93edo

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← 92edo93edo94edo →
Prime factorization 3 × 31
Step size 12.9032¢ 
Fifth 54\93 (696.774¢) (→18\31)
Semitones (A1:m2) 6:9 (77.42¢ : 116.1¢)
Dual sharp fifth 55\93 (709.677¢)
Dual flat fifth 54\93 (696.774¢) (→18\31)
Dual major 2nd 16\93 (206.452¢)
Consistency limit 7
Distinct consistency limit 7

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

Approximation of odd harmonics in 93edo
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)

See also