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
93 = 3 × 31, and 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 superpine (43 & 50) temperament. It is the 13th no-3s zeta peak edo.
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) |
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 | 12.903 | ^D, vvE♭♭ | ^D, v3E♭ | |
2 | 25.806 | ^^D, vE♭♭ | ^^D, vvE♭ | |
3 | 38.71 | ^3D, E♭♭ | ^3D, vE♭ | |
4 | 51.613 | 33/32, 34/33, 35/34 | ^4D, v5E♭ | ^4D, E♭ |
5 | 64.516 | ^5D, v4E♭ | ^5D, v12E | |
6 | 77.419 | 23/22 | D♯, v3E♭ | ^6D, v11E |
7 | 90.323 | 20/19, 39/37 | ^D♯, vvE♭ | ^7D, v10E |
8 | 103.226 | 17/16, 35/33 | ^^D♯, vE♭ | ^8D, v9E |
9 | 116.129 | 31/29 | ^3D♯, E♭ | ^9D, v8E |
10 | 129.032 | 14/13, 41/38 | ^4D♯, v5E | ^10D, v7E |
11 | 141.935 | 25/23, 38/35 | ^5D♯, v4E | ^11D, v6E |
12 | 154.839 | 35/32 | D𝄪, v3E | ^12D, v5E |
13 | 167.742 | 32/29 | ^D𝄪, vvE | D♯, v4E |
14 | 180.645 | ^^D𝄪, vE | ^D♯, v3E | |
15 | 193.548 | 19/17, 28/25 | E | ^^D♯, vvE |
16 | 206.452 | ^E, vvF♭ | ^3D♯, vE | |
17 | 219.355 | 17/15, 25/22, 42/37 | ^^E, vF♭ | E |
18 | 232.258 | 8/7 | ^3E, F♭ | ^E, v3F |
19 | 245.161 | 15/13, 38/33 | ^4E, v5F | ^^E, vvF |
20 | 258.065 | 29/25 | ^5E, v4F | ^3E, vF |
21 | 270.968 | E♯, v3F | F | |
22 | 283.871 | 20/17, 33/28 | ^E♯, vvF | ^F, v3G♭ |
23 | 296.774 | 19/16 | ^^E♯, vF | ^^F, vvG♭ |
24 | 309.677 | F | ^3F, vG♭ | |
25 | 322.581 | 41/34 | ^F, vvG♭♭ | ^4F, G♭ |
26 | 335.484 | 17/14, 40/33 | ^^F, vG♭♭ | ^5F, v12G |
27 | 348.387 | ^3F, G♭♭ | ^6F, v11G | |
28 | 361.29 | 16/13, 37/30 | ^4F, v5G♭ | ^7F, v10G |
29 | 374.194 | 31/25, 41/33 | ^5F, v4G♭ | ^8F, v9G |
30 | 387.097 | 5/4 | F♯, v3G♭ | ^9F, v8G |
31 | 400 | 29/23 | ^F♯, vvG♭ | ^10F, v7G |
32 | 412.903 | 33/26 | ^^F♯, vG♭ | ^11F, v6G |
33 | 425.806 | 32/25 | ^3F♯, G♭ | ^12F, v5G |
34 | 438.71 | 40/31 | ^4F♯, v5G | F♯, v4G |
35 | 451.613 | 13/10 | ^5F♯, v4G | ^F♯, v3G |
36 | 464.516 | 17/13 | F𝄪, v3G | ^^F♯, vvG |
37 | 477.419 | 25/19, 29/22 | ^F𝄪, vvG | ^3F♯, vG |
38 | 490.323 | ^^F𝄪, vG | G | |
39 | 503.226 | G | ^G, v3A♭ | |
40 | 516.129 | 31/23, 35/26 | ^G, vvA♭♭ | ^^G, vvA♭ |
41 | 529.032 | 19/14 | ^^G, vA♭♭ | ^3G, vA♭ |
42 | 541.935 | 26/19, 41/30 | ^3G, A♭♭ | ^4G, A♭ |
