User:MisterShafXen/3edo
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Prime factorization
3 (prime)
Step size
400 ¢
Fifth
2\3 (800 ¢)
(semiconvergent)
Semitones (A1:m2)
2:-1 (800 ¢ : -400 ¢)
Consistency limit
5
Distinct consistency limit
3
| ← 2edo | 3edo | 4edo → |
(semiconvergent)
3 equal divisions of the octave (abbreviated 3edo or 3ed2), also called 3-tone equal temperament (3tet) or 3 equal temperament (3et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 3 equal parts of exactly 400 ¢ each. Each step represents a frequency ratio of 21/3, or the 3rd root of 2.
Theory
3edo is interesting as it is the first edo to have a triad, although it is augmented. This tuning notably tempers out 128/125, supporting augmented temperament.
Notation
A B C. All A, B, C.
Intervals
| Steps | Cents | Approximate ratios | Ups and downs notation |
|---|---|---|---|
| 0 | 0 | 1/1, 14/13, 16/15, 17/16, 19/18, 20/19 | D, F, B |
| 1 | 400 | 4/3, 5/4, 6/5, 13/10, 13/11, 14/11, 16/13, 17/13, 17/14, 19/15, 19/16, 20/17 | E, G |
| 2 | 800 | 3/2, 5/3, 8/5, 11/7, 13/8, 17/10, 17/11, 19/12, 20/13 | A, C |
| 3 | 1200 | 2/1, 13/7, 15/8, 17/8, 19/9, 19/10 | D |
Harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | 83 | 89 | 97 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0 | +98 | +14 | -169 | -151 | -41 | -105 | +102 | +172 | +170 | +55 | +149 | -29 | -112 | +134 | -74 | +141 | +83 | -79 | -180 | +172 | +35 | -50 | -171 | +80 |
| Relative (%) | +0.0 | +24.5 | +3.4 | -42.2 | -37.8 | -10.1 | -26.2 | +25.6 | +42.9 | +42.6 | +13.7 | +37.2 | -7.3 | -27.9 | +33.6 | -18.4 | +35.2 | +20.8 | -19.8 | -44.9 | +43.1 | +8.9 | -12.5 | -42.7 | +20.0 | |
| Steps (reduced) |
3 (0) |
5 (2) |
7 (1) |
8 (2) |
10 (1) |
11 (2) |
12 (0) |
13 (1) |
14 (2) |
15 (0) |
15 (0) |
16 (1) |
16 (1) |
16 (1) |
17 (2) |
17 (2) |
18 (0) |
18 (0) |
18 (0) |
18 (0) |
19 (1) |
19 (1) |
19 (1) |
19 (1) |
20 (2) | |
| Harmonic | 101 | 103 | 107 | 109 | 113 | 127 | 131 | 137 | 139 | 149 | 151 | 157 | 163 | 167 | 173 | 179 | 181 | 191 | 193 | 197 | 199 | 211 | 223 | 227 | 229 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +10 | -24 | -90 | -122 | -184 | +14 | -40 | -118 | -143 | +137 | +114 | +46 | -18 | -60 | -122 | -181 | -200 | +107 | +89 | +54 | +36 | -65 | -161 | -192 | +193 |
| Relative (%) | +2.5 | -6.0 | -22.4 | -30.5 | -46.1 | +3.4 | -10.0 | -29.4 | -35.7 | +34.2 | +28.5 | +11.6 | -4.6 | -15.1 | -30.4 | -45.1 | -50.0 | +26.8 | +22.3 | +13.4 | +9.0 | -16.3 | -40.3 | -48.0 | +48.2 | |
| Steps (reduced) |
20 (2) |
20 (2) |
20 (2) |
20 (2) |
20 (2) |
21 (0) |
21 (0) |
21 (0) |
21 (0) |
22 (1) |
22 (1) |
22 (1) |
22 (1) |
22 (1) |
22 (1) |
22 (1) |
22 (1) |
23 (2) |
23 (2) |
23 (2) |
23 (2) |
23 (2) |
23 (2) |
23 (2) |
24 (0) | |
| Harmonic | |
|---|---|
| Error | Absolute (¢) |
| Relative (%) | |
| Steps (reduced) | |