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, with a tuning which is slightly sharp of 1/6-comma (the tuning favored by Mozart and contemporaries, though they suggested the flatter & composite 55edo as an approximation). It is indistinguishable from 4/25=0.16-comma meantone. In the 7-limit the patent val tempers out 1029/1024 and 1728/1715, so that it supports mothra. 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 the orgone temperament.
It is 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 and an orgone temperament (26edo could be called meantone, but it is more of a flattone). 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.
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 (%) | +0.0 | -19.2 | +43.1 | -9.3 | +21.8 | +7.1 | +14.0 | +38.9 | -7.9 | -48.5 | +6.9 | -3.3 | +4.4 | |
| 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) | |
Subsets and supersets
67edo is the 19th prime edo, following 61edo and before 71edo.
Intervals
| Steps | Cents | Approximate Ratios | Ups and Downs Notation |
|---|---|---|---|
| 0 | 0 | 1/1 | D |
| 1 | 17.91 | ^D, E♭♭ | |
| 2 | 35.821 | ^^D, v4E♭ | |
| 3 | 53.731 | 31/30, 32/31, 33/32, 34/33, 35/34 | ^3D, v3E♭ |
| 4 | 71.642 | 24/23 | ^4D, vvE♭ |
| 5 | 89.552 | 20/19 | D♯, vE♭ |
| 6 | 107.463 | 17/16, 33/31 | ^D♯, E♭ |
| 7 | 125.373 | 14/13, 29/27 | ^^D♯, v4E |
| 8 | 143.284 | ^3D♯, v3E | |
| 9 | 161.194 | 11/10, 23/21, 34/31 | ^4D♯, vvE |
| 10 | 179.104 | 31/28 | D𝄪, vE |
| 11 | 197.015 | E | |
| 12 | 214.925 | 17/15, 26/23 | ^E, F♭ |
| 13 | 232.836 | 8/7 | ^^E, v4F |
| 14 | 250.746 | 15/13, 22/19 | ^3E, v3F |
| 15 | 268.657 | 7/6 | ^4E, vvF |
| 16 | 286.567 | 13/11, 33/28 | E♯, vF |
| 17 | 304.478 | 31/26 | F |
| 18 | 322.388 | ^F, G♭♭ | |
| 19 | 340.299 | 28/23 | ^^F, v4G♭ |
| 20 | 358.209 | 16/13 | ^3F, v3G♭ |
| 21 | 376.119 | 36/29 | ^4F, vvG♭ |
| 22 | 394.03 | F♯, vG♭ | |
| 23 | 411.94 | 19/15, 33/26 | ^F♯, G♭ |
| 24 | 429.851 | ^^F♯, v4G | |
| 25 | 447.761 | 22/17 | ^3F♯, v3G |
| 26 | 465.672 | 17/13 | ^4F♯, vvG |
| 27 | 483.582 | F𝄪, vG | |
| 28 | 501.493 | 4/3 | G |
| 29 | 519.403 | 23/17, 31/23 | ^G, A♭♭ |
| 30 | 537.313 | 15/11 | ^^G, v4A♭ |
| 31 | 555.224 | 11/8, 29/21 | ^3G, v3A♭ |
| 32 | 573.134 | 32/23 | ^4G, vvA♭ |
| 33 | 591.045 | 31/22 | G♯, vA♭ |
| 34 | 608.955 | ^G♯, A♭ | |
| 35 | 626.866 | 23/16, 33/23 | ^^G♯, v4A |
| 36 | 644.776 | 16/11 | ^3G♯, v3A |
| 37 | 662.687 | 22/15 | ^4G♯, vvA |
| 38 | 680.597 | 34/23 | G𝄪, vA |
| 39 | 698.507 | 3/2 | A |
| 40 | 716.418 | ^A, B♭♭ | |
| 41 | 734.328 | 26/17 | ^^A, v4B♭ |
| 42 | 752.239 | 17/11 | ^3A, v3B♭ |
| 43 | 770.149 | ^4A, vvB♭ | |
| 44 | 788.06 | 30/19 | A♯, vB♭ |
| 45 | 805.97 | 35/22 | ^A♯, B♭ |
| 46 | 823.881 | 29/18 | ^^A♯, v4B |
| 47 | 841.791 | 13/8 | ^3A♯, v3B |
| 48 | 859.701 | 23/14 | ^4A♯, vvB |
| 49 | 877.612 | A𝄪, vB | |
| 50 | 895.522 | B | |
| 51 | 913.433 | 22/13 | ^B, C♭ |
| 52 | 931.343 | 12/7 | ^^B, v4C |
| 53 | 949.254 | 19/11, 26/15 | ^3B, v3C |
| 54 | 967.164 | 7/4 | ^4B, vvC |
| 55 | 985.075 | 23/13, 30/17 | B♯, vC |
| 56 | 1002.985 | C | |
| 57 | 1020.896 | ^C, D♭♭ | |
| 58 | 1038.806 | 20/11, 31/17 | ^^C, v4D♭ |
| 59 | 1056.716 | 35/19 | ^3C, v3D♭ |
| 60 | 1074.627 | 13/7 | ^4C, vvD♭ |
| 61 | 1092.537 | 32/17 | C♯, vD♭ |
