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
It can be used to tune mothra temperament, and is an excellent tuning for the 2.5/3.7.11.13.17.19 just intonation subgroup. One way to describe 57 is that it has a 5-limit part consisting of three versions of 19, plus a no-threes no-fives part which is much more accurate. 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.
5-limit commas: 81/80, 3125/3072
7-limit commas: 81/80, 3125/3072, 1029/1024
11-limit commas: 99/98, 385/384, 441/440, 625/616
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 | -35 | -2 | +31 | -19 | +7 | +31 | +1 | -13 | -36 | +16 | |
| Steps (reduced) |
90 (33) |
132 (18) |
160 (46) |
181 (10) |
197 (26) |
211 (40) |
223 (52) |
233 (5) |
242 (14) |
250 (22) |
258 (30) | |
Intervals
| Degree | Cents | Ups and downs notation (flat fifth 11\19) | Ups and downs notation (sharp fifth 34\57) |
|---|---|---|---|
| 0 | 0.0000 | D | D |
| 1 | 21.0526 | ^D, vvE♭♭ | ^D, E♭ |
| 2 | 42.1053 | ^^D, vE♭♭ | ^^D, v9E |
| 3 | 63.1579 | D♯, E♭♭ | ^3D, v8E |
| 4 | 84.2105 | ^D♯, vvE♭ | ^4D, v7E |
| 5 | 105.2632 | ^^D♯, vE♭ | ^5D, v6E |
| 6 | 126.3158 | D𝄪, E♭ | ^6D, v5E |
| 7 | 147.3684 | ^D𝄪, vvE | ^7D, v4E |
| 8 | 168.42105 | ^^D𝄪, vE | ^8D, v3E |
| 9 | 189.4737 | E | ^9D, vvE |
| 10 | 210.5263 | ^E, vvF♭ | D♯, vE |
| 11 | 231.57895 | ^^E, vF♭ | E |
| 12 | 252.6316 | E♯, F♭ | F |
| 13 | 273.6842 | ^E♯, vvF | ^F, G♭ |
| 14 | 294.7368 | ^^E♯, vF | ^^F, v9G |
| 15 | 315.7895 | F | ^3F, v8G |
| 16 | 336.8421 | ^F, vvG♭♭ | ^4F, v7G |
| 17 | 357.8947 | ^^F, vG♭♭ | ^5F, v6G |
| 18 | 378.9474 | F♯, G♭♭ | ^6F, v5G |
| 19 | 400 | ^F♯, vvG♭ | ^7F, v4G |
| 20 | 421.0526 | ^^F♯, vG♭ | ^8F, v3G |
| 21 | 442.1053 | F𝄪, G♭ | ^9F, vvG |
| 22 | 463.1579 | ^F𝄪, vvG | F♯, vG |
| 23 | 484.2105 | ^^F𝄪, vG | G |
| 24 | 505.2632 | G | ^G, A♭ |
| 25 | 526.3158 | ^G, vvA♭♭ | ^^G, v9A |
| 26 | 547.3684 | ^^G, vA♭♭ | ^3G, v8A |
| 27 | 568.42105 | G♯, A♭♭ | ^4G, v7A |
| 28 | 589.4737 | ^G♯, vvA♭ | ^5G, v6A |
| 29 | 610.5263 | ^^G♯, vA♭ | ^6G, v5A |
| 30 | 631.57895 | G𝄪, A♭ | ^7G, v4A |
| 31 | 652.6316 | ^G𝄪, vvA | ^8G, v3A |
| 32 | 673.6842 | ^^G𝄪, vA | ^9G, vvA |
| 33 | 694.7368 | A | G♯, vA |
| 34 | 715.7895 | ^A, vvB♭♭ | A |
| 35 | 736.8421 | ^^A, vB♭♭ | ^A, B♭ |
| 36 | 757.8947 | A♯, B♭♭ | ^^A, v9B |
| 37 | 778.9474 | ^A♯, vvB♭ | ^3A, v8B |
| 38 | 800 | ^^A♯, vB♭ | ^4A, v7B |
| 39 | 821.0526 | A𝄪, B♭ | ^5A, v6B |
| 40 | 842.1053 | ^A𝄪, vvB | ^6A, v5B |
| 41 | 863.1579 | ^^A𝄪, vB | ^7A, v4B |
| 42 | 884.2105 | B | ^8A, v3B |
| 43 | 905.2632 | ^B, vvC♭ | ^9A, vvB |
| 44 | 926.3158 | ^^B, vC♭ | A♯, vB |
| 45 | 947.3684 | B♯, C♭ | B |
| 46 | 968.42105 | ^B♯, vvC | C |
| 47 | 989.4737 | ^^B♯, vC | ^C, D♭ |
| 48 | 1010.5263 | C | ^^C, v9D |
| 49 | 1031.57895 | ^C, vvD♭♭ | ^3C, v8D |
| 50 | 1052.6316 | ^^C, vD♭♭ | ^4C, v7D |
| 51 | 1073.6842 | C♯, D♭♭ | ^5C, v6D |
| 52 | 1094.7368 | ^C♯, vvD♭ | ^6C, v5D |
| 53 | 1115.7895 | ^^C♯, vD♭ | ^7C, v4D |
| 54 | 1136.8421 | C𝄪, D♭ | ^8C, v3D |
| 55 | 1157.8947 | ^C𝄪, vvD | ^9C, vvD |
| 56 | 1178.9474 | ^^C𝄪, vD | C♯, vD |
| 57 | 1200 | 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)