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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|87}}
{{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 also the smallest edo with [[relative interval error]]s of no greater than 25% on all of the first 16 harmonics of the [[harmonic series]]. It is also 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]], 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.  
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.  


The equal temperament [[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]].  
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 41 & 46 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 ===
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 [[Square superparticular|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.
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}}
{{Harmonics in equal|87|columns=12|start=13|collapsed=1|title=Approximation of prime harmonics in 87edo (continued)}}
{{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 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 35: Line 38:
|-
|-
| 1
| 1
| 13.793
| 13.8
| [[126/125]], [[100/99]], [[91/90]]
| [[91/90]], [[100/99]], [[126/125]]
|
|
| ^1
| ^1
Line 42: Line 45:
|-
|-
| 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 49: Line 52:
|-
|-
| 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 56: Line 59:
|-
|-
| 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 70: Line 73:
|-
|-
| 6
| 6
| 82.759
| 82.8
| [[21/20]], [[22/21]]
| [[21/20]], [[22/21]]
| [[20/19]], [[23/22]]
| [[20/19]], [[23/22]]
Line 77: Line 80:
|-
|-
| 7
| 7
| 96.552
| 96.6
| [[35/33]]
| [[35/33]]
| [[18/17]], [[19/18]]
| [[18/17]], [[19/18]]
Line 84: Line 87:
|-
|-
| 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 98: Line 101:
|-
|-
| 10
| 10
| 137.931
| 137.9
| [[13/12]], [[27/25]]
| [[13/12]], [[27/25]]
| [[25/23]]
| [[25/23]]
Line 105: Line 108:
|-
|-
| 11
| 11
| 151.724
| 151.7
| [[12/11]], [[35/32]]
| [[12/11]], [[35/32]]
|
|
Line 112: Line 115:
|-
|-
| 12
| 12
| 165.517
| 165.5
| [[11/10]]
| [[11/10]]
| [[32/29]], [[34/31]]
| [[32/29]], [[34/31]]
Line 119: Line 122:
|-
|-
| 13
| 13
| 179.310
| 179.3
| [[10/9]]
| [[10/9]]
|
|
Line 126: Line 129:
|-
|-
| 14
| 14
| 193.103
| 193.1
| [[28/25]]
| [[28/25]]
| [[19/17]], [[29/26]]
| [[19/17]], [[29/26]]
Line 133: Line 136:
|-
|-
| 15
| 15
| 206.897
| 206.9
| [[9/8]]
| [[9/8]]
| [[26/23]]
| [[26/23]]
Line 140: Line 143:
|-
|-
| 16
| 16
| 220.690
| 220.7
| [[25/22]]
| [[25/22]]
| [[17/15]], [[33/29]]
| [[17/15]], [[33/29]]
Line 147: Line 150:
|-
|-
| 17
| 17
| 234.483
| 234.5
| [[8/7]]
| [[8/7]]
| [[31/27]]
| [[31/27]]
Line 154: Line 157:
|-
|-
| 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 168: Line 171:
|-
|-
| 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 182: Line 185:
|-
|-
| 22
| 22
| 303.448
| 303.4
| [[25/21]]
| [[25/21]]
| [[19/16]], [[31/26]]
| [[19/16]], [[31/26]]
Line 189: Line 192:
|-
|-
| 23
| 23
| 317.241
| 317.2
| [[6/5]]
| [[6/5]]
|
|
Line 196: Line 199:
|-
|-
| 24
| 24
| 331.034
| 331.0
| [[40/33]]
| [[40/33]]
| [[23/19]], [[29/24]]
| [[23/19]], [[29/24]]
Line 203: Line 206:
|-
|-
| 25
| 25
| 344.828
| 344.8
| [[11/9]], [[39/32]]
| [[11/9]], [[39/32]]
|
|
Line 210: Line 213:
|-
|-
| 26
| 26
| 358.621
| 358.6
| [[27/22]], [[16/13]]
| [[16/13]], [[27/22]]
| [[38/31]]
| [[38/31]]
| ^~3
| ^~3
Line 217: Line 220:
|-
|-
| 27
| 27
| 372.414
| 372.4
| [[26/21]]
| [[26/21]]
| [[31/25]], [[36/29]]
| [[31/25]], [[36/29]]
Line 224: Line 227:
|-
|-
| 28
| 28
| 386.207
| 386.2
| [[5/4]]
| [[5/4]]
|
|
Line 231: Line 234:
|-
|-
| 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 245: Line 248:
|-
|-
| 31
| 31
| 427.586
| 427.6
| [[32/25]]
| [[32/25]]
| [[23/18]]
| [[23/18]]
Line 252: Line 255:
|-
|-
| 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 259: Line 262:
|-
|-
| 33
| 33
| 455.172
| 455.2
| [[13/10]]
| [[13/10]]
| [[30/23]]
| [[30/23]]
Line 266: Line 269:
|-
|-
| 34
| 34
| 468.966
| 469.0
| [[21/16]]
| [[21/16]]
| [[17/13]], [[25/19]], [[38/29]]
| [[17/13]], [[25/19]], [[38/29]]
Line 273: Line 276:
|-
|-
| 35
| 35
| 482.759
| 482.8
| [[33/25]]
| [[33/25]]
|
|
Line 280: Line 283:
|-
|-
| 36
| 36
| 496.552
| 496.6
| [[4/3]]
| [[4/3]]
|
|
Line 287: Line 290:
|-
|-
| 37
| 37
| 510.345
| 510.3
| [[35/26]]
| [[35/26]]
| [[31/23]]
| [[31/23]]
Line 294: Line 297:
|-
|-
| 38
| 38
| 524.138
| 524.1
| [[27/20]]
| [[27/20]]
| [[23/17]]
| [[23/17]]
Line 301: Line 304:
|-
|-
| 39
| 39
| 537.931
| 537.9
| [[15/11]]
| [[15/11]]
| [[26/19]], [[34/25]]
| [[26/19]], [[34/25]]
Line 308: Line 311:
|-
|-
| 40
| 40
| 551.724
| 551.7
| [[11/8]], [[48/35]]
| [[11/8]], [[48/35]]
|
|
Line 315: Line 318:
|-
|-
| 41
| 41
| 565.517
| 565.5
| [[18/13]]
| [[18/13]]
| [[32/23]]
| [[32/23]]
Line 322: Line 325:
|-
|-
| 42
| 42
| 579.310
| 579.3
| [[7/5]]
| [[7/5]]
| [[46/33]]
| [[46/33]]
Line 329: Line 332:
|-
|-
| 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 344: Line 347:


