Xenharmonic Wiki

87edo: Difference between revisions

← Older editNewer edit →
VisualWikitext
Jdfreivald (talk | contribs)
autopatrolled, emailconfirmed
62 edits
Added Difference in Cents column.
Xenwolf (talk | contribs)
autopatrolled, Bureaucrats, confirmaccount-notify, Interface administrators, Sysops
8,705 edits
simplified tables and links added links to interval pages
Line 1: Line 1:
The 87 equal temperament, often abbreviated 87-tET, 87-EDO, or 87-ET, is the scale derived by dividing the octave into 87 equally-sized steps, where each step represents a frequency ratio of 13.79 [[cent|cent]]s. It is solid as both a [[13-limit|13-limit]] (or 15 odd limit) and as a [[5-limit|5-limit]] system, and of course does well enough in any limit in between. It represents the [[13-limit|13-limit]] [[Tonality_diamond|tonality diamond]] both uniquely and [[consistent|consistent]]ly, and is the smallest equal temperament to do so.
The '''87 equal temperament''', often abbreviated '''87-tET''', '''87-EDO''', or '''87-ET''', is the scale derived by dividing the octave into 87 equally-sized steps, where each step represents a frequency ratio of 13.79 [[cent|cents]]. It is solid as both a [[13-limit]] (or [[15 odd limit]]) and as a [[5-limit]] system, and of course does well enough in any limit in between. It represents the [[13-limit]] [[tonality diamond]] both uniquely and [[consistent|consistently]], and is the smallest equal temperament to do so.


87et [[tempering_out|tempers out]] 196/195, 325/324, 352/351, 364/363, 385/384, 441/440, 625/624, 676/675, and 1001/1000 as well as the 29-comma, <46 -29|, the misty comma, <26 -12 -3|, the kleisma, 15625/15552, 245/243, 1029/1024, 3136/3125, and 5120/5103.
87et [[tempering out|tempers out]] 196/195, 325/324, 352/351, 364/363, 385/384, 441/440, 625/624, 676/675, and 1001/1000 as well as the 29-comma, <46 -29|, the misty comma, <26 -12 -3|, the kleisma, 15625/15552, 245/243, 1029/1024, 3136/3125, and 5120/5103.


87et is a particularly good tuning for [[Gamelismic_clan|rodan temperament]]. The 8/7 generator of 17\87 is a remarkable 0.00062 cents sharper than the 13-limit [[POTE_tuning|POTE]] generator and is close to the 11-limit POTE generator also. Also, the 32\87 generator for [[Kleismic_family|clyde temperament]] is 0.04455 cents sharp of the 7-limit POTE generator.
87et is a particularly good tuning for [[Gamelismic clan #Rodan|rodan temperament]]. The 8/7 generator of 17\87 is a remarkable 0.00062 cents sharper than the 13-limit [[POTE tuning|POTE]] generator and is close to the [[11-limit]] POTE generator also. Also, the 32\87 generator for [[Kleismic family #Clyde|clyde temperament]] is 0.04455 cents sharp of the 7-limit POTE generator.


=Rank two temperaments=
== Rank two temperaments ==


{| class="wikitable"
{| class="wikitable" style="text-align: right"
|-
|-
! | Periods
! Periods <br> per <br> octave
 
! Generator
per
! Cents
 
! Associated <br> ratio
octave
! Temperament
! | Generator
! | Cents
! | Associated
 
