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}}
'''[[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]].


== Intervals ==
{| class="wikitable"
{| class="wikitable"
|-
|-
! | degree
! | degree
! | cents value
! | cents value
!hekts
! | corresponding <br>JI intervals
! | corresponding <br>JI intervals
! | comments
! | comments
|-
|-
| | 0
! colspan="3" | 0
| | 0.0000
| | '''exact [[1/1]]'''
| | '''exact [[1/1]]'''
| |  
| |  
Line 15: Line 17:
| | 1
| | 1
| | 30.6767
| | 30.6767
|20.9677
| | 57/56, 56/55
| | 57/56, 56/55
| |  
| |  
Line 20: Line 23:
| | 2
| | 2
| | 61.3534
| | 61.3534
|41.9355
| | 57/55
| | 57/55
| |  
| |  
Line 25: Line 29:
| | 3
| | 3
| | 92.0301
| | 92.0301
|62.9032
| | 96/91
| | 96/91
| |  
| |  
Line 30: Line 35:
| | 4
| | 4
| | 122.7068
| | 122.7068
| |  
|83.871
| | 161/150, 189/176
| |  
| |  
|-
|-
| | 5
| | 5
| | 153.3835
| | 153.3835
| |  
|104.8387
| | 12/11
| |  
| |  
|-
|-
| | 6
| | 6
| | 184.0602
| | 184.0602
| | 208/187
|125.80645
| |10/9
| |  
| |  
|-
|-
| | 7
| | 7
| | 214.7369
| | 214.7369
| |  
|146.7742
| |17/15
| |  
| |  
|-
|-
| | 8
| | 8
| | 245.4135
| | 245.4135
|167.7412
| | 121/105
| | 121/105
| |  
| |  
Line 55: Line 65:
| | 9
| | 9
| | 276.0902
| | 276.0902
| |  
|188.7097
| | 20/17
| |  
| |  
|-
|-
| | 10
| | 10
| | 306.7669
| | 306.7669
| |  
|209.6774
| | 6/5
| |  
| |  
|-
|-
| | 11
| | 11
| | 337.4436
| | 337.4436
|230.6452
| | 243/200
| | 243/200
| |  
| |  
Line 70: Line 83:
| | 12
| | 12
| | 368.1203
| | 368.1203
| |  
|251.6129
| | 16/13
| |  
| |  
|-
|-
| | 13
| | 13
| | 398.7970
| | 398.797
|272.58065
| | 34/27
| | 34/27
| |  
| |  
Line 80: Line 95:
| | 14
| | 14
| | 429.4737
| | 429.4737
| |  
|293.5484
| | 9/7
| |  
| |  
|-
|-
| | 15
| | 15
| | 460.1504
| | 460.1504
| |  
|314.5161
| |21/16
| |  
| |  
|-
|-
| | 16
| | 16
| | 490.8271
| | 490.8271
| |  
|335.4839
| | [[4/3]]
| |  
| |  
|-
|-
| | 17
| | 17
| | 521.5038
| | 521.5038
|356.4516
| | 77/57
| | 77/57
| |  
| |  
Line 100: Line 119:
| | 18
| | 18
| | 552.1805
| | 552.1805
|377.49135
| | [[11/8]]
| | [[11/8]]
| |  
| |  
Line 105: Line 125:
| | 19
| | 19
| | 582.8572
| | 582.8572
|398.3871
| | [[7/5]]
| | [[7/5]]
| |  
| |  
Line 110: Line 131:
| | 20
| | 20
| | 613.5339
| | 613.5339
|419.3548
| | 57/40
| | 57/40
| |  
| |  
Line 115: Line 137:
| | 21
| | 21
| | 644.2106
| | 644.2106
| |  
|440.3226
| |16/11
| |  
| |  
|-
|-
| | 22
| | 22
| | 674.8873
| | 674.8873
|461.2903
| | 96/65
| | 96/65
| |  
| |  
|-
|-
| | 23
| | 23
| | 705.5640
| | 705.564
| |  
|482.2851
| |  
| |[[3/2]]
| |
|-
|-
| | 24
| | 24
| | 736.2406
| | 736.2406
| |  
|503.2258
| | 153/100
| |  
| |  
|-
|-
| | 25
| | 25
| | 766.9173
| | 766.9173
| |  
|524.19355
| | 81/52
| |  
| |  
|-
|-
| | 26
| | 26
| | 797.5940
| | 797.594
| |  
|545.1613
| |27/17
| |  
| |  
|-
|-
| | 27
| | 27
| | 828.2707
| | 828.2707
| |  
|566.129
| |13/8
| |  
| |  
|-
|-
| | 28
| | 28
| | 858.9474
| | 858.9474
| |  
|587.0968
| | 69/42
| |  
| |  
|-
|-
| | 29
| | 29
| | 889.6241
| | 889.6241
| |  
|608.0645
| |  
| | 117/70
| | pseudo-[[5/3]]
|-
|-
| | 30
| | 30
| | 920.3008
| | 920.3008
