83edo

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← 82edo83edo84edo →
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 (83edo), or 83-tone equal temperament (83tet), 83 equal temperament (83et) when viewed from a regular temperament perspective, is the tuning system that divides the octave into 83 equal parts of about 14.5 ¢ each.

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 (%) +45 +28 -1 -10 -13 -14 -27 -26 +42 +44 -46
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, Ebbb ^D, v3Eb 55/54, 64/63
2 28.9157 ^^D, v3Ebb ^^D, vvEb 49/48, 56/55, 65/64, 66/65, 77/75, 78/77
3 43.3735 ^3D, vvEbb ^3D, vEb 28/27, 40/39, 50/49, 65/63, 81/80
4 57.8313 D#, vEbb ^4D, Eb 26/25, 33/32, 36/35, 45/44
5 72.2892 ^D#, Ebb ^5D, v10E 22/21, 25/24, 80/77
6 86.747 ^^D#, v3Eb ^6D, v9E 21/20, 27/26
7 101.205 ^3D#, vvEb ^7D, v8E 16/15, 35/33, 52/49, 55/52, 77/72
8 115.663 Dx, vEb ^8D, v7E 81/77
9 130.12 ^Dx, Eb ^9D, v6E 13/12, 14/13, 15/14, 49/45
10 144.578 ^^Dx, v3E ^10D, v5E 27/25
11 159.036 ^3Dx, vvE D#, v4E 11/10, 12/11, 35/32
12 173.494 D#x, vE ^D#, v3E 10/9
13 187.952 E ^^D#, vvE 28/25, 39/35, 49/44, 54/49, 72/65
14 202.41 ^E, Fbb ^3D#, vE 44/39, 55/49
15 216.867 ^^E, v3Fb E 9/8
16 231.325 ^3E, vvFb ^E, v3F 8/7, 25/22, 52/45, 55/48
17 245.783 E#, vFb ^^E, vvF 63/55
18 260.241 ^E#, Fb ^3E, vF 7/6, 15/13, 64/55, 65/56
19 274.699 ^^E#, v3F F 32/27, 81/70
20 289.157 ^3E#, vvF ^F, v3Gb 13/11, 33/28, 75/64, 77/65
21 303.614 Ex, vF ^^F, vvGb 25/21, 65/54
22 318.072 F ^3F, vGb 6/5, 77/64
23 332.53 ^F, Gbbb ^4F, Gb 11/9, 40/33
24 346.988 ^^F, v3Gbb ^5F, v10G 39/32, 49/40, 63/52
25 361.446 ^3F, vvGbb ^6F, v9G 16/13, 26/21, 56/45, 60/49
26 375.904 F#, vGbb ^7F, v8G 27/22
27 390.361 ^F#, Gbb ^8F, v7G 5/4, 44/35, 49/39
28 404.819 ^^F#, v3Gb ^9F, v6G 63/50, 80/63, 81/65
29 419.277 ^3F#, vvGb ^10F, v5G 14/11, 32/25, 33/26, 77/60
30 433.735 Fx, vGb F#, v4G 35/27, 50/39, 81/64
31 448.193 ^Fx, Gb ^F#, v3G 9/7, 13/10
32 462.651 ^^Fx, v3G ^^F#, vvG 55/42, 64/49
33 477.108 ^3Fx, vvG ^3F#, vG 21/16, 33/25, 72/55
34 491.566 F#x, vG G 4/3, 65/49
35 506.024 G ^G, v3Ab
36 520.482 ^G, Abbb ^^G, vvAb 35/26, 49/36, 65/48, 66/49, 75/56
37 534.94 ^^G, v3Abb ^3G, vAb 27/20
38 549.398 ^3G, vvAbb ^4G, Ab 11/8, 15/11, 48/35
39 563.855 G#, vAbb ^5G, v10A 25/18
40 578.313 ^G#, Abb ^6G, v9A 7/5, 18/13, 39/28
41 592.771 ^^G#, v3Ab ^7G, v8A 55/39, 64/45, 77/54
42 607.229 ^3G#, vvAb ^8G, v7A 45/32, 78/55
43 621.687 Gx, vAb ^9G, v6A 10/7, 13/9, 56/39
44 636.145 ^Gx, Ab ^10G, v5A 36/25, 63/44
45 650.602 ^^Gx, v3A G#, v4A 16/11, 22/15, 35/24, 75/52
46 665.06 ^3Gx, vvA ^G#, v3A 40/27, 81/56
47 679.518 G#x, vA ^^G#, vvA 49/33, 52/35, 65/44, 72/49, 77/52
48 693.976 A ^3G#, vA 81/55
49 708.434 ^A, Bbbb A 3/2
50 722.892 ^^A, v3Bbb ^A, v3Bb 32/21, 50/33, 55/36
51 737.349 ^3A, vvBbb ^^A, vvBb 49/32, 77/50
52 751.807 A#, vBbb ^3A, vBb 14/9, 20/13, 65/42, 75/49
53 766.265 ^A#, Bbb ^4A, Bb 39/25, 54/35
54 780.723 ^^A#, v3Bb ^5A, v10B 11/7, 25/16, 52/33
55 795.181 ^3A#, vvBb ^6A, v9B 63/40, 81/52
56 809.639 Ax, vBb ^7A, v8B 8/5, 35/22, 77/48, 78/49
57 824.096 ^Ax, Bb ^8A, v7B 44/27
58 838.554 ^^Ax, v3B ^9A, v6B 13/8, 21/13, 45/28, 49/30
59 853.012 ^3Ax, vvB ^10A, v5B 64/39, 80/49, 81/50
60 867.47 A#x, vB A#, v4B 18/11, 33/20
61 881.928 B ^A#, v3B 5/3
62 896.386 ^B, Cbb ^^A#, vvB 42/25, 81/49
63 910.843 ^^B, v3Cb ^3A#, vB 22/13, 56/33, 77/45
64 925.301 ^3B, vvCb B 27/16
65 939.759 B#, vCb ^B, v3C 12/7, 26/15, 55/32, 75/44
66 954.217 ^B#, Cb ^^B, vvC
67 968.675 ^^B#, v3C ^3B, vC 7/4, 44/25, 45/26
68 983.133 ^3B#, vvC C 16/9
69 997.59 Bx, vC ^C, v3Db 39/22
70 1012.05 C ^^C, vvDb 25/14, 49/27, 65/36, 70/39
71 1026.51 ^C, Dbbb ^3C, vDb 9/5
72 1040.96 ^^C, v3Dbb ^4C, Db 11/6, 20/11, 64/35
73 1055.42 ^3C, vvDbb ^5C, v10D 50/27
74 1069.88 C#, vDbb ^6C, v9D 13/7, 24/13, 28/15
75 1084.34 ^C#, Dbb ^7C, v8D 81/44
76 1098.8 ^^C#, v3Db ^8C, v7D 15/8, 49/26, 66/35
77 1113.25 ^3C#, vvDb ^9C, v6D 40/21, 52/27
78 1127.71 Cx, vDb ^10C, v5D 21/11, 48/25, 77/40
79 1142.17 ^Cx, Db C#, v4D 25/13, 35/18, 64/33
80 1156.63 ^^Cx, v3D ^C#, v3D 27/14, 39/20, 49/25
81 1171.08 ^3Cx, vvD ^^C#, vvD 55/28, 65/33, 77/39
82 1185.54 C#x, vD ^3C#, vD 63/32
83 1200 D D 2/1
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