40edf: Difference between revisions

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'''40EDF''' is the [[EDF|equal division of the just perfect fifth]] into 40 parts of 17.5489 [[cent|cents]] each, corresponding to 68.3805 [[edo]]. It is related to the regular temperament which tempers out 2401/2400, 9801/9800, and 9453125/9437184 in the 11-limit, which is supported by [[68edo]], [[274edo]], [[342edo]], [[410edo]], 616edo, and 752edo among others.
{{Infobox ET}}
{{ED intro}}


==Intervals==
== Theory ==
{| class="wikitable"
40edf corresponds to 68.3805[[edo]]. It is related to the [[regular temperament]] which [[tempering out|tempers out]] 2401/2400, 9801/9800, and 9453125/9437184 in the [[11-limit]], which is supported by [[68edo]], [[274edo]], [[342edo]], [[410edo]], [[616edo]], and [[752edo]] among others.
 
=== Harmonics ===
{{Harmonics in equal|40|3|2}}
{{Harmonics in equal|40|3|2|start=12|collapsed=1}}
 
== Intervals ==
{| class="wikitable mw-collapsible"
|+ Intervals of 40edf
|-
|-
! | degree
! | degree
Line 9: Line 18:
! | comments
! | comments
|-
|-
| | 0
| colspan="2"| 0
| | 0.0000
| | '''exact [[1/1]]'''
| | '''exact [[1/1]]'''
| |  
| |  
Line 16: Line 24:
| | 1
| | 1
| | 17.5489
| | 17.5489
| | [[100/99]], [[99/98]]
| | [[100/99]], [[99/98]], 81/80
| |  
| |  
|-
|-
| | 2
| | 2
| | 35.0978
| | 35.09775
| | [[50/49]], [[49/48]]
| | [[50/49]], [[49/48]]
| |  
| |  
Line 40: Line 48:
|-
|-
| | 6
| | 6
| | 105.2933
| | 105.29325
| | [[17/16]]
| | [[17/16]]
| |  
| |  
Line 46: Line 54:
| | 7
| | 7
| | 122.8421
| | 122.8421
| |  
| |15/14
| |  
| |  
|-
|-
| | 8
| | 8
| | 140.3910
| | 140.391
| |  
| |13/12
| |  
| |  
|-
|-
Line 60: Line 68:
|-
|-
| | 10
| | 10
| | 175.4888
| | 175.48875
| |  
| |10/9
| |  
| |  
|-
|-
Line 71: Line 79:
| | 12
| | 12
| | 210.5865
| | 210.5865
| |  
| |17/15
| |  
| |  
|-
|-
| | 13
| | 13
| | 228.1354
| | 228.1354
| |  
| |8/7
| |  
| |  
|-
|-
| | 14
| | 14
| | 245.6843
| | 245.68425
| |  
| |15/13
| |  
| |  
|-
|-
| | 15
| | 15
| | 263.2331
| | 263.2331
| |  
| |7/6
| |  
| |  
|-
|-
| | 16
| | 16
| | 280.7820
| | 280.782
| | [[20/17]]
| | [[20/17]]
| |  
| |  
Line 100: Line 108:
|-
|-
| | 18
| | 18
| | 315.8798
| | 315.87975
| | [[6/5]]
| | [[6/5]]
| |  
| |  
Line 106: Line 114:
| | 19
| | 19
| | 333.4286
| | 333.4286
| |  
| |63/52
| |  
| |  
|-
|-
Line 116: Line 124:
| | 21
| | 21
| | 368.5264
| | 368.5264
| |  
| |26/21
| |  
| |  
|-
|-
| | 22
| | 22
| | 386.0753
| | 386.07525
