# 83edo

 ← 82edo 83edo 84edo →
Prime factorization 83 (prime)
Step size 14.4578¢
Fifth 49\83 (708.434¢)
Semitones (A1:m2) 11:4 (159¢ : 57.83¢)
Dual sharp fifth 49\83 (708.434¢)
Dual flat fifth 48\83 (693.976¢)
Dual major 2nd 14\83 (202.41¢)
Consistency limit 7
Distinct consistency limit 7

83 equal divisions of the octave (abbreviated 83edo or 83ed2), also called 83-tone equal temperament (83tet) or 83 equal temperament (83et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 83 equal parts of about 14.5 ¢ each. Each step represents a frequency ratio of 21/83, or the 83rd root of 2.

## Theory

Approximation of odd harmonics in 83edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +6.48 +4.05 -0.15 -1.50 -1.92 -1.97 -3.93 -3.75 +6.10 +6.33 -6.59
Relative (%) +44.8 +28.0 -1.0 -10.4 -13.3 -13.6 -27.2 -25.9 +42.2 +43.8 -45.6
Steps
(reduced)
132
(49)
193
(27)
233
(67)
263
(14)
287
(38)
307
(58)
324
(75)
339
(7)
353
(21)
365
(33)
375
(43)

The 3/1 is 6.5 cents sharp and the 5/1 is 4 cents sharp, with 7, 11, and 13 more accurate but a little flat. It tempers out 15625/15552 in the 5-limit and 686/675, 4000/3969 and 6144/6125 in the 7-limit, and provides the optimal patent val for the 7-limit 27&56 temperament with wedgie ⟨⟨5 18 17 17 13 -11]]. In the 11-limit it tempers out 121/120, 176/175 and 385/384, and in the 13-limit 91/90, 169/168 and 196/195, and it provides the optimal patent val for the 11-limit 22&61 temperament and the 13-limit 15&83 temperament. 83edo is the 23rd prime EDO.

Every odd harmonic between the 7th and the 17th is tuned flatly. As a consequence, this tuning provides a good approximation of the 7:9:11:13:15:17 hexad, and especially of the 9:11:13 triad.

