User:Francium/Miscellaneous rank-3 temperaments
On this page, I explore some unnamed rank-3 temperaments that catched my interest.
31 & 53 & 227
Subgroup: 2.3.5.7
Comma list: 65625/65536
Mapping:
| [⟨ | 1 | 0 | 1 | 11 | ], |
| ⟨ | 0 | 1 | 1 | -6 | ], |
| ⟨ | 0 | 0 | 2 | -10 | ] |
CTE generators: ~2 = 1200.0937, ~3 = 1901.9403, ~175/192 = -157.9391
Optimal ET sequence: 9, 19d, 22, 31, 53, 84, 87, 118, 140, 171, 1592c, 1763c, 1934c, 2074c, 2105c, 2245cd, 2416cd, 2587cd, 2758cd, 2929cd, 3100cd, 3271ccd, 3442ccd, 3613ccd, 5034bcccdd, 5205bcccdd
Badness: 0.761
11-limit
Subgroup: 2.3.5.7.11
Comma list: 385/384, 3388/3375
Mapping:
| [⟨ | 1 | 0 | 1 | 11 | -5 | ], |
| ⟨ | 0 | 1 | 1 | -6 | 6 | ], |
| ⟨ | 0 | 0 | 2 | -10 | 8 | ] |
CTE generators: ~2 = 1200.2769, ~3 = 1902.4434, ~175/192 = -157.9651
Optimal ET sequence: 9, 22, 31, 53, 84e, 87, 118, 258e, 376de, 547dee
Badness: 0.771
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 385/384, 352/351, 1001/1000
Mapping:
| [⟨ | 1 | 0 | 1 | 11 | -5 | 0 | ], |
| ⟨ | 0 | 1 | 1 | -6 | 6 | 3 | ], |
| ⟨ | 0 | 0 | 2 | -10 | 8 | 8 | ] |
CTE generators: ~2 = 1200.1146, ~3 = 1902.7798, ~175/192 = -158.3562
Optimal ET sequence: 9f, 22, 31, 53, 84e, 87, 118, 140, 258e
Badness: 0.986
31 & 311 & 431
Subgroup: 2.3.5.7.11.13
Comma list: 4096/4095, 2401/2400, 190512/190333
Mapping:
| [⟨ | 1 | 1 | 5 | 4 | 4 | 1 | ], |
| ⟨ | 0 | 2 | 5 | 3 | 7 | -12 | ], |
| ⟨ | 0 | 0 | 8 | 4 | 5 | -12 | ] |
CTE generators: ~2 = 1199.9603, ~49/40 = 350.9859, ~63/44 = 621.0427
Optimal ET sequence: 31, 58f, 89, 120, 191, 222f, 280, 311, 400, 711, 742, 1053, 1453
Badness: 2.258
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 4096/4095, 2401/2400, 14400/14399, 11016/11011
Mapping:
| [⟨ | 1 | 1 | 5 | 4 | 4 | 1 | 6 | ], |
| ⟨ | 0 | 2 | 5 | 3 | 7 | -12 | -3 | ], |
| ⟨ | 0 | 0 | 8 | 4 | 5 | -12 | 2 | ] |
CTE generators: ~2 = 1199.9677, ~49/40 = 350.9737, ~63/44 = 621.0347
Optimal ET sequence: 31, 58f, 89, 120g, 191g, 222f, 280, 311, 400, 711, 742, 1453, 2195ef
Badness: 1.410
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 2926/2925, 4096/4095, 3136/3135, 5985/5984, 11016/11011
Mapping:
| [⟨ | 1 | 1 | 5 | 4 | 4 | 1 | 6 | 4 | ], |
| ⟨ | 0 | 2 | 5 | 3 | 7 | -12 | -3 | -8 | ], |
| ⟨ | 0 | 0 | 8 | 4 | 5 | -12 | 2 | -5 | ] |
CTE generators: ~2 = 1199.9748, ~49/40 = 350.9675, ~63/44 = 621.0343
Optimal ET sequence: 31, 58f, 89, 120g, 191g, 222f, 280, 311, 400, 653f, 711, 742, 1453, 1795f, 2195ef