| 62 | 1110.448 | 19/10 | ^C♯, D♭ |
| 63 | 1128.358 | 23/12 | ^^C♯, v4D |
| 64 | 1146.269 | 31/16, 33/17 | ^3C♯, v3D |
| 65 | 1164.179 | ^4C♯, vvD | |
| 66 | 1182.09 | C𝄪, vD | |
| 67 | 1200 | 2/1 | D |
Scales
Mos scales
- Meantone[5]: 11 11 17 11 17
- Meantone[7]: 11 11 6 11 11 11 6
- Barbados[5], Bustling Docks (original/default tuning): 14 14 11 14 14
Modmos scales
- Cavernous (original/default tuning): 14 14 11 21 7
- Formicarium (original/default tuning): 14 7 18 14 14
- Negri Blues (original/default tuning): 14 14 3 8 14 14
- Negri Blues Septatonic (original/default tuning): 14 14 3 8 11 3 14
- Negri Blues Octatonic (original/default tuning): 7 14 7 11 7 11 3 7
- Understory (original/default tuning): 14 7 18 7 21
- Meantone Ionian Pentatonic: 22 6 11 22 6
- Meantone Minor Melodic: 11 6 11 11 11 11 6
- Meantone Minor Harmonic: 11 6 11 11 6 16 6
- Meantone Minor Hexatonic: 11 6 11 11 17 11
- Meantone Dorian Harmonic: 11 6 16 6 11 6 11
- Meantone Mixolydian Pentatonic: 22 6 11 17 11
- Meantone Phrygian Dominant: 6 16 6 11 6 11 11
- Meantone Phrygian Dominant Hexatonic: 6 16 6 11 6 22
- Meantone Phrygian Dominant Pentatonic: 22 6 11 6 22
- Meantone Phrygian Pentatonic: 6 11 22 6 22
- Meantone Double Harmonic: 6 16 6 11 6 16 6
Blues scales
- Lost spirit (approximated from 31edo): 17 11 6 5 13 4 11
- Blackened skies (approximated from 72edo): 18 10 5 6 5 18 5
- Blues Aeolian Hexatonic: 17 11 6 5 6 22
- Blues Aeolian Pentatonic I: 17 11 11 6 22
- Blues Aeolian Pentatonic II: 17 22 6 11 11
- Blues Bright Double Harmonic: 6 16 6 11 6 11 6 5
- Blues Dark Double Harmonic: 11 6 11 6 5 6 16 6
- Blues Dorian Hexatonic: 17 11 11 11 6 11
- Blues Dorian Pentatonic: 17 22 11 6 11
- Blues Dorian Septatonic: 17 11 6 5 11 6 11
- Blues Harmonic Hexatonic: 11 6 11 11 22 6
- Blues Harmonic Septatonic: 17 11 6 5 6 11 5 6
- Blues Leading: 17 11 6 5 17 6 5
- Blues Minor: 17 11 6 5 17 11
- Blues Minor Maj7: 17 11 6 5 22 6
- Blues Pentachordal: 11 6 11 5 6 28
- Greyed Skies (approximated from 91edo): 17 11 5 6 6 17 5
- Akebono I: 11 6 11 11 17
- Augmented: 17 6 16 6 16 6
- Dominant Pentatonic: 11 11 17 17 11
- Hirajoshi: 11 6 12 6 22
- Javanese Pentachordal: 6 11 17 4 29
Others
- Approximation of Pelog lima: 6 10 22 7 22
- Arcade (approximated from 32afdo): 22 4 13 15 13
- Cosmic (approximated from 32afdo): 29 10 6 11 11
- Mechanical (approximated from 16afdo): 17 5 17 15 13
- Moonbeam (approximated from 16afdo): 11 6 12 22 6
- Springwater (approximated from 8afdo): 11 11 17 15 13
- Volcanic (approximated from 16afdo): 6 16 17 15 13
- Deja Vu (approximated from 101afdo): 18 21 6 12 10
- Freeway (approximated from 6afdo): 15 12 11 11 9 8
- Mushroom (approximated from 30afdo): 15 12 11 4 24
- Underpass (approximated from 10afdo): 18 21 12 6 10
- Sourgummy (approximated from 51afdo): 14 12 14 14 13
- Bubblegum/Cola (approximated from 60afdo/99afdo): 14 13 13 13 14
- Spearmint/Whitechocolate (approximated from 62afdo/90afdo): 13 14 13 14 13
- Lemonade (approximated from 79afdo): 14 13 13 14 13
- Candycorn (approximated from 91afdo): 11 12 11 10 12 11
- Trailmix (approximated from 97afdo): 11 11 11 12 11 11
- Liquorice (approximated from 101afdo): 11 11 12 10 12 11
- Apple Mint (approximated from 80afdo): 9 11 9 9 10 9 10