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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.
{{Infobox ET}}
{{ED intro}}


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.
== Theory ==
87edo is solid as both a [[13-limit]] (or [[15-odd-limit]]) and as a [[5-limit]] system, and does well enough in any limit in between. It is the smallest edo that is [[distinctly consistent]] in the [[13-odd-limit]] [[tonality diamond]], and the smallest edo that is [[purely consistent]]{{idiosyncratic}} in the [[15-odd-limit]] (maintains [[relative interval error]]s of no greater than 25% on all of the first 16 [[harmonic]]s of the [[harmonic series]]). It is also a [[zeta peak integer edo]]. Since {{nowrap|87 {{=}} 3 × 29}}, 87edo shares the same perfect fifth with [[29edo]].  


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.
87edo also shows some potential in limits beyond 13. The next four prime harmonics [[17/1|17]], [[19/1|19]], [[23/1|23]], and [[29/1|29]] are all near-critically sharp, but the feature of it is that the overtones and undertones are distinct, and most intervals are usable as long as they do not combine with [[7/1|7]], which is flat. Actually, as a no-sevens system, it is consistent in the 33-odd-limit.  


=Rank two temperaments=
It [[tempering out|tempers out]] 15625/15552 ([[15625/15552|kleisma]]), {{monzo| 26 -12 -3 }} ([[misty comma]]), and {{monzo| 46 -29 }} ([[29-comma]]) in the 5-limit, in addition to [[245/243]], [[1029/1024]], [[3136/3125]], and [[5120/5103]] in the 7-limit. In the 13-limit, notably [[196/195]], [[325/324]], [[352/351]], [[364/363]], [[385/384]], [[441/440]], [[625/624]], [[676/675]], and [[1001/1000]].


{| class="wikitable"
87edo is a particularly good tuning for [[rodan]], the {{nowrap|41 & 46}} temperament. The 8/7 generator of 17\87 is a remarkable 0.00061{{c}} sharper than the 13-limit [[CWE tuning|CWE generator]]. Also, the 32\87 generator for [[Kleismic family #Clyde|clyde temperament]] is 0.01479{{c}} sharp of the 13-limit CWE generator.
|-
! | Periods


per
=== Prime harmonics ===
In higher limits it excels as a [[subgroup]] temperament, especially as an incomplete 71-limit temperament with [[128/127]] and [[129/128]] (the subharmonic and harmonic hemicomma-sized intervals, respectively) mapped accurately to a single step. Generalizing a single step of 87edo harmonically yields harmonics 115 through 138, which when detempered is the beginning of the construction of [[Ringer scale|Ringer]] 87, thus tempering [[S-expression|S116 through S137]] by patent val and corresponding to the gravity of the fact that 87edo is a circle of [[126/125]]'s, meaning ([[126/125]])<sup>87</sup> only very slightly exceeds the octave.
{{Harmonics in equal|87|columns=12}}
{{Harmonics in equal|87|columns=12|start=13|collapsed=1|title=Approximation of prime harmonics in 87edo (continued)}}


octave
=== Subsets and supersets ===
! | Generator
87edo contains [[3edo]] and [[29edo]] as subset edos.
! | Cents
! | Associated


