9539edo
| ← 9538edo | 9539edo | 9540edo → |
9539 equal divisions of the octave (abbreviated 9539edo or 9539ed2), also called 9539-tone equal temperament (9539tet) or 9539 equal temperament (9539et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 9539 equal parts of about 0.126 ¢ each. Each step represents a frequency ratio of 21/9539, or the 9539th root of 2.
Theory
9539edo is consistent to the 9-odd-limit due to its harmonics 11 and 13 being halfway between its steps. It is strong in the 2.3.5.19.31.37.43 subgroup, tempering out 5000211/5000000, 320013/320000, 3486880000/3486784401, 79456894976/79449609375, 94470144/94465625 and 30294016/30292137. Using the 2.3.5.31.41 subgroup, it tempers out 15376/15375.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0000 | +0.0054 | +0.0161 | -0.0451 | +0.0606 | -0.0622 | -0.0388 | +0.0024 | -0.0324 | -0.0353 | -0.0099 |
| Relative (%) | +0.0 | +4.3 | +12.8 | -35.9 | +48.2 | -49.4 | -30.8 | +1.9 | -25.7 | -28.1 | -7.9 | |
| Steps (reduced) |
9539 (0) |
15119 (5580) |
22149 (3071) |
26779 (7701) |
33000 (4383) |
35298 (6681) |
38990 (834) |
40521 (2365) |
43150 (4994) |
46340 (8184) |
47258 (9102) | |
Subsets and supersets
9539edo is the 1181st prime edo. 19078edo, which doubles it, gives a good correction to its harmonics 11 and 13.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [15119 -9539⟩ | [⟨9539 15119]] | −0.0017 | 0.0017 | 1.35 |
| 2.3.5 | [-17 62 -35⟩, [380 -38 -107⟩ | [⟨9539 15119 22149]] | −0.0034 | 0.0028 | 2.23 |
| 2.3.5.7 | [-5 -12 -9 16⟩, [-52 6 5 11⟩, [6 -37 13 8⟩ | [⟨9539 15119 22149 26779]] | +0.0014 | 0.0088 | 7.00 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 2566\9539 | 322.8011 | [-6 23 -13⟩ | Senior |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct