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

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 +6 -8 +20 +27 -14 -34 -13 -6 -49 +31
Steps
(reduced) 		147
(54) 		216
(30) 		261
(75) 		295
(16) 		322
(43) 		344
(65) 		363
(84) 		380
(8) 		395
(23) 		408
(36) 		421
(49)

93 = 3 * 31, and 93 is a contorted 31 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 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.

Intervals

Steps Cents Ups and downs notation
(dual flat fifth 54\93) 		Ups and downs notation
(dual sharp fifth 55\93) 		Approximate ratios
0 0 D D 1/1
1 12.9032 ^D, vvE♭♭ ^D, v3E♭
2 25.8065 ^^D, vE♭♭ ^^D, vvE♭ 65/64, 66/65
3 38.7097 ^3D, E♭♭ ^3D, vE♭ 49/48, 50/49
4 51.6129 ^4D, v5E♭ ^4D, E♭ 33/32
5 64.5161 ^5D, v4E♭ ^5D, v12E 26/25, 80/77
6 77.4194 D♯, v3E♭ ^6D, v11E
7 90.3226 ^D♯, vvE♭ ^7D, v10E
8 103.226 ^^D♯, vE♭ ^8D, v9E 35/33, 52/49
9 116.129 ^3D♯, E♭ ^9D, v8E 15/14, 16/15
10 129.032 ^4D♯, v5E ^10D, v7E 14/13
11 141.935 ^5D♯, v4E ^11D, v6E 13/12
12 154.839 D𝄪, v3E ^12D, v5E 35/32
13 167.742 ^D𝄪, vvE D♯, v4E 11/10
14 180.645 ^^D𝄪, vE ^D♯, v3E
15 193.548 E ^^D♯, vvE 28/25
16 206.452 ^E, vvF♭ ^3D♯, vE
17 219.355 ^^E, vF♭ E 25/22
18 232.258 ^3E, F♭ ^E, v3F 8/7
19 245.161 ^4E, v5F ^^E, vvF 15/13
20 258.065 ^5E, v4F ^3E, vF 64/55, 65/56
21 270.968 E♯, v3F F 7/6, 75/64
22 283.871 ^E♯, vvF ^F, v3G♭ 33/28
23 296.774 ^^E♯, vF ^^F, vvG♭ 77/65
24 309.677 F ^3F, vG♭
25 322.581 ^F, vvG♭♭ ^4F, G♭ 77/64
26 335.484 ^^F, vG♭♭ ^5F, v12G 40/33
27 348.387 ^3F, G♭♭ ^6F, v11G 49/40, 60/49
28 361.29 ^4F, v5G♭ ^7F, v10G 16/13
29 374.194 ^5F, v4G♭ ^8F, v9G 26/21
30 387.097 F♯, v3G♭ ^9F, v8G 5/4
31 400 ^F♯, vvG♭ ^10F, v7G 44/35, 49/39
32 412.903 ^^F♯, vG♭ ^11F, v6G 14/11, 33/26
33 425.806 ^3F♯, G♭ ^12F, v5G 32/25
34 438.71 ^4F♯, v5G F♯, v4G
35 451.613 ^5F♯, v4G ^F♯, v3G 13/10
36 464.516 F𝄪, v3G ^^F♯, vvG 64/49
37 477.419 ^F𝄪, vvG ^3F♯, vG 33/25
38 490.323 ^^F𝄪, vG G 65/49
39 503.226 G ^G, v3A♭ 75/56
40 516.129 ^G, vvA♭♭ ^^G, vvA♭ 35/26, 66/49
41 529.032 ^^G, vA♭♭ ^3G, vA♭ 65/48
42 541.935 ^3G, A♭♭ ^4G, A♭ 48/35
43 554.839 ^4G, v5A♭ ^5G, v12A 11/8
44 567.742 ^5G, v4A♭ ^6G, v11A
45 580.645 G♯, v3A♭ ^7G, v10A 7/5
46 593.548 ^G♯, vvA♭ ^8G, v9A
47 606.452 ^^G♯, vA♭ ^9G, v8A
48 619.355 ^3G♯, A♭ ^10G, v7A 10/7