Badness: 1.069
31 & 317 & 1193
Subgroup: 2.5.7.11
Comma list: [-29 11 0 1⟩
Mapping:
| [⟨ | 1 | 0 | 0 | 29 | ], |
| ⟨ | 0 | 2 | 0 | -22 | ], |
| ⟨ | 0 | 0 | 1 | 0 | ] |
CTE generators: ~2 = 1200.015, ~5/4 = 386.254, ~7/4 = 968.796
Optimal ET sequence: 22e, 25e, 28, 31, 56, 84, 87, 115, 146, 202, 233, 348, 379, 581, 727, 960, 2647, 3607
Badness: 0.830
31 & 431 & 1889
Subgroup: 2.3.5.7.11.13
Comma list: 655473/655360, 820125/819896, 3764768/3764475
Mapping:
| [⟨ | 1 | 0 | 18 | 7 | 11 | 0 | ], |
| ⟨ | 0 | 1 | 6 | 2 | 6 | -5 | ], |
| ⟨ | 0 | 0 | 65 | 19 | 44 | -30 | ] |
CTE generators: ~2 = 1200.008, ~3/2 = 701.925, ~78975/60368 = 465.007
Optimal ET sequence: 31, 369, 400, 831, 1058c, 1089c, 1427f, 1458, 1489, 1889, 3809, 4209, 6129, 8018
Badness: 5.755
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 14400/14399, 61965/61952, 31213/31212, 3764768/3764475
Mapping:
| [⟨ | 1 | 0 | 18 | 7 | 11 | 0 | 13 | ], |
| ⟨ | 0 | 1 | 6 | 2 | 6 | -5 | 0 | ], |
| ⟨ | 0 | 0 | 65 | 19 | 44 | -30 | 23 | ] |
CTE generators: ~2 = 1200.008, ~3/2 = 701.925, ~78975/60368 = 465.007
Optimal ET sequence: 31, 369, 400, 831, 1058cg, 1089c, 1458g, 1489, 1889, 2320, 3809, 4209, 6129, 8018
Badness: 2.373
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 5985/5984, 14400/14399, 13377/13376, 61965/61952, 3764768/3764475
Mapping:
| [⟨ | 1 | 0 | 18 | 7 | 11 | 0 | 13 | 4 | ], |
| ⟨ | 0 | 1 | 6 | 2 | 6 | -5 | 0 | -4 | ], |
| ⟨ | 0 | 0 | 65 | 19 | 44 | -30 | 23 | -17 | ] |
CTE generators: ~2 = 1200.011, ~3/2 = 701.921, ~22880/17493 = 465.008
Optimal ET sequence: 31, 369, 400, 831, 1058cg, 1089c, 1458g, 1489, 1889, 2320, 3809, 6129h, 9938h
Badness: 1.706
37 & 109 & 577
Subgroup: 2.5.7.11
Comma list: 102487/102400
Mapping:
| [⟨ | 1 | 0 | 4 | 2 | ], |
| ⟨ | 0 | 1 | 6 | -1 | ], |
| ⟨ | 0 | 0 | 8 | -2 | ] |
CTE generators: ~2 = 1200.048, ~5/4 = 386.260, ~11/8 = 550.980
Optimal ET sequence: 7, 11cd, 13, 15, 22, 35, 37, 72, 109, 233, 270, 342, 379, 612, 1333, 1603, 1945e, 2557e, 2936e, 3548ee
Badness: 0.189
13-limit
Subgroup: 2.5.7.11.13
Comma list: 1001/1000, 6656/6655
Mapping:
| [⟨ | 1 | 0 | 4 | 2 | -3 | ], |
| ⟨ | 0 | 1 | 6 | -1 | -2 | ], |
| ⟨ | 0 | 0 | 8 | -2 | -6 | ] |
CTE generators: ~2 = 1200.021, ~5/4 = 386.630, ~11/8 = 551.170
Optimal ET sequence: 11cdf, 13, 15, 22, 35f, 37, 87, 109, 124, 161, 196, 233, 270, 503, 627, 664, 897, 1167
