# 398edo

← 397edo | 398edo | 399edo → |

**398 equal divisions of the octave** (abbreviated **398edo** or **398ed2**), also called **398-tone equal temperament** (**398tet**) or **398 equal temperament** (**398et**) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 398 equal parts of about 3.02 ¢ each. Each step represents a frequency ratio of 2^{1/398}, or the 398th root of 2.

## Theory

398edo is only consistent to the 5-odd-limit, though it has a reasonable approximation to the full 13-limit using the patent val, which tempers out 10976/10935, 65625/65536, 1500625/1492992, 102760448/102515625, 102942875/102036672, and 200120949/200000000 in the 7-limit; 3025/3024, 4000/3993, 6250/6237, 59290/59049, 117649/117128, and 131072/130977 in the 11-limit; and 625/624, 1575/1573, 2080/2079, 2200/2197, 4096/4095, and 4225/4224 in the 13-limit. It supports yarman I, bisupermajor and semiquindromeda.

### Prime harmonics

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

Error | Absolute (¢) | +0.00 | +0.56 | -0.38 | -0.99 | +0.44 | +0.68 | +0.57 | +0.98 | -1.14 | -1.44 | +0.69 |

Relative (%) | +0.0 | +18.5 | -12.7 | -32.7 | +14.6 | +22.5 | +19.0 | +32.5 | -37.8 | -47.6 | +23.0 | |

Steps (reduced) |
398 (0) |
631 (233) |
924 (128) |
1117 (321) |
1377 (183) |
1473 (279) |
1627 (35) |
1691 (99) |
1800 (208) |
1933 (341) |
1972 (380) |

### Subsets and supersets

Since 398 factors into 2 × 199, 398edo has 2edo and 199edo as its subsets.

## Regular temperament properties

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

Absolute (¢) | Relative (%) | ||||

2.3 | [631 -398⟩ | [⟨398 631]] | -0.1759 | 0.1759 | 5.83 |

2.3.5 | 390625000/387420489, [-53 10 16⟩ | [⟨398 631 924]] | -0.0622 | 0.2157 | 7.15 |

2.3.5.7 | 10976/10935, 65625/65536, 200120949/200000000 | [⟨398 631 924 1117]] | +0.0412 | 0.2588 | 8.58 |

2.3.5.7.11 | 3025/3024, 4000/3993, 10976/10935, 65625/65536 | [⟨398 631 924 1117 1377]] | +0.0075 | 0.2411 | 8.00 |

2.3.5.7.11.13 | 625/624, 1575/1573, 2080/2079, 2200/2197, 10976/10935 | [⟨398 631 924 1117 1377 1473]] | -0.0243 | 0.2313 | 7.67 |

### Rank-2 temperaments

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

1 | 5\398 | 15.08 | 126/125 | Yarman I |

1 | 183\398 | 551.76 | 11/8 | Emkay |

2 | 54\398 | 162.81 | 11/10 | Bisupermajor |

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