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.
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
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 | 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)..