8ed6

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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8ed6

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Prime factorization	23
Step size	387.744 ¢
Octave	3\8ed6 (1163.23 ¢) (semiconvergent)
Twelfth	5\8ed6 (1938.72 ¢) (semiconvergent)
Consistency limit	6
Distinct consistency limit	4

8 equal divisions of the 6th harmonic (abbreviated 8ed6) is a nonoctave tuning system that divides the interval of 6/1 into 8 equal parts of about 388 ¢ each. Each step represents a frequency ratio of 6^1/8, or the 8th root of 6.

Theory

8ed6 can be thought of as a subset (where the ~5/4 generator is stacked) of the 6/1-eigenmonzo tuning of würschmidt.

Harmonics

Approximation of harmonics in 8ed6

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-37	+37	-74	-72	+0	+121	-110	+74	-109	+114	-37
	Relative (%)	-9.5	+9.5	-19.0	-18.6	+0.0	+31.2	-28.4	+19.0	-28.1	+29.4	-9.5
Steps (reduced)		3 (3)	5 (5)	6 (6)	7 (7)	8 (0)	9 (1)	9 (1)	10 (2)	10 (2)	11 (3)	11 (3)

Intervals

#	Cents	Approximate JI ratio(s)
0	0	1/1
1	388	5/4
2	775	11/7, 25/16
3	1163
4	1551	22/9
5	1939	49/16
6	2326
7	2714	24/5
8	3102	6/1