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{{Infobox ET}}
'''5ED5/4''' is the [[Equal-step tuning|equal division]] of the [[5/4|just major third]] into five parts of 77.2627 [[cent|cents]] each, corresponding to every second step of [[31edo]]. It is related to [[Carlos Alpha]] and the 7-limit temperaments which temper out 2100875/2097152 (including the [[Breedsmic temperaments|tertiaseptal temperament]] and the [[Starling temperaments|valentine temperament]]).
'''5ED5/4''' is the [[Equal-step tuning|equal division]] of the [[5/4|just major third]] into five parts of 77.2627 [[cent|cents]] each, corresponding to every second step of [[31edo]]. It is related to [[Carlos Alpha]] and the 7-limit temperaments which temper out 2100875/2097152 (including the [[Breedsmic temperaments|tertiaseptal temperament]] and the [[Starling temperaments|valentine temperament]]).


Line 272: Line 273:
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">13</font> = 1220703125/67108864
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">13</font> = 1220703125/67108864
|}
|}
== Harmonics ==
{{Harmonics in equal
| steps = 5
| num = 5
| denom = 4
}}
{{Harmonics in equal
| steps = 5
| num = 5
| denom = 4
| start = 12
| collapsed = 1
}}


==5ED5/4 as a generator==
==5ED5/4 as a generator==
Line 278: Line 293:


===Tertiaseptal===
===Tertiaseptal===
Aside from 2100875/2097152, tertiaseptal temperament tempers out 2401/2400, 65625/65536, and 703125/702464 in the 7-limit. It can be described as the 31&amp;171 temperament, and the step interval of 5ED5/4 (tuned between 256/245 and 68/65) can serve as its generator. In the 17-limit, it tempers out 243/242, 375/374, 441/440, 625/624, and 3584/3575.
Aside from 2100875/2097152, [[tertiaseptal]] temperament tempers out 2401/2400, 65625/65536, and 703125/702464 in the 7-limit. It can be described as the 31&amp;171 temperament, and the step interval of 5ED5/4 (tuned between 256/245 and 68/65) can serve as its generator. In the 17-limit, it tempers out 243/242, 375/374, 441/440, 625/624, and 3584/3575.
 
