# 433edo

← 432edo | 433edo | 434edo → |

**433 equal divisions of the octave** (abbreviated **433edo** or **433ed2**), also called **433-tone equal temperament** (**433tet**) or **433 equal temperament** (**433et**) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 433 equal parts of about 2.77 ¢ each. Each step represents a frequency ratio of 2^{1/433}, or the 433rd root of 2.

## Theory

443edo is only consistent to the 5-odd-limit since harmonic 7 is about halfway between its steps. To start with, the patent val ⟨433 686 1005 **1216**] as well as the 433d val ⟨433 686 1005 **1215**] are worth considering.

Using the patent val, the equal temperament tempers out 19683/19600 and 4096000/4084101 in the 7-limit; 3025/3024, 4000/3993, 6250/6237, 161280/161051, and 180224/180075 in the 11-limit.

### Odd harmonics

Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Error | Absolute (¢) | -0.80 | -1.09 | +1.15 | +1.17 | +0.18 | -0.80 | +0.88 | +0.36 | -0.98 | +0.35 | +0.82 |

Relative (%) | -28.9 | -39.5 | +41.5 | +42.2 | +6.6 | -29.0 | +31.6 | +12.9 | -35.3 | +12.7 | +29.8 | |

Steps (reduced) |
686 (253) |
1005 (139) |
1216 (350) |
1373 (74) |
1498 (199) |
1602 (303) |
1692 (393) |
1770 (38) |
1839 (107) |
1902 (170) |
1959 (227) |

### Subsets and supersets

433edo is the 84th prime edo.

## Regular temperament properties

Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
---|---|---|---|---|---|

Absolute (¢) | Relative (%) | ||||

2.3 | [-686 433⟩ | [⟨433 686]] | 0.2525 | 0.2525 | 9.11 |

2.3.5 | 2109375/2097152, [-29 52 -23⟩ | [⟨433 686 1005]] | 0.3254 | 0.2306 | 8.32 |

2.3.5.7 | 19683/19600, 4096000/4084101, 2109375/2097152 | [⟨433 686 1005 1216]] | 0.1414 | 0.3759 | 13.56 |

2.3.5.7.11 | 3025/3024, 6250/6237, 30375/30184, 180224/180075 | [⟨433 686 1005 1216 1498]] | 0.1026 | 0.3451 | 12.45 |

2.3.5.7.11.13 | 2080/2079, 625/624, 3025/3024, 18954/18865, 41472/41405 | [⟨433 686 1005 1216 1498 1602]] | 0.1217 | 0.3179 | 11.47 |

2.3.5.7.11.13.17 | 2080/2079, 375/374, 715/714, 936/935, 1377/1372, 76032/75803 | [⟨433 686 1005 1216 1498 1602 1770]] | 0.0919 | 0.3033 | 10.94 |

### Rank-2 temperaments

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

1 | 98\433 | 271.594 | 75/64 | Orson |

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

## Music

*sleeping as we don't know*(2023)