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
The harmonic 3 is 6.5 cents sharp and the 5 is 4 cents sharp, with 7, 11, and 13 more accurate but a little flat. Using the patent val, 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.
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.
Odd harmonics
| 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) | |
Subsets and supersets
83edo is the 23rd prime edo, following 79edo and before 89edo.
Intervals
| Steps | Cents | Approximate Ratios | Ups and Downs Notation (Dual Flat Fifth 48\83) |
Ups and Downs Notation (Dual Sharp Fifth 49\83) |
|---|---|---|---|---|
| 0 | 0 | 1/1 | D | D |
| 1 | 14.458 | ^D, E♭♭♭ | ^D, v3E♭ | |
| 2 | 28.916 | 56/55, 65/64, 66/65 | ^^D, v3E♭♭ | ^^D, vvE♭ |
| 3 | 43.373 | 40/39 | ^3D, vvE♭♭ | ^3D, vE♭ |
| 4 | 57.831 | 33/32 | D♯, vE♭♭ | ^4D, E♭ |
| 5 | 72.289 | 25/24 | ^D♯, E♭♭ | ^5D, v10E |
| 6 | 86.747 | 21/20 | ^^D♯, v3E♭ | ^6D, v9E |
| 7 | 101.205 | 35/33, 52/49, 55/52 | ^3D♯, vvE♭ | ^7D, v8E |
| 8 | 115.663 | D𝄪, vE♭ | ^8D, v7E | |
| 9 | 130.12 | 14/13 | ^D𝄪, E♭ | ^9D, v6E |
| 10 | 144.578 | ^^D𝄪, v3E | ^10D, v5E | |
| 11 | 159.036 | 35/32 | ^3D𝄪, vvE | D♯, v4E |
| 12 | 173.494 | D♯𝄪, vE | ^D♯, v3E | |
| 13 | 187.952 | 39/35, 49/44 | E | ^^D♯, vvE |
| 14 | 202.41 | 55/49 | ^E, F♭♭ | ^3D♯, vE |
| 15 | 216.867 | ^^E, v3F♭ | E | |
| 16 | 231.325 | 8/7, 55/48 | ^3E, vvF♭ | ^E, v3F |
| 17 | 245.783 | E♯, vF♭ | ^^E, vvF | |
| 18 | 260.241 | 64/55, 65/56 | ^E♯, F♭ | ^3E, vF |
| 19 | 274.699 | ^^E♯, v3F | F | |
| 20 | 289.157 | 13/11, 33/28, 77/65 | ^3E♯, vvF | ^F, v3G♭ |
| 21 | 303.614 | 25/21 | E𝄪, vF | ^^F, vvG♭ |
| 22 | 318.072 | 6/5, 77/64 | F | ^3F, vG♭ |
| 23 | 332.53 | 40/33 | ^F, G♭♭♭ | ^4F, G♭ |
| 24 | 346.988 | 39/32, 49/40 | ^^F, v3G♭♭ | ^5F, v10G |
| 25 | 361.446 | 16/13 | ^3F, vvG♭♭ | ^6F, v9G |
| 26 | 375.904 | F♯, vG♭♭ | ^7F, v8G | |
| 27 | 390.361 | 5/4, 49/39 | ^F♯, G♭♭ | ^8F, v7G |
| 28 | 404.819 | 63/50 | ^^F♯, v3G♭ | ^9F, v6G |
| 29 | 419.277 | 14/11 | ^3F♯, vvG♭ | ^10F, v5G |
| 30 | 433.735 | 50/39 | F𝄪, vG♭ | F♯, v4G |
| 31 | 448.193 | ^F𝄪, G♭ | ^F♯, v3G | |
| 32 | 462.651 | 55/42, 64/49 | ^^F𝄪, v3G | ^^F♯, vvG |
| 33 | 477.108 | 33/25 | ^3F𝄪, vvG | ^3F♯, vG |
| 34 | 491.566 | 65/49 | F♯𝄪, vG | G |
