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{{Infobox ET}}
{{ED intro}}
== Theory ==
== 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]].
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]].


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.  
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.


87et also shows some potential in limits beyond 13. The next four prime harmonics 17, 19, 23 and 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 don't combine with 7, which is flat. Actually, as a no-sevens system, it is consistent in the 33-odd-limit.  
=== 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)}}


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, &lt;46 -29|, the misty comma, &lt;26 -12 -3|, the kleisma, 15625/15552, 245/243, 1029/1024, 3136/3125, and 5120/5103.
=== Subsets and supersets ===
87edo contains [[3edo]] and [[29edo]] as subset edos.


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.
[[348edo]], which slices the edostep in four, provides a good correction of the 7th harmonic.


== Intervals ==
== Intervals ==
{| class="wikitable center-all right-2 left-3 left-4"
{| class="wikitable center-all right-2 left-3 left-4"
|-
! rowspan="2" | #
! rowspan="2" | #
! rowspan="2" | Cents
! rowspan="2" | Cents
! colspan="2" | Approximated Ratios
! colspan="2" | Approximated ratios
! colspan="2" rowspan="2" |[[Ups and Downs Notation]]
! colspan="2" rowspan="2" | [[Ups and downs notation]]
|-
|-
! 13-Limit
! 13-limit
! 31-Limit No-7s Extension
! 31-limit extension
|-
|-
| 0
| 0
| 0.000
| 0.0
| [[1/1]]
| [[1/1]]
|
|
Line 27: Line 40:
|-
|-
| 1
| 1
| 13.793
| 13.8
| [[126/125]], [[100/99]], [[91/90]]
| [[91/90]], [[100/99]], [[126/125]]
|
|
| ^1
| ^1
Line 34: Line 47:
|-
|-
| 2
| 2
| 27.586
| 27.6
| [[81/80]], [[64/63]], [[49/48]], [[55/54]], [[65/64]]
| ''[[49/48]]'', [[55/54]], [[64/63]], [[65/64]], [[81/80]]
|
|
| ^^1
| ^^1
Line 41: Line 54:
|-
|-
| 3
| 3
| 41.379
| 41.4
| [[50/49]], [[45/44]], [[40/39]]
| [[40/39]], [[45/44]], [[50/49]]
| [[39/38]]
| [[39/38]]
| ^<sup>3</sup>1
| ^<sup>3</sup>1
Line 48: Line 61:
|-
|-
| 4
| 4
| 55.172
| 55.2
| [[28/27]], [[36/35]], [[33/32]]
| ''[[28/27]]'', [[33/32]], [[36/35]]
| [[34/33]], [[30/29]], [[32/31]], [[31/30]]
| [[30/29]], [[31/30]], [[32/31]], [[34/33]]
| vvm2
| vvm2
| vvEb
| vvEb
|-
|-
| 5
| 5
| 68.966
| 69.0
| [[25/24]], [[27/26]], [[26/25]]
| [[25/24]], [[26/25]], [[27/26]]
| [[24/23]]
| [[24/23]]
| vm2
| vm2
Line 62: Line 75:
|-
|-
| 6
| 6
| 82.759
| 82.8
| [[21/20]], [[22/21]]
| [[21/20]], [[22/21]]
| [[20/19]], [[23/22]]
| [[20/19]], [[23/22]]
Line 69: Line 82:
|-
|-
| 7
| 7
| 96.552
| 96.6
| [[35/33]]
| [[35/33]]
| [[18/17]], [[19/18]]
| [[18/17]], [[19/18]]
Line 76: Line 89:
|-
|-
| 8
| 8
| 110.345
| 110.3
| [[16/15]]
| [[16/15]]
| [[17/16]], [[33/31]], [[31/29]]
| [[17/16]], [[31/29]], [[33/31]]
| ^^m2
| ^^m2
| ^^Eb
| ^^Eb
|-
|-
| 9
| 9
| 124.138
| 124.1
| [[15/14]], [[14/13]]
| [[14/13]], [[15/14]]
| [[29/27]]
| [[29/27]]
| vv~2
| vv~2
Line 90: Line 103:
|-
|-
| 10
| 10
| 137.931
| 137.9
| [[13/12]], [[27/25]]
| [[13/12]], [[27/25]]
| [[25/23]]
| [[25/23]]
Line 97: Line 110:
|-
|-
| 11
| 11
| 151.724
| 151.7
| [[12/11]], [[35/32]]
| [[12/11]], [[35/32]]
|
|
Line 104: Line 117:
|-
|-
| 12
| 12
| 165.517
| 165.5
| [[11/10]]
| [[11/10]]
| [[32/29]], [[34/31]]
| [[32/29]], [[34/31]]
Line 111: Line 124:
|-
|-
| 13
| 13
| 179.310
| 179.3
| [[10/9]]
| [[10/9]]
|
|
Line 118: Line 131:
|-
|-
| 14
| 14
| 193.103
| 193.1
| [[28/25]]
| [[28/25]]
| [[19/17]], [[29/26]]
| [[19/17]], [[29/26]]