| |  
|629.0323
| | 17/10
| |  
| |  
|-
|-
| | 31
| | 31
| | 950.9775
| | 950.9775
| |  
|650
| | 26/15
| |  
| |  
|-
|-
| | 32
| | 32
| | 981.6542
| | 981.6542
| |  
|670.9677
| | 30/17
| |  
| |  
|-
|-
| | 33
| | 33
| | 1012.3309
| | 1012.3309
| |  
|691.9355
| |  
| | 70/39
| | pseudo-[[9/5]]
|-
|-
| | 34
| | 34
| | 1043.0076
| | 1043.0076
| |  
|712.9032
| | 42/23
| |  
| |  
|-
|-
| | 35
| | 35
| | 1073.6843
| | 1073.6843
| |  
|733.871
| | 119/64
| |  
| |  
|-
|-
| | 36
| | 36
| | 1104.3610
| | 1104.361
| |  
|754.8387
| |17/9
| |  
| |  
|-
|-
| | 37
| | 37
| | 1135.0377
| | 1135.0377
| |  
|775.80645
| | 52/27
| |  
| |  
|-
|-
| | 38
| | 38
| | 1165.7144
| | 1165.7144
| |  
|796.7742
| | 100/51
| |  
| |  
|-
|-
| | 39
| | 39
| | 1196.3910
| | 1196.391
| |  
|817.7419
| |  
| | 2/1
| | pseudo-[[octave]]
|-
|-
| | 40
| | 40
| | 1227.0677
| | 1227.0677
|838.7097
| | 65/32
| | 65/32
| |  
| |  
Line 215: Line 257:
| | 41
| | 41
| | 1257.7444
| | 1257.7444
| |  
|859.6774
| |114/55
| |  
| |  
|-
|-
| | 42
| | 42
| | 1288.4211
| | 1288.4211
|880.6452
| | [[20/19|40/19]]
| | [[20/19|40/19]]
| |  
| |  
Line 225: Line 269:
| | 43
| | 43
| | 1319.0978
| | 1319.0978
|901.6129
| | [[15/7]]
| | [[15/7]]
| |  
| |  
Line 230: Line 275:
| | 44
| | 44
| | 1349.7745
| | 1349.7745
| | [[24/11]]
|922.58065
| | [[12/11|24/11]]
| |  
| |  
|-
|-
| | 45
| | 45
| | 1380.4512
| | 1380.4512
| |  
|943.5484
| | 20/9
| |  
| |  
|-
|-
| | 46
| | 46
| | 1411.1279
| | 1411.1279
| |  
|964.5161
| | 9/4
| |  
| |  
|-
|-
| | 47
| | 47
| | 1441.8046
| | 1441.8046
|985.4839
| | 23/10
| | 23/10
| |  
| |  
Line 250: Line 299:
| | 48
| | 48
| | 1472.4813
| | 1472.4813
| |  
|1006.4516
| | 7/3
| |  
| |  
|-
|-
| | 49
| | 49
| | 1503.1580
| | 1503.158
|1027.4194
| | 81/34
| | 81/34
| |  
| |  
Line 260: Line 311:
| | 50
| | 50
| | 1533.8347
| | 1533.8347
| |  
|1048.3871
| | 39/16
| |  
| |  
|-
|-
| | 51
| | 51
| | 1564.5114
| | 1564.5114
|1069.3548
| | 200/81
| | 200/81
| |  
| |  
Line 270: Line 323:
| | 52
| | 52
| | 1595.1881
| | 1595.1881
|1090.3226
| | 98/39
| | 98/39
| |  
| |  
Line 275: Line 329:
| | 53
| | 53
| | 1625.8648
| | 1625.8648
| |  
|1111.2903
| | 51/20
| |  
| |  
|-
|-
| | 54
| | 54
| | 1656.5415
| | 1656.5415
| |  
|1132.2581
| | 192/65
| |  
| |  
|-
|-
| | 55
| | 55
| | 1687.2181
| | 1687.2181
| |  
|1153.2258
| | 8/3
| |  
| |  
|-
|-
| | 56
| | 56
| | 1717.8948
| | 1717.8948
| |  
|1174.19355
| |27/10
| |  
| |  
|-
|-
| | 57
| | 57
| | 1748.5715
| | 1748.5715
| |  
|1195.1613
| |11/4
| |  
| |  
|-
|-
| | 58
| | 58
| | 1779.2482
| | 1779.2482
| |  
|1216.129
| | 176/63
| |  
| |  
|-
|-
| | 59
| | 59
| | 1809.9249
| | 1809.9249
|1237.0968
| | 91/32
| | 91/32
| |  
| |  
Line 310: Line 371:
| | 60
| | 60
| | 1840.6016
| | 1840.6016
|1258.0645
| | 55/19
| | 55/19
| |  
| |  
Line 315: Line 377:
| | 61
| | 61
| | 1871.2783
| | 1871.2783
|1279.0323
| | 56/19
| | 56/19
| |  
| |  
|-
|-
| | 62
| | 62
| | 1901.9550
| | 1901.955
|1300
| | '''exact [[3/1]]'''
| | '''exact [[3/1]]'''
| | [[3/2|just perfect fifth]] plus an octave
| | [[3/2|just perfect fifth]] plus an octave
|}
|}


