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=62 tone equal temperament=
'''62edo''' divides the octave into 62 equal parts of 19.35484 cents each.


<b>62edo</b> divides the octave into 62 equal parts of 19.35484 cents each. 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 [[31_comma_temperaments#Gallium|gallium]], [[Starling_temperaments#Valentine temperament-Semivalentine|semivalentine]] and [[Meantone_family#Septimal meantone-Unidecimal meantone aka Huygens-Hemimeantone|hemimeantone]] temperaments.
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 [[31 comma temperaments #Gallium|gallium]], [[Starling temperaments #Valentine|semivalentine]] and [[Meantone family #Septimal meantone|hemimeantone]] temperaments.


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


==='''62-EDO Intervals'''===
== Intervals ==


{| class="wikitable"
{| class="wikitable"
|-
|-
| | '''ARMODUE NOMENCLATURE 8;3 RELATION'''
! ARMODUE NOMENCLATURE 8;3 RELATION
|-
|-
| | <ul><li>'''Ɨ''' = Thick (1/8-tone up)</li><li>'''‡''' = Semisharp (1/4-tone up)</li><li>'''b''' = Flat (5/8-tone down)</li><li>'''◊''' = Node (sharp/flat blindspot 1/2-tone)</li><li>'''#''' = Sharp (5/8-tone up)</li><li>'''v''' = Semiflat (1/4-tone down)</li><li>'''⌐''' = Thin (1/8-tone down)</li></ul>
| <ul><li>'''Ɨ''' = Thick (1/8-tone up)</li><li>'''‡''' = Semisharp (1/4-tone up)</li><li>'''b''' = Flat (5/8-tone down)</li><li>'''◊''' = Node (sharp/flat blindspot 1/2-tone)</li><li>'''#''' = Sharp (5/8-tone up)</li><li>'''v''' = Semiflat (1/4-tone down)</li><li>'''⌐''' = Thin (1/8-tone down)</li></ul>
|}
|}


