4ed5

This page presents a topic of primarily mathematical interest. While it is derived from sound mathematical principles, its applications in terms of utility for actual music may be limited, highly contrived, or as yet unknown.

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4ed5

5ed5 →

Prime factorization	22
Step size	696.578 ¢
Octave	2\4ed5 (1393.16 ¢) (→ 1\2ed5)
Twelfth	3\4ed5 (2089.74 ¢) (semiconvergent)
Consistency limit	3
Distinct consistency limit	3
Special properties	Highly composite

4 equal divisions of the 5th harmonic (abbreviated 4ed5) is a nonoctave tuning system that divides the interval of 5/1 into 4 equal parts of about 697 ¢ each. Each step represents a frequency ratio of 5^1/4, or the 4th root of 5.

Theory

4ed5 is equivalent to the generator chain of quarter-comma meantone, and therefore, treated as a 3/2.5-subgroup temperament, it is just a chain of meantone fifths without octaves. By adding an independent dimension for 2, the tuning of quarter-comma meantone is attained.

Harmonics

Approximation of harmonics in 4ed5

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+193	+188	-310	+0	-316	+114	-117	-321	+193	+28	-122
	Relative (%)	+27.7	+27.0	-44.5	+0.0	-45.3	+16.4	-16.8	-46.1	+27.7	+4.0	-17.6
Steps (reduced)		2 (2)	3 (3)	3 (3)	4 (0)	4 (0)	5 (1)	5 (1)	5 (1)	6 (2)	6 (2)	6 (2)

Approximation of harmonics in 4ed5 (continued)

Harmonic		13	14	15	16	17	18	19	20	21	22	23	24
Error	Absolute (¢)	-261	+307	+188	+76	-29	-128	-221	-310	+302	+221	+144	+71
	Relative (%)	-37.5	+44.1	+27.0	+10.9	-4.1	-18.4	-31.8	-44.5	+43.3	+31.8	+20.7	+10.1
Steps (reduced)		6 (2)	7 (3)	7 (3)	7 (3)	7 (3)	7 (3)	7 (3)	7 (3)	8 (0)	8 (0)	8 (0)	8 (0)

Subsets and supersets

4ed5 is the first composite ed5, containing 2ed5 as the only nontrivial subset ed5.

Interval table

Steps	Cents	Approximate ratios
0	0	1/1, 21/20
1	696.6	3/2, 10/7, 11/7, 14/9, 17/11, 19/13
2	1393.2	7/3, 9/4, 11/5, 15/7, 20/9
3	2089.7	7/2, 10/3, 17/5
4	2786.3	5/1, 21/4