== Approximation to JI ==
== Approximation to JI ==
=== 15-odd-limit interval mappings ===
=== Interval mappings ===
{{Q-odd-limit intervals|87|note=As 87edo is consistent in the 15-odd-limit, the results by direct approximation and patent val mapping are identical.}}
{{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|Comma List]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" | Tuning error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
Line 360: Line 364:
| 2.3.5
| 2.3.5
| 15625/15552, 67108864/66430125
| 15625/15552, 67108864/66430125
| [{{val| 87 138 202 }}]
| {{mapping| 87 138 202 }}
| &minus;0.299
| −0.299
| 0.455
| 0.455
| 3.30
| 3.30
Line 367: Line 371:
| 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 378:
| 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 385:
| 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 }}
| &minus;0.011
| −0.011
| 0.625
| 0.625
| 4.53
| 4.53
Line 388: Line 392:
| 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 }}
| &minus;0.198
| −0.198
| 0.738
| 0.738
| 5.35
| 5.35
Line 395: Line 399:
| 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 }}
| &minus;0.348
| −0.348
| 0.796
| 0.796
| 5.77
| 5.77
Line 406: Line 410:
=== 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 8ve
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated Ratio<br>(Reduced)
! Associated<br>ratio*
! Temperament
! Temperament
|-
|-
Line 424: Line 428:
| 55.172
| 55.172
| 33/32
| 33/32
| [[Escapade]] / [[sensa]] / [[alphaquarter]]
| [[Escapade]] / [[escaped]] / [[alphaquarter]]
|-
|-
| 1
| 1
Line 504: Line 508:
| [[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:
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 ==
=== MOS scales ===
=== Mos scales ===
{{main|List of MOS scales in 87edo}}
{{main|List of MOS scales in 87edo}}


=== Harmonic scale ===
=== 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.  
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
Line 575: Line 581:
|}
|}


* 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 12 ====
==== (Mode 12) ====
{| class="wikitable center-all"
{| class="wikitable center-all"
|-
! Overtones
! Overtones
| 12
| 12
Line 655: Line 662:
|}
|}


* The scale in adjacent steps is 10, 9, 9, 8, 7, 7, 6, 6, 6, 6, 5.  
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.
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.
 
=== Other scales ===
* [[Sequar5m]]
 
== 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)
; [[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
* ''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
Retrieved from "https://en.xen.wiki/w/87edo"