EDe
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This page presents a topic of primarily mathematical interest.
While it is derived from sound mathematical principles, its applications in terms of utility for actual music may be limited and highly contrived. |
Equal division of the natave (EDe or EDN) is the equal division of the constant e (where e is treated as a musical interval in the same way as 2 is an octave or 1.5 is a perfect fifth).
e is of particular interest because of its relationship with logarithms, given the fact that pitch is perceived logarithmically, and the fact that equal divisions are logarithmic.
Correspondence of EDe to EDO
| Tuning | Equivalent edo | Comment |
|---|---|---|
| 2EDe | A stack of two major sixths | |
| 3EDe | 2edo | |
| 4EDe | Neither are equivalent with 3edo | |
| 5EDe | ||
| 6EDe | 4edo | With a stretch |
| 7EDe | 5edo | |
| 8EDe | Entirely misses 2/1, falling halfway between 5edo and 6edo | |
| 9EDe | 6edo | With a considerable stretch |
| 10EDe | 7edo | |
| 11EDe | Neither are equivalent to 8edo | |
| 12EDe | ||
| 13EDe | 9edo | |
| 14EDe | Neither are equivalent to 10edo | |
| 15EDe | ||
| 16EDe | 11edo | |
| 17EDe | 12edo | With a noticeable stretch, given the dominance of 12edo this is more likely to sound like out of tune 12edo than it's own tuning |
| 18EDe | Entirely misses 2/1, falling halfway between 12 and 13edo | |
| 19EDe | 13edo | Noticeably compressed |
| 20EDe | 14edo | Noticeably stretched |
| 21EDe | Entirely misses 2/1, falling halfway between 14edo and 15edo | |
| 22EDe | Cannot be considered equivalent to 15edo | |
| 23EDe | 16edo | |
| 24EDe | 17edo | Some equivalences can be spotted due to 17edo's fame but it's a heavy stretch amounting to 40% |
Zeta function and tuning
In Gene’s black magic formulas, it is mathematically more "natural" to consider the number of divisions to the natave rather than the octave, thus scaling the graph of |Z(x)| horizontally by a factor of 1 instead of 1/ln(2).
The sequence of non-stretched zeta peak EDe's are 1, 2, 3, 10, 20, 36, 39, 72, 111, 163, 202, 264, 466, 538, 740, 1349, 1887... corresponding to 1, 1, 2, 7, 14, 25, 27, 50, 77, 113, 140, 183, 323, 373, 513, 935, 1308... edos.
Selected divisions
10-EDe
| 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-EDe 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.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +11.9 | +2.4 | +23.7 | -16.3 | +14.3 | -79.5 | +35.6 | +4.8 | -4.5 | +3.6 | +26.1 |
| Relative (%) | +6.9 | +1.4 | +13.7 | -9.4 | +8.2 | -45.9 | +20.6 | +2.8 | -2.6 | +2.1 | +15.1 | |
| Steps (reduced) |
7 (7) |
11 (1) |
14 (4) |
16 (6) |
18 (8) |
19 (9) |
21 (1) |
22 (2) |
23 (3) |
24 (4) |
25 (5) | |
17-EDe
17-EDe 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.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +22.0 | +33.0 | +44.1 | -36.7 | -46.8 | -8.2 | -35.7 | -35.9 | -14.7 | +24.0 | -24.8 |
| Relative (%) | +21.7 | +32.4 | +43.3 | -36.0 | -46.0 | -8.0 | -35.0 | -35.3 | -14.4 | +23.6 | -24.3 | |
| Steps (reduced) |
12 (12) |
19 (2) |
24 (7) |
27 (10) |
30 (13) |
33 (16) |
35 (1) |
37 (3) |
39 (5) |
41 (7) |
42 (8) | |
20-EDe
20-EDe is a doubling of 10-EDe with intervals closer to semitones.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +11.9 | +2.4 | +23.7 | -16.3 | +14.3 | +7.1 | +35.6 | +4.8 | -4.5 | +3.6 | +26.1 |
| Relative (%) | +13.7 | +2.8 | +27.4 | -18.9 | +16.5 | +8.2 | +41.1 | +5.6 | -5.2 | +4.2 | +30.2 | |
| Steps (reduced) |
14 (14) |
22 (2) |
28 (8) |
32 (12) |
36 (16) |
39 (19) |
42 (2) |
44 (4) |
46 (6) |
48 (8) |
50 (10) | |
24-EDe
24-EDe has third tones so far sharp of 17-EDO that it becomes a stretched 50-ED8 (50\24 is 3606.74 cents). However, 43\24 is essentially the 6th harmonic (1514.83+1586.965=3101.79 cents).
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +26.3 | -26.5 | -19.6 | +26.9 | -0.2 | +21.5 | +6.7 | +19.2 | -18.9 | +32.5 | +26.1 |
| Relative (%) | +36.4 | -36.7 | -27.1 | +37.4 | -0.2 | +29.8 | +9.3 | +26.7 | -26.2 | +45.1 | +36.2 | |
| Steps (reduced) |
17 (17) |
26 (2) |
33 (9) |
39 (15) |
43 (19) |
47 (23) |
50 (2) |
53 (5) |
55 (7) |
58 (10) |
60 (12) | |