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m →13-limit detempering of 87et: made todo more prominent
Move the detempering table somewhere else and replace w/ a normal one
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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.
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.
== Intervals ==
{| class="wikitable center-all right-2 left-3"
! #
! Cents
! Approximated Ratios
! [[Ups and Downs Notation]]
|-
|0
|0.000
|1/1
|D
|-
|1
|13.793
|126/125, 100/99, 91/90
|^D
|-
|2
|27.586
|81/80, 64/63, 49/48, 55/54, 65/64
|^^D
|-
|3
|41.379
|50/49, 45/44
|^<sup>3</sup>D/v<sup>3</sup>Eb
|-
|4
|55.172
|28/27, 36/35, 33/32
|vvEb
|-
|5
|68.966
|25/24, 27/26, 26/25
|vEb
|-
|6
|82.759
|21/20, 22/21
|Eb
|-
|7
|96.552
|35/33
|^Eb
|-
|8
|110.345
|16/15
|^^Eb
|-
|9
|124.138
|15/14, 14/13
|^<sup>3</sup>Eb
|-
|10
|137.931
|13/12
|^<sup>4</sup>Eb
|-
|11
|151.724
|12/11
|v<sup>4</sup>E
|-
|12
|165.517
|11/10
|v<sup>3</sup>E
|-
|13
|179.310
|10/9
|vvE
|-
|14
|193.103
|28/25
|vE
|-
|15
|206.897
|9/8
|E
|-
|16
|220.690
|25/22
|^E
|-
|17
|234.483
|8/7
|^^E
|-
|18
|248.276
|15/13
|^<sup>3</sup>E/v<sup>3</sup>F
|-
|19
|262.089
|7/6
|vvF
|-
|20
|275.862
|75/64
|vF
|-
|21
|289.655
|33/28, 13/11
|F
|-
|22
|303.448
|25/21
|^F
|-
|23
|317.241
|6/5
|^^F
|-
|24
|331.034
|63/52
|^<sup>3</sup>F
|-
|25
|344.828
|11/9, 39/32
|^<sup>4</sup>F
|-
|26
|358.621
|27/22, 16/13
|v<sup>4</sup>F#
|-
|27
|372.414
|26/21
|v<sup>3</sup>F#
|-
|28
|386.207
|5/4
|vvF#
|-
|29
|400.000
|63/50, 44/35
|vF#
|-
|30
|413.793
|14/11, 33/26
|F#
|-
|31
|427.586
|32/25
|^F#
|-
|32
|441.379
|9/7
|^^F#
|-
|33
|455.172
|13/10
|^<sup>3</sup>F#/v<sup>3</sup>G
|-
|34
|468.966
|21/16
|vvG
|-
|35
|482.759
|33/25
|vG
|-
|36
|496.552
|4/3
|G
|-
|37
|510.345
|75/56
|^G
|-
|38
|524.138
|27/20
|^^G
|-
|39
|537.931
|15/11
|^<sup>3</sup>G
|-
|40
|551.724
|11/8
|^<sup>4</sup>G
|-
|41
|565.517
|18/13
|v<sup>4</sup>G#, vAb
|-
|42
|579.310
|7/5, 39/28
|v<sup>3</sup>G#, Ab
|-
|43
|593.103
|45/32
|vvG#, ^Ab
|}


== Rank two temperaments ==
== Rank two temperaments ==
Line 83: Line 311:
== 13-limit detempering of 87et ==
== 13-limit detempering of 87et ==


See [[Detempering|detempering]].
''See also: [[Detempering]]''


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


== Music ==
== Music ==
Line 533: Line 319:
* [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]]


[[Category:theory]]
[[Category:edo]]
[[Category:87edo]]
[[Category:87edo]]
[[Category:listen]]
[[Category:clyde]]
[[Category:clyde]]
[[Category:countercata]]
[[Category:countercata]]
[[Category:edo]]
[[Category:hemithirds]]
[[Category:hemithirds]]
[[Category:listen]]
[[Category:mystery]]
[[Category:mystery]]
[[Category:rodan]]
[[Category:rodan]]
[[Category:theory]]
[[Category:tritikleismic]]
[[Category:tritikleismic]]

Revision as of 14:58, 12 June 2020

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 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 consistently, and is the smallest equal temperament to do so.

87et 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 rodan temperament. The 8/7 generator of 17\87 is a remarkable 0.00062 cents sharper than the 13-limit POTE generator and is close to the 11-limit POTE generator also. Also, the 32\87 generator for clyde temperament is 0.04455 cents sharp of the 7-limit POTE generator.

Intervals

# Cents Approximated Ratios Ups and Downs Notation
0 0.000 1/1 D
1 13.793 126/125, 100/99, 91/90 ^D
2 27.586 81/80, 64/63, 49/48, 55/54, 65/64 ^^D
3 41.379 50/49, 45/44 ^3D/v3Eb
4 55.172 28/27, 36/35, 33/32 vvEb
5 68.966 25/24, 27/26, 26/25 vEb
6 82.759 21/20, 22/21 Eb
7 96.552 35/33 ^Eb
8 110.345 16/15 ^^Eb
9 124.138 15/14, 14/13 ^3Eb
10 137.931 13/12 ^4Eb
11 151.724 12/11 v4E
12 165.517 11/10 v3E
13 179.310 10/9 vvE
14 193.103 28/25 vE
15 206.897 9/8 E
16 220.690 25/22 ^E
17 234.483 8/7 ^^E
18 248.276 15/13 ^3E/v3F
19 262.089 7/6 vvF
20 275.862 75/64 vF
21 289.655 33/28, 13/11 F
22 303.448 25/21 ^F
23 317.241 6/5 ^^F
24 331.034 63/52 ^3F
25 344.828 11/9, 39/32 ^4F
26 358.621 27/22, 16/13 v4F#
27 372.414 26/21 v3F#
28 386.207 5/4 vvF#
29 400.000 63/50, 44/35 vF#
30 413.793 14/11, 33/26 F#
31 427.586 32/25 ^F#
32 441.379 9/7 ^^F#
33 455.172 13/10 ^3F#/v3G
34 468.966 21/16 vvG
35 482.759 33/25 vG
36 496.552 4/3 G
37 510.345 75/56 ^G
38 524.138 27/20 ^^G
39 537.931 15/11 ^3G
40 551.724 11/8 ^4G
41 565.517 18/13 v4G#, vAb
42 579.310 7/5, 39/28 v3G#, Ab
43 593.103 45/32 vvG#, ^Ab

Rank two temperaments

Periods
per
octave 		Generator Cents Associated
ratio 		Temperament
1 4\87 55.172 33/32 Sensa
1 10\87 137.931 13/12 Quartemka
1 14\87 193.103 28/25 Luna / Hemithirds
1 17\87 234.483 8/7 Rodan
1 23\87 317.241 6/5 Hanson / Countercata / Metakleismic
1 32\87 441.379 9/7 Clyde
1 38\87 524.138 65/48 Widefourth
1 40\87 551.724 11/8 Emkay
3 23\87 317.241 6/5 Tritikleismic
29 28\87 386.207 5/4 Mystery

87 can serve as a MOS in these:

  • 270&87 <<24 -9 -66 12 27 ... ||
  • 494&87 <<51 -1 -133 11 32 ... ||

13-limit detempering of 87et

See also: Detempering

Main article: 87edo/13-limit detempering

Music

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