552edo
| ← 551edo | 552edo | 553edo → |
Theory
552edo is distinctly consistent in the 15-odd-limit. It has a sharp tendency, with prime harmonics 3 through 13 all tuned sharp. The equal temperament tempers out [8 14 -3⟩ (parakleisma) in the 5-limit; 250047/250000 (landscape comma), 589824/588245 (hewuermera comma), 26873856/26796875, and 33554432/33480783 (garischisma) in the 7-limit; 5632/5625, 9801/9800, 46656/46585, 151263/151250, and 161280/161051 in the 11-limit; and 1716/1715, 2080/2079, 10648/10647, and 20480/20449 in the 13-limit. It supports sextile and gives a good tuning for it.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.219 | +0.643 | +0.739 | +0.856 | +0.777 | -0.608 | +0.313 | -0.013 | +0.858 | +0.617 |
| Relative (%) | +0.0 | +10.1 | +29.6 | +34.0 | +39.4 | +35.7 | -27.9 | +14.4 | -0.6 | +39.4 | +28.4 | |
| Steps (reduced) |
552 (0) |
875 (323) |
1282 (178) |
1550 (446) |
1910 (254) |
2043 (387) |
2256 (48) |
2345 (137) |
2497 (289) |
2682 (474) |
2735 (527) | |
Subsets and supersets
Since 552 factors into 23 × 3 × 23, 552edo has subset edos 2, 3, 4, 6, 8, 12, 23, 24, 46, 69, 92, 138, and 276.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [875 -552⟩ | [⟨552 875]] | -0.0691 | 0.0691 | 3.18 |
| 2.3.5 | [8 14 -13⟩, [71 -36 -6⟩ | [⟨552 875 1282]] | -0.1383 | 0.1130 | 5.20 |
| 2.3.5.7 | 250047/250000, 589824/588245, 33554432/33480783 | [⟨552 875 1282 1550]] | -0.1696 | 0.1118 | 5.15 |
| 2.3.5.7.11 | 5632/5625, 9801/9800, 151263/151250, 161280/161051 | [⟨552 875 1282 1550 1910]] | -0.1851 | 0.1048 | 4.82 |
| 2.3.5.7.11.13 | 1716/1715, 2080/2079, 5632/5625, 10648/10647, 20480/20449 | [⟨552 875 1282 1550 1910 2043]] | -0.1892 | 0.0961 | 4.42 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 145\552 | 315.22 | 6/5 | Parakleismic (5-limit) |
| 1 | 229\552 | 497.83 | 4/3 | Gary (2.3.7 subgroup) |
| 6 | 229\552 (45\552) |
497.83 (97.83) |
4/3 (128/121) |
Sextile |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct