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 | 13.953 | ^D, vE♭♭ | |
| 2 | 27.907 | ^^D, E♭♭ | |
| 3 | 41.86 | 41/40 | ^3D, v5E♭ |
| 4 | 55.814 | 30/29, 31/30, 32/31, 33/32 | ^4D, v4E♭ |
| 5 | 69.767 | ^5D, v3E♭ | |
| 6 | 83.721 | D♯, vvE♭ | |
| 7 | 97.674 | 37/35 | ^D♯, vE♭ |
| 8 | 111.628 | 16/15 | ^^D♯, E♭ |
| 9 | 125.581 | 14/13 | ^3D♯, v5E |
| 10 | 139.535 | 13/12, 38/35 | ^4D♯, v4E |
| 11 | 153.488 | 35/32 | ^5D♯, v3E |
| 12 | 167.442 | 11/10 | D𝄪, vvE |
| 13 | 181.395 | ^D𝄪, vE | |
| 14 | 195.349 | 37/33 | E |
| 15 | 209.302 | 26/23, 35/31 | ^E, vF♭ |
| 16 | 223.256 | 25/22, 33/29 | ^^E, F♭ |
| 17 | 237.209 | ^3E, v5F | |
| 18 | 251.163 | 37/32 | ^4E, v4F |
| 19 | 265.116 | 7/6 | ^5E, v3F |
| 20 | 279.07 | 20/17 | E♯, vvF |
| 21 | 293.023 | ^E♯, vF | |
| 22 | 306.977 | 31/26, 37/31 | F |
| 23 | 320.93 | ^F, vG♭♭ | |
| 24 | 334.884 | 40/33 | ^^F, G♭♭ |
| 25 | 348.837 | ^3F, v5G♭ | |
| 26 | 362.791 | 37/30 | ^4F, v4G♭ |
| 27 | 376.744 | 41/33 | ^5F, v3G♭ |
| 28 | 390.698 | F♯, vvG♭ | |
| 29 | 404.651 | 24/19 | ^F♯, vG♭ |
| 30 | 418.605 | ^^F♯, G♭ | |
| 31 | 432.558 | 9/7 | ^3F♯, v5G |
| 32 | 446.512 | 22/17 | ^4F♯, v4G |
| 33 | 460.465 | 30/23 | ^5F♯, v3G |
| 34 | 474.419 | F𝄪, vvG | |
| 35 | 488.372 | ^F𝄪, vG | |
| 36 | 502.326 | G | |
| 37 | 516.279 | 31/23, 35/26 | ^G, vA♭♭ |
| 38 | 530.233 | 19/14, 34/25 | ^^G, A♭♭ |
| 39 | 544.186 | 26/19 | ^3G, v5A♭ |
| 40 | 558.14 | 40/29 | ^4G, v4A♭ |
| 41 | 572.093 | 32/23, 39/28 | ^5G, v3A♭ |
| 42 | 586.047 | G♯, vvA♭ | |
| 43 | 600 | 41/29 | ^G♯, vA♭ |
| 44 | 613.953 | ^^G♯, A♭ | |
| 45 | 627.907 | 23/16, 33/23 | ^3G♯, v5A |
| 46 | 641.86 | 29/20 | ^4G♯, v4A |
| 47 | 655.814 | 19/13, 35/24 | ^5G♯, v3A |
| 48 | 669.767 | 25/17, 28/19 | G𝄪, vvA |
| 49 | 683.721 | ^G𝄪, vA | |
| 50 | 697.674 | A | |
| 51 | 711.628 | ^A, vB♭♭ | |
| 52 | 725.581 | 35/23 | ^^A, B♭♭ |
| 53 | 739.535 | 23/15 | ^3A, v5B♭ |
| 54 | 753.488 | 17/11 | ^4A, v4B♭ |
| 55 | 767.442 | 14/9 | ^5A, v3B♭ |
| 56 | 781.395 | A♯, vvB♭ | |
| 57 | 795.349 | 19/12 | ^A♯, vB♭ |
| 58 | 809.302 | ^^A♯, B♭ | |
| 59 | 823.256 | 37/23 | ^3A♯, v5B |
| 60 | 837.209 | ^4A♯, v4B | |
| 61 | 851.163 | ^5A♯, v3B | |
| 62 | 865.116 | 33/20 | A𝄪, vvB |
| 63 | 879.07 | ^A𝄪, vB | |
| 64 | 893.023 | B | |
| 65 | 906.977 | ^B, vC♭ | |
| 66 | 920.93 | 17/10 | ^^B, C♭ |
| 67 | 934.884 | 12/7 | ^3B, v5C |
| 68 | 948.837 | ^4B, v4C | |
| 69 | 962.791 | ^5B, v3C | |
| 70 | 976.744 | B♯, vvC | |
| 71 | 990.698 | 23/13 | ^B♯, vC |
| 72 | 1004.651 | C | |
| 73 | 1018.605 | ^C, vD♭♭ | |
| 74 | 1032.558 | 20/11 | ^^C, D♭♭ |
| 75 | 1046.512 | ^3C, v5D♭ | |
| 76 | 1060.465 | 24/13, 35/19 | ^4C, v4D♭ |
| 77 | 1074.419 | 13/7 | ^5C, v3D♭ |
| 78 | 1088.372 | 15/8 | C♯, vvD♭ |
| 79 | 1102.326 | ^C♯, vD♭ | |
| 80 | 1116.279 | ^^C♯, D♭ | |
| 81 | 1130.233 | ^3C♯, v5D | |
| 82 | 1144.186 | 29/15, 31/16 | ^4C♯, v4D |
| 83 | 1158.14 | ^5C♯, v3D | |
| 84 | 1172.093 | C𝄪, vvD | |
| 85 | 1186.047 | ^C𝄪, vD | |
| 86 | 1200 | 2/1 | D |