64edo
| ← 63edo | 64edo | 65edo → |
64 equal divisions of the octave (abbreviated 64edo or 64ed2), also called 64-tone equal temperament (64tet) or 64 equal temperament (64et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 64 equal parts of about 18.8 ¢ each. Each step represents a frequency ratio of 21/64, or the 64th root of 2.
Theory
64edo is a zeta valley edo and is very bad at approximating JI for its size. It has two options of fifth almost equally far from just. The sharp fifth from the 64b val is inherited from 32edo and produces a hard superpythagorean scale, while the slightly more accurate flat fifth from the patent val is within the meantone/flattone range. However bizarrely, the flat fifth does not support meantone or flattone in its patent val, and instead supports the obscure unnamed 7c & 12c (or 19 & 64) temperament which reaches 5/4 as a double-diminished fourth. In order to interpret it as flattone, the 64cd val must be used.
Still, the patent val tempers out 648/625 in the 5-limit and 225/224 in the 7-limit, plus 66/65, 121/120 and 441/440 in the 11-limit and 144/143 in the 13-limit. It provides the optimal patent val in the 7-, 11- and 13-limits for the 16&64 temperament.
64be val is a tuning for the beatles temperament and for the rank-3 temperaments heimlaug and vili in the 17-limit. 64bccc tunes dichotic, although that is an exotemperament. 64cdf is a tuning for vibhu.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -8.21 | +7.44 | +6.17 | +2.34 | -7.57 | +3.22 | -0.77 | +7.54 | +2.49 | -2.03 | +9.23 |
| Relative (%) | -43.8 | +39.7 | +32.9 | +12.5 | -40.4 | +17.2 | -4.1 | +40.2 | +13.3 | -10.8 | +49.2 | |
| Steps (reduced) |
101 (37) |
149 (21) |
180 (52) |
203 (11) |
221 (29) |
237 (45) |
250 (58) |
262 (6) |
272 (16) |
281 (25) |
290 (34) | |
Subsets and supersets
64edo is the 6th power of two edo, and it has subset edos 2, 4, 8, 16, 32. 128edo, which doubles it, corrects its approximation to many of the lower harmonics.
Intervals
| Steps | Cents | Approximate Ratios | Ups and Downs Notation (Dual Flat Fifth 37\64) |
Ups and Downs Notation (Dual Sharp Fifth 38\64) |
|---|---|---|---|---|
| 0 | 0 | 1/1 | D | D |
| 1 | 18.75 | ^D, E♭♭♭ | ^D, vE♭ | |
| 2 | 37.5 | ^^D, vvE♭♭ | ^^D, E♭ | |
| 3 | 56.25 | 30/29, 31/30, 32/31 | D♯, vE♭♭ | ^3D, v9E |
| 4 | 75 | ^D♯, E♭♭ | ^4D, v8E | |
| 5 | 93.75 | ^^D♯, vvE♭ | ^5D, v7E | |
