# 119edo

← 118edo | 119edo | 120edo → |

**119 equal divisions of the octave** (abbreviated **119edo** or **119ed2**), also called **119-tone equal temperament** (**119tet**) or **119 equal temperament** (**119et**) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 119 equal parts of about 10.1 ¢ each. Each step represents a frequency ratio of 2^{1/119}, or the 119th root of 2.

## Theory

119edo is inconsistent in the 5-odd-limit, with both harmonics 3 and 5 falling halfway between steps. It does have potential as a 2.7.9.15 subgroup system. In higher limits, 2.7.15.29.37 is a strong interpretation.

Nonetheless, there is a number of mappings to be considered. In the 11-limit, 119edo's provides the optimal patent val for the 11-limit androboh and quasitemp temperaments. The patent val also tunes the 11-limit quadrawell temperament. 119c val tunes treecreeper, sensus, and senator as high as the 17-limit.

### Odd harmonics

Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | 25 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Error | Absolute (¢) | +3.93 | -3.12 | -0.76 | -2.23 | +3.30 | -3.55 | +0.81 | -4.12 | +5.01 | +3.17 | -3.06 | +3.84 |

Relative (%) | +38.9 | -30.9 | -7.5 | -22.1 | +32.8 | -35.2 | +8.0 | -40.8 | +49.7 | +31.4 | -30.4 | +38.1 | |

Steps (reduced) |
189 (70) |
276 (38) |
334 (96) |
377 (20) |
412 (55) |
440 (83) |
465 (108) |
486 (10) |
506 (30) |
523 (47) |
538 (62) |
553 (77) |

Harmonic | 27 | 29 | 31 | 33 | 35 | 37 | 39 | 41 | 43 | 45 | 47 | 49 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Error | Absolute (¢) | +1.70 | -1.01 | +4.54 | -2.85 | -3.88 | +0.76 | +0.37 | +4.55 | +2.77 | +4.73 | +0.04 | -1.52 |

Relative (%) | +16.8 | -10.0 | +45.1 | -28.3 | -38.5 | +7.5 | +3.7 | +45.1 | +27.4 | +46.9 | +0.4 | -15.0 | |

Steps (reduced) |
566 (90) |
578 (102) |
590 (114) |
600 (5) |
610 (15) |
620 (25) |
629 (34) |
638 (43) |
646 (51) |
654 (59) |
661 (66) |
668 (73) |

### Subsets and supersets

Since 119edo factors as 7 × 17, it contains 7edo and 17edo as a subset. Hence it supports circles of fifths of those respective equal temperaments.

## Scales

- Approximation of 2/7 comma meantone: 19 19 19 12 19 19 19 19 12
- Approximation of half comma eventone: 23 23 2 23 23 23 2, 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2