{| class="wikitable"
|-
! | generator
! | cents value <sup>a</sup><br>(octave-reduced)
! | 17-limit ratio<br>(octave-reduced)
|-
| | 1
| | 77.2
| | 117/112, 256/245, 68/65
|-
| | 2
| | 154.4
| | 130/119, [[35/32]]
|-
| | 3
| | 231.6
| | '''[[8/7]]'''
|-
| | 4
| | 308.8
| | 117/98, 140/117
|-
| | 5
| | 386.0
| | '''[[5/4]]'''
|-
| | 6
| | 463.2
| | '''[[17/13]]'''
|-
| | 7
| | 540.4
| | 175/128
|-
| | 8
| | 617.6
| | '''[[10/7]]'''
|-
| | 9
| | 694.8
| | 112/75
|-
| | 10
| | 772.0
| | [[25/16]]
|-
| | 11
| | 849.2
| | 44/27, '''[[18/11]]'''
|-
| | 12
| | 926.4
| | 128/75
|-
| | 13
| | 1003.6
| | [[25/14]]
|-
| | 14
| | 1080.8
| | '''[[28/15]]'''
|-
| | 15
| | 1158.0
| | 39/20
|-
| | 16
| | 35.2
| | 55/54, 52/51, 51/50, [[50/49]], [[49/48]], [[45/44]]
|-
| | 17
| | 112.4
| | '''[[16/15]]'''
|-
| | 18
| | 189.6
| | 39/35
|-
| | 19
| | 266.8
| | '''[[7/6]]'''
|-
| | 20
| | 344.0
| | 39/32
|-
| | 21
| | 421.2
| | [[51/40]]
|-
| | 22
| | 498.4
| | '''[[4/3]]'''
|-
| | 23
| | 575.6
| | 39/28
|-
| | 24
| | 652.8
| | [[35/24]]
|-
| | 25
| | 730.0
| | [[32/21]]
|-
| | 26
| | 807.2
| | 51/32
|-
| | 27
| | 884.4
| | '''[[5/3]]'''
|-
| | 28
| | 961.6
| | 68/39
|-
| | 29
| | 1038.8
| | 51/28
|-
| | 30
| | 1116.0
| | [[40/21]], [[21/11]]
|-
| | 31
| | 1193.2
| |
|-
| | 32
| | 70.4
| | [[26/25]], [[25/24]]
|-
| | 33
| | 147.6
| | '''[[12/11]]'''
|-
| | 34
| | 224.8
| | 91/80
|-
| | 35
| | 302.0
| | [[25/21]]
|-
| | 36
| | 379.3
| | 96/77
|-
| | 37
| | 456.5
| | '''[[13/10]]'''
|-
| | 38
| | 533.7
| | 34/25, '''[[15/11]]'''
|-
| | 39
| | 610.9
| | [[64/45]]
|-
| | 40
| | 688.1
| | 52/35
|-
| | 41
| | 765.3
| | '''[[14/9]]'''
|-
| | 42
| | 842.5
| | '''[[13/8]]'''
|-
| | 43
| | 919.7
| | '''[[17/10]]'''
|-
| | 44
| | 996.9
| | '''[[16/9]]'''
|-
| | 45
| | 1074.1
| | '''[[13/7]]'''
|-
| | 46
| | 1151.3
| | 68/35, [[35/18]]
|-
| | 47
| | 28.5
| | [[65/64]], [[64/63]], [[56/55]]
|-
| | 48
| | 105.7
| | '''[[17/16]]'''
|-
| | 49
| | 182.9
| | '''[[10/9]]'''
|-
| | 50
| | 260.1
| | [[64/55]]
|-
| | 51
| | 337.3
| | '''[[17/14]]'''
|-
| | 52
| | 414.5
| | '''[[14/11]]'''
|-
| | 53
| | 491.7
| | 65/49
|-
| | 54
| | 568.9
| | [[25/18]]
|-
| | 55
| | 646.1
| | '''[[16/11]]'''
|-
| | 56
| | 723.3
| | 85/56
|-
| | 57
| | 800.5
| | 35/22
