86edo
| ← 85edo | 86edo | 87edo → |
86 equal divisions of the octave (abbreviated 86edo or 86ed2), also called 86-tone equal temperament (86tet) or 86 equal temperament (86et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 86 equal parts of about 14 ¢ each. Each step represents a frequency ratio of 21/86, or the 86th root of 2.
Theory
86 = 2 × 43, and the patent val is a contorted 43edo in the 5-limit. In the 7-limit the patent val tempers out 6144/6125, so that it supports the mohajira temperament. In the 11-limit it tempers out 245/242, 540/539 and 4000/3993, and in the 13-limit 144/143, 196/195 and 676/675. It provides the optimal patent val for the 13-limit 9 & 86 temperament tempering out 144/143, 196/195, 245/242 and 676/675.
It is perhaps more interesting to consider the alternative 86e val, which tempers out 121/120 and 243/242 and supports 11-limit mohajira. The 86de val is a less good entry for 11-limit migration. In any case, this tuning is between 31edo and 55edo, and replaces 43edo's lopsided placement of 11/9 and 27/22 with a true neutral third.
86edo is closely related to the delta scale, which is the equal division of the classic diatonic semitone into eight parts of 13.9664 cents each.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -4.28 | +4.38 | -6.04 | +5.39 | +6.82 | -3.32 | +0.10 | +6.67 | -4.49 | +3.64 | -0.37 |
| Relative (%) | -30.7 | +31.4 | -43.3 | +38.6 | +48.9 | -23.8 | +0.7 | +47.8 | -32.2 | +26.1 | -2.6 | |
| Steps (reduced) |
136 (50) |
200 (28) |
241 (69) |
273 (15) |
298 (40) |
318 (60) |
336 (78) |
352 (8) |
365 (21) |
378 (34) |
389 (45) | |
Subsets and supersets
86edo contains 2edo and 43edo as subsets. 258edo, which triples it, is a notable tuning.
Interval table
| Steps | Cents | Approximate ratios | Ups and downs notation |
|---|---|---|---|
| 0 | 0 | 1/1 | D |
| 1 | 14 | ^D, vE♭♭ | |
| 2 | 27.9 | ^^D, E♭♭ | |
| 3 | 41.9 | 41/40 | ^3D, ^E♭♭ |
| 4 | 55.8 | 30/29, 31/30, 32/31, 33/32 | vvD♯, ^^E♭♭ |
| 5 | 69.8 | vD♯, v3E♭ | |
| 6 | 83.7 | D♯, vvE♭ | |
| 7 | 97.7 | 37/35 | ^D♯, vE♭ |
| 8 | 111.6 | 16/15 | ^^D♯, E♭ |
| 9 | 125.6 | 14/13 | ^3D♯, ^E♭ |
| 10 | 139.5 | 13/12, 38/35 | vvD𝄪, ^^E♭ |
| 11 | 153.5 | 35/32 | vD𝄪, v3E |
| 12 | 167.4 | 11/10 | D𝄪, vvE |
| 13 | 181.4 | ^D𝄪, vE | |
| 14 | 195.3 | 37/33 | E |
