72ed5

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72ed5

73ed5 →

Prime factorization	23 × 32
Step size	38.6988¢
Octave	31\72ed5 (1199.66¢) (semiconvergent)
Twelfth	49\72ed5 (1896.24¢)
Consistency limit	12
Distinct consistency limit	8

Division of the 5th harmonic into 72 equal parts (72ed5) is related to 31 edo, but with the 5/1 rather than the 2/1 being just. The octave is slightly compressed (about 0.3372 cents) and the step size is about 38.6988 cents. This tuning has a meantone fifth as the number of divisions of the 5th harmonic is multiple of 4.

Harmonics

Approximation of harmonics in 72ed5

Harmonic		2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Error	Absolute (¢)	-0.34	-5.71	-0.67	+0.00	-6.05	-2.03	-1.01	-11.43	-0.34	-10.55	-6.39	+9.83	-2.37	-5.71	-1.35
	Relative (%)	-0.9	-14.8	-1.7	+0.0	-15.6	-5.2	-2.6	-29.5	-0.9	-27.3	-16.5	+25.4	-6.1	-14.8	-3.5
Steps (reduced)		31 (31)	49 (49)	62 (62)	72 (0)	80 (8)	87 (15)	93 (21)	98 (26)	103 (31)	107 (35)	111 (39)	115 (43)	118 (46)	121 (49)	124 (52)

31edo for comparison:

Approximation of harmonics in 31edo

Harmonic		2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Error	Absolute (¢)	+0.00	-5.18	+0.00	+0.78	-5.18	-1.08	+0.00	-10.36	+0.78	-9.38	-5.18	+11.09	-1.08	-4.40	+0.00
	Relative (%)	+0.0	-13.4	+0.0	+2.0	-13.4	-2.8	+0.0	-26.8	+2.0	-24.2	-13.4	+28.6	-2.8	-11.4	+0.0
Steps (reduced)		31 (0)	49 (18)	62 (0)	72 (10)	80 (18)	87 (25)	93 (0)	98 (5)	103 (10)	107 (14)	111 (18)	115 (22)	118 (25)	121 (28)	124 (0)

Intervals

degree	cents value	corresponding JI intervals	comments
0	0.0000	exact 1/1
1	38.6988	46/45, 45/44
2	77.3976	23/22, 68/65, 22/21
3	116.0964	15/14	pseudo-16/15
4	154.7952	35/32, 23/21
5	193.4940	19/17, 85/76
6	232.1928	8/7
7	270.8916	76/65	pseudo-7/6
8	309.5904	55/46	pseudo-6/5
9	348.2892	11/9
10	386.9880	5/4
11	425.6868	23/18
12	464.3856	17/13
13	503.0844		pseudo-4/3
14	541.7832	175/128, 26/19
15	580.4820	7/5
16	619.1808	10/7
17	657.8796	19/13
18	696.5784		meantone fifth (pseudo-3/2)
19	735.2772	55/36, 26/17
20	773.9760	25/16, 36/23
21	812.6748	8/5
22	851.3736	85/52, 18/11
23	890.0724		pseudo-5/3
24	928.7712	65/38
25	967.4700	7/4
26	1006.1688	25/14
27	1044.8676	95/52, 64/35
28	1083.5664		pseudo-15/8
29	1122.2652	21/11, 65/34, 44/23
30	1160.9640	45/23
31	1199.6628	2/1
32	1238.3617	45/22
33	1277.0605	23/11
34	1315.7593
35	1354.4581	35/16
36	1393.1569	38/17, 85/38	meantone major second plus an octave
37	1431.8557	16/7
38	1470.5545
39	1509.2533	55/23
40	1547.9521	22/9
41	1586.6509	5/2
42	1625.3497	23/9
43	1664.0485	34/13
44	1702.7473		pseudo-8/3
45	1741.4461	175/64, 52/19
46	1780.1449	14/5
47	1818.8437	20/7
48	1857.5425	38/13
49	1896.2413		pseudo-3/1
50	1934.9401	55/18, 52/17
51	1973.6389	25/8
52	2012.3377	115/36, 16/5
53	2051.0365	85/26, 36/11
54	2089.7353		meantone major sixth plus an octave (pseudo-10/3)
55	2128.4341	65/19
56	2167.1329	7/2
57	2205.8317	25/7
58	2244.5305	95/26, 128/35
59	2283.2293		pseudo-15/4
60	2321.9281	65/17
61	2360.6269	90/23
62	2399.3257	4/1
63	2438.0245	45/11
64	2476.7233	46/11
65	2515.4221
66	2554.1209	35/8
67	2592.8197	76/17, 85/19
68	2631.5185	32/7
69	2670.2173	14/3
70	2708.9161	110/23
71	2747.6149	44/9
72	2786.3137	exact 5/1	just major third plus two octaves