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Line 1: Line 1:
{{Infobox ET
{{Infobox ET}}
| Prime factorization = 3 × 29
{{ED intro}}
| Step size = 13.793¢
| Fifth = 51\87 (703.448¢) (→ [[29edo|17\29]])
| Major 2nd = 15\87 (207¢)
| Semitones = 9:6 (124¢ : 83¢)
| Consistency = 15
}}
The '''87 equal divisions of the octave''' ('''87edo'''), or the '''87(-tone) equal temperament''' ('''87tet''', '''87et''') when viewed from a [[regular temperament]] perspective, is the tuning system derived by dividing the [[octave]] into 87 [[equal]]ly-sized steps, where each step is about 13.8 [[cent]]s.


== Theory ==
== Theory ==
87edo 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-odd-limit]] [[tonality diamond]] both uniquely and [[consistent]]ly (see [[87edo/13-limit detempering]]), and is the smallest edo to do so. It is a [[zeta peak integer edo]].
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, 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.  
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.  


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]], {{val| 46 -29 }}, the [[misty comma]], {{val| 26 -12 -3 }}, the [[kleisma]], 15625/15552, in addition to [[245/243]], [[1029/1024]], [[3136/3125]], and [[5120/5103]].
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 [[Gamelismic clan #Rodan|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 [[Kleismic family #Clyde|clyde temperament]] is 0.04455 cents sharp of the 7-limit POTE generator.
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 ===
=== Prime harmonics ===
{{Primes in edo|87}}
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.
 
[[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.
 
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.
 
See also [[87th-octave temperaments]].


