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, ↓↓E♭♭ | ↑D, E♭ |
| 2 | 42.1053 | ↑↑D, ↓E♭♭ | ↑↑D, ↓9E |
| 3 | 63.1579 | D♯, E♭♭ | ↑3D, ↓8E |
| 4 | 84.2105 | ↑D♯, ↓↓E♭ | ↑4D, ↓7E |
| 5 | 105.2632 | ↑↑D♯, ↓E♭ | ↑5D, ↓6E |
| 6 | 126.3158 | D𝄪, E♭ | ↑6D, ↓5E |
| 7 | 147.3684 | ↑D𝄪, ↓↓E | ↑7D, ↓4E |
| 8 | 168.42105 | ↑↑D𝄪, ↓E | ↑8D, ↓3E |
| 9 | 189.4737 | E | ↑9D, ↓↓E |
| 10 | 210.5263 | ↑E, ↓↓F♭ | D♯, ↓E |
| 11 | 231.57895 | ↑↑E, ↓F♭ | E |
| 12 | 252.6316 | E♯, F♭ | F |
| 13 | 273.6842 | ↑E♯, ↓↓F | ↑F, G♭ |
| 14 | 294.7368 | ↑↑E♯, ↓F | ↑↑F, ↓9G |
| 15 | 315.7895 | F | ↑3F, ↓8G |
| 16 | 336.8421 | ↑F, ↓↓G♭♭ | ↑4F, ↓7G |
| 17 | 357.8947 | ↑↑F, ↓G♭♭ | ↑5F, ↓6G |
| 18 | 378.9474 | F♯, G♭♭ | ↑6F, ↓5G |
| 19 | 400 | ↑F♯, ↓↓G♭ | ↑7F, ↓4G |
| 20 | 421.0526 | ↑↑F♯, ↓G♭ | ↑8F, ↓3G |
| 21 | 442.1053 | F𝄪, G♭ | ↑9F, ↓↓G |
| 22 | 463.1579 | ↑F𝄪, ↓↓G | F♯, ↓G |
| 23 | 484.2105 | ↑↑F𝄪, ↓G | G |
| 24 | 505.2632 | G | ↑G, A♭ |
| 25 | 526.3158 | ↑G, ↓↓A♭♭ | ↑↑G, ↓9A |
| 26 | 547.3684 | ↑↑G, ↓A♭♭ | ↑3G, ↓8A |
| 27 | 568.42105 | G♯, A♭♭ | ↑4G, ↓7A |
| 28 | 589.4737 | ↑G♯, ↓↓A♭ | ↑5G, ↓6A |
| 29 | 610.5263 | ↑↑G♯, ↓A♭ | ↑6G, ↓5A |
| 30 | 631.57895 | G𝄪, A♭ | ↑7G, ↓4A |
| 31 | 652.6316 | ↑G𝄪, ↓↓A | ↑8G, ↓3A |
| 32 | 673.6842 | ↑↑G𝄪, ↓A | ↑9G, ↓↓A |
| 33 | 694.7368 | A | G♯, ↓A |
| 34 | 715.7895 | ↑A, ↓↓B♭♭ | A |
| 35 | 736.8421 | ↑↑A, ↓B♭♭ | ↑A, B♭ |
| 36 | 757.8947 | A♯, B♭♭ | ↑↑A, ↓9B |
| 37 | 778.9474 | ↑A♯, ↓↓B♭ | ↑3A, ↓8B |
| 38 | 800 | ↑↑A♯, ↓B♭ | ↑4A, ↓7B |
| 39 | 821.0526 | A𝄪, B♭ | ↑5A, ↓6B |
| 40 | 842.1053 | ↑A𝄪, ↓↓B | ↑6A, ↓5B |
| 41 | 863.1579 | ↑↑A𝄪, ↓B | ↑7A, ↓4B |
| 42 | 884.2105 | B | ↑8A, ↓3B |
| 43 | 905.2632 | ↑B, ↓↓C♭ | ↑9A, ↓↓B |
| 44 | 926.3158 | ↑↑B, ↓C♭ | A♯, ↓B |
| 45 | 947.3684 | B♯, C♭ | B |
| 46 | 968.42105 | ↑B♯, ↓↓C | C |
| 47 | 989.4737 | ↑↑B♯, ↓C | ↑C, D♭ |
| 48 | 1010.5263 | C | ↑↑C, ↓9D |
| 49 | 1031.57895 | ↑C, ↓↓D♭♭ | ↑3C, ↓8D |
| 50 | 1052.6316 | ↑↑C, ↓D♭♭ | ↑4C, ↓7D |
| 51 | 1073.6842 | C♯, D♭♭ | ↑5C, ↓6D |
| 52 | 1094.7368 | ↑C♯, ↓↓D♭ | ↑6C, ↓5D |
| 53 | 1115.7895 | ↑↑C♯, ↓D♭ | ↑7C, ↓4D |
| 54 | 1136.8421 | C𝄪, D♭ | ↑8C, ↓3D |
| 55 | 1157.8947 | ↑C𝄪, ↓↓D | ↑9C, ↓↓D |
| 56 | 1178.9474 | ↑↑C𝄪, ↓D | C♯, ↓D |
| 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)