EDN

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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.

Correspondence of EDN to EDO

Tuning Equivalent edo Comment
2edn A stack of two major sixths
3edn 2edo
4edn Neither are equivalent with 3edo
5edn
6edn 4edo With a stretch
7edn 5edo
8edn Entirely misses 2/1, falling halfway between 5edo and 6edo
9edn 6edo With a considerable stretch
10edn 7edo
11edn Neither are equivalent to 8edo
12edn
13edn 9edo
14edn Neither are equivalent to 10edo
15edn
16edn 11edo
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
19edn 13edo Noticeably compressed
20edn 14edo Noticeably stretched
21edn Entirely misses 2/1, falling halfway between 14edo and 15edo
22edn Cannot be considered equivalent to 15edo
23edn 16edo
24edn 17edo Some equivalences can be spotted due to 17edo's fame but it's a heavy stretch amounting to 40%

Selected divisions

10-EDN

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

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)..