83edo
| ← 82edo | 83edo | 84edo → |
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
| 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, 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 |