93edo
| ← 92edo | 93edo | 94edo → |
93 equal divisions of the octave (abbreviated 93edo or 93ed2), also called 93-tone equal temperament (93tet) or 93 equal temperament (93et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 93 equal parts of about 12.9 ¢ each. Each step represents a frequency ratio of 21/93, or the 93rd root of 2.
Theory
Since 93 = 3 × 31, 93edo is a contorted 31edo through the 7-limit. In the 11-limit the patent val tempers out 4000/3993 and in the 13-limit 144/143, 1188/1183, and 364/363. It provides the optimal patent val for the 11-limit prajapati and 13-limit kumhar temperaments and the 11- and 13-limit trimean (43 & 50) temperament, and is the 13th no-3s zeta peak edo. The 93bd val is close to the 9-odd limit minimax tuning for superpyth and approximates 2⁄7-septimal comma superpyth very well.
Since 93edo has good approximations of 13th, 17th and 19th harmonics unlike 31edo (as 838.710 ¢, 103.226 ¢, and 296.774 ¢ respectively, octave-reduced), it also allows one to give a clearer harmonic identity to 31edo's excellent approximation of 13:17:19.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -5.18 | +0.78 | -1.08 | +2.54 | +3.52 | -1.82 | -4.40 | -1.73 | -0.74 | -6.26 | +3.98 |
| Relative (%) | -40.2 | +6.1 | -8.4 | +19.7 | +27.3 | -14.1 | -34.1 | -13.4 | -5.7 | -48.6 | +30.9 | |
| Steps (reduced) |
147 (54) |
216 (30) |
261 (75) |
295 (16) |
322 (43) |
344 (65) |
363 (84) |
380 (8) |
395 (23) |
408 (36) |
421 (49) | |
No-3 approach
If prime 3 is ignored, 93edo represents the no-3 35-odd-limit consistently. 93edo is distinctly consistent within the no-3 19-integer-limit.
Intervals
| Steps | Cents | Approximate ratios | Ups and downs notation (Dual flat fifth 54\93) |
Ups and downs notation (Dual sharp fifth 55\93) |
|---|---|---|---|---|
| 0 | 0 | 1/1 | D | D |
| 1 | 12.9 | ^D, vvE♭♭ | ^D, v3E♭ | |
| 2 | 25.8 | ^^D, vE♭♭ | ^^D, vvE♭ | |
| 3 | 38.7 | ^3D, E♭♭ | ^3D, vE♭ | |
| 4 | 51.6 | 33/32, 34/33, 35/34 | vvD♯, ^E♭♭ | ^4D, E♭ |
| 5 | 64.5 | vD♯, ^^E♭♭ | ^5D, ^E♭ | |
| 6 | 77.4 | 23/22 | D♯, v3E♭ | ^6D, ^^E♭ |