Line 125: Line 138:
|-
|-
| 15
| 15
| 206.897
| 206.9
| [[9/8]]
| [[9/8]]
| [[26/23]]
| [[26/23]]
Line 132: Line 145:
|-
|-
| 16
| 16
| 220.690
| 220.7
| [[25/22]]
| [[25/22]]
| [[17/15]], [[33/29]]
| [[17/15]], [[33/29]]
Line 139: Line 152:
|-
|-
| 17
| 17
| 234.483
| 234.5
| [[8/7]]
| [[8/7]]
| [[31/27]]
| [[31/27]]
Line 146: Line 159:
|-
|-
| 18
| 18
| 248.276
| 248.3
| [[15/13]]
| [[15/13]]
| [[22/19]], [[38/33]], [[23/20]]
| [[22/19]], [[23/20]], [[38/33]]
| ^<sup>3</sup>M2/v<sup>3</sup>m3
| ^<sup>3</sup>M2/v<sup>3</sup>m3
| ^<sup>3</sup>E/v<sup>3</sup>F
| ^<sup>3</sup>E/v<sup>3</sup>F
|-
|-
| 19
| 19
| 262.089
| 262.1
| [[7/6]]
| [[7/6]]
| [[29/25]], [[36/31]]
| [[29/25]], [[36/31]]
Line 160: Line 173:
|-
|-
| 20
| 20
| 275.862
| 275.9
| [[75/64]]
| [[75/64]]
| [[27/23]], [[34/29]]
| [[20/17]], [[27/23]], [[34/29]]
| vm3
| vm3
| vF
| vF
|-
|-
| 21
| 21
| 289.655
| 289.7
| [[32/27]], [[33/28]], [[13/11]]
| [[13/11]], [[32/27]], [[33/28]]
|
|
| m3
| m3
Line 174: Line 187:
|-
|-
| 22
| 22
| 303.448
| 303.4
| [[25/21]]
| [[25/21]]
| [[19/16]], [[31/26]]
| [[19/16]], [[31/26]]
Line 181: Line 194:
|-
|-
| 23
| 23
| 317.241
| 317.2
| [[6/5]]
| [[6/5]]
|
|
Line 188: Line 201:
|-
|-
| 24
| 24
| 331.034
| 331.0
| [[40/33]]
| [[40/33]]
| [[23/19]], [[29/24]]
| [[23/19]], [[29/24]]
Line 195: Line 208:
|-
|-
| 25
| 25
| 344.828
| 344.8
| [[11/9]], [[39/32]]
| [[11/9]], [[39/32]]
|
|
Line 202: Line 215:
|-
|-
| 26
| 26
| 358.621
| 358.6
| [[27/22]], [[16/13]]
| [[16/13]], [[27/22]]
| [[38/31]]
| [[38/31]]
| ^~3
| ^~3
Line 209: Line 222:
|-
|-
| 27
| 27
| 372.414
| 372.4
| [[26/21]]
| [[26/21]]
| [[31/25]], [[36/29]]
| [[31/25]], [[36/29]]
Line 216: Line 229:
|-
|-
| 28
| 28
| 386.207
| 386.2
| [[5/4]]
| [[5/4]]
|
|
Line 223: Line 236:
|-
|-
| 29
| 29
| 400.000
| 400.0
| [[44/35]]
| [[44/35]]
| [[34/27]], [[24/19]], [[29/23]]
| [[24/19]], [[29/23]], [[34/27]]
| vM3
| vM3
| vF#
| vF#
|-
|-
| 30
| 30
| 413.793
| 413.8
| [[81/64]], [[14/11]], [[33/26]]
| [[14/11]], [[33/26]], [[81/64]]
| [[19/15]]
| [[19/15]]
| M3
| M3
Line 237: Line 250:
|-
|-
| 31
| 31
| 427.586
| 427.6
| [[32/25]]
| [[32/25]]
| [[23/18]]
| [[23/18]]
Line 244: Line 257:
|-
|-
| 32
| 32
| 441.379
| 441.4
| [[9/7]], [[35/27]]
| [[9/7]], [[35/27]]
| [[22/17]], [[31/24]], [[40/31]]
| [[22/17]], [[31/24]], [[40/31]]
Line 251: Line 264:
|-
|-
| 33
| 33
| 455.172
| 455.2
| [[13/10]]
| [[13/10]]
| [[30/23]]
| [[30/23]]
Line 258: Line 271:
|-
|-
| 34
| 34
| 468.966
| 469.0
| [[21/16]]
| [[21/16]]
| [[17/13]], [[25/19]], [[38/29]]
| [[17/13]], [[25/19]], [[38/29]]
Line 265: Line 278:
|-
|-
| 35
| 35
| 482.759
| 482.8
| [[33/25]]
| [[33/25]]
|
|
Line 272: Line 285:
|-
|-
| 36
| 36
| 496.552
| 496.6
| [[4/3]]
| [[4/3]]
|
|
Line 279: Line 292:
|-
|-
| 37
| 37
| 510.345
| 510.3
| [[35/26]]
| [[35/26]]
| [[31/23]]
| [[31/23]]
Line 286: Line 299:
|-
|-
| 38
| 38
| 524.138
| 524.1
| [[27/20]]
| [[27/20]]
| [[23/17]]
| [[23/17]]
Line 293: Line 306:
|-
|-
| 39
| 39
| 537.931
| 537.9
| [[15/11]]
| [[15/11]]
| [[26/19]], [[34/25]]
| [[26/19]], [[34/25]]
Line 300: Line 313:
|-
|-
| 40
| 40
| 551.724
| 551.7
| [[11/8]], [[48/35]]
| [[11/8]], [[48/35]]
|
|
Line 307: Line 320:
|-
|-
| 41
| 41
| 565.517
| 565.5
| [[18/13]]
| [[18/13]]
| [[32/23]]
| [[32/23]]
Line 314: Line 327:
|-
|-
| 42
| 42
| 579.310
| 579.3
| [[7/5]]
| [[7/5]]
| [[46/33]]
| [[46/33]]
Line 321: Line 334:
|-
|-
| 43
| 43
| 593.103
| 593.1
| [[45/32]]
| [[45/32]]
| [[24/17]], [[38/27]], [[31/22]]
| [[24/17]], [[31/22]], [[38/27]]
| vvA4, ^d5
| vvA4, ^d5
| vvG#, ^Ab
| vvG#, ^Ab
Line 335: Line 348:
|}
|}


