102edo

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← 101edo102edo103edo →
Prime factorization 2 × 3 × 17
Step size 11.7647¢
Fifth 60\102 (705.882¢) (→10\17)
Semitones (A1:m2) 12:6 (141.2¢ : 70.59¢)
Dual sharp fifth 60\102 (705.882¢) (→10\17)
Dual flat fifth 59\102 (694.118¢)
Dual major 2nd 17\102 (200¢) (→1\6)
Consistency limit 5
Distinct consistency limit 5

102 equal divisions of the octave (abbreviated 102edo or 102ed2), also called 102-tone equal temperament (102tet) or 102 equal temperament (102et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 102 equal parts of about 11.8 ¢ each. Each step represents a frequency ratio of 21/102, or the 102nd root of 2.

In the 5-limit it tempers out the same commas (2048/2025, 15625/15552, 20000/19683) as 34edo. In the 7-limit it tempers out 686/675 and 1029/1024; in the 11-limit 385/384, 441/440 and 4000/3993; in the 13-limit 91/90 and 169/168; in the 17-limit 136/135 and 154/153; and in the 19-limit 133/132 and 190/189. It is the optimal patent val for 13-limit echidnic temperament, and the rank five temperament tempering out 91/90.

Harmonics

Approximation of odd harmonics in 102edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error absolute (¢) +3.93 +1.92 -4.12 -3.91 +1.62 -5.23 +5.85 +0.93 -3.40 -0.19 -4.74
relative (%) +33 +16 -35 -33 +14 -44 +50 +8 -29 -2 -40
Steps
(reduced) 		162
(60) 		237
(33) 		286
(82) 		323
(17) 		353
(47) 		377
(71) 		399
(93) 		417
(9) 		433
(25) 		448
(40) 		461
(53)