| 7 | 90.3 | 20/19, 39/37 | ^D♯, vvE♭ | v6D♯, ^3E♭ |
| 8 | 103.2 | 17/16, 35/33 | ^^D♯, vE♭ | v5D♯, ^4E♭ |
| 9 | 116.1 | 31/29 | ^3D♯, E♭ | v4D♯, ^5E♭ |
| 10 | 129 | 14/13, 41/38 | vvD𝄪, ^E♭ | v3D♯, ^6E♭ |
| 11 | 141.9 | 25/23, 38/35 | vD𝄪, ^^E♭ | vvD♯, v6E |
| 12 | 154.8 | 35/32 | D𝄪, v3E | vD♯, v5E |
| 13 | 167.7 | 32/29 | ^D𝄪, vvE | D♯, v4E |
| 14 | 180.6 | ^^D𝄪, vE | ^D♯, v3E | |
| 15 | 193.5 | 19/17, 28/25 | E | ^^D♯, vvE |
| 16 | 206.5 | ^E, vvF♭ | ^3D♯, vE | |
| 17 | 219.4 | 17/15, 25/22, 42/37 | ^^E, vF♭ | E |
| 18 | 232.3 | 8/7 | ^3E, F♭ | ^E, v3F |
| 19 | 245.2 | 15/13, 38/33 | vvE♯, ^F♭ | ^^E, vvF |
| 20 | 258.1 | 29/25 | vE♯, ^^F♭ | ^3E, vF |
| 21 | 271 | E♯, v3F | F | |
| 22 | 283.9 | 20/17, 33/28 | ^E♯, vvF | ^F, v3G♭ |
| 23 | 296.8 | 19/16 | ^^E♯, vF | ^^F, vvG♭ |
| 24 | 309.7 | F | ^3F, vG♭ | |
| 25 | 322.6 | 41/34 | ^F, vvG♭♭ | ^4F, G♭ |
| 26 | 335.5 | 17/14, 40/33 | ^^F, vG♭♭ | ^5F, ^G♭ |
| 27 | 348.4 | ^3F, G♭♭ | ^6F, ^^G♭ | |
| 28 | 361.3 | 16/13, 37/30 | vvF♯, ^G♭♭ | v6F♯, ^3G♭ |
| 29 | 374.2 | 31/25, 41/33 | vF♯, ^^G♭♭ | v5F♯, ^4G♭ |
| 30 | 387.1 | 5/4 | F♯, v3G♭ | v4F♯, ^5G♭ |
| 31 | 400 | 29/23 | ^F♯, vvG♭ | v3F♯, ^6G♭ |
| 32 | 412.9 | 33/26 | ^^F♯, vG♭ | vvF♯, v6G |
| 33 | 425.8 | 32/25 | ^3F♯, G♭ | vF♯, v5G |
| 34 | 438.7 | 40/31 | vvF𝄪, ^G♭ | F♯, v4G |
| 35 | 451.6 | 13/10 | vF𝄪, ^^G♭ | ^F♯, v3G |
| 36 | 464.5 | 17/13 | F𝄪, v3G | ^^F♯, vvG |
| 37 | 477.4 | 25/19, 29/22 | ^F𝄪, vvG | ^3F♯, vG |
| 38 | 490.3 | ^^F𝄪, vG | G | |
| 39 | 503.2 | G | ^G, v3A♭ | |
| 40 | 516.1 | 31/23, 35/26 | ^G, vvA♭♭ | ^^G, vvA♭ |
| 41 | 529 | 19/14 | ^^G, vA♭♭ | ^3G, vA♭ |
| 42 | 541.9 | 26/19, 41/30 | ^3G, A♭♭ | ^4G, A♭ |
| 43 | 554.8 | 40/29 | vvG♯, ^A♭♭ | ^5G, ^A♭ |
| 44 | 567.7 | 43/31 | vG♯, ^^A♭♭ | ^6G, ^^A♭ |
| 45 | 580.6 | 7/5 | G♯, v3A♭ | v6G♯, ^3A♭ |
| 46 | 593.5 | 31/22 | ^G♯, vvA♭ | v5G♯, ^4A♭ |
| 47 | 606.5 | ^^G♯, vA♭ | v4G♯, ^5A♭ | |
| 48 | 619.4 | 10/7 | ^3G♯, A♭ | v3G♯, ^6A♭ |
| 49 | 632.3 | vvG𝄪, ^A♭ | vvG♯, v6A | |
| 50 | 645.2 | 29/20 | vG𝄪, ^^A♭ | vG♯, v5A |
| 51 | 658.1 | 19/13, 41/28 | G𝄪, v3A | G♯, v4A |
| 52 | 671 | 28/19 | ^G𝄪, vvA | ^G♯, v3A |