ratio
! | Temperament
|-
|-
| style="text-align:right;" | 1
| 1
| style="text-align:right;" | 4\87
| style="text-align: center" | 4\87
| style="text-align:right;" | 55.172
| 55.172
| style="text-align:center;" | 33/32
| style="text-align: center" | [[33/32]]
| | [[Sensa|Sensa]]
| style="text-align: left;" | [[Sensa]]
|-
|-
| style="text-align:right;" | 1
| 1
| style="text-align:right;" | 10\87
| style="text-align: center" | 10\87
| style="text-align:right;" | 137.931
| 137.931
| style="text-align:center;" | 13/12
| style="text-align: center" | [[13/12]]
| | [[Quartemka|Quartemka]]
| style="text-align: left" | [[Quartemka]]
|-
|-
| style="text-align:right;" | 1
| 1
| style="text-align:right;" | 14\87
| style="text-align: center" | 14\87
| style="text-align:right;" | 193.103
| 193.103
| style="text-align:center;" | 28/25
| style="text-align: center" | [[28/25]]
| | [[Luna|Luna]]/[[Hemithirds|hemithirds]]
| style="text-align: left" | [[Luna]] / [[Hemithirds]]
|-
|-
| style="text-align:right;" | 1
| 1
| style="text-align:right;" | 17\87
| style="text-align: center" | 17\87
| style="text-align:right;" | 234.483
| 234.483
| style="text-align:center;" | 8/7
| style="text-align: center" | [[8/7]]
| | [[Rodan|Rodan]]
| style="text-align: left" | [[Rodan]]
|-
|-
| style="text-align:right;" | 1
| 1
| style="text-align:right;" | 23\87
| style="text-align: center" | 23\87
| style="text-align:right;" | 317.241
| 317.241
| style="text-align:center;" | 6/5
| style="text-align: center" | [[6/5]]
| | [[Hanson|Hanson]]/[[Countercata|countercata]]/[[metakleismic|metakleismic]]
| style="text-align: left" | [[Hanson]] / [[Countercata]] / [[Metakleismic]]
|-
|-
| style="text-align:right;" | 1
| 1
| style="text-align:right;" | 32\87
| style="text-align: center" | 32\87
| style="text-align:right;" | 441.379
| 441.379
| style="text-align:center;" | 9/7
| style="text-align: center" | [[9/7]]
| | [[Clyde|Clyde]]
| style="text-align: left" | [[Clyde]]
|-
|-
| style="text-align:right;" | 1
| 1
| style="text-align:right;" | 38\87
| style="text-align: center" | 38\87
| style="text-align:right;" | 524.138
| 524.138
| style="text-align:center;" | 65/48
| style="text-align: center" | [[65/48]]
| | [[Widefourth|Widefourth]]
| style="text-align: left" | [[Widefourth]]
|-
|-
| style="text-align:right;" | 1
| 1
| style="text-align:right;" | 40\87
| style="text-align: center" | 40\87
| style="text-align:right;" | 551.724
| 551.724
| style="text-align:center;" | 11/8
| style="text-align: center" | [[11/8]]
| | [[Emkay|Emkay]]
| style="text-align: left" | [[Emkay]]
|-
|-
| style="text-align:right;" | 3
| 3
| style="text-align:right;" | 23\87
| style="text-align: center" | 23\87
| style="text-align:right;" | 317.241
| 317.241
| style="text-align:center;" | 6/5
| style="text-align: center" | [[6/5]]
| | [[Tritikleismic|Tritikleismic]]
| style="text-align: left" | [[Tritikleismic]]
|-
|-
| style="text-align:right;" | 29
| 29
| style="text-align:right;" | 28\87
| style="text-align: center" | 28\87
| style="text-align:right;" | 386.207
| 386.207
| style="text-align:center;" | 5/4
| style="text-align: center" | [[5/4]]
| | [[mystery|Mystery]]
| style="text-align: left" | [[Mystery]]
|}
|}


87 can serve as a MOS in these:
87 can serve as a MOS in these:


[[M&N_temperaments|270&amp;87]] &lt;&lt;24 -9 -66 12 27 ... ||
* [[M&N temperaments|270&amp;87]] &lt;&lt;24 -9 -66 12 27 ... ||
 
* [[M&N temperaments|494&amp;87]] &lt;&lt;51 -1 -133 11 32 ... ||
[[M&N_temperaments|494&amp;87]] &lt;&lt;51 -1 -133 11 32 ... ||


=13-limit detempering of 87et=
== 13-limit detempering of 87et ==


See [[Detempering|detempering]].
See [[Detempering|detempering]].