|-
| | 58
| | 877.7
| | 128/77
|-
| | 59
| | 954.9
| | '''[[26/15]]'''
|-
| | 60
| | 1032.1
| | '''[[20/11]]'''
|-
| | 61
| | 1109.3
| | 91/48, 256/135
|-
| | 62
| | 1186.5
| | 208/105
|-
| | 63
| | 63.7
| | [[28/27]]
|-
| | 64
| | 140.9
| | '''[[13/12]]'''
|-
| | 65
| | 218.1
| | '''[[17/15]]''', [[25/22]]
|-
| | 66
| | 295.3
| | [[32/27]]
|-
| | 67
| | 372.5
| | [[26/21]]
|-
| | 68
| | 449.7
| | [[35/27]]
|-
| | 69
| | 526.9
| | 65/48
|-
| | 70
| | 604.1
| | '''[[17/12]]'''
|-
| | 71
| | 681.3
| | [[40/27]]
|-
| | 72
| | 758.5
| | 65/42
|-
| | 73
| | 835.7
| | [[34/21]]
|-
| | 74
| | 912.9
| | 56/33
|-
| | 75
| | 990.1
| | 39/22
|-
| | 76
| | 1067.3
| | 50/27
|-
| | 77
| | 1144.5
| | 64/33
|-
| | 78
| | 21.7
| | 91/90, 85/84, 78/77
|-
| | 79
| | 98.9
| | 35/33
|-
| | 80
| | 176.1
| | 195/176
|-
| | 81
| | 253.3
| | 52/45
|-
| | 82
| | 330.5
| | 40/33
|-
| | 83
| | 407.7
| | 91/72
|-
| | 84
| | 484.9
| | 119/90
|-
| | 85
| | 562.1
| | 112/81
|-
| | 86
| | 639.3
| | '''[[13/9]]'''
|-
| | 87
| | 716.5
| | 68/45, 50/33
|-
| | 88
| | 793.7
| | 128/81
|-
| | 89
| | 870.9
| | 119/72
|-
| | 90
| | 948.1
| | 140/81
|-
| | 91
| | 1025.3
| | 65/36
|-
| | 92
| | 1102.5
| | '''[[17/9]]'''
|-
| | 93
| | 1179.7
| | [[160/81]], 196/99, 240/121
|-
| | 94
| | 56.9
| | 91/88
|-
| | 95
| | 134.1
| | 68/63
|-
| | 96
| | 211.3
| | 112/99
|-
| | 97
| | 288.5
| | '''[[13/11]]'''
|-
| | 98
| | 365.7
| | 68/55
|-
| | 99
| | 442.9
| | 128/99
|-
| | 100
| | 520.1
| | 104/77
|-
| | 101
| | 597.3
| |
|-
| | 102
| | 674.5
| |
|-
| | 103
| | 751.7
| | '''[[17/11]]'''
|-
| | 104
| | 828.9
| | 160/99
|-
| | 105
| | 906.1
| |
|-
| | 106
| | 983.3
| | 136/77
|-
| | 107
| | 1060.6
| |
|-
| | 108
| | 1137.8
| | 52/27, 85/44
|-
| | 109
| | 15.0
| | [[100/99]]
|-
| | 110
| | 92.2
| | 128/121, [[256/243]]
|-
| | 111
| | 169.4
| |
|-
| | 112
| | 246.6
| |
|-
| | 113
| | 323.8
| |
|-
| | 114
| | 401.0
| | 34/27
|-
| | 115
| | 478.2
| |
|-
| | 116
| | 555.4
| |
|-
| | 117
| | 632.6
| |
|-
| | 118
| | 709.8
| |
|-
| | 119
| | 787.0
| | 52/33
|-
| | 120
| | 864.2
| |
|-
| | 121
| | 941.4
| |
|-
| | 122
| | 1018.6
| |
|-
| | 123
| | 1095.8
| |
|-
| | 124
| | 1173.0
| | 65/33
|-
| | 125
| | 50.2
| | 34/33
|}
<sup>a</sup> in 17-limit POTE tuning


