40ed7/3

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40ed7/3

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Prime factorization	23 × 5
Step size	36.6718¢
Octave	33\40ed7/3 (1210.17¢)
Twelfth	52\40ed7/3 (1906.93¢) (→13\10ed7/3)
Consistency limit	3
Distinct consistency limit	3

40 equal divisions of 7/3 (abbreviated 40ed7/3) is a nonoctave tuning system that divides the interval of 7/3 into 40 equal parts of about 36.7⁠ ⁠¢ each. Each step represents a frequency ratio of (7/3)^1/40, or the 40th root of 7/3.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	36.7
2	73.3	22/21, 23/22, 26/25
3	110
4	146.7	25/23
5	183.4	10/9, 29/26
6	220	17/15, 25/22
7	256.7	22/19, 29/25
8	293.4	13/11
9	330	17/14, 23/19
10	366.7	21/17, 26/21
11	403.4	19/15, 29/23
12	440.1	9/7, 22/17
13	476.7	25/19, 29/22
14	513.4	27/20
15	550.1	26/19
16	586.7	7/5
17	623.4	10/7
18	660.1	19/13, 22/15
19	696.8	3/2
20	733.4	23/15, 26/17, 29/19
21	770.1	14/9
22	806.8	27/17
23	843.5
24	880.1	5/3
25	916.8	17/10, 22/13
26	953.5	26/15
27	990.1	23/13, 30/17
28	1026.8
29	1063.5
30	1100.2	17/9
31	1136.8	25/13, 27/14, 29/15
32	1173.5
33	1210.2
34	1246.8
35	1283.5	21/10, 23/11
36	1320.2	15/7
37	1356.9
38	1393.5	29/13
39	1430.2
40	1466.9	7/3

Harmonics

Approximation of harmonics in 40ed7/3

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+10.2	+5.0	-16.3	+0.7	+15.1	+5.0	-6.2	+10.0	+10.9	-7.4	-11.4
	Relative (%)	+27.7	+13.6	-44.5	+2.0	+41.3	+13.6	-16.8	+27.1	+29.7	-20.2	-31.0
Steps (reduced)		33 (33)	52 (12)	65 (25)	76 (36)	85 (5)	92 (12)	98 (18)	104 (24)	109 (29)	113 (33)	117 (37)

Approximation of harmonics in 40ed7/3

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	-3.2	+15.1	+5.7	+4.0	+9.1	-16.5	-0.1	-15.6	+10.0	+2.8	-0.9
	Relative (%)	-8.8	+41.3	+15.6	+10.9	+24.7	-45.1	-0.4	-42.5	+27.1	+7.5	-2.3
Steps (reduced)		121 (1)	125 (5)	128 (8)	131 (11)	134 (14)	136 (16)	139 (19)	141 (21)	144 (24)	146 (26)	148 (28)