323ed6
| ← 322ed6 | 323ed6 | 324ed6 → |
323 equal divisions of the 6th harmonic (abbreviated 323ed6) is a nonoctave tuning system that divides the interval of 6/1 into 323 equal parts of about 9.6 ¢ each. Each step represents a frequency ratio of 61/323, or the 323rd root of 6.
Theory
323ed6 is closely related to 125edo, but with the perfect twelfth rather than the octave being just. The octave is stretched by about 0.729 cents. Unlike 125edo, which is only consistent to the 10-integer-limit, 323ed6 is consistent to the 12-integer-limit. In particular, it improves the approximated prime harmonics 5, 11 and 13 over 125edo, though the 7, 17 and 19, which are sharp to start with, are tuned a little worse here.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.45 | -0.45 | +0.89 | -1.28 | +0.00 | +2.03 | +1.34 | -0.89 | -0.83 | -2.57 | +0.45 |
| Relative (%) | +4.7 | -4.7 | +9.3 | -13.3 | +0.0 | +21.1 | +14.0 | -9.3 | -8.6 | -26.8 | +4.7 | |
| Steps (reduced) |
125 (125) |
198 (198) |
250 (250) |
290 (290) |
323 (0) |
351 (28) |
375 (52) |
396 (73) |
415 (92) |
432 (109) |
448 (125) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -3.68 | +2.48 | -1.72 | +1.79 | +2.47 | -0.45 | +1.99 | -0.38 | +1.58 | -2.13 | -2.25 | +0.89 |
| Relative (%) | -38.3 | +25.8 | -17.9 | +18.6 | +25.7 | -4.7 | +20.7 | -4.0 | +16.5 | -22.1 | -23.5 | +9.3 | |
| Steps (reduced) |
462 (139) |
476 (153) |
488 (165) |
500 (177) |
511 (188) |
521 (198) |
531 (208) |
540 (217) |
549 (226) |
557 (234) |
565 (242) |
573 (250) | |
Subsets and supersets
Since 323 factors into primes as 17 × 19, 323ed6 contains 17ed6 and 19ed6 as subset ed6's.