67edo
| ← 66edo | 67edo | 68edo → |
67 equal divisions of the octave (abbreviated 67edo or 67ed2), also called 67-tone equal temperament (67tet) or 67 equal temperament (67et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 67 equal parts of about 17.9 ¢ each. Each step represents a frequency ratio of 21/67, or the 67th root of 2.
Theory
67edo tempers out 81/80, supporting meantone temperament, with a tuning which is approximately 1/6 comma (the tuning favored by Mozart and contemporaries), or 0.16 comma, meantone. In the 7-limit the patent val tempers out 1029/1024 and 1728/1715, so that it supports mothra temperament. In the 11-limit it tempers out 176/175 and 540/539, supporting mosura, an alternative 11-limit mothra. In the 13-limit it tempers out 144/143 and 196/195, supporting 13-limit mosura. It tempers out the orgonisma, and on the 2.7.11 subgroup it supports orgone temperament.
A promising tuning which has, as many relatively large equal temperaments do, a variety of tonal resources: it is the first edo to have both meantone (26 could be called meantone, but it's more of a flattone) and an orgone temperament. It has relatively good approximations of the 3rd, 7th, 11th, 13th, 15th, 17th harmonics, although the 5th, 9th, and 19th as well as certain higher ones are workable as well. 33+34 can be used to construct this temperament explaining some of its properties. It does well on the 2.3.7.11.13.17.23.31.37.41 subgroup.
67edo is the 19th prime EDO.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.00 | -3.45 | +7.72 | -1.66 | +3.91 | +1.26 | +2.51 | +6.96 | -1.41 | -8.68 | +1.23 | -0.60 | +0.79 |
| relative (%) | +0 | -19 | +43 | -9 | +22 | +7 | +14 | +39 | -8 | -48 | +7 | -3 | +4 | |
| Steps (reduced) |
67 (0) |
106 (39) |
156 (22) |
188 (54) |
232 (31) |
248 (47) |
274 (6) |
285 (17) |
303 (35) |
325 (57) |
332 (64) |
349 (14) |
359 (24) | |
Intervals
| Steps | Cents | Ups and downs notation | Approximate ratios |
|---|---|---|---|
| 0 | 0 | D | 1/1 |
| 1 | 17.9104 | ^D, E♭♭ | 78/77 |
| 2 | 35.8209 | ^^D, v4E♭ | 45/44, 49/48 |
| 3 | 53.7313 | ^3D, v3E♭ | 33/32 |
| 4 | 71.6418 | ^4D, vvE♭ | 80/77 |
| 5 | 89.5522 | D♯, vE♭ | |
| 6 | 107.463 | ^D♯, E♭ | 16/15, 35/33, 52/49 |
| 7 | 125.373 | ^^D♯, v4E | 14/13, 15/14 |
| 8 | 143.284 | ^3D♯, v3E | 13/12, 49/45 |
| 9 | 161.194 | ^4D♯, vvE | 11/10, 35/32, 54/49 |
| 10 | 179.104 | D𝄪, vE | |
| 11 | 197.015 | E | 9/8 |
| 12 | 214.925 | ^E, F♭ | 44/39 |
| 13 | 232.836 | ^^E, v4F | 8/7 |
| 14 | 250.746 | ^3E, v3F | 15/13, 52/45 |
| 15 | 268.657 | ^4E, vvF | 7/6 |
| 16 | 286.567 | E♯, vF | 13/11, 33/28, 77/65 |
| 17 | 304.478 | F | |
| 18 | 322.388 | ^F, G♭♭ | 77/64 |
| 19 | 340.299 | ^^F, v4G♭ | 39/32 |
| 20 | 358.209 | ^3F, v3G♭ | 16/13 |
| 21 | 376.119 | ^4F, vvG♭ | 26/21, 56/45 |
| 22 | 394.03 | F♯, vG♭ | 44/35, 49/39 |
| 23 | 411.94 | ^F♯, G♭ | 14/11, 33/26 |
| 24 | 429.851 | ^^F♯, v4G | 9/7, 77/60 |
| 25 | 447.761 | ^3F♯, v3G | 13/10 |
| 26 | 465.672 | ^4F♯, vvG | 21/16, 64/49 |
| 27 | 483.582 | F𝄪, vG | |
| 28 | 501.493 | G | 4/3 |
| 29 | 519.403 | ^G, A♭♭ | 35/26, 66/49 |
| 30 | 537.313 | ^^G, v4A♭ | 15/11, 49/36 |
| 31 | 555.224 | ^3G, v3A♭ | 11/8 |
| 32 | 573.134 | ^4G, vvA♭ | 39/28 |
| 33 | 591.045 | G♯, vA♭ | 45/32 |
| 34 | 608.955 | ^G♯, A♭ | 64/45 |
| 35 | 626.866 | ^^G♯, v4A | 56/39 |
| 36 | 644.776 | ^3G♯, v3A | 16/11 |
| 37 | 662.687 | ^4G♯, vvA | 22/15, 72/49 |
| 38 | 680.597 | G𝄪, vA | 49/33, 52/35, 65/44, 77/52 |
| 39 | 698.507 | A | 3/2 |
| 40 | 716.418 | ^A, B♭♭ | |
| 41 | 734.328 | ^^A, v4B♭ | 32/21, 49/32 |
| 42 | 752.239 | ^3A, v3B♭ | 20/13 |
| 43 | 770.149 | ^4A, vvB♭ | 14/9 |
| 44 | 788.06 | A♯, vB♭ | 11/7, 52/33 |
| 45 | 805.97 | ^A♯, B♭ | 35/22, 78/49 |
| 46 | 823.881 | ^^A♯, v4B | 21/13, 45/28, 77/48 |
| 47 | 841.791 | ^3A♯, v3B | 13/8 |
| 48 | 859.701 | ^4A♯, vvB | 64/39 |
| 49 | 877.612 | A𝄪, vB | |
| 50 | 895.522 | B | |
| 51 | 913.433 | ^B, C♭ | 22/13, 56/33 |
| 52 | 931.343 | ^^B, v4C | 12/7, 77/45 |
| 53 | 949.254 | ^3B, v3C | 26/15, 45/26 |
| 54 | 967.164 | ^4B, vvC | 7/4 |
| 55 | 985.075 | B♯, vC | 39/22 |
| 56 | 1002.99 | C | 16/9 |
| 57 | 1020.9 | ^C, D♭♭ | |
| 58 | 1038.81 | ^^C, v4D♭ | 20/11, 49/27, 64/35 |
| 59 | 1056.72 | ^3C, v3D♭ | 24/13 |
| 60 | 1074.63 | ^4C, vvD♭ | 13/7, 28/15 |
| 61 | 1092.54 | C♯, vD♭ | 15/8, 49/26, 66/35 |
| 62 | 1110.45 | ^C♯, D♭ | |
| 63 | 1128.36 | ^^C♯, v4D | 77/40 |
| 64 | 1146.27 | ^3C♯, v3D | 64/33 |
| 65 | 1164.18 | ^4C♯, vvD | |
| 66 | 1182.09 | C𝄪, vD | 77/39 |
| 67 | 1200 | D | 2/1 |