4ed6

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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4ed6

5ed6 →

Prime factorization	22
Step size	775.489 ¢
Octave	2\4ed6 (1550.98 ¢) (→ 1\2ed6)
Twelfth	2\4ed6 (1550.98 ¢) (→ 1\2ed6)
Consistency limit	2
Distinct consistency limit	2
Special properties	Highly composite

4 equal divisions of the 6th harmonic (abbreviated 4ed6) is a nonoctave tuning system that divides the interval of 6/1 into 4 equal parts of about 775 ¢ each. Each step represents a frequency ratio of 6^1/4, or the 4th root of 6.

4ed6 notably approximates the 23rd harmonic with incredible accuracy, less than 0.15 cents off.

Harmonics

Approximation of harmonics in 4ed6

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+351	-351	-74	+316	+0	-267	+277	+74	-109	-274	+351
	Relative (%)	+45.3	-45.3	-9.5	+40.7	+0.0	-34.4	+35.8	+9.5	-14.0	-35.3	+45.3
Steps (reduced)		2 (2)	2 (2)	3 (3)	4 (0)	4 (0)	4 (0)	5 (1)	5 (1)	5 (1)	5 (1)	6 (2)

Approximation of harmonics in 4ed6 (continued)

Harmonic		13	14	15	16	17	18	19	20	21	22	23	24
Error	Absolute (¢)	+212	+84	-35	-147	-252	-351	+331	+242	+158	+77	+0	-74
	Relative (%)	+27.4	+10.8	-4.6	-19.0	-32.5	-45.3	+42.7	+31.2	+20.3	+9.9	+0.0	-9.5
Steps (reduced)		6 (2)	6 (2)	6 (2)	6 (2)	6 (2)	6 (2)	7 (3)	7 (3)	7 (3)	7 (3)	7 (3)	7 (3)

Subsets and supersets

4ed6 is the first composite ed6, containing 2ed6 as the only nontrivial subset ed6.