| 15 | 209.3 | 26/23, 35/31 | ^E, vF♭ |
| 16 | 223.3 | 25/22, 33/29 | ^^E, F♭ |
| 17 | 237.2 | ^3E, ^F♭ | |
| 18 | 251.2 | 37/32 | vvE♯, ^^F♭ |
| 19 | 265.1 | 7/6 | vE♯, v3F |
| 20 | 279.1 | 20/17 | E♯, vvF |
| 21 | 293 | ^E♯, vF | |
| 22 | 307 | 31/26, 37/31 | F |
| 23 | 320.9 | ^F, vG♭♭ | |
| 24 | 334.9 | 40/33 | ^^F, G♭♭ |
| 25 | 348.8 | ^3F, ^G♭♭ | |
| 26 | 362.8 | 37/30 | vvF♯, ^^G♭♭ |
| 27 | 376.7 | 41/33 | vF♯, v3G♭ |
| 28 | 390.7 | F♯, vvG♭ | |
| 29 | 404.7 | 24/19 | ^F♯, vG♭ |
| 30 | 418.6 | ^^F♯, G♭ | |
| 31 | 432.6 | 9/7 | ^3F♯, ^G♭ |
| 32 | 446.5 | 22/17 | vvF𝄪, ^^G♭ |
| 33 | 460.5 | 30/23 | vF𝄪, v3G |
| 34 | 474.4 | F𝄪, vvG | |
| 35 | 488.4 | ^F𝄪, vG | |
| 36 | 502.3 | G | |
| 37 | 516.3 | 31/23, 35/26 | ^G, vA♭♭ |
| 38 | 530.2 | 19/14, 34/25 | ^^G, A♭♭ |
| 39 | 544.2 | 26/19 | ^3G, ^A♭♭ |
| 40 | 558.1 | 40/29 | vvG♯, ^^A♭♭ |
| 41 | 572.1 | 32/23, 39/28 | vG♯, v3A♭ |
| 42 | 586 | G♯, vvA♭ | |
| 43 | 600 | 41/29 | ^G♯, vA♭ |
| 44 | 614 | ^^G♯, A♭ | |
| 45 | 627.9 | 23/16, 33/23 | ^3G♯, ^A♭ |
| 46 | 641.9 | 29/20 | vvG𝄪, ^^A♭ |
| 47 | 655.8 | 19/13, 35/24 | vG𝄪, v3A |
| 48 | 669.8 | 25/17, 28/19 | G𝄪, vvA |
| 49 | 683.7 | ^G𝄪, vA | |
| 50 | 697.7 | A | |
| 51 | 711.6 | ^A, vB♭♭ | |
| 52 | 725.6 | 35/23 | ^^A, B♭♭ |
| 53 | 739.5 | 23/15 | ^3A, ^B♭♭ |
| 54 | 753.5 | 17/11 | vvA♯, ^^B♭♭ |
| 55 | 767.4 | 14/9 | vA♯, v3B♭ |
| 56 | 781.4 | A♯, vvB♭ | |
| 57 | 795.3 | 19/12 | ^A♯, vB♭ |
| 58 | 809.3 | ^^A♯, B♭ | |
| 59 | 823.3 | 37/23 | ^3A♯, ^B♭ |
| 60 | 837.2 | vvA𝄪, ^^B♭ | |
| 61 | 851.2 | vA𝄪, v3B | |
| 62 | 865.1 | 33/20 | A𝄪, vvB |
| 63 | 879.1 | ^A𝄪, vB | |
| 64 | 893 | B | |
| 65 | 907 | ^B, vC♭ | |
| 66 | 920.9 | 17/10 | ^^B, C♭ |
| 67 | 934.9 | 12/7 | ^3B, ^C♭ |
| 68 | 948.8 | vvB♯, ^^C♭ | |
| 69 | 962.8 | vB♯, v3C | |
| 70 | 976.7 | B♯, vvC | |
| 71 | 990.7 | 23/13 | ^B♯, vC |
| 72 | 1004.7 | C | |
| 73 | 1018.6 | ^C, vD♭♭ | |
| 74 | 1032.6 | 20/11 | ^^C, D♭♭ |
| 75 | 1046.5 | ^3C, ^D♭♭ | |
| 76 | 1060.5 | 24/13, 35/19 | vvC♯, ^^D♭♭ |
| 77 | 1074.4 | 13/7 | vC♯, v3D♭ |
| 78 | 1088.4 | 15/8 | C♯, vvD♭ |
| 79 | 1102.3 | ^C♯, vD♭ | |
| 80 | 1116.3 | ^^C♯, D♭ | |
| 81 | 1130.2 | ^3C♯, ^D♭ | |
| 82 | 1144.2 | 29/15, 31/16 | vvC𝄪, ^^D♭ |
| 83 | 1158.1 | vC𝄪, v3D | |
| 84 | 1172.1 | C𝄪, vvD | |
| 85 | 1186 | ^C𝄪, vD | |
| 86 | 1200 | 2/1 | D |