4ed5

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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