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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|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]] (see [[87edo/13-limit detempering]]), 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]], 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|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.
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.
 
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]].
 
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.
 
=== 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)}}
 
=== Subsets and supersets ===
87edo contains [[3edo]] and [[29edo]] as subset edos.


== Intervals ==
== Intervals ==
{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-all right-2 left-3 left-4"
! #
|-
! Cents
! rowspan="2" | #
! Approximated Ratios
! rowspan="2" | Cents
! [[Ups and Downs Notation]]
! colspan="2" | Approximated ratios
! colspan="2" rowspan="2" | [[Ups and downs notation]]
|-
! 13-limit
! 31-limit extension
|-
| 0
| 0.0
| [[1/1]]
|
| P1
| D
|-
| 1
| 13.8
| [[91/90]], [[100/99]], [[126/125]]
|
| ^1
| ^D
|-
| 2
| 27.6
| ''[[49/48]]'', [[55/54]], [[64/63]], [[65/64]], [[81/80]]
|
| ^^1
| ^^D
|-
| 3
| 41.4
| [[40/39]], [[45/44]], [[50/49]]
| [[39/38]]
| ^<sup>3</sup>1
| ^<sup>3</sup>D/v<sup>3</sup>Eb
|-
| 4
| 55.2
| ''[[28/27]]'', [[33/32]], [[36/35]]
| [[30/29]], [[31/30]], [[32/31]], [[34/33]]
| vvm2
| vvEb
|-
|-
|0
| 5
|0.000
| 69.0
|1/1
| [[25/24]], [[26/25]], [[27/26]]
|D
| [[24/23]]
| vm2
| vEb
|-
|-
|1
| 6
|13.793
| 82.8
|126/125, 100/99, 91/90
| [[21/20]], [[22/21]]
|^D
| [[20/19]], [[23/22]]
| m2
| Eb
|-
|-
|2
| 7
|27.586
| 96.6
|81/80, 64/63, 49/48, 55/54, 65/64
| [[35/33]]
|^^D
| [[18/17]], [[19/18]]
| ^m2
| ^Eb
|-
|-
|3
| 8
|41.379
| 110.3
|50/49, 45/44
| [[16/15]]
|^<sup>3</sup>D/v<sup>3</sup>Eb
| [[17/16]], [[31/29]], [[33/31]]
| ^^m2
| ^^Eb
|-
|-
|4
| 9
|55.172
| 124.1
|28/27, 36/35, 33/32
| [[14/13]], [[15/14]]
|vvEb
| [[29/27]]
| vv~2
| ^<sup>3</sup>Eb
|-
|-
|5
| 10
|68.966
| 137.9
|25/24, 27/26, 26/25
| [[13/12]], [[27/25]]
|vEb
| [[25/23]]
| v~2
| ^<sup>4</sup>Eb
|-
|-
|6
| 11
|82.759
| 151.7
|21/20, 22/21
| [[12/11]], [[35/32]]
|Eb
|
| ^~2
| v<sup>4</sup>E
|-
|-
|7
| 12
|96.552
| 165.5
|35/33
| [[11/10]]
|^Eb
| [[32/29]], [[34/31]]
| ^^~2
| v<sup>3</sup>E
|-
|-
|8
| 13
|110.345
| 179.3
|16/15
| [[10/9]]
|^^Eb
|
| vvM2
| vvE
|-
|-
|9
| 14
|124.138
| 193.1
|15/14, 14/13
| [[28/25]]
|^<sup>3</sup>Eb
| [[19/17]], [[29/26]]
| vM2
| vE
|-
|-
|10
| 15
|137.931
| 206.9
|13/12
| [[9/8]]
|^<sup>4</sup>Eb
| [[26/23]]
| M2
| E
|-
|-
|11
| 16
|151.724
| 220.7
|12/11
| [[25/22]]
|v<sup>4</sup>E
| [[17/15]], [[33/29]]
| ^M2
| ^E
|-
|-
|12
| 17
|165.517
| 234.5
|11/10
| [[8/7]]
|v<sup>3</sup>E