ratio
[[348edo]], which slices the edostep in four, provides a good correction of the 7th harmonic. Multiples like [[696edo]] and [[783edo]] are also of theoretical interest.
! | Temperament
|-
| style="text-align:right;" | 1
| style="text-align:right;" | 4\87
| style="text-align:right;" | 55.172
| style="text-align:center;" | 33/32
| | [[Sensa|Sensa]]
|-
| style="text-align:right;" | 1
| style="text-align:right;" | 10\87
| style="text-align:right;" | 137.931
| style="text-align:center;" | 13/12
| | [[Quartemka|Quartemka]]
|-
| style="text-align:right;" | 1
| style="text-align:right;" | 14\87
| style="text-align:right;" | 193.103
| style="text-align:center;" | 28/25
| | [[Luna|Luna]]/[[Hemithirds|hemithirds]]
|-
| style="text-align:right;" | 1
| style="text-align:right;" | 17\87
| style="text-align:right;" | 234.483
| style="text-align:center;" | 8/7
| | [[Rodan|Rodan]]
|-
| style="text-align:right;" | 1
| style="text-align:right;" | 23\87
| style="text-align:right;" | 317.241
| style="text-align:center;" | 6/5
| | [[Hanson|Hanson]]/[[Countercata|countercata]]/[[metakleismic|metakleismic]]
|-
| style="text-align:right;" | 1
| style="text-align:right;" | 32\87
| style="text-align:right;" | 441.379
| style="text-align:center;" | 9/7
| | [[Clyde|Clyde]]
|-
| style="text-align:right;" | 1
| style="text-align:right;" | 38\87
| style="text-align:right;" | 524.138
| style="text-align:center;" | 65/48
| | [[Widefourth|Widefourth]]
|-
| style="text-align:right;" | 1
| style="text-align:right;" | 40\87
| style="text-align:right;" | 551.724
| style="text-align:center;" | 11/8
| | [[Emkay|Emkay]]
|-
| style="text-align:right;" | 3
| style="text-align:right;" | 23\87
| style="text-align:right;" | 317.241
| style="text-align:center;" | 6/5
| | [[Tritikleismic|Tritikleismic]]
|-
| style="text-align:right;" | 29
| style="text-align:right;" | 28\87
| style="text-align:right;" | 386.207
| style="text-align:center;" | 5/4
| | [[mystery|Mystery]]
|}


87 can serve as a MOS in these:
One step of 87edo is ''extremely'' close to [[126/125]], with the associated comma, 87-126/125-comma ({{monzo|86 174 -261 87}}) being tempered in equal temperaments well over 100,000 and with less than 0.001{{c}} error, with the [[optimal patent val]] for the 7-limit temperament being an enormous [[425082edo]]. But more notably, 87edo itself does the mapping consistently, making it a true circle of 126/125's.


[[M&N_temperaments|270&amp;87]] &lt;&lt;24 -9 -66 12 27 ... ||
See also [[87th-octave temperaments]].


[[M&N_temperaments|494&amp;87]] &lt;&lt;51 -1 -133 11 32 ... ||
== Intervals ==
 
{| class="wikitable center-all right-2 left-3 left-4"
=13-limit detempering of 87et=
 
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.
 
