62edo

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← 61edo62edo63edo →
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 21/62, 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
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)

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 Approximate Ratios Ups and Downs Notation
0 0 1/1 D
1 19.355 65/64, 66/65, 78/77 ^D, vE♭♭
2 38.71 45/44, 49/48, 50/49, 55/54, 56/55, 77/75 ^^D, E♭♭
3 58.065 27/26, 65/63 ^3D, v3E♭
4 77.419 21/20, 22/21, 25/24 D♯, vvE♭
5 96.774 52/49, 55/52 ^D♯, vE♭
6 116.129 15/14, 16/15, 77/72 ^^D♯, E♭
7 135.484 13/12, 14/13 ^3D♯, v3E
8 154.839 12/11, 35/32, 49/45 D𝄪, vvE
9 174.194 72/65 ^D𝄪, vE
10 193.548 28/25, 49/44, 55/49 E
11 212.903 44/39 ^E, vF♭
12 232.258 8/7, 55/48, 63/55 ^^E, F♭
13 251.613 15/13, 52/45, 65/56 ^3E, v3F
14 270.968 7/6, 75/64 E♯, vvF
15 290.323 13/11, 77/65 ^E♯, vF
16 309.677 6/5 F
17 329.032 63/52 ^F, vG♭♭
18 348.387 11/9, 27/22, 49/40, 60/49 ^^F, G♭♭
19 367.742 26/21 ^3F, v3G♭
20 387.097 5/4 F♯, vvG♭
21 406.452 33/26 ^F♯, vG♭
22 425.806 32/25, 77/60 ^^F♯, G♭
23 445.161 ^3F♯, v3G
24 464.516 21/16, 55/42, 64/49, 72/55 F𝄪, vvG
25 483.871 65/49 ^F𝄪, vG
26 503.226 4/3, 75/56 G
27 522.581 65/48 ^G, vA♭♭
28 541.935 15/11, 48/35 ^^G, A♭♭
29 561.29 18/13 ^3G, v3A♭
30 580.645 7/5 G♯, vvA♭
31 600 55/39, 78/55 ^G♯, vA♭
32 619.355 10/7, 63/44, 77/54 ^^G♯, A♭
33 638.71 13/9, 75/52 ^3G♯, v3A
34 658.065 22/15, 35/24 G𝄪, vvA
35 677.419 65/44, 77/52 ^G𝄪, vA
36 696.774 3/2 A
37 716.129 ^A, vB♭♭
38 735.484 32/21, 49/32, 55/36, 75/49 ^^A, B♭♭
39 754.839 65/42 ^3A, v3B♭
40 774.194 25/16 A♯, vvB♭
41 793.548 52/33 ^A♯, vB♭
42 812.903 8/5, 77/48 ^^A♯, B♭
43 832.258 21/13 ^3A♯, v3B
44 851.613 18/11, 44/27, 49/30, 80/49 A𝄪, vvB
45 870.968 ^A𝄪, vB
46 890.323 5/3 B
47 909.677 22/13 ^B, vC♭
48 929.032 12/7, 75/44, 77/45 ^^B, C♭
49 948.387 26/15, 45/26 ^3B, v3C
50 967.742 7/4 B♯, vvC
51 987.097 39/22 ^B♯, vC
52 1006.452 25/14 C
53 1025.806 65/36 ^C, vD♭♭
54 1045.161 11/6, 64/35 ^^C, D♭♭
55 1064.516 13/7, 24/13 ^3C, v3D♭
56 1083.871 15/8, 28/15 C♯, vvD♭
57 1103.226 49/26 ^C♯, vD♭
58 1122.581 21/11, 40/21, 48/25 ^^C♯, D♭
59 1141.935 52/27 ^3C♯, v3D
60 1161.29 49/25, 55/28 C𝄪, vvD
61 1180.645 65/33, 77/39 ^C𝄪, vD
62 1200 2/1 D

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 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 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
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
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 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 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
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
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