64edo
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 7c & 12c (or 19 & 64) temperament (named meso[idiosyncratic term] by Jerdle) 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.
The patent val of 64edo is the first patent val to represent the intervals 5/4, 81/64, 14/11, 9/7, 13/10, 21/16, and 4/3 distinctly, although in 64edo they are not in the correct order. 72edo represents all of these intervals distinctly and in the correct order.
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 | 18.8 | ^D, E♭♭♭ | ^D, vE♭ | |
| 2 | 37.5 | vD♯, ^E♭♭♭ | ^^D, E♭ | |
| 3 | 56.3 | 30/29, 31/30, 32/31 | D♯, vE♭♭ | ^3D, ^E♭ |
| 4 | 75 | ^D♯, E♭♭ | ^4D, ^^E♭ | |
| 5 | 93.8 | vD𝄪, ^E♭♭ | ^5D, ^3E♭ | |
| 6 | 112.5 | 16/15, 31/29 | D𝄪, vE♭ | v4D♯, ^4E♭ |
| 7 | 131.3 | 14/13 | ^D𝄪, E♭ | v3D♯, v5E |
| 8 | 150 | 12/11 | vD♯𝄪, ^E♭ | vvD♯, v4E |
| 9 | 168.8 | 32/29 | D♯𝄪, vE | vD♯, v3E |
| 10 | 187.5 | 29/26 | E | D♯, vvE |
| 11 | 206.3 | ^E, F♭♭ | ^D♯, vE | |
| 12 | 225 | vE♯, ^F♭♭ | E | |
| 13 | 243.8 | 15/13, 23/20 | E♯, vF♭ | ^E, vF |
| 14 | 262.5 | ^E♯, F♭ | F | |
| 15 | 281.3 | 20/17 | vE𝄪, ^F♭ | ^F, vG♭ |
| 16 | 300 | 19/16 | E𝄪, vF | ^^F, G♭ |
| 17 | 318.8 | F | ^3F, ^G♭ | |
| 18 | 337.5 | 17/14, 28/23 | ^F, G♭♭♭ | ^4F, ^^G♭ |
| 19 | 356.3 | 16/13 | vF♯, ^G♭♭♭ | ^5F, ^3G♭ |
| 20 | 375 | F♯, vG♭♭ | v4F♯, ^4G♭ | |
| 21 | 393.8 | ^F♯, G♭♭ | v3F♯, v5G | |
| 22 | 412.5 | 19/15 | vF𝄪, ^G♭♭ | vvF♯, v4G |
| 23 | 431.3 | F𝄪, vG♭ | vF♯, v3G | |
| 24 | 450 | ^F𝄪, G♭ | F♯, vvG | |
| 25 | 468.8 | 21/16 | vF♯𝄪, ^G♭ | ^F♯, vG |
| 26 | 487.5 | F♯𝄪, vG | G | |
| 27 | 506.2 | G | ^G, vA♭ | |
| 28 | 525 | 19/14, 23/17 | ^G, A♭♭♭ | ^^G, A♭ |
| 29 | 543.8 | 26/19 | vG♯, ^A♭♭♭ | ^3G, ^A♭ |
| 30 | 562.5 | 29/21 | G♯, vA♭♭ | ^4G, ^^A♭ |
| 31 | 581.3 | 7/5 | ^G♯, A♭♭ | ^5G, ^3A♭ |
| 32 | 600 | vG𝄪, ^A♭♭ | v4G♯, ^4A♭ | |
| 33 | 618.8 | 10/7 | G𝄪, vA♭ | v3G♯, v5A |
| 34 | 637.5 | ^G𝄪, A♭ | vvG♯, v4A | |
| 35 | 656.3 | 19/13 | vG♯𝄪, ^A♭ | vG♯, v3A |
| 36 | 675 | 28/19, 31/21, 34/23 | G♯𝄪, vA | G♯, vvA |
| 37 | 693.8 | A | ^G♯, vA | |
| 38 | 712.5 | ^A, B♭♭♭ | A | |
| 39 | 731.3 | 29/19, 32/21 | vA♯, ^B♭♭♭ | ^A, vB♭ |
| 40 | 750 | A♯, vB♭♭ | ^^A, B♭ | |
| 41 | 768.8 | ^A♯, B♭♭ | ^3A, ^B♭ | |
| 42 | 787.5 | 30/19 | vA𝄪, ^B♭♭ | ^4A, ^^B♭ |
| 43 | 806.3 | A𝄪, vB♭ | ^5A, ^3B♭ | |
| 44 | 825 | ^A𝄪, B♭ | v4A♯, ^4B♭ | |
| 45 | 843.7 | 13/8, 31/19 | vA♯𝄪, ^B♭ | v3A♯, v5B |
| 46 | 862.5 | 23/14, 28/17 | A♯𝄪, vB | vvA♯, v4B |
| 47 | 881.2 | B | vA♯, v3B | |
| 48 | 900 | 32/19 | ^B, C♭♭ | A♯, vvB |
| 49 | 918.8 | 17/10 | vB♯, ^C♭♭ | ^A♯, vB |
| 50 | 937.5 | B♯, vC♭ | B | |
| 51 | 956.3 | 26/15 | ^B♯, C♭ | ^B, vC |
| 52 | 975 | vB𝄪, ^C♭ | C | |