{|class="wikitable"
|-
|-
| | Steps of 87
! rowspan="2" | #
| | Cents value
! rowspan="2" | Cents
| | Detempered Interval
! colspan="2" | Approximated ratios
| | Difference in Cents
! colspan="2" rowspan="2" | [[Ups and downs notation]]
|-
|-
| | 1
! 13-limit
| | 13.79310
! 31-limit extension
| | 91/90
| | -5.3
|-
|-
| | 2
| 0
| | 27.58621
| 0.0
| | 49/48
| [[1/1]]
| | -8.1
|
| P1
| D
|-
|-
| | 3
| 1
| | 41.37931
| 13.8
| | 40/39
| [[91/90]], [[100/99]], [[126/125]]
| | -2.5
|
| ^1
| ^D
|-
|-
| | 4
| 2
| | 55.17241
| 27.6
| | 28/27
| ''[[49/48]]'', [[55/54]], [[64/63]], [[65/64]], [[81/80]]
| | -7.8
|
| ^^1
| ^^D
|-
|-
| | 5
| 3
| | 68.96552
| 41.4
| | 25/24
| [[40/39]], [[45/44]], [[50/49]]
| | -1.7
| [[39/38]]
| ^<sup>3</sup>1
| ^<sup>3</sup>D/v<sup>3</sup>Eb
|-
|-
| | 6
| 4
| | 82.75862
| 55.2
| | 21/20
| ''[[28/27]]'', [[33/32]], [[36/35]]
| | -1.7
| [[30/29]], [[31/30]], [[32/31]], [[34/33]]
| vvm2
| vvEb
|-
|-
| | 7
| 5
| | 96.55172
| 69.0
| | 35/33
| [[25/24]], [[26/25]], [[27/26]]
| | -5.3
| [[24/23]]
| vm2
| vEb
|-
|-
| | 8
| 6
| | 110.34483
| 82.8
| | 16/15
| [[21/20]], [[22/21]]
| | -1.4
| [[20/19]], [[23/22]]
| m2
| Eb
|-
|-
| | 9
| 7
| | 124.13793
| 96.6
| | 14/13
| [[35/33]]
| | -4.2
| [[18/17]], [[19/18]]
| ^m2
| ^Eb
|-
|-
| | 10
| 8
| | 137.93103
| 110.3
| | 13/12
| [[16/15]]
| | -0.6
| [[17/16]], [[31/29]], [[33/31]]
| ^^m2
| ^^Eb
|-
|-
| | 11
| 9
| | 151.72414
| 124.1
| | 12/11
| [[14/13]], [[15/14]]
| | 1.1
| [[29/27]]
| vv~2
| ^<sup>3</sup>Eb
|-
|-
| | 12
| 10
| | 165.51724
| 137.9
| | 11/10
| [[13/12]], [[27/25]]
| | 0.5
| [[25/23]]
| v~2
| ^<sup>4</sup>Eb
|-
|-
| | 13
| 11
| | 179.31035
| 151.7
| | 10/9
| [[12/11]], [[35/32]]
| | -3.1
|
| ^~2
| v<sup>4</sup>E
|-
|-
| | 14
| 12
| | 193.10345
| 165.5
| | 28/25
| [[11/10]]
| | -3.1
| [[32/29]], [[34/31]]
| ^^~2
| v<sup>3</sup>E
|-
|-
| | 15
| 13
| | 206.89655
| 179.3
| | 9/8
| [[10/9]]
| | 3.0
|
| vvM2
| vvE
|-
|-
| | 16
| 14
| | 220.68966
| 193.1
| | 25/22
| [[28/25]]
| | -0.6
| [[19/17]], [[29/26]]
| vM2
| vE
|-
|-
| | 17
| 15
| | 234.48276
| 206.9
| | 8/7
| [[9/8]]
| | 3.3
| [[26/23]]
| M2
| E
|-
|-
| | 18
| 16
| | 248.27586
| 220.7
| | 15/13
| [[25/22]]
| | 0.5
| [[17/15]], [[33/29]]
| ^M2
| ^E
|-
|-
| | 19
| 17
| | 262.06897
| 234.5
| | 7/6
| [[8/7]]
| | -4.8
| [[31/27]]
| ^^M2
| ^^E
|-
|-
| | 20
| 18
| | 275.86207
| 248.3
| | 75/64
| [[15/13]]
| | 1.3
| [[22/19]], [[23/20]], [[38/33]]
| ^<sup>3</sup>M2/v<sup>3</sup>m3
| ^<sup>3</sup>E/v<sup>3</sup>F
|-
|-
| | 21
| 19
| | 289.65517
| 262.1
| | 13/11
| [[7/6]]
| | 0.4
| [[29/25]], [[36/31]]
| vvm3
| vvF
|-
|-
| | 22
| 20
| | 303.44828
| 275.9
| | 25/21
| [[75/64]]