| [[31/27]]
| ^^M2
| ^^E
|-
|-
|13
| 18
|179.310
| 248.3
|10/9
| [[15/13]]
|vvE
| [[22/19]], [[23/20]], [[38/33]]
| ^<sup>3</sup>M2/v<sup>3</sup>m3
| ^<sup>3</sup>E/v<sup>3</sup>F
|-
|-
|14
| 19
|193.103
| 262.1
|28/25
| [[7/6]]
|vE
| [[29/25]], [[36/31]]
| vvm3
| vvF
|-
|-
|15
| 20
|206.897
| 275.9
|9/8
| [[75/64]]
|E
| [[20/17]], [[27/23]], [[34/29]]
| vm3
| vF
|-
|-
|16
| 21
|220.690
| 289.7
|25/22
| [[13/11]], [[32/27]], [[33/28]]
|^E
|
| m3
| F
|-
|-
|17
| 22
|234.483
| 303.4
|8/7
| [[25/21]]
|^^E
| [[19/16]], [[31/26]]
| ^m3
| ^F
|-
|-
|18
| 23
|248.276
| 317.2
|15/13
| [[6/5]]
|^<sup>3</sup>E/v<sup>3</sup>F
|
| ^^m3
| ^^F
|-
|-
|19
| 24
|262.089
| 331.0
|7/6
| [[40/33]]
|vvF
| [[23/19]], [[29/24]]
| vv~3
| ^<sup>3</sup>F
|-
|-
|20
| 25
|275.862
| 344.8
|75/64
| [[11/9]], [[39/32]]
|vF
|
| v~3
| ^<sup>4</sup>F
|-
|-
|21
| 26
|289.655
| 358.6
|33/28, 13/11
| [[16/13]], [[27/22]]
|F
| [[38/31]]
| ^~3
| v<sup>4</sup>F#
|-
|-
|22
| 27
|303.448
| 372.4
|25/21
| [[26/21]]
|^F
| [[31/25]], [[36/29]]
| ^^3
| v<sup>3</sup>F#
|-
|-
|23
| 28
|317.241
| 386.2
|6/5
| [[5/4]]
|^^F
|
| vvM3
| vvF#
|-
|-
|24
| 29
|331.034
| 400.0
|63/52
| [[44/35]]
|^<sup>3</sup>F
| [[24/19]], [[29/23]], [[34/27]]
| vM3
| vF#
|-
|-
|25
| 30
|344.828
| 413.8
|11/9, 39/32
| [[14/11]], [[33/26]], [[81/64]]
|^<sup>4</sup>F
| [[19/15]]
| M3
| F#
|-
|-
|26
| 31
|358.621
| 427.6
|27/22, 16/13
| [[32/25]]
|v<sup>4</sup>F#
| [[23/18]]
| ^M3
| ^F#
|-
|-
|27
| 32
|372.414
| 441.4
|26/21
| [[9/7]], [[35/27]]
|v<sup>3</sup>F#
| [[22/17]], [[31/24]], [[40/31]]
| ^^M3
| ^^F#
|-
|-
|28
| 33
|386.207
| 455.2
|5/4
| [[13/10]]
|vvF#
| [[30/23]]
| ^<sup>3</sup>M3/v<sup>3</sup>4
| ^<sup>3</sup>F#/v<sup>3</sup>G
|-
|-
|29
| 34
|400.000
| 469.0
|63/50, 44/35
| [[21/16]]
|vF#
| [[17/13]], [[25/19]], [[38/29]]
| vv4
| vvG
|-
|-
|30
| 35
|413.793
| 482.8
|14/11, 33/26
| [[33/25]]
|F#
|
| v4
| vG
|-
|-
|31
| 36
|427.586
| 496.6
|32/25
| [[4/3]]
|^F#
|
| P4
| G
|-
|-
|32
| 37
|441.379
| 510.3
|9/7
| [[35/26]]
|^^F#
| [[31/23]]
| ^4
| ^G
|-
|-
|33
| 38
|455.172
| 524.1
|13/10
| [[27/20]]
|^<sup>3</sup>F#/v<sup>3</sup>G
| [[23/17]]
| ^^4
| ^^G
|-
|-
|34
| 39
|468.966
| 537.9
|21/16
| [[15/11]]
|vvG
| [[26/19]], [[34/25]]
| ^<sup>3</sup>4
| ^<sup>3</sup>G
|-
|-
|35
| 40
|482.759
| 551.7
|33/25
| [[11/8]], [[48/35]]
|vG
|
| ^<sup>4</sup>4
| ^<sup>4</sup>G
|-
|-
|36
| 41
|496.552
| 565.5
|4/3
| [[18/13]]
|G
| [[32/23]]
| v<sup>4</sup>A4, vd5
| v<sup>4</sup>G#, vAb
|-
|-
|37
| 42
|510.345
| 579.3
|75/56
| [[7/5]]
|^G
| [[46/33]]
| v<sup>3</sup>A4, d5
| v<sup>3</sup>G#, Ab
|-
|-
|38
| 43
|524.138
| 593.1
|27/20
| [[45/32]]
|^^G
| [[24/17]], [[31/22]], [[38/27]]
| vvA4, ^d5
| vvG#, ^Ab
|-
|-
|39
| …
|537.931
| …
|15/11
| …
|^<sup>3</sup>G
| …
| …
| …
|}
 
== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals|87}}
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
|-
|40
! rowspan="2" | [[Subgroup]]
|551.724
! rowspan="2" | [[Comma list]]
|11/8
! rowspan="2" | [[Mapping]]
|^<sup>4</sup>G
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
|41
! [[TE error|Absolute]] (¢)
|565.517
! [[TE simple badness|Relative]] (%)
|18/13
|v<sup>4</sup>G#, vAb
|-
|-
|42
| 2.3.5
|579.310
| 15625/15552, 67108864/66430125
|7/5, 39/28
| {{mapping| 87 138 202 }}
|v<sup>3</sup>G#, Ab
| −0.299
| 0.455
| 3.30
|-
|-
|43
| 2.3.5.7
|593.103
| 245/243, 1029/1024, 3136/3125
|45/32
| {{mapping| 87 138 202 244 }}
|vvG#, ^Ab
| +0.070
| 0.752
| 5.45
|-
| 2.3.5.7.11
| 245/243, 385/384, 441/440, 3136/3125
| {{mapping| 87 138 202 244 301 }}
| +0.033
| 0.676
| 4.90
|-
| 2.3.5.7.11.13
| 196/195, 245/243, 352/351, 364/363, 625/624
| {{mapping| 87 138 202 244 301 322 }}
| −0.011
| 0.625
| 4.53
|-
| 2.3.5.7.11.13.17
| 154/153, 196/195, 245/243, 273/272, 364/363, 375/374
| {{mapping| 87 138 202 244 301 322 356 }}
| −0.198
| 0.738
| 5.35
|-
| 2.3.5.7.11.13.17.19
| 154/153, 196/195, 210/209, 245/243, 273/272, 286/285, 364/363
| {{mapping| 87 138 202 244 301 322 356 370 }}
| −0.348
| 0.796
| 5.77
|}
|}


== Rank two temperaments ==
=== 13-limit detempering ===
{{Main|87edo/13-limit detempering}}


{| class="wikitable" style="text-align: right"
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods <br> per <br> octave
! Periods<br>per 8ve
! Generator
! Generator*
! Cents
! Cents*
! Associated <br> ratio
! Associated<br>ratio*
! Temperament
! Temperament
|-
|-
| 1
| 1
| style="text-align: center" | 4\87
| 2\87
| 27.586
| 64/63
| [[Arch]]
|-
| 1
| 4\87
| 55.172
| 55.172
| style="text-align: center" | [[33/32]]
| 33/32
| style="text-align: left;" | [[Sensa]]
| [[Escapade]] / [[escaped]] / [[alphaquarter]]
|-
|-
| 1
| 1
| style="text-align: center" | 10\87
| 10\87
| 137.931
| 137.931
| style="text-align: center" | [[13/12]]
| 13/12
| style="text-align: left" | [[Quartemka]]
| [[Quartemka]]
|-
|-
| 1
| 1
| style="text-align: center" | 14\87
| 14\87
| 193.103
| 193.103
| style="text-align: center" | [[28/25]]
| 28/25
| style="text-align: left" | [[Luna]] / [[Hemithirds]]
| [[Luna]] / [[didacus]] / [[hemithirds]]
|-
|-
| 1
| 1
| style="text-align: center" | 17\87
| 17\87
| 234.483
| 234.483
| style="text-align: center" | [[8/7]]
| 8/7
| style="text-align: left" | [[Rodan]]
| [[Slendric]] / [[rodan]]
|-
|-
| 1
| 1
| style="text-align: center" | 23\87
| 23\87
| 317.241
| 317.241
| style="text-align: center" | [[6/5]]
| 6/5
| style="text-align: left" | [[Hanson]] / [[Countercata]] / [[Metakleismic]]
| [[Hanson]] / [[countercata]] / [[metakleismic]]
|-
| 1
| 26\87
| 358.621
| 16/13
| [[Restles]]
|-
|-
| 1
| 1
| style="text-align: center" | 32\87
| 32\87
| 441.379
| 441.379
| style="text-align: center" | [[9/7]]
| 9/7
| style="text-align: left" | [[Clyde]]
| [[Clyde]]
|-
|-
| 1
| 1
| style="text-align: center" | 38\87
| 38\87
| 524.138
| 524.138
| style="text-align: center" | [[65/48]]
| 65/48
| style="text-align: left" | [[Widefourth]]
| [[Widefourth]]
|-
|-
| 1
| 1
| style="text-align: center" | 40\87
| 40\87
| 551.724
| 551.724
| style="text-align: center" | [[11/8]]
| 11/8
| style="text-align: left" | [[Emkay]]
| [[Emka]] / [[emkay]]
|-
|-
| 3
| 3
| style="text-align: center" | 23\87
| 18\87<br>(11\87)
| 317.241
| 248.276<br>(151.724)
| style="text-align: center" | [[6/5]]
| 15/13<br>(12/11)
| style="text-align: left" | [[Tritikleismic]]
| [[Hemimist]]
|-
| 3
| 23\87<br>(6\87)
| 317.241<br>(82.759)
| 6/5<br>(21/20)
| [[Tritikleismic]]
|-
| 3
| 28\87<br>(1\87)
| 386.207<br>(13.793)
| 5/4<br>(126/125)
| [[Mutt]]
|-
| 3
| 36\87<br>(7\87)
| 496.552<br>(96.552)
| 4/3<br>(18/17~19/18)
| [[Misty]]
|-
|-
| 29
| 29
| style="text-align: center" | 28\87
| 28\87<br>(1\87)
| 386.207
| 386.207<br>(13.793)
| style="text-align: center" | [[5/4]]
| 5/4<br>(121/120)
| style="text-align: left" | [[Mystery]]
| [[Mystery]]
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 
87 can serve as a mos in these:
 