In this table, "Difference in Cents" indicates whether 87 is flat (negative) or sharp (positive) of the detempered interval. For example, 15 steps, at 206.89655 cents, corresponds to 9/8 and is 3.0 cents sharp.
In this table, "Difference in Cents" indicates whether the 87-interval is flat (negative) or sharp (positive) of the detempered interval. For example, 15 steps, at 206.89655 cents, corresponds to [[9/8]] and is 3.0 cents sharp. ''<tt>todo: align cent precision of size and difference</tt>''


{|class="wikitable"
{|class="wikitable" style="text-align: right"
! Steps <br> of 87
! Size in <br> [[Cent]]s
! Detempered <br> Interval
! Difference <br> in Cents
|-
|-
| | Steps of 87
| 1
| | Cents value
| 13.79310
| | Detempered Interval
| style="text-align: center" | [[91/90]]
| | Difference in Cents
| -5.3
|-
|-
| | 1
| 2
| | 13.79310
| 27.58621
| | 91/90
| style="text-align: center" | [[49/48]]
| | -5.3
| -8.1
|-
|-
| | 2
| 3
| | 27.58621
| 41.37931
| | 49/48
| style="text-align: center" | [[40/39]]
| | -8.1
| -2.5
|-
|-
| | 3
| 4
| | 41.37931
| 55.17241
| | 40/39
| style="text-align: center" | [[28/27]]
| | -2.5
| -7.8
|-
|-
| | 4
| 5
| | 55.17241
| 68.96552
| | 28/27
| style="text-align: center" | [[25/24]]
| | -7.8
| -1.7
|-
|-
| | 5
| 6
| | 68.96552
| 82.75862
| | 25/24
| style="text-align: center" | [[21/20]]
| | -1.7
| -1.7
|-
|-
| | 6
| 7
| | 82.75862
| 96.55172
| | 21/20
| style="text-align: center" | [[35/33]]
| | -1.7
| -5.3
|-
|-
| | 7
| 8
| | 96.55172
| 110.34483
| | 35/33
| style="text-align: center" | [[16/15]]
| | -5.3
| -1.4
|-
|-
| | 8
| 9
| | 110.34483
| 124.13793
| | 16/15
| style="text-align: center" | [[14/13]]
| | -1.4
| -4.2
|-
|-
| | 9
| 10
| | 124.13793
| 137.93103
| | 14/13
| style="text-align: center" | [[13/12]]
| | -4.2
| -0.6
|-
|-
| | 10
| 11
| | 137.93103
| 151.72414
| | 13/12
| style="text-align: center" | [[12/11]]
| | -0.6
| 1.1
|-
|-
| | 11
| 12
| | 151.72414
| 165.51724
| | 12/11
| style="text-align: center" | [[11/10]]
| | 1.1
| 0.5
|-
|-
| | 12
| 13
| | 165.51724
| 179.31035
| | 11/10
| style="text-align: center" | [[10/9]]
| | 0.5
| -3.1
|-
|-
| | 13
| 14
| | 179.31035
| 193.10345
| | 10/9
| style="text-align: center" | [[28/25]]
| | -3.1
| -3.1
|-
|-
| | 14
| 15
| | 193.10345
| 206.89655
| | 28/25
| style="text-align: center" | [[9/8]]
| | -3.1
| 3.0
|-
|-
| | 15
| 16
| | 206.89655
| 220.68966
| | 9/8
| style="text-align: center" | [[25/22]]
| | 3.0
| -0.6
|-
|-
| | 16
| 17
| | 220.68966
| 234.48276
| | 25/22
| style="text-align: center" | [[8/7]]
| | -0.6
| 3.3
|-
|-
| | 17
| 18
| | 234.48276
| 248.27586
| | 8/7
| style="text-align: center" | [[15/13]]
| | 3.3
| 0.5
|-
|-
| | 18
| 19
| | 248.27586