| | [[5/4]]
| | [[5/4]]
| |  
| |  
Line 130: Line 138:
|-
|-
| | 24
| | 24
| | 421.1730
| | 421.173
| | [[51/40]]
| | [[51/40]]
| |  
| |  
Line 136: Line 144:
| | 25
| | 25
| | 438.7219
| | 438.7219
| |  
| |9/7
| |  
| |  
|-
|-
| | 26
| | 26
| | 456.2708
| | 456.27075
| |  
| |13/10
| |  
| |  
|-
|-
| | 27
| | 27
| | 473.8196
| | 473.8196
| |  
| |21/16
| |  
| |  
|-
|-
| | 28
| | 28
| | 491.3685
| | 491.3685
| |  
| |4/3
| |  
| |  
|-
|-
| | 29
| | 29
| | 508.9174
| | 508.9174
| |  
| |66/49
| |  
| |
|-
|-
| | 30
| | 30
| | 526.4663
| | 526.46625
| |  
| |200/147, 49/36, 27/20
| |  
| |
|-
|-
| | 31
| | 31
| | 544.0151
| | 544.0151
| |  
| |11/8
| |  
| |  
|-
|-
| | 32
| | 32
| | 561.5640
| | 561.564
| |  
| |25/18
| |  
| |  
|-
|-
| | 33
| | 33
| | 579.1129
| | 579.1129
| |  
| |7/5
| |  
| |  
|-
|-
| | 34
| | 34
| | 596.6618
| | 596.66175
| | [[24/17]]
| | [[24/17]]
| |  
| |  
Line 186: Line 194:
| | 35
| | 35
| | 614.2106
| | 614.2106
| |  
| |10/7
| |  
| |  
|-
|-
Line 200: Line 208:
|-
|-
| | 38
| | 38
| | 666.8573
| | 666.85725
| |  
| |147/100, 72/49
| |  
| |
|-
|-
| | 39
| | 39
| | 694.4061
| | 684.4061
| | 49/33
| | 297/200, 49/33, 40/27
| |  
| |
|-
|-
| | 40
| | 40
| | 701.9550
| | 701.955
| | '''exact [[3/2]]'''
| | '''exact [[3/2]]'''
| | just perfect fifth
| | just perfect fifth
|-
|41
|719.5039
|50/33, 297/196, 243/160, 32/21
|
|-
|42
|737.05275
|75/49, 49/32
|
|-
|43
|754.6016
|99/64
|
|-
|44
|772.1505
|25/16
|
|-
|45
|789.6994
|30/19
|
|-
|46
|807.24825
|51/32
|
|-
|47
|824.7971
|45/28
|
|-
|48
|842.346
|13/8
|
|-
|49
|859.8949
|32/14
|
|-
|50
|877.44375
|5/3
|
|-
|51
|894.9926
|57/34
|
|-
|52
|912.5415
|17/10
|
|-
|53
|930.0904
|12/7
|
|-
|54
|947.63925
|45/26
|
|-
|55
|965.1981
|7/4
|
|-
|56
|982.737
|30/17
|
|-
|57
|1000.2859
|57/32
|
|-
|58
|1017.83275
|9/5
|
|-
|59
|1035.3836
|189/104
|
|-
|60
|1052.9325
|90/49, 147/80
|
|-
|61
|1070.4814
|13/7
|
|-
|62
|1088.03025
|15/8
|
|-
|63
|1105.4791
|36/19
|
|-
|64
|1123.128
|153/80
|
|-
|65
|1140.6769
|27/14
|
|-
|66
|1158.22575
|39/20
|
|-
|67
|1175.7746
|63/32
|
|-
|68
|1193.3235
|2/1
|
|-
|69
|1210.8724
|99/49
|
|-
|70
|1228.42125
|300/147, 49/24, 81/40
|
|-
|71
|1246.9701
|33/16
|
|-
|72
|1263.519
|25/12
|
|-
|73
|1281.0679
|21/10
|
|-
|74
|1298.61675
|36/17
|
|-
|75
|1316.1756
|15/7
|
|-
|76
|1333.7145
|54/25
|
|-
|77
|1351.2634
|24/11
|
|-
|78
|1368.81375
|441/200, 108/49
|
|-
|79
|1386.3611
|891/400, 49/22, 20/9
|
|-
|80
|1403.91
|'''exact''' 9/8
|
|}
|}


