512ed6
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| ← 511ed6 | 512ed6 | 513ed6 → |
512 equal divisions of the 6th harmonic (abbreviated 512ed6) is a nonoctave tuning system that divides the interval of 6/1 into 512 equal parts of about 6.06 ¢ each. Each step represents a frequency ratio of 61/512, or the 512th root of 6.
Theory
512ed6 is related to 198edo, but with the 6th harmonic rather than the octave being just. The octave is compressed by about 0.416 cents. Like 198edo, 314edt is consistent to the 16-integer-limit. It is well optimized for the 7-limit, with the same tuning as 256ed6, but the higher harmonics tend a little flat, especially considering how flat the 11 and 19 are tuned.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.42 | +0.42 | -0.83 | +0.60 | +0.00 | -0.30 | -1.25 | +0.83 | +0.18 | -1.24 | -0.42 |
| Relative (%) | -6.9 | +6.9 | -13.7 | +9.9 | +0.0 | -4.9 | -20.6 | +13.7 | +3.0 | -20.5 | -6.9 | |
| Steps (reduced) |
198 (198) |
314 (314) |
396 (396) |
460 (460) |
512 (0) |
556 (44) |
594 (82) |
628 (116) |
658 (146) |
685 (173) |
710 (198) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.36 | -0.71 | +1.01 | -1.66 | +2.43 | +0.42 | -2.31 | -0.23 | +0.12 | -1.66 | +0.15 | -0.83 |
| Relative (%) | +5.9 | -11.8 | +16.8 | -27.5 | +40.2 | +6.9 | -38.1 | -3.8 | +2.0 | -27.4 | +2.4 | -13.7 | |
| Steps (reduced) |
733 (221) |
754 (242) |
774 (262) |
792 (280) |
810 (298) |
826 (314) |
841 (329) |
856 (344) |
870 (358) |
883 (371) |
896 (384) |
908 (396) | |
Subsets and supersets
Since 512 factors into primes as 29, 512ed6 contains subset ed6's 2, 4, 8, 16, 32, 64, 128, and 256.