| 53 | 993.8 | B𝄪, vC | ^C, vD♭ | |
| 54 | 1012.5 | C | ^^C, D♭ | |
| 55 | 1031.3 | 29/16 | ^C, D♭♭♭ | ^3C, ^D♭ |
| 56 | 1050 | 11/6 | vC♯, ^D♭♭♭ | ^4C, ^^D♭ |
| 57 | 1068.8 | 13/7 | C♯, vD♭♭ | ^5C, ^3D♭ |
| 58 | 1087.5 | 15/8 | ^C♯, D♭♭ | v4C♯, ^4D♭ |
| 59 | 1106.3 | vC𝄪, ^D♭♭ | v3C♯, v5D | |
| 60 | 1125 | C𝄪, vD♭ | vvC♯, v4D | |
| 61 | 1143.8 | 29/15, 31/16 | ^C𝄪, D♭ | vC♯, v3D |
| 62 | 1162.5 | vC♯𝄪, ^D♭ | C♯, vvD | |
| 63 | 1181.3 | C♯𝄪, vD | ^C♯, vD | |
| 64 | 1200 | 2/1 | D | D |
Notation
Stein–Zimmermann–Gould notation
64edo can be notated with Stein–Zimmermann–Gould notation:
| Step offset | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|---|
| Sharp symbol | | | | | | | | |
| Flat symbol | | | | | | | |
Here, a sharp raises by three steps, and a flat lowers by three steps, so arrows can be used to fill in the gap. If the arrows are taken to have their own layer of enharmonic spellings, some notes may be best spelled with double arrows.
Kite's ups and downs notation
Spoken as up, downsharp, sharp, upsharp, etc. Note that downsharp can be respelled as dup (double-up), and upflat as dud.
| Step offset | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|---|
| Sharp symbol | 
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| Flat symbol | |
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Sagittal notation
Best fifth notation
This notation uses the same sagittal sequence as edos 50, 57, and 71b.
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.
Second-best fifth notation
This notation is a superset of the notation for 32edo.
Evo flavor
Revo flavor
Evo-SZ flavor
Octave stretch or compression
- Stretch
Stretching the octave of 64edo by around 4.5 ¢ results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 8.99 ¢. The tuning 179ed7 does this. So does the tuning 326zpi whose octave is identical within 0.3 ¢.
Stretching the octave of 64edo by around 3 ¢ results in improved primes 3, 5, 7, 11, 13 and 17, but a worse prime 2. This approximates all harmonics up to 16 within 9.25 ¢. The tuning 165ed6 does this.
Stretching the octave of 64edo by around 2 ¢ results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 9.17 ¢. The tuning 229ed12 does this. So does the tuning 221ed11 whose octave is identical within 0.1 ¢.
- Compression
Compressing the octave of 64edo by just under 2 ¢ results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 8.02 ¢. The tuning 328zpi does this.
Compressing the octave of 64edo by just under 3 ¢ results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 8.80 ¢. The tuning 230ed12 does this.
Scales
- Gutierrez Moonglade scale: 1 4 5 1 4 2 4 6 1 3 5 1 1 4 4 1 5 1 1 3 1 1 4 1