| | 1.6
| [[20/17]], [[27/23]], [[34/29]]
| vm3
| vF
|-
|-
| | 23
| 21
| | 317.24138
| 289.7
| | 6/5
| [[13/11]], [[32/27]], [[33/28]]
| | 1.6
|
| m3
| F
|-
|-
| | 24
| 22
| | 331.03448
| 303.4
| | 40/33
| [[25/21]]
| | -2.0
| [[19/16]], [[31/26]]
| ^m3
| ^F
|-
|-
| | 25
| 23
| | 344.82759
| 317.2
| | 11/9
| [[6/5]]
| | -2.6
|
| ^^m3
| ^^F
|-
|-
| | 26
| 24
| | 358.62069
| 331.0
| | 16/13
| [[40/33]]
| | -0.9
| [[23/19]], [[29/24]]
| vv~3
| ^<sup>3</sup>F
|-
|-
| | 27
| 25
| | 372.41379
| 344.8
| | 26/21
| [[11/9]], [[39/32]]
| | 2.7
|
| v~3
| ^<sup>4</sup>F
|-
|-
| | 28
| 26
| | 386.20690
| 358.6
| | 5/4
| [[16/13]], [[27/22]]
| | -0.1
| [[38/31]]
| ^~3
| v<sup>4</sup>F#
|-
|-
| | 29
| 27
| | 400.00000
| 372.4
| | 44/35
| [[26/21]]
| | 3.8
| [[31/25]], [[36/29]]
| ^^3
| v<sup>3</sup>F#
|-
|-
| | 30
| 28
| | 413.79310
| 386.2
| | 14/11
| [[5/4]]
| | -3.7
|
| vvM3
| vvF#
|-
|-
| | 31
| 29
| | 427.58621
| 400.0
| | 32/25
| [[44/35]]
| | 0.2
| [[24/19]], [[29/23]], [[34/27]]
| vM3
| vF#
|-
|-
| | 32
| 30
| | 441.37931
| 413.8
| | 9/7
| [[14/11]], [[33/26]], [[81/64]]
| | 6.3
| [[19/15]]
| M3
| F#
|-
|-
| | 33
| 31
| | 455.17241
| 427.6
| | 13/10
| [[32/25]]
| | 1.0
| [[23/18]]
| ^M3
| ^F#
|-
|-
| | 34
| 32
| | 468.96552
| 441.4
| | 21/16
| [[9/7]], [[35/27]]
| | -1.8
| [[22/17]], [[31/24]], [[40/31]]
| ^^M3
| ^^F#
|-
|-
| | 35
| 33
| | 482.75862
| 455.2
| | 33/25
| [[13/10]]
| | 2.1
| [[30/23]]
| ^<sup>3</sup>M3/v<sup>3</sup>4
| ^<sup>3</sup>F#/v<sup>3</sup>G
|-
|-
| | 36
| 34
| | 496.55172
| 469.0
| | 4/3
| [[21/16]]
| | -1.5
| [[17/13]], [[25/19]], [[38/29]]
| vv4
| vvG
|-
|-
| | 37
| 35
| | 510.34483
| 482.8
| | 35/26
| [[33/25]]
| | -4.3
|
| v4
| vG
|-
|-
| | 38
| 36
| | 524.13793
| 496.6
| | 27/20
| [[4/3]]
| | 4.6
|
| P4
| G
|-
|-
| | 39
| 37
| | 537.93103
| 510.3
| | 15/11
| [[35/26]]
| | 1.0
| [[31/23]]
| ^4
| ^G
|-
|-
| | 40
| 38
| | 551.72414
| 524.1
| | 11/8
| [[27/20]]
| | 0.4
| [[23/17]]
| ^^4
| ^^G
|-
|-
| | 41
| 39
| | 565.51724
| 537.9
| | 18/13
| [[15/11]]
| | 2.1
| [[26/19]], [[34/25]]
| ^<sup>3</sup>4
| ^<sup>3</sup>G
|-
|-
| | 42
| 40
| | 579.31035
| 551.7
| | 7/5
| [[11/8]], [[48/35]]
| | -3.2
|
| ^<sup>4</sup>4
| ^<sup>4</sup>G
|-
|-
| | 43
| 41
| | 593.10345
| 565.5
| | 45/32
| [[18/13]]
| | 2.9
| [[32/23]]
| v<sup>4</sup>A4, vd5
| v<sup>4</sup>G#, vAb
|-
|-
| | 44
| 42
| | 606.89655
| 579.3
| | 64/45
| [[7/5]]
| | -2.9
| [[46/33]]
| v<sup>3</sup>A4, d5
| v<sup>3</sup>G#, Ab
|-
|-
| | 45
| 43
| | 620.68966
| 593.1
| | 10/7
| [[45/32]]
| | 3.2
| [[24/17]], [[31/22]], [[38/27]]
| vvA4, ^d5
| vvG#, ^Ab
|-
|-
| | 46
| …
| | 634.48276
| …
| | 13/9
| …
| | -2.1
| …
| …
| …
|}
 
