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 has two options of fifth equally far from just. The sharp fifth is inherited from 32edo and produces a hard superpythagorean scale, while the flat fifth is within the meantone/flattone range, supporting flattone temperament.
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, which would perhaps be of more interest if it was lower in badness.
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 (%) | -44 | +40 | +33 | +12 | -40 | +17 | -4 | +40 | +13 | -11 | +49 | |
| 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 1, 2, 4, 8, 16, 32.
Intervals
| Steps | Cents | Ups and downs notation (dual flat fifth 37\64) |
Ups and downs notation (dual sharp fifth 38\64) |
Approximate ratios |
|---|---|---|---|---|
| 0 | 0 | D | D | 1/1 |
| 1 | 18.75 | ^D, E♭♭♭ | ^D, vE♭ | 78/77 |
| 2 | 37.5 | ^^D, vvE♭♭ | ^^D, E♭ | 45/44, 50/49, 56/55 |
| 3 | 56.25 | D♯, vE♭♭ | ^3D, v9E | |
| 4 | 75 | ^D♯, E♭♭ | ^4D, v8E | 22/21 |
| 5 | 93.75 | ^^D♯, vvE♭ | ^5D, v7E | 55/52 |
| 6 | 112.5 | D𝄪, vE♭ | ^6D, v6E | 15/14, 16/15 |
| 7 | 131.25 | ^D𝄪, E♭ | ^7D, v5E | 14/13 |
| 8 | 150 | ^^D𝄪, vvE | ^8D, v4E | 12/11 |
| 9 | 168.75 | D♯𝄪, vE | ^9D, v3E | |
| 10 | 187.5 | E | D♯, vvE | |
| 11 | 206.25 | ^E, F♭♭ | ^D♯, vE | 44/39 |
| 12 | 225 | ^^E, vvF♭ | E | 8/7 |
| 13 | 243.75 | E♯, vF♭ | ^E, vF | 15/13 |
| 14 | 262.5 | ^E♯, F♭ | F | 64/55, 65/56 |
| 15 | 281.25 | ^^E♯, vvF | ^F, vG♭ | 75/64 |
| 16 | 300 | E𝄪, vF | ^^F, G♭ | |
| 17 | 318.75 | F | ^3F, v9G | 77/64 |
| 18 | 337.5 | ^F, G♭♭♭ | ^4F, v8G | 39/32 |
| 19 | 356.25 | ^^F, vvG♭♭ | ^5F, v7G | 16/13, 49/40 |
| 20 | 375 | F♯, vG♭♭ | ^6F, v6G | 26/21 |
| 21 | 393.75 | ^F♯, G♭♭ | ^7F, v5G | 5/4 |
| 22 | 412.5 | ^^F♯, vvG♭ | ^8F, v4G | |
| 23 | 431.25 | F𝄪, vG♭ | ^9F, v3G | 77/60 |
| 24 | 450 | ^F𝄪, G♭ | F♯, vvG | 13/10 |
| 25 | 468.75 | ^^F𝄪, vvG | ^F♯, vG | 21/16, 55/42 |
| 26 | 487.5 | F♯𝄪, vG | G | 65/49 |
| 27 | 506.25 | G | ^G, vA♭ | 75/56 |
| 28 | 525 | ^G, A♭♭♭ | ^^G, A♭ | |
| 29 | 543.75 | ^^G, vvA♭♭ | ^3G, v9A | 15/11 |
| 30 | 562.5 | G♯, vA♭♭ | ^4G, v8A | |
| 31 | 581.25 | ^G♯, A♭♭ | ^5G, v7A | 7/5 |
| 32 | 600 | ^^G♯, vvA♭ | ^6G, v6A | 55/39, 78/55 |
| 33 | 618.75 | G𝄪, vA♭ | ^7G, v5A | 10/7, 63/44 |
| 34 | 637.5 | ^G𝄪, A♭ | ^8G, v4A | 75/52 |
| 35 | 656.25 | ^^G𝄪, vvA | ^9G, v3A | 22/15 |
| 36 | 675 | G♯𝄪, vA | G♯, vvA | 77/52 |
| 37 | 693.75 | A | ^G♯, vA | |
| 38 | 712.5 | ^A, B♭♭♭ | A | |
| 39 | 731.25 | ^^A, vvB♭♭ | ^A, vB♭ | 32/21, 75/49 |
| 40 | 750 | A♯, vB♭♭ | ^^A, B♭ | 20/13 |
| 41 | 768.75 | ^A♯, B♭♭ | ^3A, v9B | |
| 42 | 787.5 | ^^A♯, vvB♭ | ^4A, v8B | |
| 43 | 806.25 | A𝄪, vB♭ | ^5A, v7B | 8/5 |
| 44 | 825 | ^A𝄪, B♭ | ^6A, v6B | 21/13, 77/48 |
| 45 | 843.75 | ^^A𝄪, vvB | ^7A, v5B | 13/8, 80/49 |
| 46 | 862.5 | A♯𝄪, vB | ^8A, v4B | 64/39 |
| 47 | 881.25 | B | ^9A, v3B | |
| 48 | 900 | ^B, C♭♭ | A♯, vvB | |
| 49 | 918.75 | ^^B, vvC♭ | ^A♯, vB | |
| 50 | 937.5 | B♯, vC♭ | B | 55/32 |
| 51 | 956.25 | ^B♯, C♭ | ^B, vC | 26/15 |
| 52 | 975 | ^^B♯, vvC | C | 7/4 |
| 53 | 993.75 | B𝄪, vC | ^C, vD♭ | 39/22 |
| 54 | 1012.5 | C | ^^C, D♭ | |
| 55 | 1031.25 | ^C, D♭♭♭ | ^3C, v9D | |
| 56 | 1050 | ^^C, vvD♭♭ | ^4C, v8D | 11/6 |
| 57 | 1068.75 | C♯, vD♭♭ | ^5C, v7D | 13/7 |
| 58 | 1087.5 | ^C♯, D♭♭ | ^6C, v6D | 15/8, 28/15 |
| 59 | 1106.25 | ^^C♯, vvD♭ | ^7C, v5D | |
| 60 | 1125 | C𝄪, vD♭ | ^8C, v4D | 21/11 |
| 61 | 1143.75 | ^C𝄪, D♭ | ^9C, v3D | |
| 62 | 1162.5 | ^^C𝄪, vvD | C♯, vvD | 49/25, 55/28 |
| 63 | 1181.25 | C♯𝄪, vD | ^C♯, vD | 77/39 |
| 64 | 1200 | D | D | 2/1 |