62edt: Difference between revisions

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'''[[Edt|Division of the third harmonic]] into 62 equal parts''' (62EDT) is related to [[39edo|39 edo]], but with the 3/1 rather than the 2/1 being just. The octave is about 3.6090 cents compressed and the step size is about 30.6767 cents. It is consistent to the [[7-odd-limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 39edo is only consistent up to the [[5-odd-limit|6-integer-limit]].
{{Infobox ET}}
{{ED intro}}


{| class="wikitable"
== Theory ==
62edt is related to [[39edo]], but with the 3/1 rather than the 2/1 being just. The octave is about 3.61 cents compressed and the step size is about 30.6767 cents. 62edt is [[consistent]] to the [[integer limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 39edo is only consistent up to the 6-integer-limit.
 
=== Harmonics ===
{{Harmonics in equal|62|3|1|columns=11}}
{{Harmonics in equal|62|3|1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 62edt (continued)}}
 
=== Subsets and supersets ===
Since 62 factors into primes as {{nowrap| 2 × 31 }}, 62edt contains [[2edt]] and [[31edt]] as subset edts.
 
== Intervals ==
{| class="wikitable center-1 right-2 right-3"
|-
|-
! | degree
! #
! | cents value
! Cents
! | corresponding <br>JI intervals
! Hekts
! | comments
! Approximate ratios
|-
|-
| | 0
| 0
| | 0.0000
| 0.0
| | '''exact [[1/1]]'''
| 0.0
| |
| [[1/1]]
|-
|-
| | 1
| 1
| | 30.6767
| 30.7
| | 57/56, 56/55
| 21.0
| |
| [[36/35]], [[50/49]], [[55/54]], [[56/55]], [[81/80]]
|-
|-
| | 2
| 2
| | 61.3534
| 61.4
| | 57/55
| 41.9
| |
| [[22/21]], [[28/27]], [[33/32]], [[45/44]], [[49/48]]
|-
|-
| | 3
| 3
| | 92.0301
| 92.0
| | 96/91
| 62.9
| |
| [[16/15]], [[19/18]], [[20/19]], [[21/20]], [[25/24]]
|-
|-
| | 4
| 4
| | 122.7068
| 122.7
| | 161/150, 189/176
| 83.9
| |
| [[15/14]]
|-
|-
| | 5
| 5
| | 153.3835
| 153.4
| |
| 104.8
| |
| [[11/10]], [[12/11]]
|-
|-
| | 6
| 6
| | 184.0602
| 184.1
| | 208/187
| 125.8
| |
| [[10/9]]
|-
|-
| | 7
| 7
| | 214.7369
| 214.7
| | 77/68
| 146.8
| |
| [[8/7]], [[9/8]]
|-
|-
| | 8
| 8
| | 245.4135
| 245.4
| | 121/105
| 167.7
| |