{| class="wikitable"
{| class="wikitable center-1 right-2"
|-
|-
! [[Degree|Degree]]
! #
![[cent|Cents]]
! [[Cent]]s
! Armodue notation
! Armodue notation
! Approximate intervals
! Approximate intervals
|-
|-
| style="text-align:center;" | 0
| 0
| style="text-align:right;" | 0.0000
| 0.000
| style="text-align:center;" | 1
| 1
| |  
|  
|-
|-
| style="text-align:center;" | 1
| 1
| style="text-align:right;" | 19.3548
| 19.355
| style="text-align:center;" | 1Ɨ
| 1Ɨ
| | 90/89
| 90/89
|-
|-
| style="text-align:center;" | 2
| 2
| style="text-align:right;" | 38.7097
| 38.710
| style="text-align:center;" | 1‡ (9#)
| 1‡ (9#)
| | 45/44
| 45/44
|-
|-
| style="text-align:center;" | 3
| 3
| style="text-align:right;" | 58.0645
| 58.065
| style="text-align:center;" | 2b
| 2b
| | 30/29
| 30/29
|-
|-
| style="text-align:center;" | 4
| 4
| style="text-align:right;" | 77.41935
| 77.419
| style="text-align:center;" | 1◊2
| 1◊2
| | 23/22
| 23/22
|-
|-
| style="text-align:center;" | 5
| 5
| style="text-align:right;" | 96.7742
| 96.774
| style="text-align:center;" | 1#
| 1#
| | 37/35, 18/17, 19/18
| 37/35, 18/17, 19/18
|-
|-
| style="text-align:center;" | 6
| 6
| style="text-align:right;" | 116.129
| 116.129
| style="text-align:center;" | 2v
| 2v
| | 31/29, 15/14, 16/15
| 31/29, 15/14, 16/15
|-
|-
| style="text-align:center;" | 7
| 7
| style="text-align:right;" | 135.4839
| 135.484
| style="text-align:center;" | 2⌐
| 2⌐
| | 27/25, 13/12, 14/13
| 27/25, 13/12, 14/13
|-
|-
| style="text-align:center;" | 8
| 8
| style="text-align:right;" | 154.8387
| 154.839
| style="text-align:center;" | 2
| 2
| | 12/11
| 12/11
|-
|-
| style="text-align:center;" | 9
| 9
| style="text-align:right;" | 174.19355
| 174.194
| style="text-align:center;" | 2Ɨ
| 2Ɨ
| | 11/10
| 11/10
|-
|-
| style="text-align:center;" | 10
| 10
| style="text-align:right;" | 193.5484
| 193.548
| style="text-align:center;" | 2‡
| 2‡
| | 19/17, 9/8, 10/9
| 19/17, 9/8, 10/9
|-
|-
| style="text-align:center;" | 11
| 11
| style="text-align:right;" | 212.9032
| 212.903
| style="text-align:center;" | 3b
| 3b
| | 17/15, 9/8
| 17/15, 9/8
|-
|-
| style="text-align:center;" | 12
| 12
| style="text-align:right;" | 232.2581
| 232.258
| style="text-align:center;" | 2◊3
| 2◊3
| | 8/7
| 8/7
|-
|-
| style="text-align:center;" | 13
| 13
| style="text-align:right;" | 251.6129
| 251.613
| style="text-align:center;" | 2#
| 2#
| | 15/13
| 15/13
|-
|-
| style="text-align:center;" | 14
| 14
| style="text-align:right;" | 270.9677
| 270.968
| style="text-align:center;" | 3v
| 3v
| | 7/6
| 7/6
|-
|-
| style="text-align:center;" | 15
| 15
| style="text-align:right;" | 290.3226
| 290.323
| style="text-align:center;" | 3⌐
| 3⌐
| |  
|  
|-
|-
| style="text-align:center;" | 16
| 16
| style="text-align:right;" | 309.6774
| 309.677
| style="text-align:center;" | 3
| 3
| | 6/5
| 6/5
|-
|-
| style="text-align:center;" | 17
| 17
| style="text-align:right;" | 329.0323
| 329.032
| style="text-align:center;" | 3Ɨ
| 3Ɨ
| |  
|  
|-
|-
| style="text-align:center;" | 18
| 18
| style="text-align:right;" | 348.3871
| 348.387
| style="text-align:center;" | 3‡
| 3‡
| | 11/9
| 11/9
|-
|-
| style="text-align:center;" | 19
| 19
| style="text-align:right;" | 367.7419
| 367.742
| style="text-align:center;" | 4b
| 4b
| | ·
| ·
|-
|-
| style="text-align:center;" | 20
| 20
| style="text-align:right;" | 387.0968
| 387.097
| style="text-align:center;" | 3◊4
| 3◊4
| | 5/4
| 5/4
|-
|-
| style="text-align:center;" | 21
| 21
| style="text-align:right;" | 406.4516
| 406.452
| style="text-align:center;" | 3#
| 3#
| |  
|  
|-
|-
| style="text-align:center;" | 22
| 22
| style="text-align:right;" | 425.80645
| 425.806
| style="text-align:center;" | 4v (5b)
| 4v (5b)
| |  
|  
|-
|-
| style="text-align:center;" | 23
| 23
| style="text-align:right;" | 445.1613
| 445.161
| style="text-align:center;" | 4⌐
| 4⌐
| |  
|  
|-
|-
| style="text-align:center;" | 24
| 24
| style="text-align:right;" | 464.5161
| 464.516
| style="text-align:center;" | 4
| 4
| |  
|  
|-
|-
| style="text-align:center;" | 25
| 25
| style="text-align:right;" | 483.871
| 483.871
| style="text-align:center;" | 4Ɨ (5v)
| 4Ɨ (5v)
| |  
|  
|-
|-
| style="text-align:center;" | 26
| 26
| style="text-align:right;" | 503.2258
| 503.226
| style="text-align:center;" | 5⌐ (4‡)
| 5⌐ (4‡)
| | 4/3
| 4/3
|-
|-
| style="text-align:center;" | 27
| 27
| style="text-align:right;" | 522.58065
| 522.581
| style="text-align:center;" | 5
| 5
| | ·
| ·
|-
|-
| style="text-align:center;" | 28
| 28
| style="text-align:right;" | 541.9355
| 541.935
| style="text-align:center;" | 5Ɨ
| 5Ɨ
| |  
|  
|-
|-
| style="text-align:center;" | 29
| 29
| style="text-align:right;" | 561.2903
| 561.290
| style="text-align:center;" | 5‡ (4#)
| 5‡ (4#)
| |  
|  
|-
|-
| style="text-align:center;" | 30
| 30
| style="text-align:right;" | 580.6452
| 580.645
| style="text-align:center;" | 6b
| 6b
| | 7/5
| 7/5
|-
|-
| style="text-align:center;" | 31
| 31