| 262.06897
| | 15/13
| style="text-align: center" | [[7/6]]
| | 0.5
| -4.8
|-
|-
| | 19
| 20
| | 262.06897
| 275.86207
| | 7/6
| style="text-align: center" | [[75/64]]
| | -4.8
| 1.3
|-
|-
| | 20
| 21
| | 275.86207
| 289.65517
| | 75/64
| style="text-align: center" | [[13/11]]
| | 1.3
| 0.4
|-
|-
| | 21
| 22
| | 289.65517
| 303.44828
| | 13/11
| style="text-align: center" | [[25/21]]
| | 0.4
| 1.6
|-
|-
| | 22
| 23
| | 303.44828
| 317.24138
| | 25/21
| style="text-align: center" | [[6/5]]
| | 1.6
| 1.6
|-
|-
| | 23
| 24
| | 317.24138
| 331.03448
| | 6/5
| style="text-align: center" | [[40/33]]
| | 1.6
| -2.0
|-
|-
| | 24
| 25
| | 331.03448
| 344.82759
| | 40/33
| style="text-align: center" | [[11/9]]
| | -2.0
| -2.6
|-
|-
| | 25
| 26
| | 344.82759
| 358.62069
| | 11/9
| style="text-align: center" | [[16/13]]
| | -2.6
| -0.9
|-
|-
| | 26
| 27
| | 358.62069
| 372.41379
| | 16/13
| style="text-align: center" | [[26/21]]
| | -0.9
| 2.7
|-
|-
| | 27
| 28
| | 372.41379
| 386.20690
| | 26/21
| style="text-align: center" | [[5/4]]
| | 2.7
| -0.1
|-
|-
| | 28
| 29
| | 386.20690
| 400.00000
| | 5/4
| style="text-align: center" | [[44/35]]
| | -0.1
| 3.8
|-
|-
| | 29
| 30
| | 400.00000
| 413.79310
| | 44/35
| style="text-align: center" | [[14/11]]
| | 3.8
| -3.7
|-
|-
| | 30
| 31
| | 413.79310
| 427.58621
| | 14/11
| style="text-align: center" | [[32/25]]
| | -3.7
| 0.2
|-
|-
| | 31
| 32
| | 427.58621
| 441.37931
| | 32/25
| style="text-align: center" | [[9/7]]
| | 0.2
| 6.3
|-
|-
| | 32
| 33
| | 441.37931
| 455.17241
| | 9/7
| style="text-align: center" | [[13/10]]
| | 6.3
| 1.0
|-
|-
| | 33
| 34
| | 455.17241
| 468.96552
| | 13/10
| style="text-align: center" | [[21/16]]
| | 1.0
| -1.8
|-
|-
| | 34
| 35
| | 468.96552
| 482.75862
| | 21/16
| style="text-align: center" | [[33/25]]
| | -1.8
| 2.1
|-
|-
| | 35
| 36
| | 482.75862
| 496.55172
| | 33/25
| style="text-align: center" | [[4/3]]
| | 2.1
| -1.5
|-
|-
| | 36
| 37
| | 496.55172
| 510.34483
| | 4/3
| style="text-align: center" | [[35/26]]
| | -1.5
| -4.3
|-
|-
| | 37
| 38
| | 510.34483
| 524.13793
| | 35/26
| style="text-align: center" | [[27/20]]
| | -4.3
| 4.6
|-
|-
| | 38
| 39
| | 524.13793
| 537.93103
| | 27/20
| style="text-align: center" | [[15/11]]
| | 4.6
| 1.0
|-
|-
| | 39
| 40
| | 537.93103
| 551.72414
| | 15/11
| style="text-align: center" | [[11/8]]
| | 1.0
| 0.4
|-
|-
| | 40
| 41
| | 551.72414
| 565.51724
| | 11/8
| style="text-align: center" | [[18/13]]
| | 0.4
| 2.1
|-
|-
| | 41
| 42
| | 565.51724
| 579.31035
| | 18/13
| style="text-align: center" | [[7/5]]
| | 2.1
| -3.2
|-
|-
| | 42
| 43
| | 579.31035
| 593.10345
| | 7/5
| style="text-align: center" | [[45/32]]
| | -3.2
| 2.9
|-
|-
