# 1323edo

← 1322edo | 1323edo | 1324edo → |

^{3}× 7^{2}**1323 equal divisions of the octave** (abbreviated **1323edo** or **1323ed2**), also called **1323-tone equal temperament** (**1323tet**) or **1323 equal temperament** (**1323et**) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1323 equal parts of about 0.907 ¢ each. Each step represents a frequency ratio of 2^{1/1323}, or the 1323rd root of 2.

## Theory

1323edo is the smallest edo distinctly consistent in the 29-odd-limit. It is enfactored in the 7-limit, sharing the same excellent 7-limit approximation with 441edo, but it makes for a great higher-limit system by splitting each step of 441edo into three.

It provides the optimal patent val for the 11-limit trinealimmal temperament, which has a period of 1\27 octave.

### Prime harmonics

Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Error | Absolute (¢) | +0.000 | +0.086 | +0.081 | -0.118 | +0.156 | +0.289 | +0.260 | -0.007 | +0.297 | -0.099 | -0.364 |

Relative (%) | +0.0 | +9.5 | +8.9 | -13.1 | +17.2 | +31.8 | +28.7 | -0.8 | +32.8 | -10.9 | -40.2 | |

Steps (reduced) |
1323 (0) |
2097 (774) |
3072 (426) |
3714 (1068) |
4577 (608) |
4896 (927) |
5408 (116) |
5620 (328) |
5985 (693) |
6427 (1135) |
6554 (1262) |

### Subsets and supersets

Since 1323 factors into 3^{3} × 7^{2}, 1323edo has subset edos 3, 7, 9, 21, 27, 49, 63, 147, 189, 441, of which 441edo is a member of the zeta edos.

## Regular temperament properties

### Rank-2 temperaments

Note: 7-limit temperaments supported by 441et are not included.

Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
---|---|---|---|---|

27 | 299\1323 (5\1323) |
271.201 (4.535) |
1375/1176 (?) |
Trinealimmal |

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