== Just approximation ==
== 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}}


=== Selected just intervals ===
== Regular temperament properties ==
{| class="wikitable center-all"
{| class="wikitable center-4 center-5 center-6"
! colspan="2" |
|-
! prime 2
! rowspan="2" | [[Subgroup]]
! prime 3
! rowspan="2" | [[Comma list]]
! prime 5
! rowspan="2" | [[Mapping]]
! prime 7
! rowspan="2" | Optimal<br>8ve stretch (¢)
! prime 11
! colspan="2" | Tuning error
! prime 13
! prime 17
! prime 19
! prime 23
! prime 29
! prime 31
|-
|-
! rowspan="2" | Error
! [[TE error|Absolute]] (¢)
! absolute (¢)
! [[TE simple badness|Relative]] (%)
| 0.00
| +1.49
| -0.11
| -3.31
| +0.41
| +0.85
| +5.39
| +5.94
| +6.21
| +4.91
| -0.21
|-
|-
! [[Relative error|relative]] (%)
| 2.3.5
| 0.0
| 15625/15552, 67108864/66430125
| +10.8
| {{Mapping| 87 138 202 }}
| -0.8
| −0.299
| -24.0
| 0.455
| +2.9
| 3.30
| +6.2
| +39.1
| +43.0
| +45.0
| +35.6
| -1.5
|}
 
