104edo
| ← 103edo | 104edo | 105edo → |
104 equal divisions of the octave (abbreviated 104edo or 104ed2), also called 104-tone equal temperament (104tet) or 104 equal temperament (104et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 104 equal parts of about 11.5 ¢ each. Each step represents a frequency ratio of 21/104, or the 104th root of 2.
Theory
104edo has two different equally viable 5-limit vals, and both are useful. The flat major third val, ⟨104 165 241] (patent val), tempers out 3125/3072, and supports magic temperament. The sharp major third val, ⟨104 165 242] (104c val), tempers out 2048/2025 and supports diaschismic temperament.
104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the optimal patent val for 11-limit magic and the 13-limit magic extension necromancy. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out 225/224, 245/243 and 875/864; and in the 11-limit, 100/99, 896/891, 385/384 and 540/539. It provides an excellent tuning also for the rank-3 temperaments pairing 100/99 with 225/224 (apollo temperament), 245/243 or 875/864, or the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val.
104 with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, 126/125 and 5120/5103 in the 7-limit, 176/175 and 896/891 in the 11-limit, 196/195, 352/351 and 364/363 in the 13-limit and 136/135 and 256/255 in the 17-limit.
104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, 2197/2187, 10648/10647, 16807/16731, 20449/20412, 21632/21609, and 26411/26364. It is the optimal patent val for the 17&87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +1.89 | -5.54 | +0.40 | +2.53 | +1.78 | -1.11 | +2.49 | -5.20 | -2.65 | -2.73 |
| Relative (%) | +0.0 | +16.4 | -48.1 | +3.5 | +21.9 | +15.4 | -9.6 | +21.6 | -45.0 | -23.0 | -23.6 | |
| Steps (reduced) |
104 (0) |
165 (61) |
241 (33) |
292 (84) |
360 (48) |
385 (73) |
425 (9) |
442 (26) |
470 (54) |
505 (89) |
515 (99) | |
Subsets and supersets
Since 104 factors into 23 × 13, it has subset edos 2, 4, 8, 13, 26, and 52.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [165 -104⟩ | [⟨104 165]] | −0.597 | 0.596 | 5.17 |
| 2.3.5 | 2048/2025, [0 22 -15⟩ | [⟨104 165 242]] (104c) | −1.258 | 1.054 | 9.14 |
| 2.3.5.7 | 126/125, 2048/2025, 117649/116640 | [⟨104 165 242 292]] (104c) | −0.980 | 1.032 | 8.95 |
| 2.3.5.7.11 | 126/125, 176/175, 896/891, 14641/14580 | [⟨104 165 242 292 360]] (104c) | −0.930 | 0.929 | 8.05 |
| 2.3.5.7.11.13 | 126/125, 176/175, 196/195, 364/363, 2197/2187 | [⟨104 165 242 292 360 385]] (104c) | −0.855 | 0.864 | 7.49 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperament |
|---|---|---|---|---|
| 1 | 33\104 | 380.77 | 5/4 | Magic / necromancy / divination |
| 1 | 51\104 | 588.46 | 7/5 | Untriton |
| 4 | 9\104 | 103.85 | 18/17 | Undim |
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperament |
|---|---|---|---|---|
| 1 | 11\104 | 126.92 | 27/25 | Mowgli |
| 1 | 21\104 | 242.31 | 147/128 | Septiquarter |
| 1 | 27\104 | 311.54 | 6/5 | Oolong |
| 1 | 47\104 | 542.31 | 15/11 | Casablanca / marrakesh |
| 2 | 21\104 | 242.31 | 121/105 | Semiseptiquarter |
| 2 | 43\104 (9\104) |
496.15 (103.85) |
4/3 (17/16) |
Diaschismic |
| 8 | 49\104 (2\104) |
565.38 (34.62) |
168/121 (55/54) |
Octowerck / octowerckis |
| 26 | 43\104 (1\104) |
496.15 (11.54) |
4/3 (225/224) |
Bosonic |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct
Intervals
| # | Cents | Approximate Ratios | ||
|---|---|---|---|---|
| Of 2.3.7.11.13.17.19.25 subgroup |
Additional ratios of 5 tending sharp (104c val) |
Additional ratios of 5 tending flat (patent val) | ||
| 0 | 0.000 | 1/1 | 126/125 | 225/224, 100/99 |
| 1 | 11.538 | 225/224, 100/99 | ||
| 2 | 23.077 | 64/63 | 81/80, 225/224 | 50/49 |
| 3 | 34.615 | 49/48, 50/49 | 81/80, 126/125 | |
| 4 | 46.154 | 36/35, 50/49 | ||
| 5 | 57.692 | 28/27, 33/32 | 25/24, 36/35 | |
| 6 | 69.231 | 25/24 | ||
| 7 | 80.769 | 22/21 | 25/24, 21/20 | 20/19 |
| 8 | 92.308 | 19/18 | 20/19 | 21/20 |
| 9 | 103.846 | 17/16, 18/17 | 16/15 | |
| 10 | 115.385 | 16/15, 15/14 | ||
| 11 | 126.923 | 14/13 | 15/14 | |
| 12 | 138.462 | 13/12 | ||
| 13 | 150.000 | 12/11 | ||
| 14 | 161.538 | 11/10 | ||
| 15 | 173.077 | 21/19 | 10/9, 11/10 | |
| 16 | 184.615 | 10/9 | ||
| 17 | 196.154 | 28/25, 19/17 | ||
| 18 | 207.692 | 9/8 | 17/15 | |
| 19 | 219.231 | 25/22 | 17/15 | |
| 20 | 230.769 | 8/7 | ||
| 21 | 242.308 | 15/13 | ||
| 22 | 253.846 | 22/19 | 15/13 | |
| 23 | 265.385 | 7/6 | ||
| 24 | 276.923 | 75/64 | 20/17 | |
| 25 | 288.462 | 32/27, 13/11 | 20/17 | |
| 26 | 300.000 | 25/21, 19/16 | ||
| 27 | 311.538 | 6/5 | ||
| 28 | 323.077 | 6/5, 40/33 | ||
| 29 | 334.615 | 17/14 | 40/33 | |
| 30 | 346.154 | 11/9, 39/32 | ||
| 31 | 357.692 | 27/22, 16/13 | ||
| 32 | 369.231 | 26/21, 21/17 | ||
| 33 | 380.769 | 5/4 | ||
| 34 | 392.308 | 5/4 | ||
| 35 | 403.846 | 63/50, 24/19 | 19/15 | |
| 36 | 415.385 | 81/64, 14/11 | 19/15 | |
| 37 | 426.923 | 32/25 | ||
| 38 | 438.462 | 9/7 | ||
| 39 | 450.000 | 22/17 | 13/10 | |
| 40 | 461.538 | 17/13 | 13/10 | |
| 41 | 473.077 | 21/16 | ||
| 42 | 484.615 | |||
| 43 | 496.154 | 4/3 | ||
| 44 | 507.692 | |||
| 45 | 519.231 | 27/20 | ||
| 46 | 530.769 | 19/14 | 27/20, 15/11 | |
| 47 | 542.308 | 26/19 | 15/11 | |
| 48 | 553.846 | 11/8 | ||
| 49 | 565.385 | 18/13 | ||
| 50 | 576.923 | 7/5 | ||
| 51 | 588.462 | 45/32, 7/5 | ||
| 52 | 600.000 | 17/12, 24/17 | 45/32, 64/45 | |
| … | … | … | … | … |