43 | 554.839 | 40/29 | ^4G, v5A♭ | ^5G, v12A |
44 | 567.742 | 43/31 | ^5G, v4A♭ | ^6G, v11A |
45 | 580.645 | 7/5 | G♯, v3A♭ | ^7G, v10A |
46 | 593.548 | 31/22 | ^G♯, vvA♭ | ^8G, v9A |
47 | 606.452 | ^^G♯, vA♭ | ^9G, v8A | |
48 | 619.355 | 10/7 | ^3G♯, A♭ | ^10G, v7A |
49 | 632.258 | ^4G♯, v5A | ^11G, v6A | |
50 | 645.161 | 29/20 | ^5G♯, v4A | ^12G, v5A |
51 | 658.065 | 19/13, 41/28 | G𝄪, v3A | G♯, v4A |
52 | 670.968 | 28/19 | ^G𝄪, vvA | ^G♯, v3A |
53 | 683.871 | 43/29 | ^^G𝄪, vA | ^^G♯, vvA |
54 | 696.774 | A | ^3G♯, vA | |
55 | 709.677 | ^A, vvB♭♭ | A | |
56 | 722.581 | 38/25 | ^^A, vB♭♭ | ^A, v3B♭ |
57 | 735.484 | 26/17 | ^3A, B♭♭ | ^^A, vvB♭ |
58 | 748.387 | 20/13, 37/24 | ^4A, v5B♭ | ^3A, vB♭ |
59 | 761.29 | 31/20 | ^5A, v4B♭ | ^4A, B♭ |
60 | 774.194 | 25/16 | A♯, v3B♭ | ^5A, v12B |
61 | 787.097 | 41/26 | ^A♯, vvB♭ | ^6A, v11B |
62 | 800 | ^^A♯, vB♭ | ^7A, v10B | |
63 | 812.903 | 8/5 | ^3A♯, B♭ | ^8A, v9B |
64 | 825.806 | ^4A♯, v5B | ^9A, v8B | |
65 | 838.71 | 13/8 | ^5A♯, v4B | ^10A, v7B |
66 | 851.613 | A𝄪, v3B | ^11A, v6B | |
67 | 864.516 | 28/17, 33/20 | ^A𝄪, vvB | ^12A, v5B |
68 | 877.419 | ^^A𝄪, vB | A♯, v4B | |
69 | 890.323 | B | ^A♯, v3B | |
70 | 903.226 | 32/19 | ^B, vvC♭ | ^^A♯, vvB |
71 | 916.129 | 17/10 | ^^B, vC♭ | ^3A♯, vB |
72 | 929.032 | 41/24 | ^3B, C♭ | B |
73 | 941.935 | ^4B, v5C | ^B, v3C | |
74 | 954.839 | 26/15, 33/19 | ^5B, v4C | ^^B, vvC |
75 | 967.742 | 7/4 | B♯, v3C | ^3B, vC |
76 | 980.645 | 30/17, 37/21 | ^B♯, vvC | C |
77 | 993.548 | ^^B♯, vC | ^C, v3D♭ | |
78 | 1006.452 | 25/14, 34/19 | C | ^^C, vvD♭ |
79 | 1019.355 | ^C, vvD♭♭ | ^3C, vD♭ | |
80 | 1032.258 | 29/16 | ^^C, vD♭♭ | ^4C, D♭ |
81 | 1045.161 | ^3C, D♭♭ | ^5C, v12D | |
82 | 1058.065 | 35/19 | ^4C, v5D♭ | ^6C, v11D |
83 | 1070.968 | 13/7 | ^5C, v4D♭ | ^7C, v10D |
84 | 1083.871 | 43/23 | C♯, v3D♭ | ^8C, v9D |
85 | 1096.774 | 32/17 | ^C♯, vvD♭ | ^9C, v8D |
86 | 1109.677 | 19/10 | ^^C♯, vD♭ | ^10C, v7D |
87 | 1122.581 | ^3C♯, D♭ | ^11C, v6D | |
88 | 1135.484 | ^4C♯, v5D | ^12C, v5D | |
89 | 1148.387 | 33/17 | ^5C♯, v4D | C♯, v4D |
90 | 1161.29 | 43/22 | C𝄪, v3D | ^C♯, v3D |
91 | 1174.194 | ^C𝄪, vvD | ^^C♯, vvD | |
92 | 1187.097 | ^^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)