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 ¢ sharp and the 5 is 4 ¢ 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 | 14.5 | ^D, E♭♭♭ | ^D, v3E♭ | |
| 2 | 28.9 | ^^D, ^E♭♭♭ | ^^D, vvE♭ | |
| 3 | 43.4 | 39/38, 40/39 | vD♯, vvE♭♭ | ^3D, vE♭ |
| 4 | 57.8 | 29/28, 32/31 | D♯, vE♭♭ | ^4D, E♭ |
| 5 | 72.3 | 25/24 | ^D♯, E♭♭ | ^5D, ^E♭ |
| 6 | 86.7 | 20/19, 21/20 | ^^D♯, ^E♭♭ | v5D♯, ^^E♭ |
| 7 | 101.2 | 35/33 | vD𝄪, vvE♭ | v4D♯, ^3E♭ |
| 8 | 115.7 | 31/29 | D𝄪, vE♭ | v3D♯, ^4E♭ |
| 9 | 130.1 | 14/13 | ^D𝄪, E♭ | vvD♯, ^5E♭ |
| 10 | 144.6 | 37/34, 38/35 | ^^D𝄪, ^E♭ | vD♯, v5E |
| 11 | 159 | 34/31 | vD♯𝄪, vvE | D♯, v4E |
| 12 | 173.5 | 21/19, 31/28, 32/29 | D♯𝄪, vE | ^D♯, v3E |
| 13 | 188 | 29/26, 39/35 | E | ^^D♯, vvE |
| 14 | 202.4 | ^E, F♭♭ | ^3D♯, vE | |
| 15 | 216.9 | ^^E, ^F♭♭ | E | |
| 16 | 231.3 | 8/7 | vE♯, vvF♭ | ^E, v3F |
| 17 | 245.8 | 38/33 | E♯, vF♭ | ^^E, vvF |
| 18 | 260.2 | ^E♯, F♭ | ^3E, vF | |
| 19 | 274.7 | 34/29 | ^^E♯, ^F♭ | F |
| 20 | 289.2 | 13/11 | vE𝄪, vvF | ^F, v3G♭ |
| 21 | 303.6 | 25/21, 31/26, 37/31 | E𝄪, vF | ^^F, vvG♭ |
| 22 | 318.1 | 6/5 | F | ^3F, vG♭ |
| 23 | 332.5 | 40/33 | ^F, G♭♭♭ | ^4F, G♭ |
| 24 | 347 | ^^F, ^G♭♭♭ | ^5F, ^G♭ | |
| 25 | 361.4 | 16/13 | vF♯, vvG♭♭ | v5F♯, ^^G♭ |
| 26 | 375.9 | F♯, vG♭♭ | v4F♯, ^3G♭ | |
| 27 | 390.4 | ^F♯, G♭♭ | v3F♯, ^4G♭ | |
| 28 | 404.8 | 24/19 | ^^F♯, ^G♭♭ | vvF♯, ^5G♭ |
| 29 | 419.3 | 14/11, 37/29 | vF𝄪, vvG♭ | vF♯, v5G |
| 30 | 433.7 | F𝄪, vG♭ | F♯, v4G | |
| 31 | 448.2 | 22/17 | ^F𝄪, G♭ | ^F♯, v3G |
| 32 | 462.7 | 17/13 | ^^F𝄪, ^G♭ | ^^F♯, vvG |
| 33 | 477.1 | 25/19, 29/22 | vF♯𝄪, vvG | ^3F♯, vG |
| 34 | 491.6 | F♯𝄪, vG | G | |
| 35 | 506 | G | ^G, v3A♭ | |
| 36 | 520.5 | 23/17 | ^G, A♭♭♭ | ^^G, vvA♭ |
| 37 | 534.9 | ^^G, ^A♭♭♭ | ^3G, vA♭ | |
| 38 | 549.4 | 11/8 | vG♯, vvA♭♭ | ^4G, A♭ |
| 39 | 563.9 | G♯, vA♭♭ | ^5G, ^A♭ | |
