34ed7
Division of the 7th harmonic into 34 equal parts (34ED7) is related to 12 EDO, but with the 7/1 rather than the 2/1 being just. The octave is about 11.0026 cents compressed and the step size is about 99.0831 cents. It is consistent to the 11-integer-limit, but not to the 12-integer-limit. In comparison, 12EDO is only consistent up to the 10-integer-limit.
| degree | cents value | corresponding JI intervals |
comments |
|---|---|---|---|
| 0 | 0.0000 | exact 1/1 | |
| 1 | 99.0831 | 18/17 | |
| 2 | 198.1662 | 28/25 | |
| 3 | 297.2493 | 19/16 | |
| 4 | 396.3325 | 49/39, 34/27 | pseudo-5/4 |
| 5 | 495.4156 | 4/3 | |
| 6 | 594.4987 | 24/17 | |
| 7 | 693.5818 | 136/91 | pseudo-3/2 |
| 8 | 792.6649 | 30/19, 128/81 | |
| 9 | 891.7480 | 77/46 | pseudo-5/3 |
| 10 | 990.8311 | 85/48, 39/22 | |
| 11 | 1089.9143 | 15/8 | |
| 12 | 1188.9974 | 143/72, 175/88 | pseudo-octave |
| 13 | 1288.0805 | 21/10, 40/19 | |
| 14 | 1387.1636 | 49/22 | |
| 15 | 1486.2467 | 33/14 | |
| 16 | 1585.3298 | 5/2 | |
| 17 | 1684.4130 | 119/45, 45/17 | pseudo-8/3 |
| 18 | 1783.4961 | 14/5 | |
| 19 | 1882.5792 | 95/32, 98/33 | pseudo-3/1 |
| 20 | 1981.6623 | 22/7 | |
| 21 | 2080.7454 | 133/40, 10/3 | |
| 22 | 2179.8285 | 88/25 | |
| 23 | 2278.9116 | 56/15 | |
| 24 | 2377.9948 | 154/39, 320/81, 336/85 | pseudo-4/1 |
| 25 | 2477.0779 | 46/11 | |
| 26 | 2576.1610 | 133/30 | |
| 27 | 2675.2441 | 169/36 | |
| 28 | 2774.3272 | 119/24 | pseudo-5/1 |
| 29 | 2873.4103 | 21/4 | pseudo-16/3 |
| 30 | 2972.4934 | 39/7 | |
| 31 | 3071.5766 | 112/19 | pseudo-6/1 |
| 32 | 3170.6597 | 25/4 | |
| 33 | 3269.7428 | 119/18 | |
| 34 | 3368.8259 | exact 7/1 | harmonic seventh plus two octaves |
34ed7 as a generator
34ED7 can also be thought of as a generator of the 11-limit temperament which tempers out 896/891, 1375/1372, and 4375/4356, which is a cluster temperament with 12 clusters of notes in an octave (quintupole temperament). This temperament is supported by 12EDO, 109EDO, and 121EDO among others.
Trisa-quingu (12&121)
5-limit
Comma: [37 -16 -5⟩
Mapping: [⟨1 2 1], ⟨0 -5 16]]
POTE generator: ~135/128 = 99.267
Vals: 12, 85, 97, 109, 121, 133, 278c, 411bc, 544bc
Badness: 0.444506
Quintupole (12&121)
7-limit
Comma list: 4000/3969, 458752/455625
Mapping: [⟨1 2 1 0], ⟨0 -5 16 34]]
POTE generator: ~135/128 = 99.175
Vals: 12, 97, 109, 121
Badness: 0.111620
11-limit
Comma list: 896/891, 1375/1372, 4375/4356
Mapping: [⟨1 2 1 0 -1], ⟨0 -5 16 34 54]]
POTE generator: ~132/125 = 99.156
Vals: 12, 109, 121, 351bde, 472bdee
Badness: 0.056501
13-limit
Comma list: 352/351, 364/363, 625/624, 2704/2695
Mapping: [⟨1 2 1 0 -1 -2], ⟨0 -5 16 34 54 69]]
POTE generator: ~55/52 = 99.165
Vals: 12f, 109, 121
Badness: 0.038431
17-limit
Comma list: 256/255, 352/351, 364/363, 375/374, 442/441
Mapping: [⟨1 2 1 0 -1 -2 5], ⟨0 -5 16 34 54 69 -11]]
POTE generator: ~18/17 = 99.172
Vals: 12f, 109, 121
Badness: 0.028721
19-limit
Comma list: 190/189, 256/255, 352/351, 361/360, 364/363, 375/374
Mapping: [⟨1 2 1 0 -1 -2 5 4], ⟨0 -5 16 34 54 69 -11 3]]
POTE generator: ~18/17 = 99.164
Vals: 12f, 109, 121
Badness: 0.023818