49 632.258 ^4G♯, v5A ^11G, v6A 75/52
50 645.161 ^5G♯, v4A ^12G, v5A 16/11
51 658.065 G𝄪, v3A G♯, v4A 35/24
52 670.968 ^G𝄪, vvA ^G♯, v3A 65/44
53 683.871 ^^G𝄪, vA ^^G♯, vvA 49/33, 52/35, 77/52
54 696.774 A ^3G♯, vA
55 709.677 ^A, vvB♭♭ A
56 722.581 ^^A, vB♭♭ ^A, v3B♭ 50/33
57 735.484 ^3A, B♭♭ ^^A, vvB♭ 49/32, 75/49
58 748.387 ^4A, v5B♭ ^3A, vB♭ 20/13, 77/50
59 761.29 ^5A, v4B♭ ^4A, B♭
60 774.194 A♯, v3B♭ ^5A, v12B 25/16
61 787.097 ^A♯, vvB♭ ^6A, v11B 11/7, 52/33
62 800 ^^A♯, vB♭ ^7A, v10B 35/22, 78/49
63 812.903 ^3A♯, B♭ ^8A, v9B 8/5
64 825.806 ^4A♯, v5B ^9A, v8B 21/13
65 838.71 ^5A♯, v4B ^10A, v7B 13/8
66 851.613 A𝄪, v3B ^11A, v6B 49/30, 80/49
67 864.516 ^A𝄪, vvB ^12A, v5B 33/20
68 877.419 ^^A𝄪, vB A♯, v4B
69 890.323 B ^A♯, v3B
70 903.226 ^B, vvC♭ ^^A♯, vvB
71 916.129 ^^B, vC♭ ^3A♯, vB 56/33
72 929.032 ^3B, C♭ B 12/7
73 941.935 ^4B, v5C ^B, v3C 55/32
74 954.839 ^5B, v4C ^^B, vvC 26/15
75 967.742 B♯, v3C ^3B, vC 7/4
76 980.645 ^B♯, vvC C 44/25
77 993.548 ^^B♯, vC ^C, v3D♭
78 1006.45 C ^^C, vvD♭ 25/14
79 1019.35 ^C, vvD♭♭ ^3C, vD♭
80 1032.26 ^^C, vD♭♭ ^4C, D♭ 20/11
81 1045.16 ^3C, D♭♭ ^5C, v12D 64/35
82 1058.06 ^4C, v5D♭ ^6C, v11D 24/13
83 1070.97 ^5C, v4D♭ ^7C, v10D 13/7
84 1083.87 C♯, v3D♭ ^8C, v9D 15/8, 28/15
85 1096.77 ^C♯, vvD♭ ^9C, v8D 49/26, 66/35
86 1109.68 ^^C♯, vD♭ ^10C, v7D
87 1122.58 ^3C♯, D♭ ^11C, v6D
88 1135.48 ^4C♯, v5D ^12C, v5D 25/13, 77/40
89 1148.39 ^5C♯, v4D C♯, v4D 64/33
90 1161.29 C𝄪, v3D ^C♯, v3D 49/25
91 1174.19 ^C𝄪, vvD ^^C♯, vvD 65/33
92 1187.1 ^^C𝄪, vD ^3C♯, vD
93 1200 D D 2/1

Scales

Meantone Chromatic

  • 116.129
  • 193.548
  • 309.677
  • 387.097
  • 503.226
  • 580.645
  • 696.774
  • 812.903
  • 890.323
  • 1006.452
  • 1083.871
  • 1200.000


Superpyth Chromatic

  • 51.613
  • 219.355
  • 270.968
  • 438.710
  • 490.323
  • 658.065
  • 709.677
  • 761.290
  • 929.032
  • 980.645
  • 1148.387
  • 1200.000


Superpyth Shailaja

  • 270.968
  • 709.677
  • 761.290
  • 980.645
  • 1200.000


Superpyth Subminor Hexatonic

  • 219.355
  • 270.968
  • 490.323
  • 709.677
  • 980.645
  • 1200.000


Superpyth Subminor Pentatonic

  • 270.968
  • 490.323
  • 709.677
  • 980.645
  • 1200.000

See Also

Retrieved from "https://en.xen.wiki/index.php?title=93edo&oldid=132164"