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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.


Intervals of 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 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 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)..