== 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 39: Line 44:
|-
|-
| 1
| 1
| 13.793
| 13.8
| [[126/125]], [[100/99]], [[91/90]]
| [[91/90]], [[100/99]], [[126/125]]
|
|
| ^1
| ^1
Line 46: Line 51:
|-
|-
| 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 53: Line 58:
|-
|-
| 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 60: Line 65:
|-
|-
| 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 74: Line 79:
|-
|-
| 6
| 6
| 82.759
| 82.8
| [[21/20]], [[22/21]]
| [[21/20]], [[22/21]]
| [[20/19]], [[23/22]]
| [[20/19]], [[23/22]]
Line 81: Line 86:
|-
|-
| 7
| 7
| 96.552
| 96.6
| [[35/33]]
| [[35/33]]
| [[18/17]], [[19/18]]
| [[18/17]], [[19/18]]
Line 88: Line 93:
|-
|-
| 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 102: Line 107:
|-
|-
| 10
| 10
| 137.931
| 137.9
| [[13/12]], [[27/25]]
| [[13/12]], [[27/25]]
| [[25/23]]
| [[25/23]]
Line 109: Line 114:
|-
|-
| 11
| 11
| 151.724
| 151.7
| [[12/11]], [[35/32]]
| [[12/11]], [[35/32]]
|
|
Line 116: Line 121:
|-
|-
| 12
| 12
| 165.517
| 165.5
| [[11/10]]
| [[11/10]]
| [[32/29]], [[34/31]]
| [[32/29]], [[34/31]]
Line 123: Line 128:
|-
|-
| 13
| 13
| 179.310
| 179.3
| [[10/9]]
| [[10/9]]
|
|
Line 130: Line 135:
|-
|-
| 14
| 14
| 193.103
| 193.1
| [[28/25]]
| [[28/25]]
| [[19/17]], [[29/26]]
| [[19/17]], [[29/26]]
Line 137: Line 142:
|-
|-
| 15
| 15
| 206.897
| 206.9
| [[9/8]]
| [[9/8]]
| [[26/23]]
| [[26/23]]
Line 144: Line 149:
|-
|-
| 16
| 16
| 220.690
| 220.7
| [[25/22]]
| [[25/22]]
| [[17/15]], [[33/29]]
| [[17/15]], [[33/29]]
Line 151: Line 156:
|-
|-
| 17
| 17
| 234.483
| 234.5
| [[8/7]]
| [[8/7]]
| [[31/27]]
| [[31/27]]
Line 158: Line 163:
|-
|-
| 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 172: Line 177:
|-
|-
| 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 186: Line 191:
|-
|-
| 22
| 22
| 303.448
| 303.4
| [[25/21]]
| [[25/21]]
| [[19/16]], [[31/26]]
| [[19/16]], [[31/26]]
Line 193: Line 198:
|-
|-
| 23
| 23
| 317.241
| 317.2
| [[6/5]]
| [[6/5]]
|
|
Line 200: Line 205:
|-
|-
| 24
| 24
| 331.034
| 331.0
| [[40/33]]
| [[40/33]]
| [[23/19]], [[29/24]]
| [[23/19]], [[29/24]]
Line 207: Line 212:
|-
|-
| 25
| 25
| 344.828
| 344.8
| [[11/9]], [[39/32]]
| [[11/9]], [[39/32]]
|
|
Line 214: Line 219:
|-
|-
| 26
| 26
| 358.621
| 358.6
| [[27/22]], [[16/13]]
| [[16/13]], [[27/22]]
| [[38/31]]
| [[38/31]]
| ^~3
| ^~3
Line 221: Line 226:
|-
|-
| 27
| 27
| 372.414
| 372.4
| [[26/21]]
| [[26/21]]
| [[31/25]], [[36/29]]
| [[31/25]], [[36/29]]
Line 228: Line 233:
|-
|-
| 28
| 28
| 386.207
| 386.2
| [[5/4]]
| [[5/4]]
|
|
Line 235: Line 240:
|-
|-
| 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 249: Line 254:
|-
|-
| 31
| 31
| 427.586
| 427.6
| [[32/25]]
| [[32/25]]
| [[23/18]]
| [[23/18]]
Line 256: Line 261:
|-
|-
| 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 263: Line 268:
|-
|-
| 33
| 33
| 455.172
| 455.2
| [[13/10]]
| [[13/10]]
| [[30/23]]
| [[30/23]]
Line 270: Line 275:
|-
|-
| 34
| 34
| 468.966
| 469.0
| [[21/16]]
| [[21/16]]
| [[17/13]], [[25/19]], [[38/29]]
| [[17/13]], [[25/19]], [[38/29]]
Line 277: Line 282:
|-
|-
| 35
| 35
| 482.759
| 482.8
| [[33/25]]
| [[33/25]]
|
|
Line 284: Line 289:
|-
|-
| 36
| 36
| 496.552
| 496.6
| [[4/3]]
| [[4/3]]
|
|
Line 291: Line 296:
|-
|-
| 37
| 37
| 510.345
| 510.3
| [[35/26]]
| [[35/26]]
| [[31/23]]
| [[31/23]]
Line 298: Line 303:
|-
|-
| 38
| 38
| 524.138
| 524.1
| [[27/20]]
| [[27/20]]
| [[23/17]]
| [[23/17]]
Line 305: Line 310:
|-
|-
| 39
| 39
| 537.931
| 537.9
| [[15/11]]
| [[15/11]]
| [[26/19]], [[34/25]]
| [[26/19]], [[34/25]]
Line 312: Line 317:
|-
|-
| 40
| 40
| 551.724
| 551.7
| [[11/8]], [[48/35]]
| [[11/8]], [[48/35]]
|
|
Line 319: Line 324:
|-
|-
| 41
| 41
| 565.517
| 565.5
| [[18/13]]
| [[18/13]]
| [[32/23]]
| [[32/23]]
Line 326: Line 331:
|-
|-
| 42
| 42
| 579.310
| 579.3
| [[7/5]]
| [[7/5]]
| [[46/33]]
| [[46/33]]
Line 333: Line 338:
|-
|-
| 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 346: Line 351:
| …
| …
|}
|}
== 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 ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
Line 360: Line 376:
| 2.3.5
| 2.3.5
| 15625/15552, 67108864/66430125
| 15625/15552, 67108864/66430125
| [{{val| 87 138 202 }}]
| {{Mapping| 87 138 202 }}
| -0.299
| −0.299
| 0.455
| 0.455
| 3.30
| 3.30
Line 367: Line 383:
| 2.3.5.7
| 2.3.5.7
| 245/243, 1029/1024, 3136/3125
| 245/243, 1029/1024, 3136/3125
| [{{val| 87 138 202 244 }}]
| {{Mapping| 87 138 202 244 }}
| +0.070
| +0.070
| 0.752
| 0.752
Line 374: Line 390:
| 2.3.5.7.11
| 2.3.5.7.11
| 245/243, 385/384, 441/440, 3136/3125
| 245/243, 385/384, 441/440, 3136/3125
| [{{val| 87 138 202 244 301 }}]
| {{Mapping| 87 138 202 244 301 }}
| +0.033
| +0.033
| 0.676
| 0.676
Line 381: Line 397:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 196/195, 245/243, 352/351, 364/363, 625/624
| 196/195, 245/243, 352/351, 364/363, 625/624
| [{{val| 87 138 202 244 301 322 }}]
| {{Mapping| 87 138 202 244 301 322 }}
| -0.011
| −0.011
| 0.625
| 0.625
| 4.53
| 4.53
Line 388: Line 404:
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 154/153, 196/195, 245/243, 273/272, 364/363, 375/374
| 154/153, 196/195, 245/243, 273/272, 364/363, 375/374
| [{{val| 87 138 202 244 301 322 356 }}]
| {{Mapping| 87 138 202 244 301 322 356 }}
| -0.198
| −0.198
| 0.738
| 0.738
| 5.35
| 5.35
Line 395: Line 411:
| 2.3.5.7.11.13.17.19
| 2.3.5.7.11.13.17.19
| 154/153, 196/195, 210/209, 245/243, 273/272, 286/285, 364/363
| 154/153, 196/195, 210/209, 245/243, 273/272, 286/285, 364/363
| [{{val| 87 138 202 244 301 322 356 370 }}]
| {{Mapping| 87 138 202 244 301 322 356 370 }}
| -0.348
| −0.348
| 0.796
| 0.796
| 5.77
| 5.77
Line 402: Line 418:


