40ed7/4

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40ed7/4

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Prime factorization	23 × 5
Step size	24.2206¢
Octave	50\40ed7/4 (1211.03¢) (→5\4ed7/4)
Twelfth	79\40ed7/4 (1913.43¢)
Consistency limit	3
Distinct consistency limit	3

40 equal divisions of 7/4 (abbreviated 40ed7/4) is a nonoctave tuning system that divides the interval of 7/4 into 40 equal parts of about 24.2 ¢ each. Each step represents a frequency ratio of (7/4)^1/40, or the 40th root of 7/4.

Intervals

Steps	Cents	Approximate Ratios
0	0	1/1
1	24.221
2	48.441
3	72.662	23/22, 26/25
4	96.883
5	121.103	15/14
6	145.324	25/23
7	169.545
8	193.765	29/26
9	217.986	17/15, 25/22, 26/23
10	242.206
11	266.427
12	290.648	13/11
13	314.868
14	339.089	17/14
15	363.31	21/17, 26/21
16	387.53
17	411.751
18	435.972
19	460.192
20	484.413	29/22
21	508.634
22	532.854
23	557.075	29/21
24	581.296	7/5
25	605.516
26	629.737
27	653.957	19/13
28	678.178
29	702.399	3/2
30	726.619
31	750.84
32	775.061
33	799.281
34	823.502
35	847.723
36	871.943
37	896.164
38	920.385	17/10, 29/17
39	944.605	19/11
40	968.826

Harmonics

Approximation of harmonics in 40ed7/4

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+11.0	+11.5	-2.2	-0.9	-1.7	-2.2	+8.9	-1.3	+10.1	-9.6	+9.3
	Relative (%)	+45.5	+47.4	-8.9	-3.9	-7.1	-8.9	+36.6	-5.2	+41.7	-39.6	+38.5
Steps (reduced)		50 (10)	79 (39)	99 (19)	115 (35)	128 (8)	139 (19)	149 (29)	157 (37)	165 (5)	171 (11)	178 (18)

Approximation of harmonics in 40ed7/4

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	-8.1	+8.9	+10.5	-4.3	+11.8	+9.8	-11.2	-3.1	+9.3	+1.4	-2.8
	Relative (%)	-33.6	+36.6	+43.5	-17.8	+48.9	+40.3	-46.1	-12.8	+38.5	+6.0	-11.8
Steps (reduced)		183 (23)	189 (29)	194 (34)	198 (38)	203 (3)	207 (7)	210 (10)	214 (14)	218 (18)	221 (21)	224 (24)