5edf
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| ← 4edf | 5edf | 6edf → |
5 equal divisions of the perfect fifth (abbreviated 5edf or 5ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 5 equal parts of about 140 ¢ each. Each step represents a frequency ratio of (3/2)1/5, or the 5th root of 3/2. It corresponds to 8.5476 edo.
Theory
5edf is close to the bleu generator chain and every second step of 17edo. 4 steps of 5edf is a fraction of a cent away to the seventh harmonic (which is 112/81 instead of 7/4 since the equave is 3/2), which is an extremely accurate approximation for the size of this scale.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +63.5 | +63.5 | -13.4 | +21.5 | -13.4 | +0.6 | +50.2 | -13.4 | -55.4 | +60.4 | +50.2 | +52.0 | +64.1 | -55.4 | -26.7 |
| Relative (%) | +45.2 | +45.2 | -9.5 | +15.3 | -9.5 | +0.4 | +35.7 | -9.5 | -39.4 | +43.0 | +35.7 | +37.0 | +45.6 | -39.4 | -19.0 | |
| Steps (reduced) |
9 (4) |
14 (4) |
17 (2) |
20 (0) |
22 (2) |
24 (4) |
26 (1) |
27 (2) |
28 (3) |
30 (0) |
31 (1) |
32 (2) |
33 (3) |
33 (3) |
34 (4) | |
Subsets and supersets
5edf is the 3rd prime edf, after 3edf and before 7edf.
Intervals
| # | Cents | Approximate ratios | Neptunian notation | |
|---|---|---|---|---|
| 0 | 0.0 | 1/1 | perfect unison | C |
| 1 | 140 | 13/12, 49/45 | augmented unison, minor second | C#, Db |
| 2 | 281 | 13/11, 20/17, 75/64 | major second, minor third | D, Eb |
| 3 | 421 | 14/11, 23/18 | major third, diminished fourth | E, Fb |
| 4 | 562 | 11/8, 18/13, 25/18 | perfect fourth | F |
| 5 | 702 | 3/2 | perfect fifth | C |
| 6 | 842 | 13/8, 18/11, 21/13 | augmented fifth, minor sixth | C#, Db |
| 7 | 983 | 7/4, 30/17 | major sixth, minor seventh | D, Eb |
| 8 | 1123 | 44/23 | major seventh, minor octave | E, Fb |
| 9 | 1264 | 83/40 | major octave | F |
| 10 | 1404 | 9/4 | major ninth | C |