428edt
| ← 427edt | 428edt | 429edt → |
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.
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 and perfect powers of 3.
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) | |
Subsets and supersets
Since 428 factors into primes as 22 × 107, 428edt has subset edts 2, 4, 107, and 214.
See also
- 270edo – relative edo