2ed5

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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2ed5

3ed5 →

Prime factorization	2 (prime)
Step size	1393.16 ¢
Octave	1\2ed5 (1393.16 ¢) (convergent)
Twelfth	1\2ed5 (1393.16 ¢) (semiconvergent)
Consistency limit	2
Distinct consistency limit	2
Special properties	Highly composite

2 equal divisions of the 5th harmonic (abbreviated 2ed5) is a nonoctave tuning system that divides the interval of 5/1 into 2 equal parts of about 1390 ¢ each. Each step represents a frequency ratio of 5^1/2, or the 2nd root of 5.

Intervals

Steps	Cents	Approximate Ratios
0	0	1/1
1	1393.157	9/4
2	2786.314	5/1

Harmonics

Approximation of harmonics in 2ed5

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+193	-509	+386	+0	-316	-583	+579	+376	+193	+28	-122
	Relative (%)	+13.9	-36.5	+27.7	+0.0	-22.7	-41.8	+41.6	+27.0	+13.9	+2.0	-8.8
Steps (reduced)		1 (1)	1 (1)	2 (0)	2 (0)	2 (0)	2 (0)	3 (1)	3 (1)	3 (1)	3 (1)	3 (1)

Approximation of harmonics in 2ed5

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	-261	-389	-509	-621	+668	+569	+475	+386	+302	+221	+144
	Relative (%)	-18.7	-27.9	-36.5	-44.5	+47.9	+40.8	+34.1	+27.7	+21.7	+15.9	+10.4
Steps (reduced)		3 (1)	3 (1)	3 (1)	3 (1)	4 (0)	4 (0)	4 (0)	4 (0)	4 (0)	4 (0)	4 (0)