| | 43
| 44
| | 593.10345
| 606.89655
| | 45/32
| style="text-align: center" | [[64/45]]
| | 2.9
| -2.9
|-
|-
| | 44
| 45
| | 606.89655
| 620.68966
| | 64/45
| style="text-align: center" | [[10/7]]
| | -2.9
| 3.2
|-
|-
| | 45
| 46
| | 620.68966
| 634.48276
| | 10/7
| style="text-align: center" | [[13/9]]
| | 3.2
| -2.1
|-
|-
| | 46
| 47
| | 634.48276
| 648.27586
| | 13/9
| style="text-align: center" | [[16/11]]
| | -2.1
| -0.4
|-
|-
| | 47
| 48
| | 648.27586
| 662.06897
| | 16/11
| style="text-align: center" | [[22/15]]
| | -0.4
| -1.0
|-
|-
| | 48
| 49
| | 662.06897
| 675.86207
| | 22/15
| style="text-align: center" | [[40/27]]
| | -1.0
| -4.6
|-
|-
| | 49
| 50
| | 675.86207
| 689.65517
| | 40/27
| style="text-align: center" | [[52/35]]
| | -4.6
| 4.3
|-
|-
| | 50
| 51
| | 689.65517
| 703.44828
| | 52/35
| style="text-align: center" | [[3/2]]
| | 4.3
| 1.5
|-
|-
| | 51
| 52
| | 703.44828
| 717.24138
| | 3/2
| style="text-align: center" | [[50/33]]
| | 1.5
| -2.1
|-
|-
| | 52
| 53
| | 717.24138
| 731.03448
| | 50/33
| style="text-align: center" | [[32/21]]
| | -2.1
| 1.8
|-
|-
| | 53
| 54
| | 731.03448
| 744.82759
| | 32/21
| style="text-align: center" | [[20/13]]
| | 1.8
| -1.0
|-
|-
| | 54
| 55
| | 744.82759
| 758.62069
| | 20/13
| style="text-align: center" | [[14/9]]
| | -1.0
| -6.3
|-
|-
| | 55
| 56
| | 758.62069
| 772.41379
| | 14/9
| style="text-align: center" | [[25/16]]
| | -6.3
| -0.2
|-
|-
| | 56
| 57
| | 772.41379
| 786.20690
| | 25/16
| style="text-align: center" | [[11/7]]
| | -0.2
| 3.7
|-
|-
| | 57
| 58
| | 786.20690
| 800.00000
| | 11/7
| style="text-align: center" | [[35/22]]
| | 3.7
| -3.8
|-
|-
| | 58
| 59
| | 800.00000
| 813.79310
| | 35/22
| style="text-align: center" | [[8/5]]
| | -3.8
| 0.1
|-
|-
| | 59
| 60
| | 813.79310
| 827.58621
| | 8/5
| style="text-align: center" | [[21/13]]
| | 0.1
| -2.7
|-
|-
| | 60
| 61
| | 827.58621
| 841.37931
| | 21/13
| style="text-align: center" | [[13/8]]
| | -2.7
| 0.9
|-
|-
| | 61
| 62
| | 841.37931
| 855.17241
| | 13/8
| style="text-align: center" | [[18/11]]
| | 0.9
| 2.6
|-
|-
| | 62
| 63
| | 855.17241
| 868.96552
| | 18/11
| style="text-align: center" | [[33/20]]
| | 2.6
| 2.0
|-
|-
| | 63
| 64
| | 868.96552
| 882.75862
| | 33/20
| style="text-align: center" | [[5/3]]
| | 2.0
| -1.6
|-
|-
| | 64
| 65
| | 882.75862
| 896.55172
| | 5/3
| style="text-align: center" | [[42/25]]
| | -1.6
| -1.6
|-
|-
| | 65
| 66
| | 896.55172
| 910.34483
| | 42/25
| style="text-align: center" | [[22/13]]
| | -1.6
| -0.4
|-
|-
| | 66
| 67
| | 910.34483
| 924.13793
| | 22/13
| style="text-align: center" | [[75/44]]
| | -0.4
| 0.9
|-
|-
| | 67
| 68
| | 924.13793
| 937.93103
| | 75/44
