5edf
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Prime factorization
5 (prime)
Step size
140.391¢
Octave
9\5edf (1263.52¢)
Twelfth
14\5edf (1965.47¢)
Consistency limit
2
Distinct consistency limit
2
| ← 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
| # | Cents | Approximate Ratios | Neptunian Notation | |
|---|---|---|---|---|
| 0 | 0.0 | 1/1 | perfect unison | C |
| 1 | 140.4 | 13/12, 49/45 | augmented unison, minor second | C#, Db |
| 2 | 280.8 | 75/64, 20/17, 13/11 | major second, minor third | D, Eb |
| 3 | 421.2 | 14/11, 23/18 | major third, diminished fourth | E, Fb |
| 4 | 561.6 | 11/8, 18/13, 25/18 | perfect fourth | F |
| 5 | 702.0 | 3/2 | perfect fifth | C |
| 6 | 842.3 | 21/13, 13/8, 18/11 | augmented fifth, minor sixth | C#, Db |
| 7 | 982.7 | 7/4, 30/17 | major sixth, minor seventh | D, Eb |
| 8 | 1123.1 | major seventh, minor octave | E, Fb | |
| 9 | 1263.5 | major octave | F | |
| 10 | 1403.9 | C | ||
| 11 | 1544.3 | C#, Db | ||
| 12 | 1684.7 | D, Eb | ||
| 13 | 1825.1 | E | ||
| 14 | 1965.5 | F | ||
| 15 | 2105.9 | C | ||
| 16 | 2246.3 | C#, Db | ||
| 17 | 2386.6 | D | ||