42edf: Difference between revisions

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'''[[EDF|Division of the just perfect fifth]] into 42 equal parts''' (42EDF) is related to [[72edo|72 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 3.3514 cents stretched and the step size is about 16.7132 cents (corresponding to 71.7995 [[edo]], practically identical to every fifth step of [[359edo]]). Unlike 72edo, it is only consistent up to the [[7-odd-limit|7-integer-limit]], with discrepancy for the 8th harmonic (three octaves).
{{Infobox ET}}
{{ED intro}}


Lookalikes: [[72edo]], [[114edt]]
42EDF is related to [[72edo]], but with the [[3/2]] rather than the [[2/1]] being [[just]], which results in [[octave]]s being [[Octave stretch|stretched]] by about 3.3514{{c}}. This corresponds to 71.7995 [[edo]], practically identical to every fifth step of [[359edo]]. Unlike 72edo, it is only consistent up to the 7-[[integer-limit]], with discrepancy for the 8th harmonic (three octaves).


[[Category:Edf]]
Lookalikes: [[72edo]], [[114edt]], [[186ed6]]
[[Category:Edonoi]]
 
== Harmonics ==
{{Harmonics in equal|42|3|2|intervals=prime}}
{{Harmonics in equal|42|3|2|intervals=prime|start=12|collapsed=1}}
 
== Intervals ==
{| class="wikitable"
|-
! Degrees
! Cents value
! Approximate ratios (11-limit)
|-
| colspan="2" | 0
| 1/1
|-
| 1
| 16.7132
| 81/80
|-
| 2
| 33.4264
| 45/44
|-
| 3
| 50.1396
| 33/32
|-
| 4
| 66.8529
| 25/24
|-
| 5
| 83.5661
| 21/20
|-
| 6
| 100.2793
| 35/33
|-
| 7
| 116.9925
| 15/14
|-
| 8
| 133.7057
| 27/25
|-
| 9
| 150.4189
| 12/11
|-
| 10
| 167.1321
| 11/10
|-
| 11
| 183.8454
| 10/9
|-
| 12
| 200.5586
| 9/8
|-
| 13
| 217.2717
| 25/22
|-
| 14
| 233.985
| 8/7
|-
| 15
| 250.6982
| 81/70
|-
| 16
| 267.4114
| 7/6
|-
| 17
| 284.1246
| 33/28
|-
| 18
| 300.8379
| 25/21
|-
| 19
| 317.5511
| 6/5
|-
| 20
| 334.2643
| 40/33
|-
| 21
| 350.9775
| 11/9
|-
| 22
| 367.6907
| 99/80
|-
| 23
| 384.4039
| 5/4
|-
| 24
| 401.1171
| 44/35
|-
| 25
| 417.8304
| 14/11
|-
| 26
| 434.5436
| 9/7
|-
| 27
| 451.2568
| 35/27
|-
| 28
| 467.97
| 21/16
|-
| 29
| 484.6832
| 33/25
|-
| 30
| 501.3964
| 4/3
|-
| 31
| 518.1096
| 27/20
|-
| 32
| 534.8229
| 15/11
|-
| 33
| 551.536
| 11/8
|-
| 34
| 568.2493
| 25/18
|-
| 35
| 584.9625
| 7/5
|-
| 36
| 601.6757
| 99/70
|-
| 37
| 618.3889
| 10/7
|-
| 38
| 635.1021
| 36/25
|-
| 39
| 651.8154
| 16/11
|-
| 40
| 668.5286
| 22/15
|-
| 41
| 685.2418
| 40/27
|-
| 42
| 701.955
| 3/2
|-
| 43
| 718.6682
| 50/33
|-
| 44
| 735.3814
| 32/21
|-
| 45
| 752.0946
| 54/35
|-
| 46
| 768.8079
| 14/9
|-
| 47
| 785.5211
| 11/7
|-
| 48
| 802.2343
| 35/22
|-
| 49
| 818.9475
| 8/5
|-
| 50
| 835.6607
| 81/50
|-
| 51
| 852.3739
| 18/11
|-
| 52
| 869.0871
| 33/20
|-
| 53
| 885.8004
| 5/3
|-
| 54
| 902.5136
| 27/16
|-
| 55
| 919.2268
| 56/33
|-
| 56
| 935.94
| 12/7
|-
| 57
| 952.6532
| 121/70
|-
| 58
| 969.3664
| 7/4
|-
| 59
| 986.0796
| 44/25
|-
| 60
| 1002.7929
| 16/9
|-
| 61
| 1019.506
| 9/5
|-
| 62
| 1036.2193
| 20/11
|-
| 63
| 1052.9235
| 11/6
|-
| 64
| 1069.6457
| 50/27
|-
| 65
| 1086.3589
| 15/8
|-
| 66
| 1103.0721
| 66/35
|-
| 67
| 1119.7854
| 21/11
|-
| 68
| 1136.4986
| 27/14
|-
| 69
| 1153.2118
| 35/18
|-
| 70
| 1169.925
| 49/25
|-
| 71
| 1186.6382
| 99/50
|-
| 72
| 1203.3514
| 2/1
|-
| 73
| 1220.0646
| 81/40
|-
| 74
| 1236.7779
| 45/22
|-
| 75
| 1253.4911
| 33/16
|-
| 76
| 1270.2043
| 56/27
|-
| 77
| 1286.9175
| 21/10
|-
| 78
| 1303.6307
| 70/33
|-
| 79
| 1320.3439
| 15/7
|-
| 80
| 1337.05715
| 54/25
|-
| 81
| 1353.7704
| 24/11
|-
| 82
| 1370.4836
| 11/5
|-
| 83
| 1387.1968
| 20/9
|-
| 84
| 1403.91
| 9/4
|}
 
{{todo|expand}}

Latest revision as of 19:23, 1 August 2025

← 41edf 42edf 43edf →
Prime factorization 2 × 3 × 7
Step size 16.7132 ¢ 
Octave 72\42edf (1203.35 ¢) (→ 12\7edf)
Twelfth 114\42edf (1905.31 ¢) (→ 19\7edf)
Consistency limit 7
Distinct consistency limit 7

42 equal divisions of the perfect fifth (abbreviated 42edf or 42ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 42 equal parts of about 16.7 ¢ each. Each step represents a frequency ratio of (3/2)1/42, or the 42nd root of 3/2.

42EDF is related to 72edo, but with the 3/2 rather than the 2/1 being just, which results in octaves being stretched by about 3.3514 ¢. This corresponds to 71.7995 edo, practically identical to every fifth step of 359edo. Unlike 72edo, it is only consistent up to the 7-integer-limit, with discrepancy for the 8th harmonic (three octaves).

Lookalikes: 72edo, 114edt, 186ed6

Harmonics

Approximation of prime harmonics in 42edf
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +3.35 +3.35 +4.79 +7.24 -6.44 +5.19 -7.98 +0.02 +3.52 +3.33 +4.87
Relative (%) +20.1 +20.1 +28.7 +43.3 -38.5 +31.0 -47.8 +0.1 +21.1 +20.0 +29.1
Steps
(reduced) 		72
(30) 		114
(30) 		167
(41) 		202
(34) 		248
(38) 		266
(14) 		293
(41) 		305
(11) 		325
(31) 		349
(13) 		356
(20)
Approximation of prime harmonics in 42edf
Harmonic 37 41 43 47 53 59 61 67 71 73 79
Error Absolute (¢) -0.60 +5.53 +6.64 +3.07 -4.37 -6.20 +2.94 +7.65 +7.54 -7.12 +6.55
Relative (%) -3.6 +33.1 +39.7 +18.3 -26.2 -37.1 +17.6 +45.8 +45.1 -42.6 +39.2
Steps
(reduced) 		374
(38) 		385
(7) 		390
(12) 		399
(21) 		411
(33) 		422
(2) 		426
(6) 		436
(16) 		442
(22) 		444
(24) 		453
(33)

Intervals

Degrees Cents value Approximate ratios (11-limit)
0 1/1
1 16.7132 81/80
2 33.4264 45/44
3 50.1396 33/32
4 66.8529 25/24
5 83.5661 21/20
6 100.2793 35/33
7 116.9925 15/14
8 133.7057 27/25
9 150.4189 12/11
10 167.1321 11/10
11 183.8454 10/9
12 200.5586 9/8
13 217.2717 25/22
14 233.985 8/7
15 250.6982 81/70
16 267.4114 7/6
17 284.1246 33/28
18 300.8379 25/21
19 317.5511 6/5
20 334.2643 40/33
21 350.9775 11/9
22 367.6907 99/80
23 384.4039 5/4
24 401.1171 44/35
25 417.8304 14/11
26 434.5436 9/7
27 451.2568 35/27
28 467.97 21/16
29 484.6832 33/25
30 501.3964 4/3
31 518.1096 27/20
32 534.8229 15/11
33 551.536 11/8
34 568.2493 25/18
35 584.9625 7/5
36 601.6757 99/70
37 618.3889 10/7
38 635.1021 36/25
39 651.8154 16/11
40 668.5286 22/15
41 685.2418 40/27
42 701.955 3/2
43 718.6682 50/33
44 735.3814 32/21
45 752.0946 54/35
46 768.8079 14/9
47 785.5211 11/7
48 802.2343 35/22
49 818.9475 8/5
50 835.6607 81/50
51 852.3739 18/11
52 869.0871 33/20
53 885.8004 5/3
54 902.5136 27/16
55 919.2268 56/33
56 935.94 12/7
57 952.6532 121/70
58 969.3664 7/4
59 986.0796 44/25
60 1002.7929 16/9
61 1019.506 9/5
62 1036.2193 20/11
63 1052.9235 11/6
64 1069.6457 50/27
65 1086.3589 15/8
66 1103.0721 66/35
67 1119.7854 21/11
68 1136.4986 27/14
69 1153.2118 35/18
70 1169.925 49/25
71 1186.6382 99/50
72 1203.3514 2/1
73 1220.0646 81/40
74 1236.7779 45/22
75 1253.4911 33/16
76 1270.2043 56/27
77 1286.9175 21/10
78 1303.6307 70/33
79 1320.3439 15/7
80 1337.05715 54/25
81 1353.7704 24/11
82 1370.4836 11/5
83 1387.1968 20/9
84 1403.91 9/4
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