62edo

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

62 equal divisions of the octave (abbreviated 62edo or 62ed2), also called 62-tone equal temperament (62tet) or 62 equal temperament (62et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 62 equal parts of about 19.4 ¢ each. Each step represents a frequency ratio of 2^1/62, or the 62nd root of 2.

Theory

62 = 2 × 31 and the patent val is a contorted 31edo through the 11-limit, but it makes for a good tuning in the higher limits. In the 13-limit it tempers out 169/168, 1188/1183, 847/845 and 676/675; in the 17-limit 221/220, 273/272, and 289/288; in the 19-limit 153/152, 171/170, 209/208, 286/285, and 361/360. Unlike 31edo, which has a sharp profile for primes 13, 17, 19 and 23, 62edo has a flat profile for these, as it removes the distinction of otonal and utonal superparticular pairs of the primes (e.g. 13/12 vs 14/13 for prime 13) by tempering out the corresponding square-particulars. Interestingly, the relative size differences of consecutive harmonics are well preserved for all first 24 harmonics, and 62edo is one of the few meantone edos that achieve this, great for those who seek higher-limit meantone harmony.

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.

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
Error	relative (%)	-27	+4	-6	+46	-48	-43	-23	-42	-37	-32	-46
Steps (reduced)		98 (36)	144 (20)	174 (50)	197 (11)	214 (28)	229 (43)	242 (56)	253 (5)	263 (15)	272 (24)	280 (32)

Miscellaneous properties

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.

The 11 & 62 temperament in the 2.9.7 subgroup tempers out 44957696/43046721, and the three generators of 17\62 correspond to 16/9. It is 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 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 an unnamed extension to valentine in the 13-limit.

Intervals

Steps	Cents	Ups and downs notation	Approximate ratios
0	0	D	1/1
1	19.3548	↑D, ↓E♭♭	65/64, 66/65, 78/77
2	38.7097	↑↑D, E♭♭	45/44, 49/48, 50/49, 55/54, 56/55, 77/75
3	58.0645	↑³D, ↓³E♭	27/26, 65/63
4	77.4194	D♯, ↓↓E♭	21/20, 22/21, 25/24
5	96.7742	↑D♯, ↓E♭	52/49, 55/52
6	116.129	↑↑D♯, E♭	15/14, 16/15, 77/72
7	135.484	↑³D♯, ↓³E	13/12, 14/13
8	154.839	D𝄪, ↓↓E	12/11, 35/32, 49/45
9	174.194	↑D𝄪, ↓E	72/65
10	193.548	E	28/25, 49/44, 55/49
11	212.903	↑E, ↓F♭	44/39
12	232.258	↑↑E, F♭	8/7, 55/48, 63/55
13	251.613	↑³E, ↓³F	15/13, 52/45, 65/56
14	270.968	E♯, ↓↓F	7/6, 75/64
15	290.323	↑E♯, ↓F	13/11, 77/65
16	309.677	F	6/5
17	329.032	↑F, ↓G♭♭	63/52
18	348.387	↑↑F, G♭♭	11/9, 27/22, 49/40, 60/49
19	367.742	↑³F, ↓³G♭	26/21
20	387.097	F♯, ↓↓G♭	5/4
21	406.452	↑F♯, ↓G♭	33/26
22	425.806	↑↑F♯, G♭	32/25, 77/60
23	445.161	↑³F♯, ↓³G
24	464.516	F𝄪, ↓↓G	21/16, 55/42, 64/49, 72/55
25	483.871	↑F𝄪, ↓G	65/49
26	503.226	G	4/3, 75/56
27	522.581	↑G, ↓A♭♭	65/48
28	541.935	↑↑G, A♭♭	15/11, 48/35
29	561.29	↑³G, ↓³A♭	18/13
30	580.645	G♯, ↓↓A♭	7/5
31	600	↑G♯, ↓A♭	55/39, 78/55
32	619.355	↑↑G♯, A♭	10/7, 63/44, 77/54
33	638.71	↑³G♯, ↓³A	13/9, 75/52
34	658.065	G𝄪, ↓↓A	22/15, 35/24
35	677.419	↑G𝄪, ↓A	65/44, 77/52
36	696.774	A	3/2
37	716.129	↑A, ↓B♭♭
38	735.484	↑↑A, B♭♭	32/21, 49/32, 55/36, 75/49
39	754.839	↑³A, ↓³B♭	65/42
40	774.194	A♯, ↓↓B♭	25/16
41	793.548	↑A♯, ↓B♭	52/33
42	812.903	↑↑A♯, B♭	8/5, 77/48
43	832.258	↑³A♯, ↓³B	21/13
44	851.613	A𝄪, ↓↓B	18/11, 44/27, 49/30, 80/49
45	870.968	↑A𝄪, ↓B
46	890.323	B	5/3
47	909.677	↑B, ↓C♭	22/13
48	929.032	↑↑B, C♭	12/7, 75/44, 77/45
49	948.387	↑³B, ↓³C	26/15, 45/26
50	967.742	B♯, ↓↓C	7/4
51	987.097	↑B♯, ↓C	39/22
52	1006.45	C	25/14
53	1025.81	↑C, ↓D♭♭	65/36
54	1045.16	↑↑C, D♭♭	11/6, 64/35
55	1064.52	↑³C, ↓³D♭	13/7, 24/13
56	1083.87	C♯, ↓↓D♭	15/8, 28/15
57	1103.23	↑C♯, ↓D♭	49/26
58	1122.58	↑↑C♯, D♭	21/11, 40/21, 48/25
59	1141.94	↑³C♯, ↓³D	52/27
60	1161.29	C𝄪, ↓↓D	49/25, 55/28
61	1180.65	↑C𝄪, ↓D	65/33, 77/39
62	1200	D	2/1

Armodue notation

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.

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Absolute (¢)	Relative (%)
2.3.5.7.11.13	81/80, 99/98, 121/120, 126/125, 169/168	[⟨62 98 144 174 214 229]]	+1.38	1.41	7.28
2.3.5.7.11.13.17	81/80, 99/98, 121/120, 126/125, 169/168, 221/220	[⟨62 98 144 174 214 229 253]]	+1.47	1.32	6.83
2.3.5.7.11.13.17.19	81/80, 99/98, 121/120, 126/125, 153/152, 169/168, 209/208	[⟨62 98 144 174 214 229 253 263]]	+1.50	1.24	6.40
2.3.5.7.11.13.17.19.23	81/80, 99/98, 121/120, 126/125, 153/152, 161/160, 169/168, 209/208	[⟨62 98 144 174 214 229 253 263 280]]	+1.55	1.18	6.09

Rank-2 temperaments

Periods per 8ve	Generator (Reduced)	Cents (Reduced)	Associated Ratio	Temperaments
1	3\62	58.06	27/26	Hemisecordite
1	7\62	135.48	13/12	Doublethink
1	13\62	251.61	15/13	Hemimeantone
1	17\62	329.03	16/11	Mabon
2	3\62	58.06	27/26	Semihemisecordite
2	4\62	77.42	21/20	Semivalentine
2	6\62	116.13	15/14	Semimiracle
2	26\62	503.22	4/3	Semimeantone
31	29\62 (1\62)	561.29 (19.35)	11/8 (196/195)	Kumhar (62e)
31	19\62 (1\62)	367.74 (19.35)	16/13 (77/76)	Gallium