== Notation ==
=== Ups and downs notation ===
87edo can be written using [[Kite's ups and downs notation]]. Note that quudsharp (quadruple-down sharp) is equivalent to quip (quintuple-up) and that quupflat (quadruple-up flat) is equivalent to quid (quintuple-down):
{{Ups and downs sharpness}}
Mapping an arrow to 2\87 rather than 1\87 is an alternative approach which takes advantage of 87edo being a tuning of akea temperament. This way, one arrow is equivalent to 81/80~64/63, and two arrows are equivalent to 33/32~1053/1024.
 
== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals|87}}
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
|-
| | 47
! rowspan="2" | [[Subgroup]]
| | 648.27586
! rowspan="2" | [[Comma list]]
| | 16/11
! rowspan="2" | [[Mapping]]
| | -0.4
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
| | 48
! [[TE error|Absolute]] (¢)
| | 662.06897
! [[TE simple badness|Relative]] (%)
| | 22/15
| | -1.0
|-
|-
| | 49
| 2.3.5
| | 675.86207
| 15625/15552, 67108864/66430125
| | 40/27
| {{Mapping| 87 138 202 }}
| | -4.6
| −0.299
| 0.455
| 3.30
|-
|-
| | 50
| 2.3.5.7
| | 689.65517
| 245/243, 1029/1024, 3136/3125
| | 52/35
| {{Mapping| 87 138 202 244 }}
| | 4.3
| +0.070
| 0.752
| 5.45
|-
|-
| | 51
| 2.3.5.7.11
| | 703.44828
| 245/243, 385/384, 441/440, 3136/3125
| | 3/2
| {{Mapping| 87 138 202 244 301 }}
| | 1.5
| +0.033
| 0.676
| 4.90
|-
|-
| | 52
| 2.3.5.7.11.13
| | 717.24138
| 196/195, 245/243, 352/351, 364/363, 625/624
| | 50/33
| {{Mapping| 87 138 202 244 301 322 }}
| | -2.1
| −0.011
| 0.625
| 4.53
|-
|-
| | 53
| 2.3.5.7.11.13.17
| | 731.03448
| 154/153, 196/195, 245/243, 273/272, 364/363, 375/374
| | 32/21
| {{Mapping| 87 138 202 244 301 322 356 }}
| | 1.8
| −0.198
| 0.738
| 5.35
|-
|-
| | 54
| 2.3.5.7.11.13.17.19
| | 744.82759
| 154/153, 196/195, 210/209, 245/243, 273/272, 286/285, 364/363
| | 20/13
| {{Mapping| 87 138 202 244 301 322 356 370 }}
| | -1.0
| −0.348
| 0.796
| 5.77
|}
 
