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) | |
Divisors
97edo is the 25th prime edo.
JI approximation
97edo has very poor approximation for superparticular intervals among edos up to 200. 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 edos like 19, 29, 31, 41, or 53 have mathematically justified harmony, 97edo is essentially "irredeemable" in terms of either modulation or approximation rationales. 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 |