3ed5

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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3ed5

4ed5 →

Prime factorization	3 (prime)
Step size	928.771 ¢
Octave	1\3ed5 (928.771 ¢) (semiconvergent)
Twelfth	2\3ed5 (1857.54 ¢) (convergent)
Consistency limit	3
Distinct consistency limit	2

3 equal divisions of the 5th harmonic (abbreviated 3ed5) is a nonoctave tuning system that divides the interval of 5/1 into 3 equal parts of about 929 ¢ each. Each step represents a frequency ratio of 5^1/3, or the cube root of 5.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1, 14/13, 15/14, 18/17, 19/18, 20/19
1	928.8	5/3, 9/5, 11/6, 17/10, 18/11, 19/11, 19/12, 20/11
2	1857.5	3/1, 11/4, 14/5, 17/6
3	2786.3	5/1, 14/3, 19/4

Harmonics

Approximation of harmonics in 3ed5

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-271	-44	+386	+0	-316	+346	+115	-89	-271	-436	+342
	Relative (%)	-29.2	-4.8	+41.6	+0.0	-34.0	+37.3	+12.4	-9.6	-29.2	-47.0	+36.8
Steps (reduced)		1 (1)	2 (2)	3 (0)	3 (0)	3 (0)	4 (1)	4 (1)	4 (1)	4 (1)	4 (1)	5 (2)

Approximation of harmonics in 3ed5

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	+203	+75	-44	-156	-261	-360	-454	+386	+302	+221	+144
	Relative (%)	+21.9	+8.1	-4.8	-16.8	-28.1	-38.8	-48.8	+41.6	+32.5	+23.8	+15.5
Steps (reduced)		5 (2)	5 (2)	5 (2)	5 (2)	5 (2)	5 (2)	5 (2)	6 (0)	6 (0)	6 (0)	6 (0)