314edt
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| ← 313edt | 314edt | 315edt → |
314 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 314edt or 314ed3), is a nonoctave tuning system that divides the interval of 3/1 into 314 equal parts of about 6.06 ¢ each. Each step represents a frequency ratio of 31/314, or the 314th root of 3.
Theory
314edt is related to 198edo, but with the perfect twelfth rather than the octave being just. The octave is compressed by about 0.678 cents. Like 198edo, 314edt is consistent to the 16-integer-limit. It has a flat tuning tendency, with prime harmonics 2, 5, 7, 11, and 13 all tuned flat.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.68 | +0.00 | -1.36 | -0.01 | -0.68 | -1.03 | -2.03 | +0.00 | -0.69 | -2.15 | -1.36 |
| Relative (%) | -11.2 | +0.0 | -22.4 | -0.2 | -11.2 | -17.1 | -33.6 | +0.0 | -11.4 | -35.5 | -22.4 | |
| Steps (reduced) |
198 (198) |
314 (0) |
396 (82) |
460 (146) |
512 (198) |
556 (242) |
594 (280) |
628 (0) |
658 (30) |
685 (57) |
710 (82) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.61 | -1.71 | -0.01 | -2.71 | +1.36 | -0.68 | +2.63 | -1.37 | -1.03 | -2.83 | -1.04 | -2.03 |
| Relative (%) | -10.1 | -28.2 | -0.2 | -44.8 | +22.5 | -11.2 | +43.5 | -22.6 | -17.1 | -46.7 | -17.2 | -33.6 | |
| Steps (reduced) |
733 (105) |
754 (126) |
774 (146) |
792 (164) |
810 (182) |
826 (198) |
842 (214) |
856 (228) |
870 (242) |
883 (255) |
896 (268) |
908 (280) | |
Subsets and supersets
Since 314 factors into primes as 2 × 157, 314edt contains 2edt and 157edt as subset edts.