Line 223: Line 431:
POTE generator: ~99/98 = 17.545
POTE generator: ~99/98 = 17.545


Map: [<2 2 4 5 8|, <0 40 22 21 -37|]
Mapping: [<2 2 4 5 8|, <0 40 22 21 -37|]


EDOs: 68, 274, 342, 410, 616, 752
EDOs: {{EDOs|68, 274, 342, 410, 616, 752}}


===2.3.5.7.11.17 subgroup 68&342===
===2.3.5.7.11.17 subgroup 68&342===
Line 232: Line 440:
POTE generator: ~99/98 = 17.546
POTE generator: ~99/98 = 17.546


Map: [<2 2 4 5 8 8|, <0 40 22 21 -37 6|]
Mapping: [<2 2 4 5 8 8|, <0 40 22 21 -37 6|]


EDOs: 68, 274, 342, 410, 616, 752
EDOs: 68, 274, 342, 410, 616, 752
Line 241: Line 449:
POTE generator: ~96/95 = 17.547
POTE generator: ~96/95 = 17.547


Map: [<2 2 4 5 8 8 8|, <0 40 22 21 -37 6 17|]
Mapping: [<2 2 4 5 8 8 8|, <0 40 22 21 -37 6 17|]


EDOs: 68, 274, 342, 410, 616, 752h
EDOs: 68, 274, 342, 410, 616, 752h
Line 250: Line 458:
POTE generator: ~96/95 = 17.546
POTE generator: ~96/95 = 17.546


Map: [<2 2 4 5 8 8 8 7|, <0 40 22 21 -37 6 17 70|]
Mapping: [<2 2 4 5 8 8 8 7|, <0 40 22 21 -37 6 17 70|]


EDOs: 68, 274, 342, 410, 616i, 752h
EDOs: 68, 274, 342, 410, 616i, 752h


[[Category:Edf]]
{{Todo|cleanup|expand}}
[[Category:Edonoi]]

Latest revision as of 17:13, 17 January 2025

← 39edf 40edf 41edf →
Prime factorization 23 × 5
Step size 17.5489 ¢ 
Octave 68\40edf (1193.32 ¢) (→ 17\10edf)
Twelfth 108\40edf (1895.28 ¢) (→ 27\10edf)
Consistency limit 3
Distinct consistency limit 3

40 equal divisions of the perfect fifth (abbreviated 40edf or 40ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 40 equal parts of about 17.5 ¢ each. Each step represents a frequency ratio of (3/2)1/40, or the 40th root of 3/2.

Theory

40edf corresponds to 68.3805edo. It is related to the regular temperament which tempers out 2401/2400, 9801/9800, and 9453125/9437184 in the 11-limit, which is supported by 68edo, 274edo, 342edo, 410edo, 616edo, and 752edo among others.

Harmonics

Approximation of harmonics in 40edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -6.68 -6.68 +4.20 +3.96 +4.20 +0.56 -2.48 +4.20 -2.72 +7.77 -2.48
Relative (%) -38.0 -38.0 +23.9 +22.6 +23.9 +3.2 -14.1 +23.9 -15.5 +44.3 -14.1
Steps
(reduced) 		68
(28) 		108
(28) 		137
(17) 		159
(39) 		177
(17) 		192
(32) 		205
(5) 		217
(17) 		227
(27) 		237
(37) 		245
(5)
Approximation of harmonics in 40edf
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) -0.66 -6.12 -2.72 +8.39 +8.73 -2.48 -8.34 +8.15 -6.12 +1.09 -5.67
Relative (%) -3.8 -34.9 -15.5 +47.8 +49.7 -14.1 -47.5 +46.5 -34.9 +6.2 -32.3
Steps
(reduced) 		253
(13) 		260
(20) 		267
(27) 		274
(34) 		280
(0) 		285
(5) 		290
(10) 		296
(16) 		300
(20) 		305
(25) 		309
(29)

