5edf

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← 4edf 5edf 6edf →
Prime factorization 5 (prime)
Step size 140.391 ¢ 
Octave 9\5edf (1263.52 ¢)
Twelfth 14\5edf (1965.47 ¢)
Consistency limit 3
Distinct consistency limit 3

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 (also known as 17ed4) 5edf has an extremely accurate approximation of the seventh harmonic for its size.

5edf is notable as a relatively basic and easy-to-use nonoctave system. Traditional harmony using major and minor triads is accessible in 5edf, although they are not 5-limit but rather septimal/undecimal in flavor. One must be wary of the 3/2-equivalence paradigm-there is no dominant, and major and minor triads, seventh chords, ninth chords, etc. are all merely voicings of major and minor dyads. Diminished chords also play a more important role than they do traditionally, as unlike the conventional triads, they are not equivalent to dyads, and are somewhat more consonant than in 12edo due to the laxer subtritone.

Harmonics

Approximation of harmonics in 5edf
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