62edo

← 61edo

62edo

63edo →

Prime factorization	2 × 31
Step size	19.3548 ¢
Fifth	36\62 (696.774 ¢) (→ 18\31)
Semitones (A1:m2)	4:6 (77.42 ¢ : 116.1 ¢)
Consistency limit	7
Distinct consistency limit	7

Theory

62 = 2 × 31 and the patent val is a contorted 31edo through the 11-limit; in the 13-limit it tempers out 169/168, 1188/1183, 847/845 and 676/675. It provides the optimal patent val for gallium, semivalentine and hemimeantone temperaments.

Using the 35\62 generator, which leads to the ⟨62 97 143 173] val, 62edo is also an excellent tuning for septimal mavila temperament; alternatively ⟨62 97 143 172] supports hornbostel.

Relation to a calendar reform

62 years is the amount of years in a leap week calendar cycle which corresponds to a year of 365 days 5 hours 48 minutes 23 seconds, meaning it is both a simple cycle for a calendar, and 62 being a multiple of 31 makes it a harmonically useful and playable cycle. The corresponding maximal evenness scales are 15 & 62 and 11 & 62.

11 & 62 is best interpreted in the 2.9.7 subgroup, where it tempers out 44957696/43046721, and the three generators of 17\62 correspond to 16/9. It's possible to extend this to the 11-limit with comma basis {896/891, 1331/1296}, where 17\62 is mapped to 11/9 and two of them make a 16/11. In addition, three generators make the patent val 9/8, which is also created by combining the flat patent val fifth from 31edo with the sharp 37\62 fifth.

The 15 & 62 temperament, corresponding to the leap day cycle, is just contorted valentine, order 2.

Odd harmonics

Approximation of odd harmonics in 62edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	-5.18	+0.78	-1.08	+8.99	-9.38	-8.27	-4.40	-8.18	-7.19	-6.26	-8.92
	Relative (%)	-26.8	+4.0	-5.6	+46.5	-48.5	-42.7	-22.7	-42.3	-37.2	-32.4	-46.1
Steps (reduced)		98 (36)	144 (20)	174 (50)	197 (11)	214 (28)	229 (43)	242 (56)	253 (5)	263 (15)	272 (24)	280 (32)

Intervals

Armodue Nomenclature 8;3 Relation
Ɨ = Thick (1/8-tone up) ‡ = Semisharp (1/4-tone up) b = Flat (5/8-tone down) ◊ = Node (sharp/flat blindspot 1/2-tone) # = Sharp (5/8-tone up) v = Semiflat (1/4-tone down) ⌐ = Thin (1/8-tone down)

#	Cents	Armodue notation	Approximate intervals
0	0.000	1
1	19.355	1Ɨ	90/89
2	38.710	1‡ (9#)	45/44
3	58.065	2b	30/29
4	77.419	1◊2	23/22
5	96.774	1#	37/35, 18/17, 19/18
6	116.129	2v	31/29, 15/14, 16/15
7	135.484	2⌐	27/25, 13/12, 14/13
8	154.839	2	12/11
9	174.194	2Ɨ	11/10
10	193.548	2‡	19/17, 9/8, 10/9
11	212.903	3b	17/15, 9/8
12	232.258	2◊3	8/7
13	251.613	2#	15/13
14	270.968	3v	7/6
15	290.323	3⌐
16	309.677	3	6/5
17	329.032	3Ɨ
18	348.387	3‡	11/9
19	367.742	4b	·
20	387.097	3◊4	5/4
21	406.452	3#
22	425.806	4v (5b)
23	445.161	4⌐
24	464.516	4
25	483.871	4Ɨ (5v)
26	503.226	5⌐ (4‡)	4/3
27	522.581	5	·
28	541.935	5Ɨ
29	561.290	5‡ (4#)
30	580.645	6b	7/5
31	600.000	5◊6
32	619.355	5#	10/7
33	638.710	6v
34	658.065	6⌐
35	677.419	6	·
36	696.774	6Ɨ	3/2
37	716.129	6‡
38	735.484	7b
39	754.839	6◊7
40	774.194	6#
41	793.548	7v
42	812.903	7⌐	8/5
43	832.258	7	·
44	851.613	7Ɨ	18/11
45	870.968	7‡
46	890.323	8b	5/3
47	909.677	7◊8
48	929.032	7#	12/7
49	948.387	8v	26/15
50	967.742	8⌐	7/4
51	987.097	8	16/9
52	1006.452	8Ɨ
53	1025.806	8‡
54	1045.161	9b
55	1064.516	8◊9
56	1083.871	8#
57	1103.226	9v (1b)
58	1122.581	9⌐
59	1141.936	9
60	1161.290	9Ɨ (1v)
61	1180.645	1⌐ (9‡)
62	1200.000	1

Regular temperament properties

62edo is contorted 31edo through the 11-limit.

Rank-2 temperaments

Periods per 8ve	Generator (Reduced)	Cents (Reduced)	Associated Ratio	Temperaments
1	13\62	251.61	15/13	Hemimeantone
1	17\62	329.03	16/11	Mabon
2	4\62	77.42	21/20	Semivalentine
31	1\62	19.35	196/195	Kumhar
31	1\62	19.35	16807/16640	Gallium