97edo
| ← 96edo | 97edo | 98edo → |
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) | |
Divisors
97edo is the 25th prime edo.
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 |
Scales
Since 97edo has a step of 12.371 cents, it also allows one to use its mos scales as circulating temperaments[clarification needed]. It is the first prime edo which does this and the first edo which allows one to use an mos scale with a step 20 degrees or larger as a circulating temperament.
| Tones | Pattern | L:s |
|---|---|---|
| 5 | 2L 3s | 20:19 |
| 6 | 1L 5s | 17:16 |
| 7 | 6L 1s | 14:13 |
| 8 | 1L 7s | 13:12 |
| 9 | 7L 2s | 11:10 |
| 10 | 7L 3s | 10:9 |
| 11 | 9L 2s | 9:8 |
| 12 | 1L 11s | |
| 13 | 6L 7s | 8:7 |
| 14 | 13L 1s | 7:6 |
| 15 | 7L 8s | |
| 16 | 1L 15s | |
| 17 | 12L 5s | 6:5 |
| 18 | 7L 11s | |
| 19 | 2L 17s | |
| 20 | 17L 3s | 5:4 |
| 21 | 13L 8s | |
| 22 | 9L 13s | |
| 23 | 5L 18s | |
| 24 | 1L 23s | |
| 25 | 22L 3s | 4:3 |
| 26 | 19L 7s | |
| 27 | 16L 11s | |
| 28 | 13L 15s | |
| 29 | 10L 19s | |
| 30 | 7L 23s | |
| 31 | 4L 27s | |
| 32 | 1L 31s | |
| 33 | 31L 2s | 3:2 |
| 34 | 29L 5s | |
| 35 | 27L 8s | |
| 36 | 25L 11s | |
| 37 | 23L 14s | |
| 38 | 21L 17s | |
| 39 | 19L 20s | |
| 40 | 17L 23s | |
| 41 | 15L 26s | |
| 42 | 13L 29s | |
| 43 | 11L 32s | |
| 44 | 9L 35s | |
| 45 | 7L 38s | |
| 46 | 5L 41s | |
| 47 | 3L 44s | |
| 48 | 1L 47s | |
| 49 | 48L 1s | 2:1 |
| 50 | 47L 3s | |
| 51 | 46L 5s | |
| 52 | 45L 7s | |
| 53 | 44L 9s | |
| 54 | 43L 11s | |
| 55 | 42L 13s | |
| 56 | 41L 15s | |
| 57 | 40L 17s | |
| 58 | 39L 19s | |
| 59 | 38L 21s | |
| 60 | 37L 23s | |
| 61 | 36L 25s | |
| 62 | 35L 27s | |
| 63 | 34L 29s | |
| 64 | 33L 31s | |
| 65 | 32L 33s | |
| 66 | 31L 35s | |
| 67 | 30L 37s | |
| 68 | 29L 39s | |
| 69 | 28L 41s | |
| 70 | 27L 43s | |
| 71 | 26L 45s | |
| 72 | 25L 47s | |
| 73 | 24L 49s | |
| 74 | 23L 51s | |
| 75 | 22L 53s | |
| 76 | 21L 55s | |
| 77 | 20L 57s |