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


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+ Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br> per octave
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated Ratio<br>(Reduced)
! Associated<br>ratio*
! Temperament
! Temperament
|-
|-
Line 426: Line 440:
| 55.172
| 55.172
| 33/32
| 33/32
| [[Escapade]] / [[sensa]] / [[alphaquarter]]
| [[Escapade]] / [[escaped]] / [[alphaquarter]]
|-
|-
| 1
| 1
Line 506: Line 520:
| [[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:


* [[Avicenna (temperament)|Avicenna]] ([[Breed|87&amp;270]]) {{multival| 24 -9 -66 12 27 … }}
* [[Avicenna (temperament)|Avicenna]] ([[Breed|87 & 270]])  
* [[Breed|87&amp;494]] {{multival| 51 -1 -133 11 32 … }}
* [[Breed|87 & 494]]  


== Scales ==
== Scales ==
=== 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 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"
|-
! Overtones
| 12
| 14
| 16
| 18
| 20
| 22
| 24
|-
! JI Ratios
| 1/1
| 7/6
| 4/3
| 3/2
| 5/3
| 11/6
| 2/1
|-
! … in cents
| 0.0
| 266.9
| 498.0
| 702.0
| 884.4
| 1049.4
| 1200.0
|-
! Degrees in 87edo
| 0
| 19
| 36
| 51
| 64
| 76
| 87
|-
! … in cents
| 0.0
| 262.1
| 496.6
| 703.4
| 882.8
| 1048.3
| 1200.0
|}
 
The scale in adjacent steps is '''19 17 15 13 12 11'''.  


==== Mode 8 ====
==== (Mode 8) ====
{| class="wikitable center-all"
{| class="wikitable center-all"
| Overtones
|-
! Overtones
| 8
| 8
| 9
| 9
Line 529: Line 596:
| 16
| 16
|-
|-
| JI Ratios
! JI Ratios
| 1/1
| 1/1
| 9/8
| 9/8
Line 540: Line 607:
| 2/1
| 2/1
|-
|-
| … in cents
! … in cents
| 0.0
| 0.0
| 203.9
| 203.9
Line 551: Line 618:
| 1200.0
| 1200.0
|-
|-
| Degrees in 87edo
! Degrees in 87edo
| 0
| 0
| 15
| 15
Line 562: Line 629:
| 87
| 87
|-
|-
| … in cents
! … in cents
| 0.0
| 0.0
| 206.9
| 206.9
Line 574: Line 641:
|}
|}


* The scale in adjacent steps is 15, 13, 12, 11, 10, 9, 9, 8.  
The scale in adjacent steps is '''15 13 12 11 10 9 9 8'''.  


==== Mode 16 ====
==== (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.
The scale in adjacent steps is '''10 9 9 8 7 7 6 6 6 6 5'''.
* 25 and 31 are close matches.  
 
* 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.
 
=== 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 ==
== Music ==
* [http://www.archive.org/details/Pianodactyl Pianodactyl] [http://www.archive.org/download/Pianodactyl/pianodactyl.mp3 play] by [[Gene Ward Smith]]
=== 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


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Retrieved from "https://en.xen.wiki/w/87edo"