=== 13-limit detempering ===
{{Main|87edo/13-limit detempering}}
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
| | 55
! Periods<br>per 8ve
| | 758.62069
! Generator*
| | 14/9
! Cents*
| | -6.3
! Associated<br>ratio*
! Temperament
|-
|-
| | 56
| 1
| | 772.41379
| 2\87
| | 25/16
| 27.586
| | -0.2
| 64/63
| [[Arch]]
|-
|-
| | 57
| 1
| | 786.20690
| 4\87
| | 11/7
| 55.172
| | 3.7
| 33/32
| [[Escapade]] / [[escaped]] / [[alphaquarter]]
|-
|-
| | 58
| 1
| | 800.00000
| 10\87
| | 35/22
| 137.931
| | -3.8
| 13/12
| [[Quartemka]]
|-
|-
| | 59
| 1
| | 813.79310
| 14\87
| | 8/5
| 193.103
| | 0.1
| 28/25
| [[Luna]] / [[didacus]] / [[hemithirds]]
|-
|-
| | 60
| 1
| | 827.58621
| 17\87
| | 21/13
| 234.483
| | -2.7
| 8/7
| [[Slendric]] / [[rodan]]
|-
|-
| | 61
| 1
| | 841.37931
| 23\87
| | 13/8
| 317.241
| | 0.9
| 6/5
| [[Hanson]] / [[countercata]] / [[metakleismic]]
|-
|-
| | 62
| 1
| | 855.17241
| 26\87
| | 18/11
| 358.621
| | 2.6
| 16/13
| [[Restles]]
|-
|-
| | 63
| 1
| | 868.96552
| 32\87
| | 33/20
| 441.379
| | 2.0
| 9/7
| [[Clyde]]
|-
|-
| | 64
| 1
| | 882.75862
| 38\87
| | 5/3
| 524.138
| | -1.6
| 65/48
| [[Widefourth]]
|-
|-
| | 65
| 1
| | 896.55172
| 40\87
| | 42/25
| 551.724
| | -1.6
| 11/8
| [[Emka]] / [[emkay]]
|-
|-
| | 66
| 3
| | 910.34483
| 18\87<br>(11\87)
| | 22/13
| 248.276<br>(151.724)
| | -0.4
| 15/13<br>(12/11)
| [[Hemimist]]
|-
|-
| | 67
| 3
| | 924.13793
| 23\87<br>(6\87)
| | 75/44
| 317.241<br>(82.759)
| | 0.9
| 6/5<br>(21/20)
| [[Tritikleismic]]
|-
|-
| | 68
| 3
| | 937.93103
| 28\87<br>(1\87)
| | 12/7
| 386.207<br>(13.793)
| | 4.8
| 5/4<br>(126/125)
| [[Mutt]]
|-
|-
| | 69
| 3
| | 951.72414
| 36\87<br>(7\87)
| | 26/15
| 496.552<br>(96.552)
| | -0.5
| 4/3<br>(18/17~19/18)
| [[Misty]]
|-
|-
| | 70
| 29
| | 965.51724
| 28\87<br>(1\87)
| | 7/4
| 386.207<br>(13.793)
| | -3.3
| 5/4<br>(121/120)
| [[Mystery]]
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 
87 can serve as a mos in these:
 
* [[Avicenna (temperament)|Avicenna]] ([[Breed|87 & 270]])
* [[Breed|87 & 494]]
 
== Scales ==
=== Harmonic scales ===
87edo very accurately approximates [[Overtone scale|modes]] up to 8 of the [[harmonic series]], and the only interval pair not distinct is 14/13 and 15/14. It can also do mode 12 decently.  
 
==== (Mode 6) ====
{| class="wikitable center-all"
|-
|-
| | 71
! Overtones
| | 979.31035
| 12
| | 44/25
| 14
| | 0.6
| 16
| 18
| 20
| 22
| 24
|-
|-
| | 72
! JI Ratios
| | 993.10345
| 1/1
| | 16/9
| 7/6
| | -3.0
| 4/3
| 3/2
| 5/3
| 11/6
| 2/1
|-
|-
| | 73
! … in cents
| | 1006.89655
| 0.0
| | 25/14
| 266.9
| | 3.1
| 498.0
| 702.0
| 884.4
| 1049.4
| 1200.0
|-
|-
| | 74
! Degrees in 87edo
| | 1020.68966
| 0
| | 9/5
| 19
| | 3.1
| 36
| 51
| 64
| 76
| 87
|-
|-
| | 75
! … in cents
| | 1034.48276
| 0.0
| | 20/11
| 262.1
| | -0.5
| 496.6
| 703.4
| 882.8
| 1048.3
| 1200.0
|}
 
The scale in adjacent steps is '''19 17 15 13 12 11'''.
 
