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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 68&342===
===2.3.5.7.11.17 subgroup 68&342===
Commas: 1089/1088, 1225/1224, 2401/2400, 24576/24565
Commas: 1089/1088, 1225/1224, 2401/2400, 24576/24565


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


===2.3.5.7.11.17.19 68&342===
===2.3.5.7.11.17.19 subgroup 68&342===
Commas: 1089/1088, 1225/1224, 1445/1444, 1617/1615, 2401/2400
Commas: 1089/1088, 1225/1224, 1445/1444, 1617/1615, 2401/2400


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


===2.3.5.7.11.17.19.23 68&342===
===2.3.5.7.11.17.19.23 subgroup 68&342===
Commas: 875/874, 1089/1088, 1225/1224, 1445/1444, 1617/1615, 2024/2023
Commas: 875/874, 1089/1088, 1225/1224, 1445/1444, 1617/1615, 2024/2023


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]]
Retrieved from "https://en.xen.wiki/w/40edf"