42ed7/4

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42ed7/4

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Prime factorization	2 × 3 × 7
Step size	23.0673¢
Octave	52\42ed7/4 (1199.5¢) (→26\21ed7/4)
Twelfth	82\42ed7/4 (1891.52¢) (→41\21ed7/4)
Consistency limit	4
Distinct consistency limit	3

42 equal divisions of 7/4 (abbreviated 42ed7/4) is a nonoctave tuning system that divides the interval of 7/4 into 42 equal parts of about 23.1 ¢ each. Each step represents a frequency ratio of (7/4)^1/42, or the 42nd root of 7/4.

Intervals

Steps	Cents	Approximate Ratios
0	0	1/1
1	23.067
2	46.135
3	69.202	24/23, 26/25
4	92.269	20/19
5	115.336	15/14, 16/15
6	138.404
7	161.471	11/10, 23/21
8	184.538	29/26
9	207.606
10	230.673	8/7
11	253.74	22/19, 29/25
12	276.807	20/17
13	299.875	19/16
14	322.942
15	346.009	28/23
16	369.077
17	392.144	5/4
18	415.211	14/11, 19/15
19	438.278
20	461.346	17/13, 30/23
21	484.413	29/22
22	507.48
23	530.548	15/11, 19/14
24	553.615	11/8
25	576.682	7/5
26	599.749
27	622.817	10/7, 23/16
28	645.884	16/11, 29/20
29	668.951	22/15, 25/17, 28/19
30	692.019
31	715.086
32	738.153	23/15, 26/17, 29/19
33	761.22
34	784.288	11/7, 30/19
35	807.355	8/5
36	830.422
37	853.489	23/14
38	876.557
39	899.624
40	922.691	17/10, 29/17
41	945.759	19/11
42	968.826	7/4

Harmonics

Approximation of harmonics in 42ed7/4

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-0.5	-10.4	-1.0	+4.8	-10.9	-1.0	-1.5	+2.2	+4.3	+0.8	-11.4
	Relative (%)	-2.2	-45.2	-4.3	+20.9	-47.4	-4.3	-6.5	+9.5	+18.8	+3.4	-49.6
Steps (reduced)		52 (10)	82 (40)	104 (20)	121 (37)	134 (8)	146 (20)	156 (30)	165 (39)	173 (5)	180 (12)	186 (18)

Approximation of harmonics in 42ed7/4

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	+11.5	-1.5	-5.6	-2.0	+8.4	+1.7	+0.4	+3.8	-11.4	+0.3	-7.5
	Relative (%)	+49.7	-6.5	-24.3	-8.7	+36.3	+7.3	+1.5	+16.6	-49.6	+1.3	-32.4
Steps (reduced)		193 (25)	198 (30)	203 (35)	208 (40)	213 (3)	217 (7)	221 (11)	225 (15)	228 (18)	232 (22)	235 (25)