Intervals

Intervals of 40edf
degree cents value corresponding
JI intervals 		comments
0 exact 1/1
1 17.5489 100/99, 99/98, 81/80
2 35.09775 50/49, 49/48
3 52.6466 33/32
4 70.1955 25/24
5 87.7444 20/19
6 105.29325 17/16
7 122.8421 15/14
8 140.391 13/12
9 157.9399 23/21
10 175.48875 10/9
11 193.0376 19/17
12 210.5865 17/15
13 228.1354 8/7
14 245.68425 15/13
15 263.2331 7/6
16 280.782 20/17
17 298.3309 19/16
18 315.87975 6/5
19 333.4286 63/52
20 350.9775 60/49, 49/40
21 368.5264 26/21
22 386.07525 5/4
23 403.6241 24/19
24 421.173 51/40
25 438.7219 9/7
26 456.27075 13/10
27 473.8196 21/16
28 491.3685 4/3
29 508.9174 66/49
30 526.46625 200/147, 49/36, 27/20
31 544.0151 11/8
32 561.564 25/18
33 579.1129 7/5
34 596.66175 24/17
35 614.2106 10/7
36 631.7595 36/25
37 649.3084 16/11
38 666.85725 147/100, 72/49
39 684.4061 297/200, 49/33, 40/27
40 701.955 exact 3/2 just perfect fifth
41 719.5039 50/33, 297/196, 243/160, 32/21
42 737.05275 75/49, 49/32
43 754.6016 99/64
44 772.1505 25/16
45 789.6994 30/19
46 807.24825 51/32
47 824.7971 45/28
48 842.346 13/8
49 859.8949 32/14
50 877.44375 5/3
51 894.9926 57/34
52 912.5415 17/10
53 930.0904 12/7
54 947.63925 45/26
55 965.1981 7/4
56 982.737 30/17
57 1000.2859 57/32
58 1017.83275 9/5
59 1035.3836 189/104
60 1052.9325 90/49, 147/80
61 1070.4814 13/7
62 1088.03025 15/8
63 1105.4791 36/19
64 1123.128 153/80
65 1140.6769 27/14
66 1158.22575 39/20
67 1175.7746 63/32
68 1193.3235 2/1
69 1210.8724 99/49
70 1228.42125 300/147, 49/24, 81/40
71 1246.9701 33/16
72 1263.519 25/12
73 1281.0679 21/10
74 1298.61675 36/17
75 1316.1756 15/7
76 1333.7145 54/25
77 1351.2634 24/11
78 1368.81375 441/200, 108/49
79 1386.3611 891/400, 49/22, 20/9
80 1403.91 exact 9/8

Related regular temperaments

Adding one half of the octave as a generator, 40EDF leads the regular temperament which tempers out 2401/2400, 9801/9800, and 9453125/9437184 in the 11-limit.

11-limit 68&342

Commas: 2401/2400, 9801/9800, 9453125/9437184

POTE generator: ~99/98 = 17.545

Mapping: [<2 2 4 5 8|, <0 40 22 21 -37|]

EDOs: 68, 274, 342, 410, 616, 752

2.3.5.7.11.17 subgroup 68&342

Commas: 1089/1088, 1225/1224, 2401/2400, 24576/24565

POTE generator: ~99/98 = 17.546

Mapping: [<2 2 4 5 8 8|, <0 40 22 21 -37 6|]

EDOs: 68, 274, 342, 410, 616, 752

2.3.5.7.11.17.19 subgroup 68&342

Commas: 1089/1088, 1225/1224, 1445/1444, 1617/1615, 2401/2400

POTE generator: ~96/95 = 17.547

Mapping: [<2 2 4 5 8 8 8|, <0 40 22 21 -37 6 17|]

EDOs: 68, 274, 342, 410, 616, 752h

2.3.5.7.11.17.19.23 subgroup 68&342

Commas: 875/874, 1089/1088, 1225/1224, 1445/1444, 1617/1615, 2024/2023

POTE generator: ~96/95 = 17.546

Mapping: [<2 2 4 5 8 8 8 7|, <0 40 22 21 -37 6 17 70|]

EDOs: 68, 274, 342, 410, 616i, 752h

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