97edo
| ← 96edo | 97edo | 98edo → |
97 equal divisions of the octave (abbreviated 97edo or 97ed2), also called 97-tone equal temperament (97tet) or 97 equal temperament (97et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 97 equal parts of about 12.4 ¢ each. Each step represents a frequency ratio of 21/97, or the 97th root of 2.
Theory
97edo is only consistent to the 5-odd-limit. The patent val of 97edo tempers out 875/864, 1029/1024, and 4000/3969 in the 7-limit, 100/99, 245/242, 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, following 89edo and before 101edo.
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.
Approximation to JI
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. The following tables show how 15-odd-limit intervals are represented in 97edo. Prime harmonics are in bold; inconsistent intervals are in italics.
| Interval and complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 1/1, 2/1 | 0.000 | 0.0 |
| 15/13, 26/15 | 0.318 | 2.6 |
| 15/8, 16/15 | 0.391 | 3.2 |
| 13/8, 16/13 | 0.709 | 5.7 |
| 11/9, 18/11 | 1.016 | 8.2 |
| 7/5, 10/7 | 1.069 | 8.6 |
| 9/7, 14/9 | 2.094 | 16.9 |
| 11/6, 12/11 | 2.183 | 17.6 |
| 13/12, 24/13 | 2.490 | 20.1 |
| 5/4, 8/5 | 2.809 | 22.7 |
| 11/7, 14/11 | 3.111 | 25.1 |
| 9/5, 10/9 | 3.163 | 25.6 |
| 3/2, 4/3 | 3.200 | 25.9 |
| 13/10, 20/13 | 3.518 | 28.4 |
| 7/4, 8/7 | 3.877 | 31.3 |
| 11/10, 20/11 | 4.179 | 33.8 |
| 15/14, 28/15 | 4.269 | 34.5 |
| 13/7, 14/13 | 4.587 | 37.1 |
| 13/11, 22/13 | 4.674 | 37.8 |
| 15/11, 22/15 | 4.992 | 40.4 |
| 7/6, 12/7 | 5.294 | 42.8 |
| 11/8, 16/11 | 5.383 | 43.5 |
| 13/9, 18/13 | 5.690 | 46.0 |
| 9/8, 16/9 | 5.972 | 48.3 |
| 5/3, 6/5 | 6.008 | 48.6 |
| Interval and complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 1/1, 2/1 | 0.000 | 0.0 |
| 15/13, 26/15 | 0.318 | 2.6 |
| 15/8, 16/15 | 0.391 | 3.2 |
| 13/8, 16/13 | 0.709 | 5.7 |
| 11/9, 18/11 | 1.016 | 8.2 |
| 7/5, 10/7 | 1.069 | 8.6 |
| 11/6, 12/11 | 2.183 | 17.6 |
| 13/12, 24/13 | 2.490 | 20.1 |
| 5/4, 8/5 | 2.809 | 22.7 |
| 3/2, 4/3 | 3.200 | 25.9 |
| 13/10, 20/13 | 3.518 | 28.4 |
| 7/4, 8/7 | 3.877 | 31.3 |
| 15/14, 28/15 | 4.269 | 34.5 |
| 13/7, 14/13 | 4.587 | 37.1 |
| 13/11, 22/13 | 4.674 | 37.8 |
| 15/11, 22/15 | 4.992 | 40.4 |
| 11/8, 16/11 | 5.383 | 43.5 |
| 13/9, 18/13 | 5.690 | 46.0 |
| 5/3, 6/5 | 6.008 | 48.6 |
| 9/8, 16/9 | 6.399 | 51.7 |
| 7/6, 12/7 | 7.077 | 57.2 |
| 11/10, 20/11 | 8.192 | 66.2 |
| 9/5, 10/9 | 9.208 | 74.4 |
| 11/7, 14/11 | 9.261 | 74.9 |
| 9/7, 14/9 | 10.277 | 83.1 |
Intervals
| Steps | Cents | Approximate ratios | Ups and downs notation |
|---|---|---|---|
| 0 | 0 | 1/1 | D |
| 1 | 12.4 | ^D, v5E♭ | |
| 2 | 24.7 | ^^D, v4E♭ | |
| 3 | 37.1 | ^3D, v3E♭ | |
| 4 | 49.5 | 35/34 | ^4D, vvE♭ |
| 5 | 61.9 | 29/28 | ^5D, vE♭ |
| 6 | 74.2 | 24/23 | v5D♯, E♭ |
| 7 | 86.6 | 20/19, 21/20, 41/39 | v4D♯, ^E♭ |
| 8 | 99 | v3D♯, ^^E♭ | |
| 9 | 111.3 | 16/15 | vvD♯, ^3E♭ |
| 10 | 123.7 | 43/40, 44/41 | vD♯, ^4E♭ |