===Tertia===
===Tertia===
Aside from 2100875/2097152, tertia temperament tempers out 385/384, 1331/1323, and 1375/1372 in the 11-limit. It can be described as the 31&amp;140 temperament, and the step interval of 5ED5/4 (tuned between 256/245 and 22/21) can serve as its generator. In the 17-limit, it tempers out 352/351, 385/384, 561/560, 625/624, and 715/714.
Aside from 2100875/2097152, [[tertiaseptal|tertia]] temperament tempers out 385/384, 1331/1323, and 1375/1372 in the 11-limit. It can be described as the 31&amp;140 temperament, and the step interval of 5ED5/4 (tuned between 256/245 and 68/65) can serve as its generator. In the 17-limit, it tempers out 352/351, 385/384, 561/560, 625/624, and 715/714.
 
{| class="wikitable"
|-
! | generator
! | cents value<br>(octave-reduced)
! | 11-limit ratio<br>(octave-reduced)
|-
| | 1
| | 77.2
| | 256/245
|-
| | 2
| | 154.3
| | '''[[12/11]]''', [[35/32]]
|-
| | 3
| | 231.5
| | '''[[8/7]]'''
|-
| | 4
| | 308.6
| |
|-
| | 5
| | 385.8
| | '''[[5/4]]'''
|-
| | 6
| | 463.0
| |
|-
| | 7
| | 540.1
| | 175/128
|-
| | 8
| | 617.3
| | '''[[10/7]]'''
|-
| | 9
| | 694.5
| | 112/75
|-
| | 10
| | 771.6
| | [[25/16]]
|-
| | 11
| | 848.8
| |
|-
| | 12
| | 925.9
| | 128/75
|-
| | 13
| | 1003.1
| | [[25/14]]
|-
| | 14
| | 1080.3
| | '''[[28/15]]'''
|-
| | 15
| | 1157.4
| |
|-
| | 16
| | 34.6
| | [[50/49]], [[49/48]]
|-
| | 17
| | 111.8
| | '''[[16/15]]'''
|-
| | 18
| | 188.9
| |
|-
| | 19
| | 266.1
| | '''[[7/6]]'''
|-
| | 20
| | 343.2
| | '''[[11/9]]'''
|-
| | 21
| | 420.4
| | '''[[14/11]]'''
|-
| | 22
| | 497.6
| | '''[[4/3]]'''
|-
| | 23
| | 574.7
| |
|-
| | 24
| | 651.9
| | '''[[16/11]]''', [[35/24]]
|-
| | 25
| | 729.1
| | [[32/21]]
|-
| | 26
| | 806.2
| |
|-
| | 27
| | 883.4
| | '''[[5/3]]'''
|-
| | 28
| | 960.5
| |
|-
| | 29
| | 1037.7
| | '''[[20/11]]'''
|-
| | 30
| | 1114.9
| |
|-
| | 31
| | 1192.0
| |
|-
| | 32
| | 69.2
| |
|-
| | 33
| | 146.4
| |
|-
| | 34
| | 223.5
| |
|-
| | 35
| | 300.7
| |
|-
| | 36
| | 377.8
| |
|-
| | 37
| | 455.0
| |
|-
| | 38
| | 532.2
| |
|-
| | 39
| | 609.3
| |
|-
| | 40
| | 686.5
| |
|-
| | 41
| | 763.7
| | '''[[14/9]]'''
|-
| | 42
| | 840.8
| | '''[[13/8]]'''
|-
| | 43
| | 918.0
| |
|-
| | 44
| | 995.1
| | '''[[16/9]]'''
|-
| | 45
| | 1072.3
| |
|-
| | 46
| | 1149.5
| |
|-
| | 47
| | 26.6
| |
|-
| | 48
| | 103.8
| | '''[[17/16]]'''
|-
| | 49
| | 180.9
| | '''[[10/9]]'''
|-
| | 50
| | 258.1
| |
|-
| | 51
| | 335.3
| |
|-
| | 52
| | 412.4
| |
|-
| | 53
| | 489.6
| |
|-
| | 54
| | 566.8
| | [[25/18]]
|-
| | 55
| | 643.9
| |
|-
| | 56
| | 721.1
| |
|-
| | 57
| | 798.2
| |
|-
| | 58
| | 875.4
| |
|-
| | 59
| | 952.6
| |
|-
| | 60
| | 1029.7
| |
|-
| | 61
| | 1106.9
| |
|-
| | 62
| | 1184.1
| |
|-
| | 63
| | 61.2
| |
|-
| | 64
| | 138.4
| |
|-
| | 65
| | 215.5
| |
|-
| | 66
| | 292.7
| | [[32/27]]
|-
| | 67
| | 369.9
| |
|-
| | 68
| | 447.0
| |
|-
| | 69
| | 524.2
| |
|-
| | 70
| | 601.4
| |
|-
| | 71
| | 678.5
| |
|-
| | 72
| | 755.7
| |
|-
| | 73
| | 832.8
| |
|-
| | 74
| | 910.0
| |
|-
| | 75
| | 987.2
| |
|-
| | 76
| | 1064.3
| |
|-
| | 77
| | 1141.5
| |
|-
| | 78
| | 18.7
| |
|-
| | 79
| | 95.8
| |
|-
| | 80
| | 173.0
| |
|-
| | 81
| | 250.1
| |
|-
| | 82
| | 327.3
| |
|-
| | 83
| | 404.5
| |
|-
| | 84
| | 481.6
| |
|-
| | 85
| | 558.8
| |
|-
| | 86
| | 636.0
| |
|-
| | 87
| | 713.1
| |
|-
| | 88
| | 790.3
| |
|-
| | 89
| | 867.4
| |
|-
| | 90
| | 944.6
| |
|-
| | 91
| | 1021.8
| |
|-
| | 92
| | 1098.9
| |
|-
| | 93
| | 1176.1
| |
|-
| | 94
| | 53.2
| |
|-
| | 95
| | 130.4
| |
|-
| | 96
| | 207.6
| |
|-
| | 97
| | 284.7
| |
|-
| | 98
| | 361.9
| |
|-
| | 99
| | 439.1
| |
|-
| | 100
| | 516.2
| |
|-
| | 101
| | 593.4
| |
|-
| | 102
| | 670.5
| |
|-
| | 103
| | 747.7
| |
|-
| | 104
| | 824.9
| |
|-
| | 105
| | 902.0
| |
|-
| | 106
| | 979.2
| |
|-
| | 107
| | 1056.4
| |
|-
| | 108
| | 1133.5
| |
|-
| | 109
| | 10.7
| |
|-
| | 110
| | 87.8
| |
|-
| | 111
| | 165.0
| |
|-
| | 112
| | 242.2
| |
|-
| | 113
| | 319.3
| |
|-
| | 114
| | 396.5
| | 34/27
|}


[[Category:5/4]]
[[Category:Major third]]
[[Category:Equal-step tuning]]
[[Category:Equal-step tuning]]
[[Category:Edonoi]]
Retrieved from "https://en.xen.wiki/w/5ed5/4"