=== Temperament Measures ===
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 11-limit
! 13-limit
|-
|-
! colspan="2" |Octave stretch (¢)
| 2.3.5.7
| -0.471
| 245/243, 1029/1024, 3136/3125
| -0.299
| {{Mapping| 87 138 202 244 }}
| +0.070
| +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
| +0.033
| -0.011
| 0.676
| 4.90
|-
|-
! rowspan="2" |Error
| 2.3.5.7.11.13
! [[TE error|absolute]] (¢)
| 196/195, 245/243, 352/351, 364/363, 625/624
| 0.471
| {{Mapping| 87 138 202 244 301 322 }}
| 0.455
| −0.011
| 0.752
| 0.676
| 0.625
| 0.625
| 4.53
|-
|-
! [[TE simple badness|relative]] (%)
| 2.3.5.7.11.13.17
| 3.42
| 154/153, 196/195, 245/243, 273/272, 364/363, 375/374
| 3.30
| {{Mapping| 87 138 202 244 301 322 356 }}
| 5.45
| −0.198
| 4.90
| 0.738
| 4.53
| 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
|}
|}


== 13-limit detempering of 87et ==
=== 13-limit detempering ===
{{Main|87edo/13-limit detempering}}


:''See also: [[Detempering]]''
=== Rank-2 temperaments ===
 
{| class="wikitable center-all left-5"
{{main|87edo/13-limit detempering}}
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
 
== Rank two temperaments ==
 
{| class="wikitable center-all right-3 left-5"
|-
|-
! Periods<br> per octave
! Periods<br>per 8ve
! Generator
! Generator*
! Cents
! Cents*
! Associated<br>ratio
! Associated<br>ratio*
! Temperament
! Temperament
|-
| 1
| 2\87
| 27.586
| 64/63
| [[Arch]]
|-
|-
| 1
| 1
| 4\87
| 4\87
| 55.172
| 55.172
| [[33/32]]
| 33/32
| [[Sensa]]
| [[Escapade]] / [[escaped]] / [[alphaquarter]]
|-
|-
| 1
| 1
| 10\87
| 10\87
| 137.931
| 137.931
| [[13/12]]
| 13/12
| [[Quartemka]]
| [[Quartemka]]
|-
|-
Line 443: Line 447:
| 14\87
| 14\87
| 193.103
| 193.103
| [[28/25]]
| 28/25
| [[Luna]] / [[Hemithirds]]
| [[Luna]] / [[didacus]] / [[hemithirds]]
|-
|-
| 1
| 1
| 17\87
| 17\87
| 234.483
| 234.483
| [[8/7]]
| 8/7
| [[Rodan]]
| [[Slendric]] / [[rodan]]
|-
|-
| 1
| 1
| 23\87
| 23\87
| 317.241
| 317.241
| [[6/5]]
| 6/5
| [[Hanson]] / [[Countercata]] / [[Metakleismic]]
| [[Hanson]] / [[countercata]] / [[metakleismic]]
|-
|-
| 1
| 1
| 26\87
| 26\87
| 358.621
| 358.621
| [[16/13]]
| 16/13
| [[Restles]]
| [[Restles]]
|-
|-
Line 467: Line 471:
| 32\87
| 32\87
| 441.379
| 441.379
| [[9/7]]
| 9/7
| [[Clyde]]
| [[Clyde]]
|-
|-
Line 473: Line 477:
| 38\87
| 38\87
| 524.138
| 524.138
| [[65/48]]
| 65/48
| [[Widefourth]]
| [[Widefourth]]
|-
|-
Line 479: Line 483:
| 40\87
| 40\87
| 551.724
| 551.724
| [[11/8]]
| 11/8
| [[Emkay]]
| [[Emka]] / [[emkay]]
|-
|-
| 3
| 3
| 23\87
| 18\87<br>(11\87)
| 317.241
| 248.276<br>(151.724)
| [[6/5]]
| 15/13<br>(12/11)
| [[Hemimist]]
|-
| 3
| 23\87<br>(6\87)
| 317.241<br>(82.759)
| 6/5<br>(21/20)
| [[Tritikleismic]]
| [[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
| 28\87
| 28\87<br>(1\87)
| 386.207
| 386.207<br>(13.793)
| [[5/4]]
| 5/4<br>(121/120)
| [[Mystery]]
| [[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:
87 can serve as a mos in these:


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


== Scales ==
== Scales ==
=== Mos scales ===
{{main|List of MOS scales in 87edo}}


