40edf

← 39edf

40edf

41edf →

Prime factorization	23 × 5
Step size	17.5489¢
Octave	68\40edf (1193.32¢) (→17\10edf)
Twelfth	108\40edf (1895.28¢) (→27\10edf)
Consistency limit	2
Distinct consistency limit	2

40EDF is the equal division of the just perfect fifth into 40 parts of 17.5489 cents each, corresponding to 68.3805 edo.

It is related to the regular temperament which tempers out 2401/2400, 9801/9800, and 9453125/9437184 in the 11-limit, which is supported by 68edo, 274edo, 342edo, 410edo, 616edo, and 752edo among others.

Harmonics

Approximation of harmonics in 40edf

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-6.68	-6.68	+4.20	+3.96	+4.20	+0.56	-2.48	+4.20	-2.72	+7.77	-2.48
	Relative (%)	-38.0	-38.0	+23.9	+22.6	+23.9	+3.2	-14.1	+23.9	-15.5	+44.3	-14.1
Steps (reduced)		68 (28)	108 (28)	137 (17)	159 (39)	177 (17)	192 (32)	205 (5)	217 (17)	227 (27)	237 (37)	245 (5)

Approximation of harmonics in 40edf

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	-0.66	-6.12	-2.72	+8.39	+8.73	-2.48	-8.34	+8.15	-6.12	+1.09	-5.67
	Relative (%)	-3.8	-34.9	-15.5	+47.8	+49.7	-14.1	-47.5	+46.5	-34.9	+6.2	-32.3
Steps (reduced)		253 (13)	260 (20)	267 (27)	274 (34)	280 (0)	285 (5)	290 (10)	296 (16)	300 (20)	305 (25)	309 (29)

Intervals

Intervals of 40edf

degree	cents value	corresponding JI intervals	comments
0		exact 1/1
1	17.5489	100/99, 99/98, 81/80
2	35.09775	50/49, 49/48
3	52.6466	33/32
4	70.1955	25/24
5	87.7444	20/19
6	105.29325	17/16
7	122.8421	15/14
8	140.391	13/12
9	157.9399	23/21
10	175.48875	10/9
11	193.0376	19/17
12	210.5865	17/15
13	228.1354	8/7
14	245.68425	15/13
15	263.2331	7/6
16	280.782	20/17
17	298.3309	19/16
18	315.87975	6/5
19	333.4286	63/52
20	350.9775	60/49, 49/40
21	368.5264	26/21
22	386.07525	5/4
23	403.6241	24/19
24	421.173	51/40
25	438.7219	9/7
26	456.27075	13/10
27	473.8196	21/16
28	491.3685	4/3
29	508.9174	66/49
30	526.46625	200/147, 49/36, 27/20
31	544.0151	11/8
32	561.564	25/18
33	579.1129	7/5
34	596.66175	24/17
35	614.2106	10/7
36	631.7595	36/25
37	649.3084	16/11
38	666.85725	147/100, 72/49
39	684.4061	297/200, 49/33, 40/27
40	701.955	exact 3/2	just perfect fifth
41	719.5039	50/33, 297/196, 243/160, 32/21
42	737.05275	75/49, 49/32
43	754.6016	99/64
44	772.1505	25/16
45	789.6994	30/19
46	807.24825	51/32
47	824.7971	45/28
48	842.346	13/8
49	859.8949	32/14
50	877.44375	5/3
51	894.9926	57/34
52	912.5415	17/10
53	930.0904	12/7
54	947.63925	45/26
55	965.1981	7/4
56	982.737	30/17
57	1000.2859	57/32
58	1017.83275	9/5
59	1035.3836	189/104
60	1052.9325	90/49, 147/80
61	1070.4814	13/7
62	1088.03025	15/8
63	1105.4791	36/19
64	1123.128	153/80
65	1140.6769	27/14
66	1158.22575	39/20
67	1175.7746	63/32
68	1193.3235	2/1
69	1210.8724	99/49
70	1228.42125	300/147, 49/24, 81/40
71	1246.9701	33/16
72	1263.519	25/12
73	1281.0679	21/10
74	1298.61675	36/17
75	1316.1756	15/7
76	1333.7145	54/25
77	1351.2634	24/11
78	1368.81375	441/200, 108/49
79	1386.3611	891/400, 49/22, 20/9
80	1403.91	exact 9/8

Related regular temperaments

Adding one half of the octave as a generator, 40EDF leads the regular temperament which tempers out 2401/2400, 9801/9800, and 9453125/9437184 in the 11-limit.

11-limit 68&342

Commas: 2401/2400, 9801/9800, 9453125/9437184

POTE generator: ~99/98 = 17.545

Mapping: [<2 2 4 5 8|, <0 40 22 21 -37|]

EDOs: 68, 274, 342, 410, 616, 752

2.3.5.7.11.17 subgroup 68&342

Commas: 1089/1088, 1225/1224, 2401/2400, 24576/24565

POTE generator: ~99/98 = 17.546

Mapping: [<2 2 4 5 8 8|, <0 40 22 21 -37 6|]

EDOs: 68, 274, 342, 410, 616, 752

2.3.5.7.11.17.19 subgroup 68&342

Commas: 1089/1088, 1225/1224, 1445/1444, 1617/1615, 2401/2400

POTE generator: ~96/95 = 17.547

Mapping: [<2 2 4 5 8 8 8|, <0 40 22 21 -37 6 17|]

EDOs: 68, 274, 342, 410, 616, 752h

2.3.5.7.11.17.19.23 subgroup 68&342

Commas: 875/874, 1089/1088, 1225/1224, 1445/1444, 1617/1615, 2024/2023

POTE generator: ~96/95 = 17.546

Mapping: [<2 2 4 5 8 8 8 7|, <0 40 22 21 -37 6 17 70|]

EDOs: 68, 274, 342, 410, 616i, 752h