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 | e1/10 | 11/10 | flat whole tone |
| 2 | 346.25 | e1/5 | 11/9 | neutral third |
| 3 | 519.37 | e3/10 | 43/32 | sharp fourth |
| 4 | 692.49 | e2/5 | 3/2 | flat fifth |
| 5 | 865.62 | e1/2 | 5/3 | flat major sixth |
| 6 | 1038.74 | e3/5 | 117/64 | neutral seventh |
| 7 | 1211.86 | e7/10 | 2/1 | stretched octave |
| 8 | 1384.99 | e4/5 | 20/9 | flat major ninth |
| 9 | 1558.11 | e9/10 | 22/9 | neutral tenth |
| 10 | 1731.23 | e | 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 (1222.05 cents; essentially double a Pythagorean large tritone) and gives it a rather pleasant sharp fifth of 712.86 cents.
24-EDN
24-EDN has third tones so far sharp of 17-EDO that it becomes a stretched 50-ED8 (50\24 is 2406.74 cents). However, 43\24 is essentially the 6th harmonic (1514.83+1586.965=3101.79 cents)..