5ed7

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5ed7

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Prime factorization	5 (prime)
Step size	673.765 ¢
Octave	2\5ed7 (1347.53 ¢) (semiconvergent)
Twelfth	3\5ed7 (2021.3 ¢) (semiconvergent)
Consistency limit	4
Distinct consistency limit	3

5 equal divisions of the 7th harmonic (abbreviated 5ed7) is a nonoctave tuning system that divides the interval of 7/1 into 5 equal parts of about 674 ¢ each. Each step represents a frequency ratio of 7^1/5, or the 5th root of 7.

Theory

5ed7 can be treated as an 11-limit exotemperament. It performs very well at approximating the 11-limit for its size, having under 25 relative cents of relative error on harmonics 2, 3, 5, 7 and 11. Of course, that's only treating things as relative. It still has roughly 100 cents of absolute error on all those harmonics except 7, and worst of all is the octave having 148 cents of absolute error. But this could still be theoretically interesting since it does a lot better than other sub-5edo equal temperaments.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	673.8	3/2, 10/7, 13/9, 19/13, 20/13
2	1347.5	9/4, 11/5, 13/6, 15/7, 19/9, 20/9, 21/10
3	2021.3	10/3, 13/4, 19/6
4	2695.1	14/3, 19/4
5	3368.8	7/1

Harmonics

Approximation of prime harmonics in 5ed7

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+148	+119	-91	+0	-109	+276	-189	+293	-38	+234	+119
	Relative (%)	+21.9	+17.7	-13.5	+0.0	-16.1	+40.9	-28.0	+43.4	-5.7	+34.8	+17.6
Steps (reduced)		2 (2)	3 (3)	4 (4)	5 (0)	6 (1)	7 (2)	7 (2)	8 (3)	8 (3)	9 (4)	9 (4)

Approximation of prime harmonics in 5ed7

Harmonic		37	41	43	47	53	59	61	67	71	73	79
Error	Absolute (¢)	-187	+309	+226	+72	-136	-322	+295	+132	+32	-16	-153
	Relative (%)	-27.8	+45.8	+33.6	+10.7	-20.2	-47.7	+43.7	+19.6	+4.7	-2.4	-22.7
Steps (reduced)		9 (4)	10 (0)	10 (0)	10 (0)	10 (0)	10 (0)	11 (1)	11 (1)	11 (1)	11 (1)	11 (1)

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