57edo
| ← 56edo | 57edo | 58edo → |
57 equal divisions of the octave (abbreviated 57edo or 57ed2), also called 57-tone equal temperament (57tet) or 57 equal temperament (57et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 57 equal parts of about 21.1 ¢ each. Each step represents a frequency ratio of 21/57, or the 57th root of 2.
Theory
57edo is an excellent tuning for the 2.5/3.7.11.13.17.19 just intonation subgroup. One way to describe 57edo is that it has a 5-limit part consisting of three rings of 19edo, plus a no-threes no-fives part which is much more accurate.
Using the full prime-limit patent val, the equal temperament tempers out 81/80, 1029/1024, and 3125/3072 in the 7-limit; and 99/98, 385/384, 441/440, and 625/616 in the 11-limit. A good generator to exploit the 2.5/3.7.11.13.17.19 aspect of 57 is the approximate 11/8, which is 26\57. This gives the 19-limit 46 & 57 temperament heinz. It can also be used to tune mothra as well as trismegistus.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -7.22 | -7.37 | -0.40 | +6.62 | -3.95 | +1.58 | +6.47 | +0.31 | -2.78 | -7.62 | +3.30 |
| Relative (%) | -34.3 | -35.0 | -1.9 | +31.4 | -18.8 | +7.5 | +30.7 | +1.5 | -13.2 | -36.2 | +15.7 | |
| Steps (reduced) |
90 (33) |
132 (18) |
160 (46) |
181 (10) |
197 (26) |
211 (40) |
223 (52) |
233 (5) |
242 (14) |
250 (22) |
258 (30) | |
Subsets and supersets
57edo contains 3edo and 19edo as subsets.
Intervals
| # | Cents | Ups and Downs Notation (Flat Fifth 11\19) |
Ups and Downs Notation (Sharp Fifth 34\57) |
|---|---|---|---|
| 0 | 0.00 | D | D |
| 1 | 21.05 | ^D, vvE♭♭ | ^D, E♭ |
| 2 | 42.11 | ^^D, vE♭♭ | ^^D, v9E |
| 3 | 63.16 | D♯, E♭♭ | ^3D, v8E |
| 4 | 84.21 | ^D♯, vvE♭ | ^4D, v7E |
| 5 | 105.26 | ^^D♯, vE♭ | ^5D, v6E |
| 6 | 126.32 | D𝄪, E♭ | ^6D, v5E |
| 7 | 147.37 | ^D𝄪, vvE | ^7D, v4E |
| 8 | 168.42 | ^^D𝄪, vE | ^8D, v3E |
| 9 | 189.47 | E | ^9D, vvE |
| 10 | 210.53 | ^E, vvF♭ | D♯, vE |
| 11 | 231.58 | ^^E, vF♭ | E |
| 12 | 252.63 | E♯, F♭ | F |
| 13 | 273.68 | ^E♯, vvF | ^F, G♭ |
| 14 | 294.74 | ^^E♯, vF | ^^F, v9G |
| 15 | 315.79 | F | ^3F, v8G |
| 16 | 336.84 | ^F, vvG♭♭ | ^4F, v7G |
| 17 | 357.89 | ^^F, vG♭♭ | ^5F, v6G |
| 18 | 378.95 | F♯, G♭♭ | ^6F, v5G |
| 19 | 400.00 | ^F♯, vvG♭ | ^7F, v4G |
| 20 | 421.05 | ^^F♯, vG♭ | ^8F, v3G |
| 21 | 442.11 | F𝄪, G♭ | ^9F, vvG |
| 22 | 463.16 | ^F𝄪, vvG | F♯, vG |
| 23 | 484.21 | ^^F𝄪, vG | G |
| 24 | 505.26 | G | ^G, A♭ |
| 25 | 526.32 | ^G, vvA♭♭ | ^^G, v9A |
| 26 | 547.37 | ^^G, vA♭♭ | ^3G, v8A |
| 27 | 568.42 | G♯, A♭♭ | ^4G, v7A |
| 28 | 589.47 | ^G♯, vvA♭ | ^5G, v6A |
| 29 | 610.53 | ^^G♯, vA♭ | ^6G, v5A |
| 30 | 631.58 | G𝄪, A♭ | ^7G, v4A |
| 31 | 652.63 | ^G𝄪, vvA | ^8G, v3A |
| 32 | 673.68 | ^^G𝄪, vA | ^9G, vvA |
| 33 | 694.74 | A | G♯, vA |
| 34 | 715.79 | ^A, vvB♭♭ | A |
| 35 | 736.84 | ^^A, vB♭♭ | ^A, B♭ |
| 36 | 757.89 | A♯, B♭♭ | ^^A, v9B |
| 37 | 778.95 | ^A♯, vvB♭ | ^3A, v8B |
| 38 | 800.00 | ^^A♯, vB♭ | ^4A, v7B |
| 39 | 821.05 | A𝄪, B♭ | ^5A, v6B |
| 40 | 842.11 | ^A𝄪, vvB | ^6A, v5B |
| 41 | 863.16 | ^^A𝄪, vB | ^7A, v4B |
| 42 | 884.21 | B | ^8A, v3B |
| 43 | 905.26 | ^B, vvC♭ | ^9A, vvB |
| 44 | 926.32 | ^^B, vC♭ | A♯, vB |
| 45 | 947.37 | B♯, C♭ | B |
| 46 | 968.42 | ^B♯, vvC | C |
| 47 | 989.47 | ^^B♯, vC | ^C, D♭ |
| 48 | 1010.53 | C | ^^C, v9D |
| 49 | 1031.58 | ^C, vvD♭♭ | ^3C, v8D |
| 50 | 1052.63 | ^^C, vD♭♭ | ^4C, v7D |
| 51 | 1073.68 | C♯, D♭♭ | ^5C, v6D |
| 52 | 1094.74 | ^C♯, vvD♭ | ^6C, v5D |
| 53 | 1115.79 | ^^C♯, vD♭ | ^7C, v4D |
| 54 | 1136.84 | C𝄪, D♭ | ^8C, v3D |
| 55 | 1157.89 | ^C𝄪, vvD | ^9C, vvD |
| 56 | 1178.95 | ^^C𝄪, vD | C♯, vD |
| 57 | 1200.00 | D | D |
Scales
- 2 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 2 1 1 - 3mos of type 18L 21s (augene)