## Intervals

Steps Cents Ups and downs notation
(dual flat fifth 48\83)
Ups and downs notation
(dual sharp fifth 49\83)
Approximate ratios
0 0 D D 1/1
1 14.4578 ^D, E♭♭♭ ^D, v3E♭
2 28.9157 ^^D, v3E♭♭ ^^D, vvE♭ 56/55, 65/64, 66/65
3 43.3735 ^3D, vvE♭♭ ^3D, vE♭ 40/39
4 57.8313 D♯, vE♭♭ ^4D, E♭ 33/32
5 72.2892 ^D♯, E♭♭ ^5D, v10E 25/24
6 86.747 ^^D♯, v3E♭ ^6D, v9E 21/20
7 101.205 ^3D♯, vvE♭ ^7D, v8E 35/33, 52/49, 55/52
8 115.663 D𝄪, vE♭ ^8D, v7E
9 130.12 ^D𝄪, E♭ ^9D, v6E 14/13
10 144.578 ^^D𝄪, v3E ^10D, v5E
11 159.036 ^3D𝄪, vvE D♯, v4E 35/32
12 173.494 D♯𝄪, vE ^D♯, v3E
13 187.952 E ^^D♯, vvE 39/35, 49/44
14 202.41 ^E, F♭♭ ^3D♯, vE 55/49
15 216.867 ^^E, v3F♭ E
16 231.325 ^3E, vvF♭ ^E, v3F 8/7, 55/48
17 245.783 E♯, vF♭ ^^E, vvF
18 260.241 ^E♯, F♭ ^3E, vF 64/55, 65/56
19 274.699 ^^E♯, v3F F
20 289.157 ^3E♯, vvF ^F, v3G♭ 13/11, 33/28, 77/65
21 303.614 E𝄪, vF ^^F, vvG♭ 25/21
22 318.072 F ^3F, vG♭ 6/5, 77/64
23 332.53 ^F, G♭♭♭ ^4F, G♭ 40/33
24 346.988 ^^F, v3G♭♭ ^5F, v10G 39/32, 49/40
25 361.446 ^3F, vvG♭♭ ^6F, v9G 16/13
26 375.904 F♯, vG♭♭ ^7F, v8G
27 390.361 ^F♯, G♭♭ ^8F, v7G 5/4, 49/39
28 404.819 ^^F♯, v3G♭ ^9F, v6G 63/50
29 419.277 ^3F♯, vvG♭ ^10F, v5G 14/11
30 433.735 F𝄪, vG♭ F♯, v4G 50/39
31 448.193 ^F𝄪, G♭ ^F♯, v3G
32 462.651 ^^F𝄪, v3G ^^F♯, vvG 55/42, 64/49
33 477.108 ^3F𝄪, vvG ^3F♯, vG 33/25
34 491.566 F♯𝄪, vG G 65/49
35 506.024 G ^G, v3A♭
36 520.482 ^G, A♭♭♭ ^^G, vvA♭ 65/48, 66/49
37 534.94 ^^G, v3A♭♭ ^3G, vA♭
38 549.398 ^3G, vvA♭♭ ^4G, A♭ 11/8, 48/35
39 563.855 G♯, vA♭♭ ^5G, v10A 25/18
40 578.313 ^G♯, A♭♭ ^6G, v9A 7/5, 39/28
41 592.771 ^^G♯, v3A♭ ^7G, v8A 55/39
42 607.229 ^3G♯, vvA♭ ^8G, v7A 78/55
43 621.687 G𝄪, vA♭ ^9G, v6A 10/7, 56/39
44 636.145 ^G𝄪, A♭ ^10G, v5A 36/25
45 650.602 ^^G𝄪, v3A G♯, v4A 16/11, 35/24
46 665.06 ^3G𝄪, vvA ^G♯, v3A
47 679.518 G♯𝄪, vA ^^G♯, vvA 49/33, 65/44, 77/52
48 693.976 A ^3G♯, vA
49 708.434 ^A, B♭♭♭ A
50 722.892 ^^A, v3B♭♭ ^A, v3B♭ 50/33
51 737.349 ^3A, vvB♭♭ ^^A, vvB♭ 49/32
52 751.807 A♯, vB♭♭ ^3A, vB♭ 65/42
53 766.265 ^A♯, B♭♭ ^4A, B♭ 39/25
54 780.723 ^^A♯, v3B♭ ^5A, v10B 11/7
55 795.181 ^3A♯, vvB♭ ^6A, v9B
56 809.639 A𝄪, vB♭ ^7A, v8B 8/5, 78/49
57 824.096 ^A𝄪, B♭ ^8A, v7B
58 838.554 ^^A𝄪, v3B ^9A, v6B 13/8
59 853.012 ^3A𝄪, vvB ^10A, v5B 64/39, 80/49
60 867.47 A♯𝄪, vB A♯, v4B 33/20
61 881.928 B ^A♯, v3B 5/3
62 896.386 ^B, C♭♭ ^^A♯, vvB 42/25
63 910.843 ^^B, v3C♭ ^3A♯, vB 22/13, 56/33
64 925.301 ^3B, vvC♭ B
65 939.759 B♯, vC♭ ^B, v3C 55/32
66 954.217 ^B♯, C♭ ^^B, vvC
67 968.675 ^^B♯, v3C ^3B, vC 7/4
68 983.133 ^3B♯, vvC C
69 997.59 B𝄪, vC ^C, v3D♭
70 1012.05 C ^^C, vvD♭ 70/39
71 1026.51 ^C, D♭♭♭ ^3C, vD♭
72 1040.96 ^^C, v3D♭♭ ^4C, D♭ 64/35
73 1055.42 ^3C, vvD♭♭ ^5C, v10D
74 1069.88 C♯, vD♭♭ ^6C, v9D 13/7
75 1084.34 ^C♯, D♭♭ ^7C, v8D
76 1098.8 ^^C♯, v3D♭ ^8C, v7D 49/26, 66/35
77 1113.25 ^3C♯, vvD♭ ^9C, v6D 40/21
78 1127.71 C𝄪, vD♭ ^10C, v5D 48/25
79 1142.17 ^C𝄪, D♭ C♯, v4D 64/33
80 1156.63 ^^C𝄪, v3D ^C♯, v3D 39/20
81 1171.08 ^3C𝄪, vvD ^^C♯, vvD 55/28, 65/33
82 1185.54 C♯𝄪, vD ^3C♯, vD
83 1200 D D 2/1