Intervals

Steps Cents Ups and downs notation
(dual flat fifth 59\102) 		Ups and downs notation
(dual sharp fifth 60\102) 		Approximate ratios
0 0 D D 1/1
1 11.7647 ^D, E♭♭♭ ^D, v5E♭
2 23.5294 ^^D, v4E♭♭ ^^D, v4E♭ 64/63, 65/64, 78/77
3 35.2941 ^3D, v3E♭♭ ^3D, v3E♭
4 47.0588 ^4D, vvE♭♭ ^4D, vvE♭ 40/39
5 58.8235 D♯, vE♭♭ ^5D, vE♭
6 70.5882 ^D♯, E♭♭ ^6D, E♭ 25/24, 80/77
7 82.3529 ^^D♯, v4E♭ ^7D, v11E 21/20, 22/21
8 94.1176 ^3D♯, v3E♭ ^8D, v10E
9 105.882 ^4D♯, vvE♭ ^9D, v9E 52/49
10 117.647 D𝄪, vE♭ ^10D, v8E
11 129.412 ^D𝄪, E♭ ^11D, v7E 14/13
12 141.176 ^^D𝄪, v4E D♯, v6E
13 152.941 ^3D𝄪, v3E ^D♯, v5E 12/11, 35/32
14 164.706 ^4D𝄪, vvE ^^D♯, v4E 11/10
15 176.471 D♯𝄪, vE ^3D♯, v3E
16 188.235 E ^4D♯, vvE 39/35
17 200 ^E, F♭♭ ^5D♯, vE
18 211.765 ^^E, v4F♭ E 44/39
19 223.529 ^3E, v3F♭ ^E, v5F 25/22
20 235.294 ^4E, vvF♭ ^^E, v4F 8/7, 55/48, 63/55
21 247.059 E♯, vF♭ ^3E, v3F
22 258.824 ^E♯, F♭ ^4E, vvF 64/55, 65/56
23 270.588 ^^E♯, v4F ^5E, vF
24 282.353 ^3E♯, v3F F
25 294.118 ^4E♯, vvF ^F, v5G♭ 77/65
26 305.882 E𝄪, vF ^^F, v4G♭ 25/21
27 317.647 F ^3F, v3G♭ 6/5, 77/64
28 329.412 ^F, G♭♭♭ ^4F, vvG♭ 40/33
29 341.176 ^^F, v4G♭♭ ^5F, vG♭ 39/32
30 352.941 ^3F, v3G♭♭ ^6F, G♭
31 364.706 ^4F, vvG♭♭ ^7F, v11G
32 376.471 F♯, vG♭♭ ^8F, v10G
33 388.235 ^F♯, G♭♭ ^9F, v9G 5/4
34 400 ^^F♯, v4G♭ ^10F, v8G 44/35, 63/50
35 411.765 ^3F♯, v3G♭ ^11F, v7G 80/63
36 423.529 ^4F♯, vvG♭ F♯, v6G 32/25
37 435.294 F𝄪, vG♭ ^F♯, v5G
38 447.059 ^F𝄪, G♭ ^^F♯, v4G
39 458.824 ^^F𝄪, v4G ^3F♯, v3G
40 470.588 ^3F𝄪, v3G ^4F♯, vvG 21/16, 55/42, 72/55
41 482.353 ^4F𝄪, vvG ^5F♯, vG 33/25
42 494.118 F♯𝄪, vG G 4/3
43 505.882 G ^G, v5A♭
44 517.647 ^G, A♭♭♭ ^^G, v4A♭ 35/26
45 529.412 ^^G, v4A♭♭ ^3G, v3A♭
46 541.176 ^3G, v3A♭♭ ^4G, vvA♭ 15/11
47 552.941 ^4G, vvA♭♭ ^5G, vA♭ 11/8
48 564.706 G♯, vA♭♭ ^6G, A♭ 25/18
49 576.471 ^G♯, A♭♭ ^7G, v11A 39/28
50 588.235 ^^G♯, v4A♭ ^8G, v10A
51 600 ^3G♯, v3A♭ ^9G, v9A
52 611.765 ^4G♯, vvA♭ ^10G, v8A
53 623.529 G𝄪, vA♭ ^11G, v7A 56/39, 63/44
54 635.294 ^G𝄪, A♭ G♯, v6A 36/25
55 647.059 ^^G𝄪, v4A ^G♯, v5A 16/11
56 658.824 ^3G𝄪, v3A ^^G♯, v4A 22/15
57 670.588 ^4G𝄪, vvA ^3G♯, v3A
58 682.353 G♯𝄪, vA ^4G♯, vvA 52/35, 77/52
59 694.118 A ^5G♯, vA
60 705.882 ^A, B♭♭♭ A 3/2
61 717.647 ^^A, v4B♭♭ ^A, v5B♭ 50/33
62 729.412 ^3A, v3B♭♭ ^^A, v4B♭ 32/21, 55/36
63 741.176 ^4A, vvB♭♭ ^3A, v3B♭
64 752.941 A♯, vB♭♭ ^4A, vvB♭ 65/42
65 764.706 ^A♯, B♭♭ ^5A, vB♭
66 776.471 ^^A♯, v4B♭ ^6A, B♭ 25/16
67 788.235 ^3A♯, v3B♭ ^7A, v11B 63/40
68 800 ^4A♯, vvB♭ ^8A, v10B 35/22
69 811.765 A𝄪, vB♭ ^9A, v9B 8/5
70 823.529 ^A𝄪, B♭ ^10A, v8B
71 835.294 ^^A𝄪, v4B ^11A, v7B
72 847.059 ^3A𝄪, v3B A♯, v6B
73 858.824 ^4A𝄪, vvB ^A♯, v5B 64/39
74 870.588 A♯𝄪, vB ^^A♯, v4B 33/20
75 882.353 B ^3A♯, v3B 5/3
76 894.118 ^B, C♭♭ ^4A♯, vvB 42/25
77 905.882 ^^B, v4C♭ ^5A♯, vB
78 917.647 ^3B, v3C♭ B
79 929.412 ^4B, vvC♭ ^B, v5C
80 941.176 B♯, vC♭ ^^B, v4C 55/32
81 952.941 ^B♯, C♭ ^3B, v3C
82 964.706 ^^B♯, v4C ^4B, vvC 7/4
83 976.471 ^3B♯, v3C ^5B, vC 44/25
84 988.235 ^4B♯, vvC C 39/22
85 1000 B𝄪, vC ^C, v5D♭
86 1011.76 C ^^C, v4D♭ 70/39
87 1023.53 ^C, D♭♭♭ ^3C, v3D♭
88 1035.29 ^^C, v4D♭♭ ^4C, vvD♭ 20/11
89 1047.06 ^3C, v3D♭♭ ^5C, vD♭ 11/6, 64/35
90 1058.82 ^4C, vvD♭♭ ^6C, D♭
91 1070.59 C♯, vD♭♭ ^7C, v11D 13/7
92 1082.35 ^C♯, D♭♭ ^8C, v10D
93 1094.12 ^^C♯, v4D♭ ^9C, v9D 49/26
94 1105.88 ^3C♯, v3D♭ ^10C, v8D
95 1117.65 ^4C♯, vvD♭ ^11C, v7D 21/11, 40/21
96 1129.41 C𝄪, vD♭ C♯, v6D 48/25, 77/40
97 1141.18 ^C𝄪, D♭ ^C♯, v5D
98 1152.94 ^^C𝄪, v4D ^^C♯, v4D 39/20
99 1164.71 ^3C𝄪, v3D ^3C♯, v3D
100 1176.47 ^4C𝄪, vvD ^4C♯, vvD 63/32, 77/39
101 1188.24 C♯𝄪, vD ^5C♯, vD
102 1200 D D 2/1

13-limit Echidnic

Degree Cents Difference from 46edo
2 23.529 -2.5575¢
4 47.059 -5.115¢
7 82.353 4.092¢
9 105.882 1.5345¢
11 129.412 -1.023¢
13 152.941 8.184¢
16 188.235 5.627¢
18 211.765 3.069¢
20 235.294 0.511¢
22 258.824 -2.046¢
24 282.353 -4.604¢
27 317.647 4.604¢
29 341.176 2.046¢
31 364.706 -0.5115¢
33 388.235 -3.069¢
35 411.765 -5.627¢
38 447.059 3.581¢
40 470.588 1.023¢
42 494.117 -1.5345¢
44 517.647 -4.092¢
47 552.941 5.115¢
49 576.471 2.5575¢
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