==== (Mode 8) ====
{| class="wikitable center-all"
|-
|-
| | 76
! Overtones
| | 1048.27586
| 8
| | 11/6
| 9
| | -1.1
| 10
| 11
| 12
| 13
| 14
| 15
| 16
|-
|-
| | 77
! JI Ratios
| | 1062.06897
| 1/1
| | 24/13
| 9/8
| | 0.6
| 5/4
| 11/8
| 3/2
| 13/8
| 7/4
| 15/8
| 2/1
|-
|-
| | 78
! … in cents
| | 1075.86207
| 0.0
| | 13/7
| 203.9
| | 4.2
| 386.3
| 551.3
| 702.0
| 840.5
| 968.8
| 1088.3
| 1200.0
|-
|-
| | 79
! Degrees in 87edo
| | 1089.65517
| 0
| | 15/8
| 15
| | 1.4
| 28
| 40
| 51
| 61
| 70
| 79
| 87
|-
|-
| | 80
! … in cents
| | 1103.44828
| 0.0
| | 66/35
| 206.9
| | 5.3
| 386.2
|-
| 551.7
| | 81
| 703.5
| | 1117.24138
| 841.4
| | 21/11
| 965.5
| | -2.2
| 1089.7
| 1200.0
|}
 
The scale in adjacent steps is '''15 13 12 11 10 9 9 8'''.
 
==== (Mode 12) ====
{| class="wikitable center-all"
|-
|-
| | 82
! Overtones
| | 1131.03448
| 12
| | 25/13
| 13
| | -1.1
| 14
| 15
| 16
| 17
| 18
| 19
| 20
| 21
| 22
| 23
| 24
|-
|-
| | 83
! JI Ratios
| | 1144.82759
| 1/1
| | 27/14
| 13/12
| | 7.8
| 7/6
| 5/4
| 4/3
| 17/12
| 3/2
| 19/12
| 5/3
| 7/4
| 11/6
| 23/12
| 2/1
|-
|-
| | 84
! … in cents
| | 1158.62069
| 0.0
| | 39/20
| 138.6
| | 2.5
| 266.9
| 386.3
| 498.0
| 603.0
| 702.0
| 795.6
| 884.4
| 968.8
| 1049.4
| 1126.3
| 1200.0
|-
|-
| | 85
! Degrees in 87edo
| | 1172.41379
| 0
| | 55/28
| 10
| | 3.6
| 19
| 28
| 36
| 44
| 51
| 58
| 64
| 70
| 76
| 82
| 87
|-
|-
| | 86
! … in cents
| | 1186.20690
| 0.0
| | 99/50
| 137.9
| | 3.6
| 262.1
|-
| 386.2
| | 87
| 496.6
| | 1200.00000
| 606.9
| | 2/1
| 703.4
| | 0.0
| 800.0
| 882.8
| 965.5
| 1048.3
| 1131.0
| 1200.0
|}
|}


=Music=
The scale in adjacent steps is '''10 9 9 8 7 7 6 6 6 6 5'''.
 
13, 15, 16, 18, 20, and 22 are close matches.
 
14 and 21 are flat; 17, 19, and 23 are sharp. Still decent all things considered.
 