Badness: 0.177
37 & 307 & 457
Subgroup: 2.5.7.11.13.17
Comma list: 6656/6655, 140800/140777, 10985/10976
Mapping:
| [⟨ | 1 | 2 | 1 | 3 | 2 | 10 | ], |
| ⟨ | 0 | 3 | 1 | -1 | 0 | 3 | ], |
| ⟨ | 0 | 0 | 3 | 1 | 3 | -11 | ] |
CTE generators: ~2 = 1199.992, ~14/13 = 128.753, ~77/52 = 680.105
Optimal ET sequence: 30cg, 37, 113, 150, 187, 270, 307, 457, 764, 1221, 1715, 2479d
Badness: 1.001
19-limit
Subgroup: 2.5.7.11.13.17.19
Comma list: 2432/2431, 10241/10240, 12103/12100, 64220/64141
Mapping:
| [⟨ | 1 | 2 | 1 | 3 | 2 | 10 | 8 | ], |
| ⟨ | 0 | 3 | 1 | -1 | 0 | 3 | 2 | ], |
| ⟨ | 0 | 0 | 3 | 1 | 3 | -11 | -7 | ] |
CTE generators: ~2 = 1200.016, ~14/13 = 128.743, ~77/52 = 680.094
Optimal ET sequence: 30cg, 37, 83g, 113, 120g, 150, 187, 270, 307, 420, 457, 764, 1184h, 1221h
Badness: 0.899
41 & 53 & 167
Subgroup: 2.3.5.7
Comma list: 20480000/20253807
Mapping:
| [⟨ | 1 | 0 | 0 | 5 | ], |
| ⟨ | 0 | 1 | 1 | -2 | ], |
| ⟨ | 0 | 0 | 3 | 4 | ] |
CTE generators: ~2 = 1199.544, ~3/2 = 703.173, ~189/160 = 294.314
Optimal ET sequence: 9c, 12, 29, 41, 53, 94, 167, 179, 208, 220, 481bd, 493bd, 534bd, 754bdd
Badness: 12.361
11-limit
Subgroup: 2.3.5.7.11
Comma list: 2200/2187, 102400/101871
Mapping:
| [⟨ | 1 | 0 | 0 | 5 | -3 | ], |
| ⟨ | 0 | 1 | 1 | -2 | 5 | ], |
| ⟨ | 0 | 0 | 3 | 4 | -6 | ] |
CTE generators: ~2 = 1199.676, ~3/2 = 703.329, ~189/160 = 294.194
Optimal ET sequence: 9c, 12e, 20ce, 32c, 41, 53, 94, 167, 208, 220, 261
Badness: 4.132
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 352/351, 41600/41503
Mapping:
| [⟨ | 1 | 0 | 0 | 5 | -3 | 2 | ], |
| ⟨ | 0 | 1 | 1 | -2 | 5 | 2 | ], |
| ⟨ | 0 | 0 | 3 | 4 | -6 | -6 | ] |
CTE generators: ~2 = 1199.700, ~3/2 = 703.310, ~189/160 = 294.172
Optimal ET sequence: 12e, 20ce, 41, 53, 94, 167, 208, 261
Badness: 2.413
311 & 373 & 1193
Subgroup: 2.3.5.7.11.13
Comma list: 3025/3024, 2401/2400, 142884/142805
Mapping:
| [⟨ | 1 | 3 | 2 | 3 | 6 | 6 | ], |
| ⟨ | 0 | 8 | 0 | 2 | 13 | 13 | ], |
| ⟨ | 0 | 0 | 4 | 2 | -3 | 0 | ] |
CTE generators: ~2 = 1200.048, ~147/130 = 212.272, ~55/52 = 96.548
Optimal ET sequence: 23bcf, 39bceff, 51e, 62, 85c, 113, 175f, 198, 311, 373, 571, 684, 1566
Badness: 1.605
311 & 593 & 1063
Subgroup: 2.3.5.7
Comma list: [31 -17 -9 6⟩
Mapping:
| [⟨ | 1 | 2 | 1 | 2 | ], |
| ⟨ | 0 | 3 | 1 | 10 | ], |
| ⟨ | 0 | 0 | 2 | 3 | ]] |
Optimal tuning (CTE): ~2 = 1\1, ~100352/91125 = 165.981, ~224/135 = 876.211
