87edo: Difference between revisions

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.  

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.

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]])⁸⁷ 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 ===

! colspan="2" rowspan="2" | [[Ups and downs notation]]

| [[1/1]]

| [[91/90]], [[100/99]], [[126/125]]

| 27.6

| ''[[49/48]]'', [[55/54]], [[64/63]], [[65/64]], [[81/80]]

| 41.4

| [[40/39]], [[45/44]], [[50/49]]

| [[39/38]]

| ^³1

| ^³D/v³Eb

|-

|-

| [[25/24]], [[26/25]], [[27/26]]

|-

| [[21/20]], [[22/21]]

| [[20/19]], [[23/22]]

|-

| [[18/17]], [[19/18]]

|-

| [[17/16]], [[31/29]], [[33/31]]

| 124.1

| [[14/13]], [[15/14]]

|-

| [[13/12]], [[27/25]]

|-

| [[12/11]], [[35/32]]

| v⁴E

|-

| 165.5

| [[32/29]], [[34/31]]

| v³E

|-

| 13

|-

| [[19/17]], [[29/26]]

| [[9/8]]

| [[26/23]]

| [[25/22]]

| [[17/15]], [[33/29]]

| 234.5

| [[8/7]]

| [[31/27]]

| [[15/13]]

| [[22/19]], [[23/20]], [[38/33]]

| ^³E/v³F

| [[7/6]]

| [[29/25]], [[36/31]]

| [[75/64]]

| [[20/17]], [[27/23]], [[34/29]]

|-

| [[13/11]], [[32/27]], [[33/28]]

|-

| [[19/16]], [[31/26]]

|-

|-

| ^³F

|-

| [[11/9]], [[39/32]]

| [[16/13]], [[27/22]]

| [[31/25]], [[36/29]]

| [[44/35]]

| [[24/19]], [[29/23]], [[34/27]]

| [[14/11]], [[33/26]], [[81/64]]

| [[19/15]]

| [[32/25]]

| [[23/18]]

| 441.4

| [[9/7]], [[35/27]]

| [[22/17]], [[31/24]], [[40/31]]

| 455.2

| [[13/10]]

| ^³F#/v³G

|-

| 469.0

| [[17/13]], [[25/19]], [[38/29]]

| [[33/25]]

| 496.6

|-

| 510.3

| [[35/26]]

| [[31/23]]

|-

|-

|-

|-

| v⁴G#, vAb

|-

|-

|-

== Approximation to JI ==

=== Interval mappings ===

{{Q-odd-limit intervals|87}}

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | Optimal<br>8ve stretch (¢)

| 245/243, 1029/1024, 3136/3125

| 154/153, 196/195, 210/209, 245/243, 273/272, 286/285, 364/363

| {{mapping| 87 138 202 244 301 322 356 370 }}

 

 

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

! Periods<br>per 8ve

| 33/32

| [[Escapade]] / [[escaped]] / [[alphaquarter]]

| 10\87

|-

|-

| 234.483

| 317.241

| 358.621

| 16/13

| 1

| 9/7

| 38\87

|-

| 1

|-

| 18\87<br>(11\87)

| 248.276<br>(151.724)

| 317.241<br>(82.759)

| [[Tritikleismic]]

|-

| 4/3<br>(18/17~19/18)

| 5/4<br>(121/120)

 

 

* [[Avicenna (temperament)|Avicenna]] ([[Breed|87 & 270]])

* [[Breed|87 & 494]]

 

 

 

! Overtones

|-

|-

|-

|-

 

 

==== (Mode 12) ====

{| class="wikitable center-all"

| 2/1

! … in cents

|-

|-

| 386.2

 

 

 

14 and 21 are flat; 17, 19, and 23 are sharp. Still decent all things considered.

 

 

 

== Music ==

; [[Bryan Deister]]

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

 

; [[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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