| [[22/19]]
|-
|-
| | 9
| 9
| | 276.0902
| 276.1
| |
| 188.7
| |
| [[7/6]]
|-
|-
| | 10
| 10
| | 306.7669
| 306.8
| |
| 209.7
| |
| [[6/5]]
|-
|-
| | 11
| 11
| | 337.4436
| 337.4
| | 243/200
| 230.6
| |
| [[11/9]]
|-
|-
| | 12
| 12
| | 368.1203
| 368.1
| |
| 251.6
| |
| [[27/22]]
|-
|-
| | 13
| 13
| | 398.7970
| 398.8
| | 34/27
| 272.6
| |
| [[5/4]]
|-
|-
| | 14
| 14
| | 429.4737
| 429.5
| |
| 293.5
| |
| [[9/7]], [[14/11]]
|-
|-
| | 15
| 15
| | 460.1504
| 460.2
| |
| 314.5
| |
| [[35/27]], [[57/44]]
|-
|-
| | 16
| 16
| | 490.8271
| 490.8
| |
| 335.5
| |
| [[4/3]]
|-
|-
| | 17
| 17
| | 521.5038
| 521.5
| | 77/57
| 356.5
| |
| [[19/14]], [[27/20]]
|-
|-
| | 18
| 18
| | 552.1805
| 552.2
| | [[11/8]]
| 377.5
| |
| [[11/8]], [[15/11]]
|-
|-
| | 19
| 19
| | 582.8572
| 582.9
| | [[7/5]]
| 398.4
| |
| [[7/5]]
|-
|-
| | 20
| 20
| | 613.5339
| 613.5
| | 57/40
| 419.4
| |
| [[10/7]]
|-
|-
| | 21
| 21
| | 644.2106
| 644.2
| |
| 440.3
| |
| [[16/11]], [[22/15]]
|-
|-
| | 22
| 22
| | 674.8873
| 674.9
| | 96/65
| 461.3
| |
| [[28/19]], [[35/24]]
|-
|-
| | 23
| 23
| | 705.5640
| 705.6
| |
| 482.3
| | pseudo-[[3/2]]
| [[3/2]]
|-
|-
| | 24
| 24
| | 736.2406
| 736.2
| | 153/100
| 503.2
| |
| [[54/35]], [[88/57]]
|-
|-
| | 25
| 25
| | 766.9173
| 766.9
| | 81/52
| 524.2
| |
| [[11/7]], [[14/9]]
|-
|-
| | 26
| 26
| | 797.5940
| 797.6
| |
| 545.2
| |
| [[8/5]], [[19/12]]
|-
|-
| | 27
| 27
| | 828.2707
| 828.3
| |
| 566.1
| |
| [[44/27]]
|-
|-
| | 28
| 28
| | 858.9474
| 858.9
| | 69/42
| 587.1
| |
| [[18/11]]
|-
|-
| | 29
| 29
| | 889.6241
| 889.6
| | 117/70
| 608.1
| | pseudo-[[5/3]]
| [[5/3]]
|-
|-
| | 30
| 30
| | 920.3008
| 920.3
| |
| 629.0
| |
| [[12/7]]
|-
|-
| | 31
| 31
| | 950.9775
| 951.0
| |
| 650.0
| |
| [[19/11]]
|-
|-
| | 32
| 32
| | 981.6542
| 981.7
| |
| 671.0
| |
| [[7/4]], [[16/9]]
|-
|-
| | 33
| 33
| | 1012.3309
| 1012.3
| | 70/39
| 691.9
| | pseudo-[[9/5]]
| [[9/5]]
|-
|-
| | 34
| 34
| | 1043.0076
| 1043.0
| | 42/23
| 712.9
| |
| [[11/6]], [[20/11]]
|-
|-
| | 35
| 35
| | 1073.6843
| 1073.7
| | 119/64
| 733.9
| |
| [[28/15]]
|-
|-
| | 36
| 36
| | 1104.3610
| 1104.4
| |
| 754.8
| |
| [[15/8]], [[19/10]]
|-
|-