| style="text-align: center" | [[12/7]]
| | 0.9
| 4.8
|-
|-
| | 68
| 69
| | 937.93103
| 951.72414
| | 12/7
| style="text-align: center" | [[26/15]]
| | 4.8
| -0.5
|-
|-
| | 69
| 70
| | 951.72414
| 965.51724
| | 26/15
| style="text-align: center" | [[7/4]]
| | -0.5
| -3.3
|-
|-
| | 70
| 71
| | 965.51724
| 979.31035
| | 7/4
| style="text-align: center" | [[44/25]]
| | -3.3
| 0.6
|-
|-
| | 71
| 72
| | 979.31035
| 993.10345
| | 44/25
| style="text-align: center" | [[16/9]]
| | 0.6
| -3.0
|-
|-
| | 72
| 73
| | 993.10345
| 1006.89655
| | 16/9
| style="text-align: center" | [[25/14]]
| | -3.0
| 3.1
|-
|-
| | 73
| 74
| | 1006.89655
| 1020.68966
| | 25/14
| style="text-align: center" | [[9/5]]
| | 3.1
| 3.1
|-
|-
| | 74
| 75
| | 1020.68966
| 1034.48276
| | 9/5
| style="text-align: center" | [[20/11]]
| | 3.1
| -0.5
|-
|-
| | 75
| 76
| | 1034.48276
| 1048.27586
| | 20/11
| style="text-align: center" | [[11/6]]
| | -0.5
| -1.1
|-
|-
| | 76
| 77
| | 1048.27586
| 1062.06897
| | 11/6
| style="text-align: center" | [[24/13]]
| | -1.1
| 0.6
|-
|-
| | 77
| 78
| | 1062.06897
| 1075.86207
| | 24/13
| style="text-align: center" | [[13/7]]
| | 0.6
| 4.2
|-
|-
| | 78
| 79
| | 1075.86207
| 1089.65517
| | 13/7
| style="text-align: center" | [[15/8]]
| | 4.2
| 1.4
|-
|-
| | 79
| 80
| | 1089.65517
| 1103.44828
| | 15/8
| style="text-align: center" | [[66/35]]
| | 1.4
| 5.3
|-
|-
| | 80
| 81
| | 1103.44828
| 1117.24138
| | 66/35
| style="text-align: center" | [[21/11]]
| | 5.3
| -2.2
|-
|-
| | 81
| 82
| | 1117.24138
| 1131.03448
| | 21/11
| style="text-align: center" | [[25/13]]
| | -2.2
| -1.1
|-
|-
| | 82
| 83
| | 1131.03448
| 1144.82759
| | 25/13
| style="text-align: center" | [[27/14]]
| | -1.1
| 7.8
|-
|-
| | 83
| 84
| | 1144.82759
| 1158.62069
| | 27/14
| style="text-align: center" | [[39/20]]
| | 7.8
| 2.5
|-
|-
| | 84
| 85
| | 1158.62069
| 1172.41379
| | 39/20
| style="text-align: center" | [[55/28]]
| | 2.5
| 3.6
|-
|-
| | 85
| 86
| | 1172.41379
| 1186.20690
| | 55/28
| style="text-align: center" | [[99/50]]
| | 3.6
| 3.6
|-
|-
| | 86
| 87
| | 1186.20690
| 1200.00000
| | 99/50
| style="text-align: center" | [[2/1]]
| | 3.6
| 0.0
|-
| | 87
| | 1200.00000
| | 2/1
| | 0.0
|}
|}


=Music=
== Music ==
 
* [http://www.archive.org/details/Pianodactyl Pianodactyl] [http://www.archive.org/download/Pianodactyl/pianodactyl.mp3 play] by [[Gene Ward Smith]]


[http://www.archive.org/details/Pianodactyl Pianodactyl] [http://www.archive.org/download/Pianodactyl/pianodactyl.mp3 play] by [[Gene_Ward_Smith|Gene Ward Smith]]      [[Category:87edo]]
[[Category:87edo]]
[[Category:clyde]]
[[Category:clyde]]
[[Category:countercata]]
[[Category:countercata]]
Retrieved from "https://en.xen.wiki/w/87edo"