=== Harmonic Scale ===
=== Harmonic scales ===
87edo accurately approximates the mode 8 of [[harmonic series]], and the only intervals not distinct are 14/13 and 15/14. It does mode 16 fairly decent, with the only anomaly at 28/27 (4 steps) and 29/28 (5 steps).  
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 ====
==== (Mode 8) ====
{| class="wikitable center-all"
{| class="wikitable center-all"
| Overtones
|-
! Overtones
| 8
| 8
| 9
| 9
Line 518: Line 544:
| 16
| 16
|-
|-
| JI Ratios
! JI Ratios
| 1/1
| 1/1
| 9/8
| 9/8
Line 529: Line 555:
| 2/1
| 2/1
|-
|-
| … in cents
! … in cents
| 0.0
| 0.0
| 203.9
| 203.9
Line 540: Line 566:
| 1200.0
| 1200.0
|-
|-
| Degrees in 87edo
! Degrees in 87edo
| 0
| 0
| 15
| 15
Line 551: Line 577:
| 87
| 87
|-
|-
| … in cents
! … in cents
| 0.0
| 0.0
| 206.9
| 206.9
Line 562: Line 588:
| 1200.0
| 1200.0
|}
|}
* The scale in adjacent steps is 15, 13, 12, 11, 10, 9, 9, 8.


==== Mode 16 ====
The scale in adjacent steps is 15, 13, 12, 11, 10, 9, 9, 8.
 
==== (Mode 12) ====
{| class="wikitable center-all"
{| class="wikitable center-all"
| Odd overtones
|-
! Overtones
| 12
| 13
| 14
| 15
| 16
| 17
| 17
| 18
| 19
| 19
| 20
| 21
| 21
| 22
| 23
| 23
| 25
| 24
| 27
| 29
| 31
|-
|-
| JI Ratios
! JI Ratios
| 17/16
| 1/1
| 19/16
| 13/12
| 21/16
| 7/6
| 23/16
| 5/4
| 25/16
| 4/3
| 27/16
| 17/12
| 29/16
| 3/2
| 31/16
| 19/12
| 5/3
| 7/4
| 11/6
| 23/12
| 2/1
|-
|-
| … in cents
! … in cents
| 105.0
| 0.0
| 297.5
| 138.6
| 470.8
| 266.9
| 628.3
| 386.3
| 772.6
| 498.0
| 905.9
| 603.0
| 1029.6
| 702.0
| 1145.0
| 795.6
| 884.4
| 968.8
| 1049.4
| 1126.3
| 1200.0
|-
|-
| Degrees in 87edo
! Degrees in 87edo
| 8
| 0
| 22
| 10
| 34
| 19
| 46
| 28
| 56
| 36
| 66
| 44
| 75
| 51
| 83
| 58
| 64
| 70
| 76
| 82
| 87
|-
|-
| … in cents
! … in cents
| 110.3
| 0.0
| 303.4
| 137.9
| 469.0
| 262.1
| 634.5
| 386.2
| 772.4
| 496.6
| 910.3
| 606.9
| 1034.5
| 703.4
| 1144.8
| 800.0
| 882.8
| 965.5
| 1048.3
| 1131.0
| 1200.0
|}
|}
* The scale in adjacent steps is 8, 7, 7, 6, 6, 6, 6, 5, 5, 5, 5, 4, 5, 4, 4, 4.


* 25 and 31 are close matches.  
The scale in adjacent steps is 10, 9, 9, 8, 7, 7, 6, 6, 6, 6, 5.  


* 21 is a little bit flat, but still decent.  
13, 15, 16, 18, 20, and 22 are close matches.  


* The others (17, 19, 23, 27 and 29) are extremely sharp, but the intervals between them are close.  
14 and 21 are flat; 17, 19, and 23 are sharp. Still decent all things considered.
 
=== Other scales ===
* [[Sequar5m]]
 
== Instruments ==
* [[Lumatone mapping for 87edo]]
* [[Skip fretting system 87 2 17]]


== Music ==
== Music ==
; [[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/rINJKiMQE78 ''Circuit Bent - Stomach Book (microtonal cover 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]]
[[Category:Listen]]
[[Category:edo]]
[[Category:Clyde]]
[[Category:87edo]]
[[Category:Countercata]]
[[Category:listen]]
[[Category:Hemithirds]]
[[Category:clyde]]
[[Category:Mystery]]
[[Category:countercata]]
[[Category:Rodan]]
[[Category:hemithirds]]
[[Category:Tritikleismic]]
[[Category:mystery]]
[[Category:rodan]]
[[Category:tritikleismic]]
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