Optimal ET sequence: 7d, 15d, 22, 51, 73c, 79c, 86, 101, 108, 130, 152, 282, 289, 311, 441, 1475, 1916, 2357, 2487, 2928, 3962c, 4403c, 4844c, 9247bcc
Badness: 13.801
311 & 653 & 1759
Subgroup: 2.3.5.7.11.13.17
Comma list: 3025/3024, 2500/2499, 56595/56576, 4100625/4100096
Mapping:
| [⟨ | 1 | 1 | 12 | 15 | -1 | 30 | 19 | ], |
| ⟨ | 0 | 1 | 2 | 3 | 1 | 12 | 1 | ], |
| ⟨ | 0 | 0 | -14 | -8 | 5 | -43 | 20 | ]] |
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.984, ~77/45 = 929.843
Optimal ET sequence: 142fg, 311, 653, 1106, 1448, 1759, 2865
Badness: 2.321
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 3025/3024, 2500/2499, 5985/5984, 6175/6174, 1549184/1549125
Mapping:
| [⟨ | 1 | 1 | 12 | 15 | -1 | 30 | 19 | -6 | ], |
| ⟨ | 0 | 1 | 2 | 3 | 1 | 12 | 1 | -5 | ], |
| ⟨ | 0 | 0 | -14 | -8 | 5 | -43 | 20 | 17 | ]] |
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.982, ~77/45 = 929.843
Optimal ET sequence: 142fg, 311, 653, 795, 1106, 1448, 1759
Badness: 1.798
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 3025/3024, 2500/2499, 5985/5984, 6175/6174, 3520/3519, 76545/76544
Mapping:
| [⟨ | 1 | 1 | 12 | 15 | -1 | 30 | 19 | -6 | -4 | ], |
| ⟨ | 0 | 1 | 2 | 3 | 1 | 12 | 1 | -5 | 0 | ], |
| ⟨ | 0 | 0 | -14 | -8 | 5 | -43 | 20 | 17 | 11 | ]] |
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.980, ~77/45 = 929.843
Optimal ET sequence: 142fg, 311, 653, 795, 1106, 1448, 1759
Badness: 1.428
29-limit
Subgroup: 2.3.5.7.11.13.17.19.23.29
Comma list: 3025/3024, 2500/2499, 5985/5984, 6175/6174, 3520/3519, 7425/7424, 76545/76544
Mapping:
| [⟨ | 1 | 1 | 12 | 15 | -1 | 30 | 19 | -6 | -4 | 18 | ], |
| ⟨ | 0 | 1 | 2 | 3 | 1 | 12 | 1 | -5 | 0 | 8 | ], |
| ⟨ | 0 | 0 | -14 | -8 | 5 | -43 | 20 | 17 | 11 | -23 | ]] |
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.983, ~77/45 = 929.843
Optimal ET sequence: 142fg, 311, 653, 795, 1106, 1448, 1759, 2865
Badness: 1.223
31-limit
Subgroup: 2.3.5.7.11.13.17.19.23.29.31
Comma list: 3025/3024, 2500/2499, 5985/5984, 6175/6174, 3520/3519, 5104/5103, 12397/12393, 6076/6075
Mapping:
| [⟨ | 1 | 1 | 12 | 15 | -1 | 30 | 19 | -6 | -4 | 18 | -3 | ], |
| ⟨ | 0 | 1 | 2 | 3 | 1 | 12 | 1 | -5 | 0 | 8 | 3 | ], |
| ⟨ | 0 | 0 | -14 | -8 | 5 | -43 | 20 | 17 | 11 | -23 | 8 | ]] |
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.995, ~77/45 = 929.846
Optimal ET sequence: 311, 653, 1106, 1417, 1448k, 1759
Badness: 1.368