[[Category:Edt]]
== Harmonics ==
[[Category:Edonoi]]
{{Harmonics in equal
| steps = 62
| num = 3
| denom = 1
| intervals = integer
}}
{{Harmonics in equal
| steps = 62
| num = 3
| denom = 1
| start = 12
| collapsed = 1
| intervals = integer
}}

Latest revision as of 19:23, 1 August 2025

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

Division of the third harmonic into 62 equal parts (62EDT) is related to 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-integer-limit, but not to the 8-integer-limit. In comparison, 39edo is only consistent up to the 6-integer-limit.

Intervals

degree cents value hekts corresponding
JI intervals 		comments
0 exact 1/1
1 30.6767 20.9677 57/56, 56/55
2 61.3534 41.9355 57/55
3 92.0301 62.9032 96/91
4 122.7068 83.871 161/150, 189/176
5 153.3835 104.8387 12/11
6 184.0602 125.80645 10/9
7 214.7369 146.7742 17/15
8 245.4135 167.7412 121/105
9 276.0902 188.7097 20/17
10 306.7669 209.6774 6/5
11 337.4436 230.6452 243/200
12 368.1203 251.6129 16/13
13 398.797 272.58065 34/27
14 429.4737 293.5484 9/7
15 460.1504 314.5161 21/16
16 490.8271 335.4839 4/3
17 521.5038 356.4516 77/57
18 552.1805 377.49135 11/8
19 582.8572 398.3871 7/5
20 613.5339 419.3548 57/40
21 644.2106 440.3226 16/11
22 674.8873 461.2903 96/65
23 705.564 482.2851 3/2
24 736.2406 503.2258 153/100
25 766.9173 524.19355 81/52
26 797.594 545.1613 27/17
27 828.2707 566.129 13/8
28 858.9474 587.0968 69/42
29 889.6241 608.0645 117/70 pseudo-5/3
30 920.3008 629.0323 17/10
31 950.9775 650 26/15
32 981.6542 670.9677 30/17
33 1012.3309 691.9355 70/39 pseudo-9/5
34 1043.0076 712.9032 42/23
35 1073.6843 733.871 119/64
36 1104.361 754.8387 17/9
37 1135.0377 775.80645 52/27
38 1165.7144 796.7742 100/51
39 1196.391 817.7419 2/1 pseudo-octave
40 1227.0677 838.7097 65/32
41 1257.7444 859.6774 114/55
42 1288.4211 880.6452 40/19
43 1319.0978 901.6129 15/7
44 1349.7745 922.58065 24/11
45 1380.4512 943.5484 20/9
46 1411.1279 964.5161 9/4
47 1441.8046 985.4839 23/10
48 1472.4813 1006.4516 7/3
49 1503.158 1027.4194 81/34
50 1533.8347 1048.3871 39/16
51 1564.5114 1069.3548 200/81
52 1595.1881 1090.3226 98/39
53 1625.8648 1111.2903 51/20
54 1656.5415 1132.2581 192/65
55 1687.2181 1153.2258 8/3
56 1717.8948 1174.19355 27/10
57 1748.5715 1195.1613 11/4
58 1779.2482 1216.129 176/63
59 1809.9249 1237.0968 91/32
60 1840.6016 1258.0645 55/19
61 1871.2783 1279.0323 56/19
62 1901.955 1300 exact 3/1 just perfect fifth plus an octave

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
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) +7.6 +2.0 +5.3 -14.4 +3.3 -3.6 -5.2 -2.0 +5.6 -13.6 +1.5
Relative (%) +24.8 +6.5 +17.2 -47.1 +10.8 -11.8 -16.9 -6.4 +18.3 -44.2 +4.9
Steps
(reduced) 		145
(21) 		149
(25) 		153
(29) 		156
(32) 		160
(36) 		163
(39) 		166
(42) 		169
(45) 		172
(48) 		174
(50) 		177
(53)
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