97edo
Theory
In the patent val, 97edo tempers out 875/864, 4000/3969 and 1029/1024 in the 7-limit, 245/242, 100/99, 385/384 and 441/440 in the 11-limit, and 196/195, 352/351 and 676/675 in the 13-limit. It provides the optimal patent val for the 13-limit 41&97 temperament tempering out 100/99, 196/195, 245/242 and 385/384.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +3.20 | -2.81 | -3.88 | -5.97 | +5.38 | +0.71 | +0.39 | -5.99 | -0.61 | -0.68 | +2.65 |
| Relative (%) | +25.9 | -22.7 | -31.3 | -48.3 | +43.5 | +5.7 | +3.2 | -48.4 | -4.9 | -5.5 | +21.4 | |
| Steps (reduced) |
154 (57) |
225 (31) |
272 (78) |
307 (16) |
336 (45) |
359 (68) |
379 (88) |
396 (8) |
412 (24) |
426 (38) |
439 (51) | |
Subsets and supersets
97edo is the 25th prime edo.
388edo and 2619edo, which contain 97edo as a subset, have very high consistency limits - 37 and 33 respectively. 3395edo, which divides the edostep in 35, is a zeta edo. The berkelium temperament realizes some relationships between them through a regular temperament perspective.
JI approximation
97edo has very poor direct approximation for superparticular intervals among edos up to 200, and the worst for intervals up to 9/8 among edos up to 100. It has errors of well above one standard deviation (about 15.87%) in superparticular intervals with denominators up to 14. The first good approximation is the 16/15 semitone using the 9th note, with an error of 3%, meaning 97edo can be used as a rough version of 16/15 equal-step tuning.
Since 97edo is a prime edo, it lacks specific modulation circles, symmetrical chords or sub-edos that are present in composite edos. When notable equal divisions like 19, 31, 41, or 53 have strong JI-based harmony, 97edo does not have easily representable modulation because of its inability to represent superparticulars. However, this might result in interest in this tuning through JI-agnostic approaches.
| Interval | Error (Relative, r¢) |
|---|---|
| 3/2 | 25.9 |
| 4/3 | 25.8 |
| 5/4 | 22.7 |
| 6/5 | 48.6 |
| 7/6 | 42.8 |
| 8/7 | 31.4 |
| 9/8 | 48.2 |
| 10/9 | 25.6 |
| 11/10 | 33.7 |
| 12/11 | 17.6 |
| 13/12 | 20.1 |
| 14/13 | 37.0 |
| 15/14 | 34.6 |
| 16/15 | 3.1 |
| 17/16 | 48.3 |