list: 121/120, 441/440, 4375/4374 | ||
Mapping: | Mapping: {{mapping| 1 2 3 3 4 | 0 -19 -31 -9 -25 }} | ||
POTE | Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.286 | ||
{{Optimal ET sequence|legend=1| 45e, 46, 91e, 137de }} | {{Optimal ET sequence|legend=1| 45e, 46, 91e, 137de }} | ||
| Line 2,095: | Line 2,121: | ||
Comma list: 121/120, 169/168, 325/324, 441/440 | Comma list: 121/120, 169/168, 325/324, 441/440 | ||
Mapping: | Mapping: {{mapping| 1 2 3 3 4 4 | 0 -19 -31 -9 -25 -14 }} | ||
POTE | Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.310 | ||
{{Optimal ET sequence|legend=1| 45ef, 46, 91ef, 137def }} | {{Optimal ET sequence|legend=1| 45ef, 46, 91ef, 137def }} | ||
| Line 2,108: | Line 2,134: | ||
Comma list: 385/384, 2401/2376, 4375/4374 | Comma list: 385/384, 2401/2376, 4375/4374 | ||
Mapping: | Mapping: {{mapping| 1 2 3 3 3 | 0 -19 -31 -9 21 }} | ||
POTE | Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.246 | ||
{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }} | {{Optimal ET sequence|legend=1| 45, 46, 91, 137d }} | ||
| Line 2,121: | Line 2,147: | ||
Comma list: 196/195, 364/363, 385/384, 4375/4374 | Comma list: 196/195, 364/363, 385/384, 4375/4374 | ||
Mapping: | Mapping: {{mapping| 1 2 3 3 3 3 | 0 -19 -31 -9 21 32 }} | ||
POTE | Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.239 | ||
{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }} | {{Optimal ET sequence|legend=1| 45, 46, 91, 137d }} | ||
| Line 2,130: | Line 2,156: | ||
== Trideci == | == Trideci == | ||
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Tridecatonic]].'' | |||
The | The trideci temperament (26 & 65) has a period of 1/13 octave and tempers out 245/242 and 385/384 in the 11-limit. It tempers out the same 5-limit comma as the [[Octagar temperaments #Tridecatonic|tridecatonic temperament]], but with the ragisma (4375/4374) rather than the octagar (4000/3969) tempered out. The name ''trideci'' comes from "tridecim" (Latin for "[[wikipedia:13|thirteen]]"). | ||
Subgroup: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 4375/4374, 83349/81920 | [[Comma list]]: 4375/4374, 83349/81920 | ||
{{Mapping|legend=1| 13 21 31 36 | 0 -1 -2 1 }} | |||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~256/245 = 1\13, ~3/2 = 699.1410 | ||
{{Optimal ET sequence|legend=1| 26, 65, 91, 156d, 247cdd }} | {{Optimal ET sequence|legend=1| 26, 65, 91, 156d, 247cdd }} | ||
| Line 2,151: | Line 2,177: | ||
Comma list: 245/242, 385/384, 4375/4374 | Comma list: 245/242, 385/384, 4375/4374 | ||
Mapping: | Mapping: {{mapping| 13 21 31 36 45 | 0 -1 -2 1 0 }} | ||
POTE | Optimal tuning (POTE): ~22/21 = 1\13, ~3/2 = 699.6179 | ||
{{Optimal ET sequence|legend=1| 26, 65, 91, 156d, 247cdde }} | {{Optimal ET sequence|legend=1| 26, 65, 91, 156d, 247cdde }} | ||
| Line 2,164: | Line 2,190: | ||
Comma list: 169/168, 245/242, 325/324, 385/384 | Comma list: 169/168, 245/242, 325/324, 385/384 | ||
Mapping: | Mapping: {{mapping| 13 21 31 36 45 48 | 0 -1 -2 1 0 0 }} | ||
POTE | Optimal tuning (POTE): ~22/21 = 1\13, ~3/2 = 699.2969 | ||
{{Optimal ET sequence|legend=1| 26, 65f, 91f, 156dff }} | {{Optimal ET sequence|legend=1| 26, 65f, 91f, 156dff }} | ||
| Line 2,173: | Line 2,199: | ||
== Counterorson == | == Counterorson == | ||
Counterorson tempers out the {{monzo|147 -103 7}} comma in the 5-limit. It uses a generator that reaches the 3rd harmonic in 7 steps, but unlike the [[semicomma family]], 5th harmonic is 103 generators up and not 3 generators down. The two mappings converge on [[53edo]]. | Counterorson tempers out the {{monzo| 147 -103 7 }} comma in the 5-limit. It uses a generator that reaches the 3rd harmonic in 7 steps, but unlike the [[semicomma family]], 5th harmonic is 103 generators up and not 3 generators down. The two mappings converge on [[53edo]]. | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
Comma list: 4375/4374, {{monzo|154 -54 -21 -7}} | Comma list: 4375/4374, {{monzo| 154 -54 -21 -7 }} | ||
Mapping: {{mapping| 1 0 -21 85 | 0 7 103 -363 }} | |||
Optimal tuning (CTE): ~2 = 1\1, ~{{monzo| 66 -23 -9 -3 }} = 271.7113 | |||
Optimal | {{Optimal ET sequence|legend=1| 53, …, 1612, 1665, 1718 }} | ||
Badness: 0.313 | |||
== Notes == | == Notes == | ||
| Line 2,189: | Line 2,217: | ||
[[Category:Temperament collections]] | [[Category:Temperament collections]] | ||
[[Category:Ragismic microtemperaments| ]] <!-- main article --> | [[Category:Ragismic microtemperaments| ]] <!-- main article --> | ||
[[Category:Ragismic| ]] <!-- key article --> | |||
[[Category:Rank 2]] | [[Category:Rank 2]] | ||
[[Category:Microtemperaments]] | [[Category:Microtemperaments]] | ||