6ed6

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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6ed6

7ed6 →

Prime factorization	2 × 3
Step size	516.993 ¢
Octave	2\6ed6 (1033.99 ¢) (→ 1\3ed6)
Twelfth	4\6ed6 (2067.97 ¢) (→ 2\3ed6)
Consistency limit	2
Distinct consistency limit	2
Special properties	Highly composite

6 equal divisions of the 6th harmonic (abbreviated 6ed6) is a nonoctave tuning system that divides the interval of 6/1 into 6 equal parts of about 517 ¢ each. Each step represents a frequency ratio of 6^1/6, or the 6th root of 6.

Theory

6ed6 corresponds to 2.3211…edo. It is the generator chain for the harry temperament.

Harmonics

Approximation of harmonics in 6ed6

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-166	+166	+185	-201	+0	+250	+19	-185	+150	-15	-166
	Relative (%)	-32.1	+32.1	+35.8	-38.9	+0.0	+48.4	+3.7	-35.8	+28.9	-3.0	-32.1
Steps (reduced)		2 (2)	4 (4)	5 (5)	5 (5)	6 (0)	7 (1)	7 (1)	7 (1)	8 (2)	8 (2)	8 (2)

Approximation of harmonics in 6ed6

Harmonic		13	14	15	16	17	18	19	20	21	22	23	24
Error	Absolute (¢)	+212	+84	-35	-147	-252	+166	+72	-16	-101	-181	-258	+185
	Relative (%)	+41.1	+16.3	-6.8	-28.4	-48.7	+32.1	+14.0	-3.2	-19.5	-35.1	-50.0	+35.8
Steps (reduced)		9 (3)	9 (3)	9 (3)	9 (3)	9 (3)	10 (4)	10 (4)	10 (4)	10 (4)	10 (4)	10 (4)	11 (5)

Subsets and supersets

Since 6 factors into primes as 2 × 3, 6ed6 contains 2ed6 and 3ed6 as subset ed6's.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	517	15/11, 18/13, 19/14
2	1034	11/6, 13/7
3	1551	5/2, 12/5
4	2068	17/5
5	2585	13/3
6	3102	6/1