428edt: Difference between revisions
Created page with "{{Infobox ET}} {{ED intro}} == 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 com..." |
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== Theory == | == Theory == | ||
428edt is related to [[270edo]], but with the [[3/1|twelfth]] rather than the [[2/1|octave]] being just. The octave is about 0.169 cents compressed. 428edt is [[consistent]] to the [[integer limit|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 [[harmonic]]s 1–22 all tuned flat except for [[17/1|17]]. | 428edt is related to [[270edo]], but with the [[3/1|twelfth]] rather than the [[2/1|octave]] being just. The octave is about 0.169 cents compressed. 428edt is [[consistent]] to the [[integer limit|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 [[harmonic]]s 1–22 all tuned flat except for [[17/1|17]] and perfect powers of 3. | ||
=== Harmonics === | === Harmonics === | ||
{{Harmonics in equal|428|3|1|columns=11}} | {{Harmonics in equal|428|3|1|columns=11}} | ||
{{Harmonics in equal|428|3|1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 428edt (continued)}} | {{Harmonics in equal|428|3|1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 428edt (continued)}} | ||
=== Subsets and supersets === | |||
Since 428 factors into primes as {{nowrap| 2<sup>2</sup> × 107 }}, 428edt has subset edts {{EDTs| 2, 4, 107, and 214 }}. | |||
== See also == | == See also == | ||
* [[270edo]] – relative edo | * [[270edo]] – relative edo | ||
Latest revision as of 21:03, 31 January 2026
| ← 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