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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox ET}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{ED intro}}
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-06-17 18:22:27 UTC</tt>.<br>
: The original revision id was <tt>149422583</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">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 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 consistently, and is the smallest equal temperament to do so.


87et 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.  
== 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]].  


Some 13-limit rank two temperaments supported by 87et are these:
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.


tritikleismic 72&amp;87 &lt;&lt;18 15 -6 9 42 ... ||
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]].  
rodan 41&amp;87 &lt;&lt;3 17 -1 -13 -22 ... ||
mystery 29&amp;58 &lt;&lt;0 29 29 29 29 ... ||
countercata 34&amp;87 &lt;&lt;6 5 -31 32 14 ... ||
luna 31&amp;87 &lt;&lt;15 -2 -5 22 -23 ... ||


87 can serve as a MOS in these:
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.


270&amp;87 &lt;&lt;24 -9 -66 12 27 ... ||
=== Prime harmonics ===
494&amp;87 &lt;&lt;51 -1 -133 11 32 ... ||
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.
224&amp;87 &lt;&lt;27 8 -67 -1 5 ...||
{{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 is a particularly good tuning for rodan temperament.</pre></div>
=== Subsets and supersets ===
<h4>Original HTML content:</h4>
87edo contains [[3edo]] and [[29edo]] as subset edos.
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;87edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;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 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 consistently, and is the smallest equal temperament to do so.&lt;br /&gt;
 
&lt;br /&gt;
== Intervals ==
87et 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, &amp;lt;46 -29|, the misty comma, &amp;lt;26 -12 -3|, the kleisma, 15625/15552, 245/243, 1029/1024, 3136/3125, and 5120/5103. &lt;br /&gt;
{| class="wikitable center-all right-2 left-3 left-4"
&lt;br /&gt;
|-
Some 13-limit rank two temperaments supported by 87et are these:&lt;br /&gt;
! rowspan="2" | #
&lt;br /&gt;
! rowspan="2" | Cents
tritikleismic 72&amp;amp;87 &amp;lt;&amp;lt;18 15 -6 9 42 ... ||&lt;br /&gt;
! colspan="2" | Approximated ratios
rodan 41&amp;amp;87 &amp;lt;&amp;lt;3 17 -1 -13 -22 ... ||&lt;br /&gt;
! colspan="2" rowspan="2" | [[Ups and downs notation]]
mystery 29&amp;amp;58 &amp;lt;&amp;lt;0 29 29 29 29 ... ||&lt;br /&gt;
|-
countercata 34&amp;amp;87 &amp;lt;&amp;lt;6 5 -31 32 14 ... ||&lt;br /&gt;
! 13-limit
luna 31&amp;amp;87 &amp;lt;&amp;lt;15 -2 -5 22 -23 ... ||&lt;br /&gt;
! 31-limit extension
&lt;br /&gt;
|-
87 can serve as a MOS in these:&lt;br /&gt;
| 0
&lt;br /&gt;
| 0.0
270&amp;amp;87 &amp;lt;&amp;lt;24 -9 -66 12 27 ... ||&lt;br /&gt;
| [[1/1]]
494&amp;amp;87 &amp;lt;&amp;lt;51 -1 -133 11 32 ... ||&lt;br /&gt;
|
224&amp;amp;87 &amp;lt;&amp;lt;27 8 -67 -1 5 ...||&lt;br /&gt;
| P1
&lt;br /&gt;
| D
87et is a particularly good tuning for rodan temperament.&lt;/body&gt;&lt;/html&gt;</pre></div>
|-
| 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
|-
| 5
| 69.0
| [[25/24]], [[26/25]], [[27/26]]
| [[24/23]]
| vm2
| vEb
|-
| 6
| 82.8
| [[21/20]], [[22/21]]
| [[20/19]], [[23/22]]
| m2
| Eb
|-
| 7
| 96.6
| [[35/33]]
| [[18/17]], [[19/18]]
| ^m2
| ^Eb
|-
| 8
| 110.3
| [[16/15]]
| [[17/16]], [[31/29]], [[33/31]]
| ^^m2
| ^^Eb
|-
| 9
| 124.1
| [[14/13]], [[15/14]]
| [[29/27]]
| vv~2
| ^<sup>3</sup>Eb
|-
| 10
| 137.9
| [[13/12]], [[27/25]]
| [[25/23]]
| v~2
| ^<sup>4</sup>Eb
|-
| 11
| 151.7
| [[12/11]], [[35/32]]
|
| ^~2
| v<sup>4</sup>E
|-
| 12
| 165.5
| [[11/10]]
| [[32/29]], [[34/31]]
| ^^~2
| v<sup>3</sup>E
|-
| 13
| 179.3
| [[10/9]]
|
| vvM2
| vvE
|-
| 14
| 193.1
| [[28/25]]
| [[19/17]], [[29/26]]
| vM2
| vE
|-
| 15
| 206.9
| [[9/8]]
| [[26/23]]
| M2
| E
|-
| 16
| 220.7
| [[25/22]]
| [[17/15]], [[33/29]]
| ^M2
| ^E
|-
| 17
| 234.5
| [[8/7]]
| [[31/27]]
| ^^M2
| ^^E
|-
| 18
| 248.3
| [[15/13]]
| [[22/19]], [[23/20]], [[38/33]]
| ^<sup>3</sup>M2/v<sup>3</sup>m3
| ^<sup>3</sup>E/v<sup>3</sup>F
|-
| 19
| 262.1
| [[7/6]]
| [[29/25]], [[36/31]]
| vvm3
| vvF
|-
| 20
| 275.9
| [[75/64]]
| [[20/17]], [[27/23]], [[34/29]]
| vm3
| vF
|-
| 21
| 289.7
| [[13/11]], [[32/27]], [[33/28]]
|
| m3
| F
|-
| 22
| 303.4
| [[25/21]]
| [[19/16]], [[31/26]]
| ^m3
| ^F
|-
| 23
| 317.2
| [[6/5]]
|
| ^^m3
| ^^F
|-
| 24
| 331.0
| [[40/33]]
| [[23/19]], [[29/24]]
| vv~3
| ^<sup>3</sup>F
|-
| 25
| 344.8
| [[11/9]], [[39/32]]
|
| v~3
| ^<sup>4</sup>F
|-
| 26
| 358.6
| [[16/13]], [[27/22]]
| [[38/31]]
| ^~3
| v<sup>4</sup>F#
|-
| 27
| 372.4
| [[26/21]]
| [[31/25]], [[36/29]]
| ^^3
| v<sup>3</sup>F#
|-
| 28
| 386.2
| [[5/4]]
|
| vvM3
| vvF#
|-
| 29
| 400.0
| [[44/35]]
| [[24/19]], [[29/23]], [[34/27]]
| vM3
| vF#
|-
| 30
| 413.8
| [[14/11]], [[33/26]], [[81/64]]
| [[19/15]]
| M3
| F#
|-
| 31
| 427.6
| [[32/25]]
| [[23/18]]
| ^M3
| ^F#
|-
| 32
| 441.4
| [[9/7]], [[35/27]]
| [[22/17]], [[31/24]], [[40/31]]
| ^^M3
| ^^F#
|-
| 33
| 455.2
| [[13/10]]
| [[30/23]]
| ^<sup>3</sup>M3/v<sup>3</sup>4
| ^<sup>3</sup>F#/v<sup>3</sup>G
|-
| 34
| 469.0
| [[21/16]]
| [[17/13]], [[25/19]], [[38/29]]
| vv4
| vvG
|-
| 35
| 482.8
| [[33/25]]
|
| v4
| vG
|-
| 36
| 496.6
| [[4/3]]
|
| P4
| G
|-
| 37
| 510.3
| [[35/26]]
| [[31/23]]
| ^4
| ^G
|-
| 38
| 524.1
| [[27/20]]
| [[23/17]]
| ^^4
| ^^G
|-
| 39
| 537.9
| [[15/11]]
| [[26/19]], [[34/25]]
| ^<sup>3</sup>4
| ^<sup>3</sup>G
|-
| 40
| 551.7
| [[11/8]], [[48/35]]
|
| ^<sup>4</sup>4
| ^<sup>4</sup>G
|-
| 41
| 565.5
| [[18/13]]
| [[32/23]]
| v<sup>4</sup>A4, vd5
| v<sup>4</sup>G#, vAb
|-
| 42
| 579.3
| [[7/5]]
| [[46/33]]
| v<sup>3</sup>A4, d5
| v<sup>3</sup>G#, Ab
|-
| 43
| 593.1
| [[45/32]]
| [[24/17]], [[31/22]], [[38/27]]
| vvA4, ^d5
| vvG#, ^Ab
|-
| …
| …
| …
| …
| …
| …
|}
 
== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals|87}}
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 15625/15552, 67108864/66430125
| {{mapping| 87 138 202 }}
| −0.299
| 0.455
| 3.30
|-
| 2.3.5.7
| 245/243, 1029/1024, 3136/3125
| {{mapping| 87 138 202 244 }}
| +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
|}
 
=== 13-limit detempering ===
{{Main|87edo/13-limit detempering}}
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>ratio*
! Temperament
|-
| 1
| 2\87
| 27.586
| 64/63
| [[Arch]]
|-
| 1
| 4\87
| 55.172
| 33/32
| [[Escapade]] / [[escaped]] / [[alphaquarter]]
|-
| 1
| 10\87
| 137.931
| 13/12
| [[Quartemka]]
|-
| 1
| 14\87
| 193.103
| 28/25
| [[Luna]] / [[didacus]] / [[hemithirds]]
|-
| 1
| 17\87
| 234.483
| 8/7
| [[Slendric]] / [[rodan]]
|-
| 1
| 23\87
| 317.241
| 6/5
| [[Hanson]] / [[countercata]] / [[metakleismic]]
|-
| 1
| 26\87
| 358.621
| 16/13
| [[Restles]]
|-
| 1
| 32\87
| 441.379
| 9/7
| [[Clyde]]
|-
| 1
| 38\87
| 524.138
| 65/48
| [[Widefourth]]
|-
| 1
| 40\87
| 551.724
| 11/8
| [[Emka]] / [[emkay]]
|-
| 3
| 18\87<br>(11\87)
| 248.276<br>(151.724)
| 15/13<br>(12/11)
| [[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
| 28\87<br>(1\87)
| 386.207<br>(13.793)
| 5/4<br>(121/120)
| [[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
|}
 
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.
 
=== Other scales ===
* [[Sequar5m]]
 
== Instruments ==
* [[Lumatone mapping for 87edo]]
* [[Skip fretting system 87 2 17]]
 
== 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
 
[[Category:Zeta|##]] <!-- 2-digit number -->
[[Category:Listen]]
[[Category:Clyde]]
[[Category:Countercata]]
[[Category:Hemithirds]]
[[Category:Mystery]]
[[Category:Rodan]]
[[Category:Tritikleismic]]
Retrieved from "https://en.xen.wiki/w/87edo"