21ed5

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21ed5

22ed5 →

Prime factorization	3 × 7
Step size	132.682¢
Octave	9\21ed5 (1194.13¢) (→3\7ed5)
Twelfth	14\21ed5 (1857.54¢) (→2\3ed5)
Consistency limit	8
Distinct consistency limit	6

21 equal divisions of the 5th harmonic (abbreviated 21ed5) is a nonoctave tuning system that divides the interval of 5/1 into 21 equal parts of about 133 ¢ each. Each step represents a frequency ratio of 5^1/21, or the 21st root of 5.

21ed5 is a strong 2.5.17.23 tuning. It is also a strong no-3s, no-7s, no-11s, no-19s 37-limit tuning.

Intervals

Steps	Cents	Approximate Ratios
0	0	1/1
1	132.682	13/12, 14/13, 15/14, 25/23
2	265.363	7/6, 22/19
3	398.045	5/4, 19/15, 24/19
4	530.726	15/11, 19/14, 23/17
5	663.408	16/11, 19/13, 22/15, 25/17
6	796.09	11/7, 19/12
7	928.771	12/7, 17/10
8	1061.453	11/6, 13/7, 24/13
9	1194.134	2/1
10	1326.816	13/6, 15/7
11	1459.498	7/3
12	1592.179	5/2
13	1724.861	19/7
14	1857.542
15	1990.224	19/6, 22/7
16	2122.906	17/5, 24/7
17	2255.587	11/3
18	2388.269	4/1
19	2520.951
20	2653.632	14/3, 23/5
21	2786.314	5/1

Harmonics

Approximation of prime harmonics in 21ed5

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	-5.9	-44.4	+0.0	-51.8	-38.2	-62.0	+4.3	-55.6	+11.7	+8.4	+25.6
	Relative (%)	-4.4	-33.5	+0.0	-39.0	-28.8	-46.8	+3.2	-41.9	+8.8	+6.3	+19.3
Steps (reduced)		9 (9)	14 (14)	21 (0)	25 (4)	31 (10)	33 (12)	37 (16)	38 (17)	41 (20)	44 (2)	45 (3)

Approximation of prime harmonics in 21ed5

Harmonic		37	41	43	47	53	59	61	67	71	73	79
Error	Absolute (¢)	-15.3	-60.3	-10.1	-31.4	+25.9	-27.0	+47.9	+18.2	+50.5	+2.4	-1.7
	Relative (%)	-11.5	-45.5	-7.6	-23.7	+19.5	-20.4	+36.1	+13.7	+38.0	+1.8	-1.3
Steps (reduced)		47 (5)	48 (6)	49 (7)	50 (8)	52 (10)	53 (11)	54 (12)	55 (13)	56 (14)	56 (14)	57 (15)