| 40 | 578.3 | ^G♯, A♭♭ | v5G♯, ^^A♭ | |
| 41 | 592.8 | 31/22 | ^^G♯, ^A♭♭ | v4G♯, ^3A♭ |
| 42 | 607.2 | vG𝄪, vvA♭ | v3G♯, ^4A♭ | |
| 43 | 621.7 | G𝄪, vA♭ | vvG♯, ^5A♭ | |
| 44 | 636.1 | ^G𝄪, A♭ | vG♯, v5A | |
| 45 | 650.6 | 16/11, 35/24 | ^^G𝄪, ^A♭ | G♯, v4A |
| 46 | 665.1 | vG♯𝄪, vvA | ^G♯, v3A | |
| 47 | 679.5 | 34/23 | G♯𝄪, vA | ^^G♯, vvA |
| 48 | 694 | A | ^3G♯, vA | |
| 49 | 708.4 | ^A, B♭♭♭ | A | |
| 50 | 722.9 | 38/25 | ^^A, ^B♭♭♭ | ^A, v3B♭ |
| 51 | 737.3 | 26/17 | vA♯, vvB♭♭ | ^^A, vvB♭ |
| 52 | 751.8 | 17/11 | A♯, vB♭♭ | ^3A, vB♭ |
| 53 | 766.3 | ^A♯, B♭♭ | ^4A, B♭ | |
| 54 | 780.7 | 11/7 | ^^A♯, ^B♭♭ | ^5A, ^B♭ |
| 55 | 795.2 | 19/12 | vA𝄪, vvB♭ | v5A♯, ^^B♭ |
| 56 | 809.6 | A𝄪, vB♭ | v4A♯, ^3B♭ | |
| 57 | 824.1 | 37/23 | ^A𝄪, B♭ | v3A♯, ^4B♭ |
| 58 | 838.6 | 13/8 | ^^A𝄪, ^B♭ | vvA♯, ^5B♭ |
| 59 | 853 | vA♯𝄪, vvB | vA♯, v5B | |
| 60 | 867.5 | 33/20 | A♯𝄪, vB | A♯, v4B |
| 61 | 881.9 | 5/3 | B | ^A♯, v3B |
| 62 | 896.4 | ^B, C♭♭ | ^^A♯, vvB | |
| 63 | 910.8 | 22/13 | ^^B, ^C♭♭ | ^3A♯, vB |
| 64 | 925.3 | 29/17 | vB♯, vvC♭ | B |
| 65 | 939.8 | B♯, vC♭ | ^B, v3C | |
| 66 | 954.2 | 33/19 | ^B♯, C♭ | ^^B, vvC |
| 67 | 968.7 | 7/4 | ^^B♯, ^C♭ | ^3B, vC |
| 68 | 983.1 | vB𝄪, vvC | C | |
| 69 | 997.6 | B𝄪, vC | ^C, v3D♭ | |
| 70 | 1012 | C | ^^C, vvD♭ | |
| 71 | 1026.5 | 29/16, 38/21 | ^C, D♭♭♭ | ^3C, vD♭ |
| 72 | 1041 | 31/17 | ^^C, ^D♭♭♭ | ^4C, D♭ |
| 73 | 1055.4 | 35/19 | vC♯, vvD♭♭ | ^5C, ^D♭ |
| 74 | 1069.9 | 13/7 | C♯, vD♭♭ | v5C♯, ^^D♭ |
| 75 | 1084.3 | ^C♯, D♭♭ | v4C♯, ^3D♭ | |
| 76 | 1098.8 | ^^C♯, ^D♭♭ | v3C♯, ^4D♭ | |
| 77 | 1113.3 | 19/10, 40/21 | vC𝄪, vvD♭ | vvC♯, ^5D♭ |
| 78 | 1127.7 | C𝄪, vD♭ | vC♯, v5D | |
| 79 | 1142.2 | 31/16 | ^C𝄪, D♭ | C♯, v4D |
| 80 | 1156.6 | 39/20 | ^^C𝄪, ^D♭ | ^C♯, v3D |
| 81 | 1171.1 | vC♯𝄪, vvD | ^^C♯, vvD | |
| 82 | 1185.5 | C♯𝄪, vD | ^3C♯, vD | |
| 83 | 1200 | 2/1 | D | D |