# 552edo

← 551edo | 552edo | 553edo → |

^{3}× 3 × 23**552 equal divisions of the octave** (abbreviated **552edo** or **552ed2**), also called **552-tone equal temperament** (**552tet**) or **552 equal temperament** (**552et**) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 552 equal parts of about 2.174 ¢ each. Each step represents a frequency ratio of 2^{1/552}, or the 552nd root of 2.

## 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.

It is also consistent in the no-17 23-odd-limit and the no-17 no-25 33-odd-limit. In the 2.3.5.7.11.13.19 subgroup, it tempers out 1216/1215, 2376/2375, 2926/2925, 3136/3135, 3328/3325, 3971/3969 among other commas.

### 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 | +10 | +30 | +34 | +39 | +36 | -28 | +14 | -1 | +39 | +28 | |

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 2^{3} × 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 |

2.3.5.7.11.13.19 | 1216/1215, 1716/1715, 2080/2079, 2376/2375, 9633/9625, 15390/15379 | [⟨552 875 1282 1550 1910 2043 2345]] | -0.1727 | 0.0977 | 4.50 |

- 552et is notable for being the first equal temperament to beat 270 in the 2.3.5.7.11.13.19 subgroup in terms of absolute error. The next equal temperament that does better in this subgroup is 581.

### 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 |

24 | 232\552 (2\552) |
504.348 (4/348) |
7/5 (?) |
Chromium |

46 | 229\552 (1\552) |
497.83 (97.83) |
4/3 (?) |
Palladium (5-limit) |

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct