51edf: Difference between revisions

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'''[[EDF|Division of the just perfect fifth]] into 51 equal parts''' (51EDF) is related to [[87edo|87 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 2.5474 cents compressed and the step size is about 13.7638 cents (corresponding to 87.1851 edo). Unlike 87edo, it is only consistent up to the [[5-odd-limit|6-integer-limit]], with discrepancy for the 7th harmonic.
{{Infobox ET}}
'''[[EDF|Division of the just perfect fifth]] into 51 equal parts''' (51EDF) is related to [[87edo|87 edo]], but with the [[3/2]] rather than the [[2/1]] being [[just]]. The octave is [[Octave shrinking|compressed]] by about 2.5474 [[cents]] and the step size is about 13.7638 cents (corresponding to 87.1851 edo).  
 
Unlike 87edo, it is only [[consistent]] up to the 6-[[integer-limit]], with discrepancy for the 7th harmonic.


Lookalikes: [[87edo]], [[138edt]]
Lookalikes: [[87edo]], [[138edt]]
== Harmonics ==
{{Harmonics in equal|51|3|2|intervals=prime}}
{{Harmonics in equal|51|3|2|intervals=prime|start=12|collapsed=1}}


== Intervals ==
== Intervals ==
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! |Cents
! |Cents
|-
|-
| style="text-align:center;" |0
| colspan="2" style="text-align:center;" |0
| style="text-align:center;" |0
|-
|-
|1
|1
Line 18: Line 24:
|27.52765
|27.52765
|-
|-
| style="text-align:center;" |3
|3
| style="text-align:center;" |41.2915
|41.2915
|-
|-
|4
|4
Line 27: Line 33:
|68.8191
|68.8191
|-
|-
| style="text-align:center;" |6
|6
| style="text-align:center;" |82.5829
|82.5829
|-
|-
|7·
|7·
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|110.1106
|110.1106
|-
|-
| style="text-align:center;" |9
|9
| style="text-align:center;" |123.8744
|123.8744
|-
|-
|10
|10
Line 45: Line 51:
|151.0206
|151.0206
|-
|-
| style="text-align:center;" |12·
|12·
| style="text-align:center;" |165.1659
|165.1659
|-
|-
|13
|13
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|192.6935
|192.6935
|-
|-
| style="text-align:center;" |15
|15
| style="text-align:center;" |206.45735
|206.45735
|-
|-
|16
|16
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|233.985
|233.985
|-
|-
| style="text-align:center;" |18
|18
| style="text-align:center;" |248.7488
|248.7488
|-
|-
|19
|19
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|275.2765
|275.2765
|-
|-
| style="text-align:center;" |21
|21
| style="text-align:center;" |289.0403
|289.0403
|-
|-
|22·
|22·
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|316.5679
|316.5679
|-
|-
| style="text-align:center;" |24
|24
| style="text-align:center;" |330.3318
|330.3318
|-
|-
|25
|25
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|357.8594
|357.8594
|-
|-
| style="text-align:center;" |27
|27
| style="text-align:center;" |371.6232
|371.6232
|-
|-
|28
|28
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|399.1509
|399.1509
|-
|-
| style="text-align:center;" |30
|30
| style="text-align:center;" |412.9147
|412.9147
|-
|-
|31
|31
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|440.44253
|440.44253
|-
|-
| style="text-align:center;" |33
|33
| style="text-align:center;" |455.2062
|455.2062
|-
|-
|34
|34
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|481.7338
|481.7338
|-
|-
| style="text-align:center;" |36
|36
| style="text-align:center;" |495.49765
|495.49765
|-
|-
|37
|37
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|523.0253
|523.0253
|-
|-
| style="text-align:center;" |39
|39
| style="text-align:center;" |536.7891
|536.7891
|-
|-
|40
|40
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|564.3168
|564.3168
|-
|-
| style="text-align:center;" |42
|42
| style="text-align:center;" |578.0806
|578.0806
|-
|-
|43
|43
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|605.6082
|605.6082
|-
|-
| style="text-align:center;" |45
|45
| style="text-align:center;" |619.3721
|619.3721
|-
|-
|46
|46
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|646.8997
|646.8997
|-
|-
| style="text-align:center;" |48
|48
| style="text-align:center;" |660.6635
|660.6635
|-
|-
|49
|49
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|688.1912
|688.1912
|-
|-
| style="text-align:center;" |51
|51
| style="text-align:center;" |701.955
|701.955
|-
|-
|52
|52
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|729.48365
|729.48365
|-
|-
| style="text-align:center;" |54
|54
| style="text-align:center;" |743.2465
|743.2465
|-
|-
|55
|55
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|770.7741
|770.7741
|-
|-
| style="text-align:center;" |57
|57
| style="text-align:center;" |784.5379
|784.5379
|-
|-
|58
|58
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|812.0656
|812.0656
|-
|-
| style="text-align:center;" |60
|60
| style="text-align:center;" |825.8294
|825.8294
|-
|-
|61
|61
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|853.3571
|853.3571
|-
|-
| style="text-align:center;" |63
|63
| style="text-align:center;" |867.1209
|867.1209
|-
|-
|64
|64
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|894.6485
|894.6485
|-
|-
| style="text-align:center;" |66
|66
| style="text-align:center;" |908.41235
|908.41235
|-
|-
|67
|67
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|935.94
|935.94
|-
|-
| style="text-align:center;" |69
|69
| style="text-align:center;" |949.7038
|949.7038
|-
|-
|70
|70
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|977.2315
|977.2315
|-
|-
| style="text-align:center;" |72
|72
| style="text-align:center;" |990.9952
|990.9952
|-
|-
|73
|73
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|1018.5229
|1018.5229
|-
|-
| style="text-align:center;" |75
|75
| style="text-align:center;" |1032.32868
|1032.32868
|-
|-
|76
|76
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|1059.8144
|1059.8144
|-
|-
| style="text-align:center;" |78
|78
| style="text-align:center;" |1073.5782
|1073.5782
|-
|-
|79
|79
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|1101.1059
|1101.1059
|-
|-
| style="text-align:center;" |81
|81
| style="text-align:center;" |1114.8697
|1114.8697
|-
|-
|82
|82
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|1142.39735
|1142.39735
|-
|-
| style="text-align:center;" |84
|84
| style="text-align:center;" |1156.1612
|1156.1612
|-
|-
|85
|85
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|1183.6888
|1183.6888
|-
|-
| style="text-align:center;" |87
|87
| style="text-align:center;" |1197.45265
|1197.45265
|-
|-
|88
|88
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|1403.91
|1403.91
|}
|}
[[Category:Edf]]
 
[[Category:Edonoi]]
{{todo|inline=1|complete table|text=Add a third column that comments on the intervals, either what [[JI]] they approximate, what they are named, or how they can be used musically.}}
 
{{todo|expand}}

Latest revision as of 19:23, 1 August 2025

← 50edf 51edf 52edf →
Prime factorization 3 × 17
Step size 13.7638 ¢ 
Octave 87\51edf (1197.45 ¢) (→ 29\17edf)
Twelfth 138\51edf (1899.41 ¢) (→ 46\17edf)
Consistency limit 6
Distinct consistency limit 6

Division of the just perfect fifth into 51 equal parts (51EDF) is related to 87 edo, but with the 3/2 rather than the 2/1 being just. The octave is compressed by about 2.5474 cents and the step size is about 13.7638 cents (corresponding to 87.1851 edo).

Unlike 87edo, it is only consistent up to the 6-integer-limit, with discrepancy for the 7th harmonic.

Lookalikes: 87edo, 138edt

Harmonics

Approximation of prime harmonics in 51edf
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -2.55 -2.55 -6.02 +3.31 +5.36 +5.19 -5.03 -4.90 -5.33 +6.28 +0.94
Relative (%) -18.5 -18.5 -43.7 +24.1 +38.9 +37.7 -36.6 -35.6 -38.7 +45.7 +6.8
Steps
(reduced) 		87
(36) 		138
(36) 		202
(49) 		245
(41) 		302
(47) 		323
(17) 		356
(50) 		370
(13) 		394
(37) 		424
(16) 		432
(24)
Approximation of prime harmonics in 51edf
Harmonic 37 41 43 47 53 59 61 67 71 73 79
Error Absolute (¢) -2.57 -1.36 -1.23 -3.82 -5.36 +1.67 -0.99 +1.76 -2.29 +4.68 +5.57
Relative (%) -18.7 -9.9 -8.9 -27.7 -38.9 +12.1 -7.2 +12.8 -16.6 +34.0 +40.4
Steps
(reduced) 		454
(46) 		467
(8) 		473
(14) 		484
(25) 		499
(40) 		513
(3) 		517
(7) 		529
(19) 		536
(26) 		540
(30) 		550
(40)

Intervals

Degree Cents
0
1 13.7638
2 27.52765
3 41.2915
4 55.0555
5 68.8191
6 82.5829
7· 96.3468
8 110.1106
9 123.8744
10 137.6382
11 151.0206
12· 165.1659
13 178.9297
14 192.6935
15 206.45735
16 220.2212
17· 233.985
18 248.7488
19 261.51265
20 275.2765
21 289.0403
22· 302.8041
23 316.5679
24 330.3318
25 344.0955
26 357.8594
27 371.6232
28 385.3871
29 399.1509
30 412.9147
31 426.6785
32 440.44253
33 455.2062
34 467.97
35 481.7338
36 495.49765
37 509.2615
38 523.0253
39 536.7891
40 550.5529
41 564.3168
42 578.0806
43 591.8444
44 605.6082
45 619.3721
46 633.1359
47 646.8997
48 660.6635
49 674.42735
50 688.1912
51 701.955
52 715.7188
53 729.48365
54 743.2465
55 757.0103
56 770.7741
57 784.5379
58 798.3018
59 812.0656
60 825.8294
61 839.5932
62 853.3571
63 867.1209
64 880.8847
65 894.6485
66 908.41235
67 922.1762
68 935.94
69 949.7038
70 963.46765
71 977.2315
72 990.9952
73 1004.7591
74 1018.5229
75 1032.32868
76 1046.0506
77 1059.8144
78 1073.5782
79 1087.3421
80 1101.1059
81 1114.8697
82 1128.6335
83 1142.39735
84 1156.1612
85 1169.925
86 1183.6888
87 1197.45265
88 1211.2165
89 1224.9803
90 1238.7441
91 1252.5079
92 1266.2718
93 1280.0356
94 1293.7994
95 1307.5632
96 1321.3271
97 1335.0909
98 1348.8547
99 1362.6185
100 1376.3824
101 1390.1462
102 1403.91
Todo: complete table

Add a third column that comments on the intervals, either what JI they approximate, what they are named, or how they can be used musically.

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