5edf

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5edf

6edf →

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

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

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		major seventh, minor octave	E, Fb
9	1264		major octave	F
10	1404	9/4		C