* [[Avicenna (temperament)|Avicenna]] ([[Breed|87 & 270]])
* [[Breed|87 & 494]]
 
== Scales ==
=== Mos scales ===
{{main|List of MOS scales in 87edo}}
 
=== Harmonic scales ===
87edo accurately approximates the mode 8 of [[harmonic series]], and the only interval pair not distinct is 14/13 and 15/14. It can also do mode 12 decently.
 
==== (Mode 8) ====
{| class="wikitable center-all"
|-
! Overtones
| 8
| 9
| 10
| 11
| 12
| 13
| 14
| 15
| 16
|-
! JI Ratios
| 1/1
| 9/8
| 5/4
| 11/8
| 3/2
| 13/8
| 7/4
| 15/8
| 2/1
|-
! … in cents
| 0.0
| 203.9
| 386.3
| 551.3
| 702.0
| 840.5
| 968.8
| 1088.3
| 1200.0
|-
! Degrees in 87edo
| 0
| 15
| 28
| 40
| 51
| 61
| 70
| 79
| 87
|-
! … in cents
| 0.0
| 206.9
| 386.2
| 551.7
| 703.5
| 841.4
| 965.5
| 1089.7
| 1200.0
|}
 
The scale in adjacent steps is 15, 13, 12, 11, 10, 9, 9, 8.
 
==== (Mode 12) ====
{| class="wikitable center-all"
|-
! Overtones
| 12
| 13
| 14
| 15
| 16
| 17
| 18
| 19
| 20
| 21
| 22
| 23
| 24
|-
! JI Ratios
| 1/1
| 13/12
| 7/6
| 5/4
| 4/3
| 17/12
| 3/2
| 19/12
| 5/3
| 7/4
| 11/6
| 23/12
| 2/1
|-
! … in cents
| 0.0
| 138.6
| 266.9
| 386.3
| 498.0
| 603.0
| 702.0
| 795.6
| 884.4
| 968.8
| 1049.4
| 1126.3
| 1200.0
|-
! Degrees in 87edo
| 0
| 10
| 19
| 28
| 36
| 44
| 51
| 58
| 64
| 70
| 76
| 82
| 87
|-
! … in cents
| 0.0
| 137.9
| 262.1
| 386.2
| 496.6
| 606.9
| 703.4
| 800.0
| 882.8
| 965.5
| 1048.3
| 1131.0
| 1200.0
|}
|}


87 can serve as a MOS in these:
The scale in adjacent steps is 10, 9, 9, 8, 7, 7, 6, 6, 6, 6, 5.


* [[M&N temperaments|270&amp;87]] &lt;&lt;24 -9 -66 12 27 ... ||
13, 15, 16, 18, 20, and 22 are close matches.  
* [[M&N temperaments|494&amp;87]] &lt;&lt;51 -1 -133 11 32 ... ||


== 13-limit detempering of 87et ==
14 and 21 are flat; 17, 19, and 23 are sharp. Still decent all things considered.


''See also: [[Detempering]]''
=== Other scales ===
* [[Sequar5m]]


''Main article: [[87edo/13-limit detempering]]''
== Instruments ==
* [[Lumatone mapping for 87edo]]
* [[Skip fretting system 87 2 17]]


== Music ==
== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/ecxELXmkYAs ''microtonal improvisation in 87edo''] (2025)


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


[[Category:theory]]
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[[Category:edo]]
[[Category:Listen]]
[[Category:87edo]]
[[Category:Clyde]]
[[Category:listen]]
[[Category:Countercata]]
[[Category:clyde]]
[[Category:Hemithirds]]
[[Category:countercata]]
[[Category:Mystery]]
[[Category:hemithirds]]
[[Category:Rodan]]
[[Category:mystery]]
[[Category:Tritikleismic]]
[[Category:rodan]]
[[Category:tritikleismic]]
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