252edt: Difference between revisions
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== Theory == | == Theory == | ||
252edt is nearly identical to [[159edo]], but with the [[3/1|perfect twelfth]] instead of the [[2/1|octave]] tuned just. Like 159edo, 252edt is [[consistent]] to the [[integer limit|18-integer-limit]]. | 252edt is nearly identical to [[159edo]], but with the [[3/1|perfect twelfth]] instead of the [[2/1|octave]] tuned just. The octave is [[stretched and compressed tuning|stretched]] by about 0.0430 cents. Like 159edo, 252edt is [[consistent]] to the [[integer limit|18-integer-limit]]. The stretch is so subtle that most of the [[prime harmonic]]s tuned flat in 159edo remain flat. | ||
=== Harmonics === | === Harmonics === | ||
Latest revision as of 14:52, 18 April 2025
| ← 251edt | 252edt | 253edt → |
252 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 252edt or 252ed3), is a nonoctave tuning system that divides the interval of 3/1 into 252 equal parts of about 7.55 ¢ each. Each step represents a frequency ratio of 31/252, or the 252nd root of 3.
Theory
252edt is nearly identical to 159edo, but with the perfect twelfth instead of the octave tuned just. The octave is stretched by about 0.0430 cents. Like 159edo, 252edt is consistent to the 18-integer-limit. The stretch is so subtle that most of the prime harmonics tuned flat in 159edo remain flat.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.04 | +0.00 | +0.09 | -1.31 | +0.04 | -2.67 | +0.13 | +0.00 | -1.27 | -0.23 | +0.09 |
| Relative (%) | +0.6 | +0.0 | +1.1 | -17.3 | +0.6 | -35.3 | +1.7 | +0.0 | -16.8 | -3.0 | +1.1 | |
| Steps (reduced) |
159 (159) |
252 (0) |
318 (66) |
369 (117) |
411 (159) |
446 (194) |
477 (225) |
504 (0) |
528 (24) |
550 (46) |
570 (66) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.63 | -2.62 | -1.31 | +0.17 | +0.88 | +0.04 | -2.99 | -1.22 | -2.67 | -0.18 | -1.66 | +0.13 |
| Relative (%) | -34.9 | -34.8 | -17.3 | +2.3 | +11.7 | +0.6 | -39.6 | -16.2 | -35.3 | -2.4 | -22.1 | +1.7 | |
| Steps (reduced) |
588 (84) |
605 (101) |
621 (117) |
636 (132) |
650 (146) |
663 (159) |
675 (171) |
687 (183) |
698 (194) |
709 (205) |
719 (215) |
729 (225) | |
Subsets and supersets
Since 252 factors into primes as 22 × 32 × 7, 252edt has subset edts 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 28, 36, 42, 63, 84, and 126.