21ed5/4

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21ed5/4

22ed5/4 →

Prime factorization	3 × 7
Step size	18.3959¢
Octave	65\21ed5/4 (1195.73¢)
Twelfth	103\21ed5/4 (1894.78¢)
Consistency limit	4
Distinct consistency limit	2

21 equal divisions of 5/4 (abbreviated 21ed5/4) is a nonoctave tuning system that divides the interval of 5/4 into 21 equal parts of about 18.4 ¢ each. Each step represents a frequency ratio of (5/4)^1/21, or the 21st root of 5/4.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	18.4
2	36.8
3	55.2
4	73.6	23/22, 25/24
5	92	21/20, 22/21
6	110.4	15/14, 16/15
7	128.8	14/13
8	147.2	13/12
9	165.6	21/19, 23/21
10	184	10/9, 19/17
11	202.4	9/8
12	220.8	8/7
13	239.1	15/13
14	257.5	22/19
15	275.9	7/6, 13/11
16	294.3	25/21
17	312.7	6/5
18	331.1	23/19
19	349.5	16/13, 17/14
20	367.9
21	386.3	5/4

Harmonics

Approximation of harmonics in 21ed5/4

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-4.27	-7.18	-8.53	-8.53	+6.95	-2.38	+5.59	+4.04	+5.59	+6.15	+2.68
	Relative (%)	-23.2	-39.0	-46.4	-46.4	+37.8	-12.9	+30.4	+22.0	+30.4	+33.5	+14.6
Steps (reduced)		65 (2)	103 (19)	130 (4)	151 (4)	169 (1)	183 (15)	196 (7)	207 (18)	217 (7)	226 (16)	234 (3)

Approximation of harmonics in 21ed5/4

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	-7.12	-6.64	+2.68	+1.33	+6.75	-0.23	-1.85	+1.33	+8.84	+1.89	-1.49
	Relative (%)	-38.7	-36.1	+14.6	+7.2	+36.7	-1.2	-10.1	+7.2	+48.1	+10.3	-8.1
Steps (reduced)		241 (10)	248 (17)	255 (3)	261 (9)	267 (15)	272 (20)	277 (4)	282 (9)	287 (14)	291 (18)	295 (1)