# EDN

Equal divisions of the natave, which is the mathematical constant e used as a musical interval. e is of particular interest because of its relationship with logarithms, the fact that pitch is perceived logarithmically, and the fact that equal divisions are logarithmic.

## 10-EDN

Step | Cents | Ratio | JI approximation(s) | Interval |
---|---|---|---|---|

0 | 0.0 | 1/1 | 1/1 | unison |

1 | 173.12 | e^(1/10) | 11/10 | flat whole tone |

2 | 346.25 | e^(1/5) | 11/9 | neutral third |

3 | 519.37 | e^(3/10) | 43/32 | sharp fourth |

4 | 692.49 | e^(2/5) | 3/2 | flat fifth |

5 | 865.62 | e^(1/2) | 5/3 | flat major sixth |

6 | 1038.74 | e^(3/5) | 117/64 | neutral seventh |

7 | 1211.86 | e^(7/10) | 2/1 | stretched octave |

8 | 1384.99 | e^(4/5) | 20/9 | flat major ninth |

9 | 1558.11 | e^(9/10) | 22/9 | neutral tenth |

10 | 1731.23 | e/1 | 43/16 | natave |

Beyond the natave, some particularly pleasant JI intervals can be found: 11\10 is only 2 cents sharp from 3/1; 13\10 is very close to 11/2; and 23\10 is very close to 10/1. This last approximation in particular makes this equal division almost equivalent to 23-ed(10/1).

10-EDN is similar to 7-EDO in that its step size is roughly 1/7 of an octave, therefore roughly corresponding to the diatonic scale, but with warped, equal-size steps. However, the octave is stretched, which simultaneously helps the extremely flat fifth of 7-EDO.

20-EDN is a doubling of 10-EDN with intervals closer to semitones.

## 17-EDN

17-EDN is very close to 12-EDO but with slightly sharp semitones (101.84 cents). This causes the octave to be far too sharp (1222 cents) and gives it a rather pleasant sharp fifth of 712 cents.