5ed5

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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5ed5

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Prime factorization	5 (prime)
Step size	557.263 ¢
Octave	2\5ed5 (1114.53 ¢) (semiconvergent)
Twelfth	3\5ed5 (1671.79 ¢)
Consistency limit	5
Distinct consistency limit	3

5 equal divisions of the 5th harmonic (abbreviated 5ed5) is a nonoctave tuning system that divides the interval of 5/1 into 5 equal parts of about 557 ¢ each. Each step represents a frequency ratio of 5^1/5, or the 5th root of 5.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	557.3	4/3, 7/5, 10/7, 11/8, 15/11, 19/14
2	1114.5	11/6, 13/7, 15/8, 19/10, 21/11
3	1671.8	8/3, 13/5, 19/7, 21/8
4	2229.1	7/2, 11/3, 15/4
5	2786.3	5/1

Harmonics

Approximation of harmonics in 5ed5

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-85	-230	-171	+0	+242	-25	-256	+97	-85	-250	+156
	Relative (%)	-15.3	-41.3	-30.7	+0.0	+43.4	-4.5	-46.0	+17.4	-15.3	-44.9	+28.0
Steps (reduced)		2 (2)	3 (3)	4 (4)	5 (0)	6 (1)	6 (1)	6 (1)	7 (2)	7 (2)	7 (2)	8 (3)

Approximation of harmonics in 5ed5

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	+18	-111	-230	+215	+110	+11	-82	-171	-255	+221	+144
	Relative (%)	+3.2	-19.9	-41.3	+38.6	+19.8	+2.1	-14.7	-30.7	-45.8	+39.7	+25.9
Steps (reduced)		8 (3)	8 (3)	8 (3)	9 (4)	9 (4)	9 (4)	9 (4)	9 (4)	9 (4)	10 (0)	10 (0)