323ed6: Difference between revisions
m →Theory |
Tristanbay (talk | contribs) →Theory: Someone forgot to change the octave stretch cent value after copying the text from the 198edt page Tags: Mobile edit Mobile web edit |
||
| (One intermediate revision by one other user not shown) | |||
| Line 3: | Line 3: | ||
== Theory == | == Theory == | ||
323ed6 is closely related to [[125edo]], but with the | 323ed6 is closely related to [[125edo]], but with the 6th harmonic rather than the [[2/1|octave]] being just. The octave is [[stretched and compressed tuning|stretched]] by about 0.447 cents. Unlike 125edo, which is only [[consistent]] to the [[integer limit|10-integer-limit]], 323ed6 is consistent to the 12-integer-limit. In particular, it improves the approximated [[prime harmonic]]s [[5/1|5]], [[11/1|11]] and [[13/1|13]] over 125edo, though the [[7/1|7]], [[17/1|17]] and [[19/1|19]], which are sharp to start with, are tuned a little worse here. | ||
=== Harmonics === | === Harmonics === | ||
Latest revision as of 14:33, 27 February 2026
| ← 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 6th harmonic rather than the octave being just. The octave is stretched by about 0.447 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.