| | 37
| 37
| | 1135.0377
| 1135.0
| | 52/27
| 775.8
| |
| [[21/11]], [[27/14]]
|-
|-
| | 38
| 38
| | 1165.7144
| 1165.7
| | 100/51
| 796.8
| |
| [[35/18]], [[49/25]], [[55/28]]
|-
|-
| | 39
| 39
| | 1196.3910
| 1196.4
| |
| 817.7
| | pseudo-[[octave]]
| [[2/1]]
|-
|-
| | 40
| 40
| | 1227.0677
| 1227.1
| | 65/32
| 838.7
| |
| [[55/27]], [[81/40]]
|-
|-
| | 41
| 41
| | 1257.7444
| 1257.7
| |
| 859.7
| |
| [[33/16]], [[45/22]], [[49/24]]
|-
|-
| | 42
| 42
| | 1288.4211
| 1288.4
| | [[20/19|40/19]]
| 880.6
| |
| [[19/9]], [[21/10]], [[25/12]]
|-
|-
| | 43
| 43
| | 1319.0978
| 1319.1
| | [[15/7]]
| 901.6
| |
| [[15/7]]
|-
|-
| | 44
| 44
| | 1349.7745
| 1349.8
| | [[12/11|24/11]]
| 922.6
| |
| [[11/5]]
|-
|-
| | 45
| 45
| | 1380.4512
| 1380.5
| |
| 943.5
| |
| [[20/9]]
|-
|-
| | 46
| 46
| | 1411.1279
| 1411.1
| |
| 964.5
| |
| [[9/4]]
|-
|-
| | 47
| 47
| | 1441.8046
| 1441.8
| | 23/10
| 985.5
| |
| [[44/19]]
|-
|-
| | 48
| 48
| | 1472.4813
| 1472.5
| |
| 1006.5
| |
| [[7/3]]
|-
|-
| | 49
| 49
| | 1503.1580
| 1503.2
| | 81/34
| 1027.4
| |
| [[12/5]], [[19/8]]
|-
|-
| | 50
| 50
| | 1533.8347
| 1533.8
| |
| 1048.4
| |
| [[22/9]]
|-
|-
| | 51
| 51
| | 1564.5114
| 1564.5
| | 200/81
| 1069.4
| |
| [[27/11]]
|-
|-
| | 52
| 52
| | 1595.1881
| 1595.2
| | 98/39
| 1090.3
| |
| [[5/2]]
|-
|-
| | 53
| 53
| | 1625.8648
| 1625.9
| |
| 1111.3
| |
| [[18/7]], [[28/11]]
|-
|-
| | 54
| 54
| | 1656.5415
| 1656.5
| |
| 1132.3
| |
| [[57/22]], [[70/27]]
|-
|-
| | 55
| 55
| | 1687.2181
| 1687.2
| |
| 1153.2
| |
| [[8/3]]
|-
|-
| | 56
| 56
| | 1717.8948
| 1717.9
| |
| 1174.2
| |
| [[19/7]], [[27/10]]
|-
|-
| | 57
| 57
| | 1748.5715
| 1748.6
| |
| 1195.2
| |
| [[11/4]]
|-
|-
| | 58
| 58
| | 1779.2482
| 1779.2
| | 176/63
| 1216.1
| |
| [[14/5]]
|-
|-
| | 59
| 59
| | 1809.9249
| 1809.9
| | 91/32
| 1237.1
| |
| [[20/7]]
|-
|-
| | 60
| 60
| | 1840.6016
| 1840.6
| | 55/19
| 1258.1
| |
| [[32/11]]
|-
|-
| | 61
| 61
| | 1871.2783
| 1871.3
| | 56/19
| 1279.0
| |
| [[35/12]]
|-
|-
| | 62
| 62
| | 1901.9550
| 1902.0
| | '''exact [[3/1]]'''
| 1300.0
| | [[3/2|just perfect fifth]] plus an octave
| [[3/1]]
|}
|}
[[Category:Edt]]
[[Category:Edonoi]]