=== Mos scales ===
{{main|List of MOS scales in 87edo}}
* [[Kleismic]][7]: 5 18 5 18 5 18 18
* Kleismic[15]: 5 8 5 5 5 8 5 5 5 8 5 5 5 8 5
* Kleismic[19]: 5 5 3 5 5 5 5 3 5 5 5 3 5 5 5 5 3 5 5
* Kleismic[34]: 3 2 3 2 3 2 3 2 3 3 2 3 2 3 2 3 2 3 3 2 3 2 3 2 3 3 2 3 2 3 2 3 2 3
* [[Misty]][6]: 22 7 22 7 22 7
* Misty[12]: 7 8 7 7 7 8 7 7 7 8 7 7
* Misty[15]: 7 7 1 7 7 7 7 1 7 7 7 7 1 7 7
* Misty[27]: 1 6 1 6 1 6 1 6 1 1 6 1 6 1 6 1 6 1 1 6 1 6 1 6 1 6 1
* Misty[39]: 1 5 1 1 5 1 1 1 5 1 1 5 1 1 5 1 1 5 1 1 1 5 1 1 5 1 1 5 1 1 5 1 1 1 5 1 1 5 1
* [[Luna]][6]: 17 14 14 14 14 14
* Luna[13]: 3 11 3 11 3 11 3 11 3 11 3 11 3
* Luna[19]: 3 8 3 3 8 3 3 8 3 3 3 8 3 3 8 3 3 8 3
* Luna[25]: 3 3 5 3 3 3 5 3 3 3 5 3 3 3 5 3 3 3 5 3 3 3 5 3 3
* Luna[31]: 3 3 2 3 3 3 3 2 3 3 3 3 2 3 3 3 3 3 2 3 3 3 3 2 3 3 3 3 2 3 3
* Luna[56]: 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2
* [[Rodan]][5]: 17 17 17 17 19
* Rodan[11]: 2 15 2 15 2 15 2 15 2 15 2
* Rodan[16]: 2 13 2 2 13 2 2 2 13 2 2 13 2 2 13 2
* Rodan[21]: 2 2 11 2 2 2 11 2 2 2 11 2 2 2 11 2 2 2 11 2 2
* Rodan[36]: 2 2 2 5 2 2 2 2 2 2 5 2 2 2 2 2 2 2 5 2 2 2 2 2 2 5 2 2 2 2 2 2 5 2 2 2
* Rodan[41]: 2 2 2 2 3 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 3 2 2 2 2
 
=== Other scales ===
* [[Sequar5m]]
 
== Instruments ==
* [[Lumatone mapping for 87edo]]
* [[Skip fretting system 87 2 17]]
 
== Music ==
=== Modern renderings ===
; {{W|Stomach Book}}
* [https://www.youtube.com/shorts/rINJKiMQE78 ''Circuit Bent''] (2024) – microtonal cover in 87edo by [[Bryan Deister]] (2025)
 
=== 21st century ===
; [[ALLY195]]
* [https://www.bilibili.com/video/BV16h411g7QM/ ''Root note and subharmonic series cadence – 103EDO, 87EDO, 94EDO''] (2023)
* [https://www.bilibili.com/video/BV1N84y1T792/ ''A comparison between 87edo and 12edo''] (2023)
 
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/ecxELXmkYAs ''microtonal improvisation in 87edo''] (2025)
* [https://www.youtube.com/shorts/5OH9OOGeuX4 ''87edo waltz''] (2025)
* [https://www.youtube.com/shorts/8mHBYBfRjy4 ''87edo improv''] (2026-07-11)
* [https://www.youtube.com/shorts/wHPyCHQn55A ''87edo improv''] (2026-09-28)
 
; [[Gene Ward Smith]]
* ''Pianodactyl'' (archived 2010) – [https://soundcloud.com/genewardsmith/pianodactyl SoundCloud] | [http://www.archive.org/details/Pianodactyl detail] | [http://www.archive.org/download/Pianodactyl/pianodactyl.mp3 play] – rodan[26] in 87edo tuning


[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:Listen]]
[[Category:clyde]]
[[Category:Clyde]]
[[Category:countercata]]
[[Category:Countercata]]
[[Category:edo]]
[[Category:Hemithirds]]
[[Category:hemithirds]]
[[Category:Mystery]]
[[Category:listen]]
[[Category:Rodan]]
[[Category:mystery]]
[[Category:Tritikleismic]]
[[Category:rodan]]
[[Category:theory]]
[[Category:tritikleismic]]
Retrieved from "https://en.xen.wiki/w/87edo"