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.
Correspondence of EDN to EDO
|2edn||A stack of two major sixths|
|4edn||Neither are equivalent with 3edo|
|6edn||4edo||With a stretch|
|8edn||Entirely misses 2/1, falling halfway between 5edo and 6edo|
|9edn||6edo||With a considerable stretch|
|11edn||Neither are equivalent to 8edo|
|14edn||Neither are equivalent to 10edo|
|17edn||12edo||With a noticeable stretch, given the dominance of 12edo this is more likely to sound like like out of tune 12edo than it's own tuning|
|18edn||Entirely misses 2/1, falling halfway between 12 and 13edo|
|21edn||Entirely misses 2/1, falling halfway between 14edo and 15edo|
|22edn||Cannot be considered equivalent to 15edo|
|24edn||17edo||Some equivalences can be spotted due to 17edo's fame but it's a heavy stretch amounting to 40%|
|1||173.12||e1/10||11/10||flat whole tone|
|5||865.62||e1/2||5/3||flat major sixth|
|8||1384.99||e4/5||20/9||flat major ninth|
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)..