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 | 17.9 | ^D, E♭♭ | |
| 2 | 35.8 | ^^D, ^E♭♭ | |
| 3 | 53.7 | 31/30, 32/31, 33/32, 34/33, 35/34 | vvD♯, ^^E♭♭ |
| 4 | 71.6 | 24/23 | vD♯, vvE♭ |
| 5 | 89.6 | 20/19 | D♯, vE♭ |
| 6 | 107.5 | 17/16, 33/31 | ^D♯, E♭ |
| 7 | 125.4 | 14/13, 29/27 | ^^D♯, ^E♭ |
| 8 | 143.3 | vvD𝄪, ^^E♭ | |
| 9 | 161.2 | 11/10, 23/21, 34/31 | vD𝄪, vvE |
| 10 | 179.1 | 31/28 | D𝄪, vE |
| 11 | 197 | E | |
| 12 | 214.9 | 17/15, 26/23 | ^E, F♭ |
| 13 | 232.8 | 8/7 | ^^E, ^F♭ |
| 14 | 250.7 | 15/13, 22/19 | vvE♯, ^^F♭ |
| 15 | 268.7 | 7/6 | vE♯, vvF |
| 16 | 286.6 | 13/11, 33/28 | E♯, vF |
| 17 | 304.5 | 31/26 | F |
| 18 | 322.4 | ^F, G♭♭ | |
| 19 | 340.3 | 28/23 | ^^F, ^G♭♭ |
| 20 | 358.2 | 16/13 | vvF♯, ^^G♭♭ |
| 21 | 376.1 | 36/29 | vF♯, vvG♭ |
| 22 | 394 | F♯, vG♭ | |
| 23 | 411.9 | 19/15, 33/26 | ^F♯, G♭ |
| 24 | 429.9 | ^^F♯, ^G♭ | |
| 25 | 447.8 | 22/17 | vvF𝄪, ^^G♭ |
| 26 | 465.7 | 17/13 | vF𝄪, vvG |
| 27 | 483.6 | F𝄪, vG | |
| 28 | 501.5 | 4/3 | G |
| 29 | 519.4 | 23/17, 31/23 | ^G, A♭♭ |
| 30 | 537.3 | 15/11 | ^^G, ^A♭♭ |
| 31 | 555.2 | 11/8, 29/21 | vvG♯, ^^A♭♭ |
| 32 | 573.1 | 32/23 | vG♯, vvA♭ |
| 33 | 591 | 31/22 | G♯, vA♭ |
| 34 | 609 | ^G♯, A♭ | |
| 35 | 626.9 | 23/16, 33/23 | ^^G♯, ^A♭ |
| 36 | 644.8 | 16/11 | vvG𝄪, ^^A♭ |
| 37 | 662.7 | 22/15 | vG𝄪, vvA |
| 38 | 680.6 | 34/23 | G𝄪, vA |
| 39 | 698.5 | 3/2 | A |
| 40 | 716.4 | ^A, B♭♭ | |
| 41 | 734.3 | 26/17 | ^^A, ^B♭♭ |
| 42 | 752.2 | 17/11 | vvA♯, ^^B♭♭ |
| 43 | 770.1 | vA♯, vvB♭ | |
| 44 | 788.1 | 30/19 | A♯, vB♭ |
| 45 | 806 | 35/22 | ^A♯, B♭ |
| 46 | 823.9 | 29/18 | ^^A♯, ^B♭ |
| 47 | 841.8 | 13/8 | vvA𝄪, ^^B♭ |
| 48 | 859.7 | 23/14 | vA𝄪, vvB |
| 49 | 877.6 | A𝄪, vB | |
| 50 | 895.5 | B | |
| 51 | 913.4 | 22/13 | ^B, C♭ |
| 52 | 931.3 | 12/7 | ^^B, ^C♭ |
| 53 | 949.3 | 19/11, 26/15 | vvB♯, ^^C♭ |
| 54 | 967.2 | 7/4 | vB♯, vvC |
| 55 | 985.1 | 23/13, 30/17 | B♯, vC |
| 56 | 1003 | C | |
| 57 | 1020.9 | ^C, D♭♭ | |
| 58 | 1038.8 | 20/11, 31/17 | ^^C, ^D♭♭ |
| 59 | 1056.7 | 35/19 | vvC♯, ^^D♭♭ |
| 60 | 1074.6 | 13/7 | vC♯, vvD♭ |
| 61 | 1092.5 | 32/17 | C♯, vD♭ |
| 62 | 1110.4 | 19/10 | ^C♯, D♭ |
| 63 | 1128.4 | 23/12 | ^^C♯, ^D♭ |
| 64 | 1146.3 | 31/16, 33/17 | vvC𝄪, ^^D♭ |
| 65 | 1164.2 | vC𝄪, vvD | |
| 66 | 1182.1 | C𝄪, vD | |
| 67 | 1200 | 2/1 | D |
Notation
Sagittal notation
Evo flavor
Revo flavor
In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's primary comma (the comma it exactly represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it approximately represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.
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