| 6 | 112.5 | 16/15, 31/29 | D𝄪, vE♭ | ^6D, v6E |
| 7 | 131.25 | 14/13 | ^D𝄪, E♭ | ^7D, v5E |
| 8 | 150 | 12/11 | ^^D𝄪, vvE | ^8D, v4E |
| 9 | 168.75 | 32/29 | D♯𝄪, vE | ^9D, v3E |
| 10 | 187.5 | 29/26 | E | D♯, vvE |
| 11 | 206.25 | ^E, F♭♭ | ^D♯, vE | |
| 12 | 225 | ^^E, vvF♭ | E | |
| 13 | 243.75 | 15/13, 23/20 | E♯, vF♭ | ^E, vF |
| 14 | 262.5 | ^E♯, F♭ | F | |
| 15 | 281.25 | 20/17 | ^^E♯, vvF | ^F, vG♭ |
| 16 | 300 | 19/16 | E𝄪, vF | ^^F, G♭ |
| 17 | 318.75 | F | ^3F, v9G | |
| 18 | 337.5 | 17/14, 28/23 | ^F, G♭♭♭ | ^4F, v8G |
| 19 | 356.25 | 16/13 | ^^F, vvG♭♭ | ^5F, v7G |
| 20 | 375 | F♯, vG♭♭ | ^6F, v6G | |
| 21 | 393.75 | ^F♯, G♭♭ | ^7F, v5G | |
| 22 | 412.5 | 19/15 | ^^F♯, vvG♭ | ^8F, v4G |
| 23 | 431.25 | F𝄪, vG♭ | ^9F, v3G | |
| 24 | 450 | ^F𝄪, G♭ | F♯, vvG | |
| 25 | 468.75 | 21/16 | ^^F𝄪, vvG | ^F♯, vG |
| 26 | 487.5 | F♯𝄪, vG | G | |
| 27 | 506.25 | G | ^G, vA♭ | |
| 28 | 525 | 19/14, 23/17 | ^G, A♭♭♭ | ^^G, A♭ |
| 29 | 543.75 | 26/19 | ^^G, vvA♭♭ | ^3G, v9A |
| 30 | 562.5 | 29/21 | G♯, vA♭♭ | ^4G, v8A |
| 31 | 581.25 | 7/5 | ^G♯, A♭♭ | ^5G, v7A |
| 32 | 600 | ^^G♯, vvA♭ | ^6G, v6A | |
| 33 | 618.75 | 10/7 | G𝄪, vA♭ | ^7G, v5A |
| 34 | 637.5 | ^G𝄪, A♭ | ^8G, v4A | |
| 35 | 656.25 | 19/13 | ^^G𝄪, vvA | ^9G, v3A |
| 36 | 675 | 28/19, 31/21, 34/23 | G♯𝄪, vA | G♯, vvA |
| 37 | 693.75 | A | ^G♯, vA | |
| 38 | 712.5 | ^A, B♭♭♭ | A | |
| 39 | 731.25 | 29/19, 32/21 | ^^A, vvB♭♭ | ^A, vB♭ |
| 40 | 750 | A♯, vB♭♭ | ^^A, B♭ | |
| 41 | 768.75 | ^A♯, B♭♭ | ^3A, v9B | |
| 42 | 787.5 | 30/19 | ^^A♯, vvB♭ | ^4A, v8B |
| 43 | 806.25 | A𝄪, vB♭ | ^5A, v7B | |
| 44 | 825 | ^A𝄪, B♭ | ^6A, v6B | |
| 45 | 843.75 | 13/8, 31/19 | ^^A𝄪, vvB | ^7A, v5B |
| 46 | 862.5 | 23/14, 28/17 | A♯𝄪, vB | ^8A, v4B |
| 47 | 881.25 | B | ^9A, v3B | |
| 48 | 900 | 32/19 | ^B, C♭♭ | A♯, vvB |
| 49 | 918.75 | 17/10 | ^^B, vvC♭ | ^A♯, vB |
| 50 | 937.5 | B♯, vC♭ | B | |
| 51 | 956.25 | 26/15 | ^B♯, C♭ | ^B, vC |
| 52 | 975 | ^^B♯, vvC | C | |
| 53 | 993.75 | B𝄪, vC | ^C, vD♭ | |
| 54 | 1012.5 | C | ^^C, D♭ | |
| 55 | 1031.25 | 29/16 | ^C, D♭♭♭ | ^3C, v9D |
| 56 | 1050 | 11/6 | ^^C, vvD♭♭ | ^4C, v8D |
| 57 | 1068.75 | 13/7 | C♯, vD♭♭ | ^5C, v7D |
| 58 | 1087.5 | 15/8 | ^C♯, D♭♭ | ^6C, v6D |
| 59 | 1106.25 | ^^C♯, vvD♭ | ^7C, v5D | |
| 60 | 1125 | C𝄪, vD♭ | ^8C, v4D | |
| 61 | 1143.75 | 29/15, 31/16 | ^C𝄪, D♭ | ^9C, v3D |
| 62 | 1162.5 | ^^C𝄪, vvD | C♯, vvD | |
| 63 | 1181.25 | C♯𝄪, vD | ^C♯, vD | |
| 64 | 1200 | 2/1 | D | D |