13ed4/3
13 equal divisions of the perfect fourth (13ed4/3) is the tuning system that divides the fourth into 13 steps of 38.3112 cents each. It can be thought of as a 4/3.5/3.7/6 subgroup analogue to 9edf or Carlos Alpha. It very closely approximates the intervals of 8/7 (at 6 steps) and 7/6 (at 7 steps), along with 5/4 (at 10 steps); these approximations are related to the slender temperament, tempering out 225/224 and 589824/588245 in the 7-limit. This tuning is close to every three steps of 94edo or to 88ed7.
Intervals
These are the intervals up to a perfect fourth up.
| Degrees | Cents | Approximation by 2.3.5.7.11.23 subgroup |
|---|---|---|
| 1 | 38.31 | 46/45, 45/44 |
| 2 | 76.62 | 24/23, 23/22, 22/21 |
| 3 | 114.93 | 16/15, 15/14 |
| 4 | 153.24 | 12/11 |
| 5 | 191.56 | |
| 6 | 229.87 | 8/7 |
| 7 | 268.18 | 7/6 |
| 8 | 306.49 | |
| 9 | 344.80 | 11/9 |
| 10 | 383.11 | 5/4 |
| 11 | 421.42 | 23/18 |
| 12 | 459.73 | 30/23 |
| 13 | 498.04 | 4/3 |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -12.4 | +13.6 | +13.6 | +10.4 | +1.2 | +2.6 | +1.2 | -11.1 | -2.0 | -13.7 | -11.1 |
| Relative (%) | -32.2 | +35.5 | +35.5 | +27.1 | +3.3 | +6.7 | +3.3 | -29.0 | -5.1 | -35.8 | -29.0 | |
| Steps (reduced) |
31 (5) |
50 (11) |
63 (11) |
73 (8) |
81 (3) |
88 (10) |
94 (3) |
99 (8) |
104 (0) |
108 (4) |
112 (8) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +3.6 | -9.8 | -14.3 | -11.1 | -1.1 | +14.9 | -2.1 | -14.3 | +16.2 | +12.2 | +11.9 |
| Relative (%) | +9.3 | -25.6 | -37.3 | -29.0 | -2.9 | +38.8 | -5.6 | -37.3 | +42.2 | +32.0 | +31.1 | |
| Steps (reduced) |
116 (12) |
119 (2) |
122 (5) |
125 (8) |
128 (11) |
131 (1) |
133 (3) |
135 (5) |
138 (8) |
140 (10) |
142 (12) | |