| 11 | 136.1 | 13/12, 40/37 | D♯, ^5E♭ |
| 12 | 148.5 | 12/11, 37/34 | ^D♯, v5E |
| 13 | 160.8 | ^^D♯, v4E | |
| 14 | 173.2 | 21/19 | ^3D♯, v3E |
| 15 | 185.6 | ^4D♯, vvE | |
| 16 | 197.9 | 28/25 | ^5D♯, vE |
| 17 | 210.3 | 26/23, 44/39 | E |
| 18 | 222.7 | 41/36 | ^E, v5F |
| 19 | 235.1 | ^^E, v4F | |
| 20 | 247.4 | 15/13 | ^3E, v3F |
| 21 | 259.8 | 36/31, 43/37 | ^4E, vvF |
| 22 | 272.2 | ^5E, vF | |
| 23 | 284.5 | F | |
| 24 | 296.9 | 19/16 | ^F, v5G♭ |
| 25 | 309.3 | ^^F, v4G♭ | |
| 26 | 321.6 | ^3F, v3G♭ | |
| 27 | 334 | 17/14 | ^4F, vvG♭ |
| 28 | 346.4 | 11/9 | ^5F, vG♭ |
| 29 | 358.8 | 16/13, 43/35 | v5F♯, G♭ |
| 30 | 371.1 | 26/21 | v4F♯, ^G♭ |
| 31 | 383.5 | v3F♯, ^^G♭ | |
| 32 | 395.9 | 39/31 | vvF♯, ^3G♭ |
| 33 | 408.2 | 19/15, 43/34 | vF♯, ^4G♭ |
| 34 | 420.6 | 37/29 | F♯, ^5G♭ |
| 35 | 433 | ^F♯, v5G | |
| 36 | 445.4 | 31/24 | ^^F♯, v4G |
| 37 | 457.7 | 30/23 | ^3F♯, v3G |
| 38 | 470.1 | 21/16, 38/29 | ^4F♯, vvG |
| 39 | 482.5 | 37/28, 41/31 | ^5F♯, vG |
| 40 | 494.8 | G | |
| 41 | 507.2 | ^G, v5A♭ | |
| 42 | 519.6 | ^^G, v4A♭ | |
| 43 | 532 | 34/25 | ^3G, v3A♭ |
| 44 | 544.3 | 26/19 | ^4G, vvA♭ |
| 45 | 556.7 | 29/21, 40/29 | ^5G, vA♭ |
| 46 | 569.1 | v5G♯, A♭ | |
| 47 | 581.4 | 7/5 | v4G♯, ^A♭ |
| 48 | 593.8 | 31/22 | v3G♯, ^^A♭ |
| 49 | 606.2 | 44/31 | vvG♯, ^3A♭ |
| 50 | 618.6 | 10/7 | vG♯, ^4A♭ |
| 51 | 630.9 | G♯, ^5A♭ | |
| 52 | 643.3 | 29/20, 42/29 | ^G♯, v5A |
| 53 | 655.7 | 19/13 | ^^G♯, v4A |
| 54 | 668 | 25/17 | ^3G♯, v3A |
| 55 | 680.4 | 37/25, 43/29 | ^4G♯, vvA |
| 56 | 692.8 | ^5G♯, vA | |
| 57 | 705.2 | A | |
| 58 | 717.5 | ^A, v5B♭ | |
| 59 | 729.9 | 29/19, 32/21 | ^^A, v4B♭ |
| 60 | 742.3 | 23/15, 43/28 | ^3A, v3B♭ |
| 61 | 754.6 | ^4A, vvB♭ | |
| 62 | 767 | ^5A, vB♭ | |
| 63 | 779.4 | v5A♯, B♭ | |
| 64 | 791.8 | 30/19 | v4A♯, ^B♭ |
| 65 | 804.1 | v3A♯, ^^B♭ | |
| 66 | 816.5 | vvA♯, ^3B♭ | |
| 67 | 828.9 | 21/13 | vA♯, ^4B♭ |
| 68 | 841.2 | 13/8 | A♯, ^5B♭ |
| 69 | 853.6 | 18/11 | ^A♯, v5B |
| 70 | 866 | 28/17 | ^^A♯, v4B |
| 71 | 878.4 | ^3A♯, v3B | |
| 72 | 890.7 | ^4A♯, vvB | |
| 73 | 903.1 | 32/19 | ^5A♯, vB |
| 74 | 915.5 | 39/23 | B |
| 75 | 927.8 | 41/24 | ^B, v5C |
| 76 | 940.2 | 31/18, 43/25 | ^^B, v4C |
| 77 | 952.6 | 26/15 | ^3B, v3C |
| 78 | 964.9 | ^4B, vvC | |
| 79 | 977.3 | ^5B, vC | |
| 80 | 989.7 | 23/13, 39/22 | C |
| 81 | 1002.1 | 25/14, 41/23 | ^C, v5D♭ |
| 82 | 1014.4 | ^^C, v4D♭ | |
| 83 | 1026.8 | 38/21 | ^3C, v3D♭ |
| 84 | 1039.2 | ^4C, vvD♭ | |
| 85 | 1051.5 | 11/6 | ^5C, vD♭ |
| 86 | 1063.9 | 24/13, 37/20 | v5C♯, D♭ |
| 87 | 1076.3 | 41/22 | v4C♯, ^D♭ |
| 88 | 1088.7 | 15/8 | v3C♯, ^^D♭ |
| 89 | 1101 | vvC♯, ^3D♭ | |
| 90 | 1113.4 | 19/10, 40/21 | vC♯, ^4D♭ |
| 91 | 1125.8 | 23/12 | C♯, ^5D♭ |
| 92 | 1138.1 | ^C♯, v5D | |
| 93 | 1150.5 | ^^C♯, v4D | |
| 94 | 1162.9 | ^3C♯, v3D | |
| 95 | 1175.3 | ^4C♯, vvD | |
| 96 | 1187.6 | ^5C♯, vD | |
| 97 | 1200 | 2/1 | D |
Music
- Joyous Stellaris (2023) – semiquartal in 97edo tuning