83edo
← 82edo | 83edo | 84edo → |
83 equal divisions of the octave (abbreviated 83edo), or 83-tone equal temperament (83tet), 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 83 root of 2.
Theory
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 | |
2 | 28.9157 | ^^D, v3Ebb | ^^D, vvEb | 56/55, 65/64, 66/65 |
3 | 43.3735 | ^3D, vvEbb | ^3D, vEb | 40/39 |
4 | 57.8313 | D#, vEbb | ^4D, Eb | 33/32 |
5 | 72.2892 | ^D#, Ebb | ^5D, v10E | 25/24 |
6 | 86.747 | ^^D#, v3Eb | ^6D, v9E | 21/20 |
7 | 101.205 | ^3D#, vvEb | ^7D, v8E | 35/33, 52/49, 55/52 |
8 | 115.663 | Dx, vEb | ^8D, v7E | |
9 | 130.12 | ^Dx, Eb | ^9D, v6E | 14/13 |
10 | 144.578 | ^^Dx, v3E | ^10D, v5E | |
11 | 159.036 | ^3Dx, vvE | D#, v4E | 35/32 |
12 | 173.494 | D#x, vE | ^D#, v3E | |
13 | 187.952 | E | ^^D#, vvE | 39/35, 49/44 |
14 | 202.41 | ^E, Fbb | ^3D#, vE | 55/49 |
15 | 216.867 | ^^E, v3Fb | E | |
16 | 231.325 | ^3E, vvFb | ^E, v3F | 8/7, 55/48 |
17 | 245.783 | E#, vFb | ^^E, vvF | |
18 | 260.241 | ^E#, Fb | ^3E, vF | 64/55, 65/56 |
19 | 274.699 | ^^E#, v3F | F | |
20 | 289.157 | ^3E#, vvF | ^F, v3Gb | 13/11, 33/28, 77/65 |
21 | 303.614 | Ex, vF | ^^F, vvGb | 25/21 |
22 | 318.072 | F | ^3F, vGb | 6/5, 77/64 |
23 | 332.53 | ^F, Gbbb | ^4F, Gb | 40/33 |
24 | 346.988 | ^^F, v3Gbb | ^5F, v10G | 39/32, 49/40 |
25 | 361.446 | ^3F, vvGbb | ^6F, v9G | 16/13 |
26 | 375.904 | F#, vGbb | ^7F, v8G | |
27 | 390.361 | ^F#, Gbb | ^8F, v7G | 5/4, 49/39 |
28 | 404.819 | ^^F#, v3Gb | ^9F, v6G | 63/50 |
29 | 419.277 | ^3F#, vvGb | ^10F, v5G | 14/11 |
30 | 433.735 | Fx, vGb | F#, v4G | 50/39 |
31 | 448.193 | ^Fx, Gb | ^F#, v3G | |
32 | 462.651 | ^^Fx, v3G | ^^F#, vvG | 55/42, 64/49 |
33 | 477.108 | ^3Fx, vvG | ^3F#, vG | 33/25 |
34 | 491.566 | F#x, vG | G | 65/49 |
35 | 506.024 | G | ^G, v3Ab | |
36 | 520.482 | ^G, Abbb | ^^G, vvAb | 65/48, 66/49 |
37 | 534.94 | ^^G, v3Abb | ^3G, vAb | |
38 | 549.398 | ^3G, vvAbb | ^4G, Ab | 11/8, 48/35 |
39 | 563.855 | G#, vAbb | ^5G, v10A | 25/18 |
40 | 578.313 | ^G#, Abb | ^6G, v9A | 7/5, 39/28 |
41 | 592.771 | ^^G#, v3Ab | ^7G, v8A | 55/39 |
42 | 607.229 | ^3G#, vvAb | ^8G, v7A | 78/55 |
43 | 621.687 | Gx, vAb | ^9G, v6A | 10/7, 56/39 |
44 | 636.145 | ^Gx, Ab | ^10G, v5A | 36/25 |
45 | 650.602 | ^^Gx, v3A | G#, v4A | 16/11, 35/24 |
46 | 665.06 | ^3Gx, vvA | ^G#, v3A | |
47 | 679.518 | G#x, vA | ^^G#, vvA | 49/33, 65/44, 77/52 |
48 | 693.976 | A | ^3G#, vA | |
49 | 708.434 | ^A, Bbbb | A | |
50 | 722.892 | ^^A, v3Bbb | ^A, v3Bb | 50/33 |
51 | 737.349 | ^3A, vvBbb | ^^A, vvBb | 49/32 |
52 | 751.807 | A#, vBbb | ^3A, vBb | 65/42 |
53 | 766.265 | ^A#, Bbb | ^4A, Bb | 39/25 |
54 | 780.723 | ^^A#, v3Bb | ^5A, v10B | 11/7 |
55 | 795.181 | ^3A#, vvBb | ^6A, v9B | |
56 | 809.639 | Ax, vBb | ^7A, v8B | 8/5, 78/49 |
57 | 824.096 | ^Ax, Bb | ^8A, v7B | |
58 | 838.554 | ^^Ax, v3B | ^9A, v6B | 13/8 |
59 | 853.012 | ^3Ax, vvB | ^10A, v5B | 64/39, 80/49 |
60 | 867.47 | A#x, vB | A#, v4B | 33/20 |
61 | 881.928 | B | ^A#, v3B | 5/3 |
62 | 896.386 | ^B, Cbb | ^^A#, vvB | 42/25 |
63 | 910.843 | ^^B, v3Cb | ^3A#, vB | 22/13, 56/33 |
64 | 925.301 | ^3B, vvCb | B | |
65 | 939.759 | B#, vCb | ^B, v3C | 55/32 |
66 | 954.217 | ^B#, Cb | ^^B, vvC | |
67 | 968.675 | ^^B#, v3C | ^3B, vC | 7/4 |
68 | 983.133 | ^3B#, vvC | C | |
69 | 997.59 | Bx, vC | ^C, v3Db | |
70 | 1012.05 | C | ^^C, vvDb | 70/39 |
71 | 1026.51 | ^C, Dbbb | ^3C, vDb | |
72 | 1040.96 | ^^C, v3Dbb | ^4C, Db | 64/35 |
73 | 1055.42 | ^3C, vvDbb | ^5C, v10D | |
74 | 1069.88 | C#, vDbb | ^6C, v9D | 13/7 |
75 | 1084.34 | ^C#, Dbb | ^7C, v8D | |
76 | 1098.8 | ^^C#, v3Db | ^8C, v7D | 49/26, 66/35 |
77 | 1113.25 | ^3C#, vvDb | ^9C, v6D | 40/21 |
78 | 1127.71 | Cx, vDb | ^10C, v5D | 48/25 |
79 | 1142.17 | ^Cx, Db | C#, v4D | 64/33 |
80 | 1156.63 | ^^Cx, v3D | ^C#, v3D | 39/20 |
81 | 1171.08 | ^3Cx, vvD | ^^C#, vvD | 55/28, 65/33 |
82 | 1185.54 | C#x, vD | ^3C#, vD | |
83 | 1200 | D | D | 2/1 |