| style="text-align:right;" | 600
| 600.000
| style="text-align:center;" | 5◊6
| 5◊6
| |  
|  
|-
|-
| style="text-align:center;" | 32
| 32
| style="text-align:right;" | 619.3548
| 619.355
| style="text-align:center;" | 5#
| 5#
| | 10/7
| 10/7
|-
|-
| style="text-align:center;" | 33
| 33
| style="text-align:right;" | 638.7097
| 638.710
| style="text-align:center;" | 6v
| 6v
| |  
|  
|-
|-
| style="text-align:center;" | 34
| 34
| style="text-align:right;" | 658.0645
| 658.065
| style="text-align:center;" | 6⌐
| 6⌐
| |  
|  
|-
|-
| style="text-align:center;" | 35
| 35
| style="text-align:right;" | 677.41935
| 677.419
| style="text-align:center;" | 6
| 6
| | ·
| ·
|-
|-
| style="text-align:center;" | 36
| 36
| style="text-align:right;" | 696.7742
| 696.774
| style="text-align:center;" | 6Ɨ
| 6Ɨ
| |3/2
| |3/2
|-
|-
| style="text-align:center;" | 37
| 37
| style="text-align:right;" | 716.129
| 716.129
| style="text-align:center;" | 6‡
| 6‡
| |  
|  
|-
|-
| style="text-align:center;" | 38
| 38
| style="text-align:right;" | 735.4839
| 735.484
| style="text-align:center;" | 7b
| 7b
| |  
|  
|-
|-
| style="text-align:center;" | 39
| 39
| style="text-align:right;" | 754.8387
| 754.839
| style="text-align:center;" | 6◊7
| 6◊7
| |  
|  
|-
|-
| style="text-align:center;" | 40
| 40
| style="text-align:right;" | 774.19355
| 774.194
| style="text-align:center;" | 6#
| 6#
| |  
|  
|-
|-
| style="text-align:center;" | 41
| 41
| style="text-align:right;" | 793.5484
| 793.548
| style="text-align:center;" | 7v
| 7v
| |  
|  
|-
|-
| style="text-align:center;" | 42
| 42
| style="text-align:right;" | 812.9032
| 812.903
| style="text-align:center;" | 7⌐
| 7⌐
| | 8/5
| 8/5
|-
|-
| style="text-align:center;" | 43
| 43
| style="text-align:right;" | 832.2581
| 832.258
| style="text-align:center;" | 7
| 7
| | ·
| ·
|-
|-
| style="text-align:center;" | 44
| 44
| style="text-align:right;" | 851.6129
| 851.613
| style="text-align:center;" | 7Ɨ
| 7Ɨ
| | 18/11
| 18/11
|-
|-
| style="text-align:center;" | 45
| 45
| style="text-align:right;" | 870.9677
| 870.968
| style="text-align:center;" | 7‡
| 7‡
| |  
|  
|-
|-
| style="text-align:center;" | 46
| 46
| style="text-align:right;" | 890.3226
| 890.323
| style="text-align:center;" | 8b
| 8b
| | 5/3
| 5/3
|-
|-
| style="text-align:center;" | 47
| 47
| style="text-align:right;" | 909.6774
| 909.677
| style="text-align:center;" | 7◊8
| 7◊8
| |  
|  
|-
|-
| style="text-align:center;" | 48
| 48
| style="text-align:right;" | 929.0323
| 929.032
| style="text-align:center;" | 7#
| 7#
| | 12/7
| 12/7
|-
|-
| style="text-align:center;" | 49
| 49
| style="text-align:right;" | 948.3871
| 948.387
| style="text-align:center;" | 8v
| 8v
| | 26/15
| 26/15
|-
|-
| style="text-align:center;" | 50
| 50
| style="text-align:right;" | 967.7419
| 967.742
| style="text-align:center;" | 8⌐
| 8⌐
| | 7/4
| 7/4
|-
|-
| style="text-align:center;" | 51
| 51
| style="text-align:right;" | 987.0968
| 987.097
| style="text-align:center;" | 8
| 8
| | 16/9
| 16/9
|-
|-
| style="text-align:center;" | 52
| 52
| style="text-align:right;" | 1006.4516
| 1006.452
| style="text-align:center;" | 8Ɨ
| 8Ɨ
| |  
|  
|-
|-
| style="text-align:center;" | 53
| 53
| style="text-align:right;" | 1025.80645
| 1025.806
| style="text-align:center;" | 8‡
| 8‡
| |  
|  
|-
|-
| style="text-align:center;" | 54
| 54
| style="text-align:right;" | 1045.1613
| 1045.161
| style="text-align:center;" | 9b
| 9b
| |  
|  
|-
|-
| style="text-align:center;" | 55
| 55
| style="text-align:right;" | 1064.5161
| 1064.516
| style="text-align:center;" | 8◊9
| 8◊9
| |  
|  
|-
|-
| style="text-align:center;" | 56
| 56
| style="text-align:right;" | 1083.871
| 1083.871
| style="text-align:center;" | 8#
| 8#
| |  
|  
|-
|-
| style="text-align:center;" | 57
| 57
| style="text-align:right;" | 1103.2258
| 1103.226
| style="text-align:center;" | 9v (1b)
| 9v (1b)
| |  
|  
|-
|-
| style="text-align:center;" | 58
| 58
| style="text-align:right;" | 1122.58065
| 1122.581
| style="text-align:center;" | 9⌐
| 9⌐
| |  
|  
|-
|-
| style="text-align:center;" | 59
| 59
| style="text-align:right;" | 1141.9355
| 1141.936
| style="text-align:center;" | 9
| 9
| |  
|  
|-
|-
| style="text-align:center;" | 60
| 60
| style="text-align:right;" | 1161.2903
| 1161.290
| style="text-align:center;" | 9Ɨ (1v)
| 9Ɨ (1v)
| |  
|  
|-
|-
| style="text-align:center;" | 61
| 61
| style="text-align:right;" | 1180.6452
| 1180.645
| style="text-align:center;" | 1⌐ (9‡)
| 1⌐ (9‡)
| |  
|  
|-
|-
| style="text-align:center;" | 62
| 62
| style="text-align:right;" | 1200
| 1200.000
| style="text-align:center;" | 1
| 1
| |  
|  
|}
|}
[[Category:Equal divisions of the octave]]

Revision as of 08:23, 11 May 2021

62edo divides the octave into 62 equal parts of 19.35484 cents each.

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.

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
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