| 53 | 683.9 | 43/29 | ^^G𝄪, vA | ^^G♯, vvA |
| 54 | 696.8 | A | ^3G♯, vA | |
| 55 | 709.7 | ^A, vvB♭♭ | A | |
| 56 | 722.6 | 38/25 | ^^A, vB♭♭ | ^A, v3B♭ |
| 57 | 735.5 | 26/17 | ^3A, B♭♭ | ^^A, vvB♭ |
| 58 | 748.4 | 20/13, 37/24 | vvA♯, ^B♭♭ | ^3A, vB♭ |
| 59 | 761.3 | 31/20 | vA♯, ^^B♭♭ | ^4A, B♭ |
| 60 | 774.2 | 25/16 | A♯, v3B♭ | ^5A, ^B♭ |
| 61 | 787.1 | 41/26 | ^A♯, vvB♭ | ^6A, ^^B♭ |
| 62 | 800 | ^^A♯, vB♭ | v6A♯, ^3B♭ | |
| 63 | 812.9 | 8/5 | ^3A♯, B♭ | v5A♯, ^4B♭ |
| 64 | 825.8 | vvA𝄪, ^B♭ | v4A♯, ^5B♭ | |
| 65 | 838.7 | 13/8 | vA𝄪, ^^B♭ | v3A♯, ^6B♭ |
| 66 | 851.6 | A𝄪, v3B | vvA♯, v6B | |
| 67 | 864.5 | 28/17, 33/20 | ^A𝄪, vvB | vA♯, v5B |
| 68 | 877.4 | ^^A𝄪, vB | A♯, v4B | |
| 69 | 890.3 | B | ^A♯, v3B | |
| 70 | 903.2 | 32/19 | ^B, vvC♭ | ^^A♯, vvB |
| 71 | 916.1 | 17/10 | ^^B, vC♭ | ^3A♯, vB |
| 72 | 929 | 41/24 | ^3B, C♭ | B |
| 73 | 941.9 | vvB♯, ^C♭ | ^B, v3C | |
| 74 | 954.8 | 26/15, 33/19 | vB♯, ^^C♭ | ^^B, vvC |
| 75 | 967.7 | 7/4 | B♯, v3C | ^3B, vC |
| 76 | 980.6 | 30/17, 37/21 | ^B♯, vvC | C |
| 77 | 993.5 | ^^B♯, vC | ^C, v3D♭ | |
| 78 | 1006.5 | 25/14, 34/19 | C | ^^C, vvD♭ |
| 79 | 1019.4 | ^C, vvD♭♭ | ^3C, vD♭ | |
| 80 | 1032.3 | 29/16 | ^^C, vD♭♭ | ^4C, D♭ |
| 81 | 1045.2 | ^3C, D♭♭ | ^5C, ^D♭ | |
| 82 | 1058.1 | 35/19 | vvC♯, ^D♭♭ | ^6C, ^^D♭ |
| 83 | 1071 | 13/7 | vC♯, ^^D♭♭ | v6C♯, ^3D♭ |
| 84 | 1083.9 | 43/23 | C♯, v3D♭ | v5C♯, ^4D♭ |
| 85 | 1096.8 | 32/17 | ^C♯, vvD♭ | v4C♯, ^5D♭ |
| 86 | 1109.7 | 19/10 | ^^C♯, vD♭ | v3C♯, ^6D♭ |
| 87 | 1122.6 | ^3C♯, D♭ | vvC♯, v6D | |
| 88 | 1135.5 | vvC𝄪, ^D♭ | vC♯, v5D | |
| 89 | 1148.4 | 33/17 | vC𝄪, ^^D♭ | C♯, v4D |
| 90 | 1161.3 | 43/22 | C𝄪, v3D | ^C♯, v3D |
| 91 | 1174.2 | ^C𝄪, vvD | ^^C♯, vvD | |
| 92 | 1187.1 | ^^C𝄪, vD | ^3C♯, vD | |
| 93 | 1200 | 2/1 | D | D |
Scales
- Superpyth[5]: 21 17 17 21 17 ((21 38 55 76 93)\93)
- Superpyth[12]: 4 13 4 13 4 13 4 4 13 4 13 4 ((4 17 21 34 38 51 55 59 72 76 89 93)\93)
- Superpyth Shailaja: 21 34 4 17 17 ((21 55 59 76 93)\93)
- Superpyth Subminor Hexatonic: 17 4 17 17 21 17 ((17 21 38 55 76 93)\93)
Instruments
A Lumatone mapping for 93edo is available.