5edf
| ← 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 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +63.5 | +63.5 | -13.4 | +21.5 | -13.4 | +0.6 | +50.2 | -13.4 | -55.4 | +60.4 | +50.2 |
| Relative (%) | +45.2 | +45.2 | -9.5 | +15.3 | -9.5 | +0.4 | +35.7 | -9.5 | -39.4 | +43.0 | +35.7 | |
| Steps (reduced) |
9 (4) |
14 (4) |
17 (2) |
20 (0) |
22 (2) |
24 (4) |
26 (1) |
27 (2) |
28 (3) |
30 (0) |
31 (1) | |
Intervals
| degree | cents value | octave-reduced cents value | approximate ratios | Neptunian notation | ||
|---|---|---|---|---|---|---|
| 0 | 1/1 | perfect unison | C | |||
| 1 | 140.391 | 13/12, 49/45 | augmented unison, minor second | C#, Db | ||
| 2 | 280.782 | 75/64, 20/17, 13/11 | major second, minor third | D, Eb | ||
| 3 | 421.173 | 14/11, 23/18 | major third, diminished fourth | E, Fb | ||
| 4 | 561.564 | 11/8, 18/13, 25/18 | perfect fourth | F | ||
| 5 | 701.955 | 3/2 | perfect fifth | C | ||
| 6 | 842.346 | 21/13, 13/8, 18/11 | augmented fifth, minor sixth | C#, Db | ||
| 7 | 982.737 | 7/4, 30/17 | major sixth, minor seventh | D, Eb | ||
| 8 | 1123.128 | major seventh, minor octave | E, Fb | |||
| 9 | 1263.519 | 63.519 | major octave | F | ||
| 10 | 1403.910 | 203.910 | C | |||
| 11 | 1544.301 | 344.301 | C#, Db | |||
| 12 | 1684.692 | 484.692 | D, Eb | |||
| 13 | 1825.083 | 625.083 | E | |||
| 14 | 1965.474 | 765.474 | F | |||
| 15 | 2105.865 | 905.865 | C | |||
| 16 | 2246.256 | 1046.256 | C#, Db | |||
| 17 | 2386.647 | 1186.647 | D | |||