428edt
428 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 428edt or 428ed3), is a nonoctave tuning system that divides the interval of 3/1 into 428 equal parts of about 4.44 ¢ each. Each step represents a frequency ratio of 31/428, or the 428th root of 3.
| ← 427edt | 428edt | 429edt → |
Theory
428edt is related to 270edo, but with the twelfth rather than the octave being just. The octave is about 0.169 cents compressed. 428edt is consistent to the 22-integer-limit; in comparison, 270edo is only consistent up to the 16-integer-limit. It fixes 270edo's inconsistently mapped 17/13, which is 270edo's only inconsistently mapped interval in the 21-odd-limit. However, this comes at the cost of a flat-tending tuning profile, with harmonics 1–22 all tuned flat except for 17.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.17 | +0.00 | -0.34 | -0.04 | -0.17 | -0.41 | -0.51 | +0.00 | -0.21 | -0.79 | -0.34 |
| Relative (%) | -3.8 | +0.0 | -7.6 | -0.9 | -3.8 | -9.2 | -11.4 | +0.0 | -4.7 | -17.8 | -7.6 | |
| Steps (reduced) |
270 (270) |
428 (0) |
540 (112) |
627 (199) |
698 (270) |
758 (330) |
810 (382) |
856 (0) |
897 (41) |
934 (78) |
968 (112) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.15 | -0.58 | -0.04 | -0.67 | +1.02 | -0.17 | -0.45 | -0.38 | -0.41 | -0.96 | +2.07 | -0.51 |
| Relative (%) | -25.9 | -13.0 | -0.9 | -15.2 | +23.0 | -3.8 | -10.2 | -8.5 | -9.2 | -21.6 | +46.7 | -11.4 | |
| Steps (reduced) |
999 (143) |
1028 (172) |
1055 (199) |
1080 (224) |
1104 (248) |
1126 (270) |
1147 (291) |
1167 (311) |
1186 (330) |
1204 (348) |
1222 (366) |
1238 (382) | |
See also
- 270edo – relative edo