Latest revision as of 08:31, 30 May 2026

← 61edt 62edt 63edt →
Prime factorization 2 × 31
Step size 30.6767 ¢ 
Octave 39\62edt (1196.39 ¢)
Consistency limit 7
Distinct consistency limit 7

62 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 62edt or 62ed3), is a nonoctave tuning system that divides the interval of 3/1 into 62 equal parts of about 30.7 ¢ each. Each step represents a frequency ratio of 31/62, or the 62nd root of 3.

Theory

62edt is related to 39edo, but with the 3/1 rather than the 2/1 being just. The octave is about 3.61 cents compressed and the step size is about 30.6767 cents. 62edt is consistent to the 7-integer-limit, but not to the 8-integer-limit. In comparison, 39edo is only consistent up to the 6-integer-limit.

Harmonics

Approximation of harmonics in 62edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -3.6 +0.0 -7.2 +5.3 -3.6 +5.6 -10.8 +0.0 +1.7 -10.0 -7.2
Relative (%) -11.8 +0.0 -23.5 +17.2 -11.8 +18.3 -35.3 +0.0 +5.4 -32.5 -23.5
Steps
(reduced) 		39
(39) 		62
(0) 		78
(16) 		91
(29) 		101
(39) 		110
(48) 		117
(55) 		124
(0) 		130
(6) 		135
(11) 		140
(16)
Approximation of harmonics in 62edt (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +7.6 +2.0 +5.3 -14.4 +3.3 -3.6 -5.2 -2.0 +5.6 -13.6 +1.5 -10.8
Relative (%) +24.8 +6.5 +17.2 -47.1 +10.8 -11.8 -16.9 -6.4 +18.3 -44.2 +4.9 -35.3
Steps
(reduced) 		145
(21) 		149
(25) 		153
(29) 		156
(32) 		160
(36) 		163
(39) 		166
(42) 		169
(45) 		172
(48) 		174
(50) 		177
(53) 		179
(55)

Subsets and supersets

Since 62 factors into primes as 2 × 31, 62edt contains 2edt and 31edt as subset edts.

Intervals

# Cents Hekts Approximate ratios
0 0.0 0.0 1/1
1 30.7 21.0 36/35, 50/49, 55/54, 56/55, 81/80
2 61.4 41.9 22/21, 28/27, 33/32, 45/44, 49/48
3 92.0 62.9 16/15, 19/18, 20/19, 21/20, 25/24
4 122.7 83.9 15/14
5 153.4 104.8 11/10, 12/11
6 184.1 125.8 10/9
7 214.7 146.8 8/7, 9/8
8 245.4 167.7 22/19
9 276.1 188.7 7/6
10 306.8 209.7 6/5
11 337.4 230.6 11/9
12 368.1 251.6 27/22
13 398.8 272.6 5/4
14 429.5 293.5 9/7, 14/11
15 460.2 314.5 35/27, 57/44
16 490.8 335.5 4/3
17 521.5 356.5 19/14, 27/20
18 552.2 377.5 11/8, 15/11
19 582.9 398.4 7/5
20 613.5 419.4 10/7
21 644.2 440.3 16/11, 22/15
22 674.9 461.3 28/19, 35/24
23 705.6 482.3 3/2
24 736.2 503.2 54/35, 88/57
25 766.9 524.2 11/7, 14/9
26 797.6 545.2 8/5, 19/12
27 828.3 566.1 44/27
28 858.9 587.1 18/11
29 889.6 608.1 5/3
30 920.3 629.0 12/7
31 951.0 650.0 19/11
32 981.7 671.0 7/4, 16/9
33 1012.3 691.9 9/5
34 1043.0 712.9 11/6, 20/11
35 1073.7 733.9 28/15
36 1104.4 754.8 15/8, 19/10
37 1135.0 775.8 21/11, 27/14
38 1165.7 796.8 35/18, 49/25, 55/28
39 1196.4 817.7 2/1
40 1227.1 838.7 55/27, 81/40
41 1257.7 859.7 33/16, 45/22, 49/24
42 1288.4 880.6 19/9, 21/10, 25/12
43 1319.1 901.6 15/7
44 1349.8 922.6 11/5
45 1380.5 943.5 20/9
46 1411.1 964.5 9/4
47 1441.8 985.5 44/19
48 1472.5 1006.5 7/3
49 1503.2 1027.4 12/5, 19/8
50 1533.8 1048.4 22/9
51 1564.5 1069.4 27/11
52 1595.2 1090.3 5/2
53 1625.9 1111.3 18/7, 28/11
54 1656.5 1132.3 57/22, 70/27
55 1687.2 1153.2 8/3
56 1717.9 1174.2 19/7, 27/10
57 1748.6 1195.2 11/4
58 1779.2 1216.1 14/5
59 1809.9 1237.1 20/7
60 1840.6 1258.1 32/11
61 1871.3 1279.0 35/12
62 1902.0 1300.0 3/1
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