| 35 | 506.024 | G | ^G, v3A♭ | |
| 36 | 520.482 | 65/48, 66/49 | ^G, A♭♭♭ | ^^G, vvA♭ |
| 37 | 534.94 | ^^G, v3A♭♭ | ^3G, vA♭ | |
| 38 | 549.398 | 11/8, 48/35 | ^3G, vvA♭♭ | ^4G, A♭ |
| 39 | 563.855 | 25/18 | G♯, vA♭♭ | ^5G, v10A |
| 40 | 578.313 | 7/5, 39/28 | ^G♯, A♭♭ | ^6G, v9A |
| 41 | 592.771 | 55/39 | ^^G♯, v3A♭ | ^7G, v8A |
| 42 | 607.229 | 78/55 | ^3G♯, vvA♭ | ^8G, v7A |
| 43 | 621.687 | 10/7, 56/39 | G𝄪, vA♭ | ^9G, v6A |
| 44 | 636.145 | 36/25 | ^G𝄪, A♭ | ^10G, v5A |
| 45 | 650.602 | 16/11, 35/24 | ^^G𝄪, v3A | G♯, v4A |
| 46 | 665.06 | ^3G𝄪, vvA | ^G♯, v3A | |
| 47 | 679.518 | 49/33, 65/44, 77/52 | G♯𝄪, vA | ^^G♯, vvA |
| 48 | 693.976 | A | ^3G♯, vA | |
| 49 | 708.434 | ^A, B♭♭♭ | A | |
| 50 | 722.892 | 50/33 | ^^A, v3B♭♭ | ^A, v3B♭ |
| 51 | 737.349 | 49/32 | ^3A, vvB♭♭ | ^^A, vvB♭ |
| 52 | 751.807 | 65/42 | A♯, vB♭♭ | ^3A, vB♭ |
| 53 | 766.265 | 39/25 | ^A♯, B♭♭ | ^4A, B♭ |
| 54 | 780.723 | 11/7 | ^^A♯, v3B♭ | ^5A, v10B |
| 55 | 795.181 | ^3A♯, vvB♭ | ^6A, v9B | |
| 56 | 809.639 | 8/5, 78/49 | A𝄪, vB♭ | ^7A, v8B |
| 57 | 824.096 | ^A𝄪, B♭ | ^8A, v7B | |
| 58 | 838.554 | 13/8 | ^^A𝄪, v3B | ^9A, v6B |
| 59 | 853.012 | 64/39, 80/49 | ^3A𝄪, vvB | ^10A, v5B |
| 60 | 867.47 | 33/20 | A♯𝄪, vB | A♯, v4B |
| 61 | 881.928 | 5/3 | B | ^A♯, v3B |
| 62 | 896.386 | 42/25 | ^B, C♭♭ | ^^A♯, vvB |
| 63 | 910.843 | 22/13, 56/33 | ^^B, v3C♭ | ^3A♯, vB |
| 64 | 925.301 | ^3B, vvC♭ | B | |
| 65 | 939.759 | 55/32 | B♯, vC♭ | ^B, v3C |
| 66 | 954.217 | ^B♯, C♭ | ^^B, vvC | |
| 67 | 968.675 | 7/4 | ^^B♯, v3C | ^3B, vC |
| 68 | 983.133 | ^3B♯, vvC | C | |
| 69 | 997.59 | B𝄪, vC | ^C, v3D♭ | |
| 70 | 1012.048 | 70/39 | C | ^^C, vvD♭ |
| 71 | 1026.506 | ^C, D♭♭♭ | ^3C, vD♭ | |
| 72 | 1040.964 | 64/35 | ^^C, v3D♭♭ | ^4C, D♭ |
| 73 | 1055.422 | ^3C, vvD♭♭ | ^5C, v10D | |
| 74 | 1069.88 | 13/7 | C♯, vD♭♭ | ^6C, v9D |
| 75 | 1084.337 | ^C♯, D♭♭ | ^7C, v8D | |
| 76 | 1098.795 | 49/26, 66/35 | ^^C♯, v3D♭ | ^8C, v7D |
| 77 | 1113.253 | 40/21 | ^3C♯, vvD♭ | ^9C, v6D |
| 78 | 1127.711 | 48/25 | C𝄪, vD♭ | ^10C, v5D |
| 79 | 1142.169 | 64/33 | ^C𝄪, D♭ | C♯, v4D |
| 80 | 1156.627 | 39/20 | ^^C𝄪, v3D | ^C♯, v3D |
| 81 | 1171.084 | 55/28, 65/33 | ^3C𝄪, vvD | ^^C♯, vvD |
| 82 | 1185.542 | C♯𝄪, vD | ^3C♯, vD | |
| 83 | 1200 | 2/1 | D | D |