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'''75edo''' divides the octave into 75 equal parts of exactly 16 cents each. In the 5-limit, it tempers out the tetracot comma, 20000/19683 and the semicomma 2109375/2097152, and provides a good tuning for [[Tetracot_family|tetracot temperament]]. In the 7-limit, it tenpers 225/224 and 1728/1715. In the 11-limit, 75e  val scores lower in [[badness]] than the [[patent val]], tempers 100/99 and 243/242, whereas the patent val tempers 99/98 and 121/120. In the 13-limit, it tempers 325/324 and 512/507, 17-limit 120/119 and 256/255 and 19-limit 190/189 and 250/247.
The '''75 equal divisions of the octave''' ('''75edo'''), or the '''75-tone equal temperament''' ('''75tet'''), '''75 equal temperament''' ('''75et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 75 [[equal]] parts of exactly 16 [[cent]]s each.  


It provides the optimal patent val for the [[M&N_temperaments|12&51 temperament]] in the 7-limit and the [[M&N_temperaments|31&44 temperament]] in the 19-limit.
== Theory ==
{{harmonics in equal|75}}
In the 5-limit, 75et tempers out 20000/19683 ([[tetracot comma]]) and 2109375/2097152 ([[semicomma]]), and provides a good tuning for the [[tetracot]] temperament. In the 7-limit, it tenpers [[225/224]] and [[1728/1715]]. In the 11-limit, 75e val scores lower in [[badness]] than the [[patent val]], tempers [[100/99]] and [[243/242]], whereas the patent val tempers [[99/98]] and [[121/120]]. In the 13-limit, it tempers [[325/324]] and [[512/507]], 17-limit [[120/119]] and [[256/255]] and 19-limit 190/189 and 250/247.


Since it's part of the Fibonacci sequence beginning with 5 and 12, it closely approximates peppermint temperament. The size of its fifth is exactly 704c, which is very close to the peppermint fifth of 704.096c. This makes it suitable for neo-Gothic tunings.
It provides the optimal patent val for the [[M&N_temperaments|12&51 temperament]] in the 7-limit and the [[M&N_temperaments|31&44 temperament]] in the 19-limit.
 
Since 75 is part of the Fibonacci sequence beginning with 5 and 12, it closely approximates peppermint temperament. The size of its fifth is exactly 704c, which is very close to the peppermint fifth of 704.096c. This makes it suitable for neo-Gothic tunings.
 
=== Prime harmonics ===
{{Harmonics in equal|75}}


== Intervals ==
== Intervals ==
 
{| class="wikitable center-all right-2"
{| class="wikitable"
|-
|-
| '''Step'''
! '''#'''
|'''Cents'''
! '''Cents'''
|-
|-
| 0
| 0
|0
|0
|-
|-
| | 1
| 1
| | 16
| 16
|-
|-
| | 2
| 2
| | 32
| 32
|-
|-
| | 3
| 3
| | 48
| 48
|-
|-
| | 4
| 4
| | 64
| 64
|-
|-
| | 5
| 5
| | 80
| 80
|-
|-
| | 6
| 6
| | 96
| 96
|-
|-
| | 7
| 7
| | 112
| 112
|-
|-
| | 8
| 8
| | 128
| 128
|-
|-
| | 9
| 9
| | 144
| 144
|-
|-
| | 10
| 10
| | 160
| 160
|-
|-
| | 11
| 11
| | 176
| 176
|-
|-
| | 12
| 12
| | 192
| 192
|-
|-
| | 13
| 13
| | 208
| 208
|-
|-
| | 14
| 14
| | 224
| 224
|-
|-
| | 15
| 15
| | 240
| 240
|-
|-
| | 16
| 16
| | 256
| 256
|-
|-
| | 17
| 17
| | 272
| 272
|-
|-
| | 18
| 18
| | 288
| 288
|-
|-
| | 19
| 19
| | 304
| 304
|-
|-
| | 20
| 20
| | 320
| 320
|-
|-
| | 21
| 21
| | 336
| 336
|-
|-
| | 22
| 22
| | 352
| 352
|-
|-
| | 23
| 23
| | 368
| 368
|-
|-
| | 24
| 24
| | 384
| 384
|-
|-
| | 25
| 25
| | 400
| 400
|-
|-
| | 26
| 26
| | 416
| 416
|-
|-
| | 27
| 27
| | 432
| 432
|-
|-
| | 28
| 28
| | 448
| 448
|-
|-
| | 29
| 29
| | 464
| 464
|-
|-
| | 30
| 30
| | 480
| 480
|-
|-
| | 31
| 31
| | 496
| 496
|-
|-
| | 32
| 32
| | 512
| 512
|-
|-
| | 33
| 33
| | 528
| 528
|-
|-
| | 34
| 34
| | 544
| 544
|-
|-
| | 35
| 35
| | 560
| 560
|-
|-
| | 36
| 36
| | 576
| 576
|-
|-
| | 37
| 37
| | 592
| 592
|-
|-
| | 38
| 38
| | 608
| 608
|-
|-
| | 39
| 39
| | 624
| 624
|-
|-
| | 40
| 40
| | 640
| 640
|-
|-
| | 41
| 41
| | 656
| 656
|-
|-
| | 42
| 42
| | 672
| 672
|-
|-
| | 43
| 43
| | 688
| 688
|-
|-
| | 44
| 44
| | 704
| 704
|-
|-
| | 45
| 45
| | 720
| 720
|-
|-
| | 46
| 46
| | 736
| 736
|-
|-
| | 47
| 47
| | 752
| 752
|-
|-
| | 48
| 48
| | 768
| 768
|-
|-
| | 49
| 49
| | 784
| 784
|-
|-
| | 50
| 50
| | 800
| 800
|-
|-
| | 51
| 51
| | 816
| 816
|-
|-
| | 52
| 52
| | 832
| 832
|-
|-
| | 53
| 53
| | 848
| 848
|-
|-
| | 54
| 54
| | 864
| 864
|-
|-
| | 55
| 55
| | 880
| 880
|-
|-
| | 56
| 56
| | 896
| 896
|-
|-
| | 57
| 57
| | 912
| 912
|-
|-
| | 58
| 58
| | 928
| 928
|-
|-
| | 59
| 59
| | 944
| 944
|-
|-
| | 60
| 60
| | 960
| 960
|-
|-
| | 61
| 61
| | 976
| 976
|-
|-
| | 62
| 62
| | 992
| 992
|-
|-
| | 63
| 63
| | 1008
| 1008
|-
|-
| | 64
| 64
| | 1024
| 1024
|-
|-
| | 65
| 65
| | 1040
| 1040
|-
|-
| | 66
| 66
| | 1056
| 1056
|-
|-
| | 67
| 67
| | 1072
| 1072
|-
|-
| | 68
| 68
| | 1088
| 1088
|-
|-
| | 69
| 69
| | 1104
| 1104
|-
|-
| | 70
| 70
| | 1120
| 1120
|-
|-
| | 71
| 71
| | 1136
| 1136
|-
|-
| | 72
| 72
| | 1152
| 1152
|-
|-
| | 73
| 73
| | 1168
| 1168
|-
|-
| | 74
| 74
| | 1184
| 1184
|-
|-
|75
| 75
|1200
| 1200
|}
|}
[[Category:Equal divisions of the octave]]
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