15ed4/3
| ← 14ed4/3 | 15ed4/3 | 16ed4/3 → |
15 equal divisions of the perfect fourth (15ed4/3) is the tuning system that divides the fourth into 15 steps of 33.203 cents each. It can be thought of as a 4/3.8/7.32/13.32/17.19/16 subgroup analogue to 11edf or Carlos Beta. It very closely approximates the intervals of 8/7 (at 7 steps) and 7/6 (at 8 steps), along with 18/17 (at 3 steps), 19/16 (at 9 steps) and 17/13 (at 14 steps); these approximations are related to the temperament which tempers out 343/342, 833/832, 1729/1728 and 4624/4617 in the 2.3.7.13.17.19 subgroup (36 & 181), along with 4914/4913, 8281/8262 and 4747561509943/4696546738176 ([-31 -7 0 15⟩, laquintrizo comma). This tuning is close to every seven steps of 253edo or every other step of 19ed6/5.
Intervals
These are the intervals up to a perfect fourth up.
| Degrees | Cents | Approximation by 2.3.7.11/5.13.17.19.23 subgroup |
|---|---|---|
| 1 | 33.20 | 52/51 |
| 2 | 66.41 | 27/26 |
| 3 | 99.61 | 18/17 |
| 4 | 132.81 | |
| 5 | 166.01 | 11/10 |
| 6 | 199.22 | |
| 7 | 232.42 | 8/7 |
| 8 | 265.62 | 7/6 |
| 9 | 298.83 | 19/16 |
| 10 | 332.03 | 23/19 |
| 11 | 365.23 | 21/17 |
| 12 | 398.44 | 34/27 |
| 13 | 431.64 | |
| 14 | 464.84 | 17/13 |
| 15 | 498.04 | 4/3 |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -4.7 | -9.4 | -9.4 | +2.7 | -14.1 | -15.3 | -14.1 | +14.4 | -2.0 | -0.9 | +14.4 |
| Relative (%) | -14.1 | -28.3 | -28.3 | +8.2 | -42.4 | -46.1 | -42.4 | +43.5 | -5.9 | -2.8 | +43.5 | |
| Steps (reduced) |
36 (6) |
57 (12) |
72 (12) |
84 (9) |
93 (3) |
101 (11) |
108 (3) |
115 (10) |
120 (0) |
125 (5) |
130 (10) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +8.7 | +13.2 | -6.6 | +14.4 | +9.1 | +9.7 | +15.7 | -6.6 | +8.5 | -5.6 | -16.2 |
| Relative (%) | +26.1 | +39.7 | -20.0 | +43.5 | +27.4 | +29.3 | +47.4 | -20.0 | +25.6 | -17.0 | -48.7 | |
| Steps (reduced) |
134 (14) |
138 (3) |
141 (6) |
145 (10) |
148 (13) |
151 (1) |
154 (4) |
156 (6) |
159 (9) |
161 (11) |
163 (13) | |