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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:bootmii|bootmii]] and made on <tt>2016-12-23 21:33:22 UTC</tt>.<br>
: The original revision id was <tt>602790088</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 [[cent]]s. 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]]ly, and is the smallest equal temperament to do so.


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.
== 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]].  


87et is a particularly good tuning for [[Gamelismic clan|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 temperament]] is 0.04455 cents sharp of the 7-limit POTE generator.
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.  


=Rank two temperaments=
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]].
||~ Periods
per
octave ||~ Generator ||~ Cents ||~ Associated
ratio ||~ Temperament ||
||&gt; 1 ||&gt; 4\87 ||&gt; 55.172 ||= 33/32 ||&lt; [[Sensa]] ||
||&gt; 1 ||&gt; 10\87 ||&gt; 137.931 ||= 13/12 || [[Quartemka]] ||
||&gt; 1 ||&gt; 14\87 ||&gt; 193.103 ||= 28/25 || [[Luna]]/[[Hemithirds|hemithirds]] ||
||&gt; 1 ||&gt; 17\87 ||&gt; 234.483 ||= 8/7 || [[Rodan]] ||
||&gt; 1 ||&gt; 23\87 ||&gt; 317.241 ||= 6/5 || [[Hanson]]/[[countercata]]/[[metakleismic]] ||
||&gt; 1 ||&gt; 32\87 ||&gt; 441.379 ||= 9/7 || [[Clyde]] ||
||&gt; 1 ||&gt; 38\87 ||&gt; 524.138 ||= 65/48 ||&lt; [[Widefourth]] ||
||&gt; 1 ||&gt; 40\87 ||&gt; 551.724 ||= 11/8 || [[Emkay]] ||
||&gt; 3 ||&gt; 23\87 ||&gt; 317.241 ||= 6/5 || [[Tritikleismic]] ||
||&gt; 29 ||&gt; 28\87 ||&gt; 386.207 ||= 5/4 || [[Mystery]] ||


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.


[[M&amp;N temperaments|270&amp;87]] &lt;&lt;24 -9 -66 12 27 ... ||
=== Prime harmonics ===
[[M&amp;N temperaments|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.
{{Harmonics in equal|87|columns=12}}
{{Harmonics in equal|87|columns=12|start=13|collapsed=1|title=Approximation of prime harmonics in 87edo (continued)}}


=13-limit detempering of 87et=  
=== Subsets and supersets ===
[91/90, 49/48, 40/39, 28/27, 25/24, 21/20, 35/33, 16/15, 14/13, 13/12, 12/11, 11/10, 10/9, 28/25, 9/8, 25/22, 8/7, 15/13, 7/6, 75/64, 13/11, 25/21, 6/5, 40/33, 11/9, 16/13, 26/21, 5/4, 44/35, 14/11, 32/25, 9/7, 13/10, 21/16, 33/25, 4/3, 35/26, 27/20, 15/11, 11/8, 18/13, 7/5, 45/32, 64/45, 10/7, 13/9, 16/11, 22/15, 40/27, 52/35, 3/2, 50/33, 32/21, 20/13, 14/9, 25/16, 11/7, 35/22, 8/5, 21/13, 13/8, 18/11, 33/20, 5/3, 42/25, 22/13, 75/44, 12/7, 26/15, 7/4, 44/25, 16/9, 25/14, 9/5, 20/11, 11/6, 24/13, 13/7, 15/8, 66/35, 21/11, 25/13, 27/14, 39/20, 55/28, 99/50, 2]
87edo contains [[3edo]] and [[29edo]] as subset edos.


=Music=  
== Intervals ==
{| class="wikitable center-all right-2 left-3 left-4"
|-
! rowspan="2" | #
! rowspan="2" | Cents
! colspan="2" | Approximated ratios
! colspan="2" rowspan="2" | [[Ups and downs notation]]
|-
! 13-limit
! 31-limit extension
|-
| 0
| 0.0
| [[1/1]]
|
| P1
| D
|-
| 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
|-
| …
| …
| …
| …
| …
| …
|}


[[http://www.archive.org/details/Pianodactyl|Pianodactyl]] [[http://www.archive.org/download/Pianodactyl/pianodactyl.mp3|play]] by [[Gene Ward Smith]]</pre></div>
== Approximation to JI ==
<h4>Original HTML content:</h4>
=== Interval mappings ===
<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 &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s. It is solid as both a &lt;a class="wiki_link" href="/13-limit"&gt;13-limit&lt;/a&gt; (or 15 odd limit) and as a &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt; system, and of course does well enough in any limit in between. It represents the &lt;a class="wiki_link" href="/13-limit"&gt;13-limit&lt;/a&gt; &lt;a class="wiki_link" href="/tonality%20diamond"&gt;tonality diamond&lt;/a&gt; both uniquely and &lt;a class="wiki_link" href="/consistent"&gt;consistent&lt;/a&gt;ly, and is the smallest equal temperament to do so.&lt;br /&gt;
{{Q-odd-limit intervals|87}}
&lt;br /&gt;
87et &lt;a class="wiki_link" href="/tempering%20out"&gt;tempers out&lt;/a&gt; 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;
&lt;br /&gt;
87et is a particularly good tuning for &lt;a class="wiki_link" href="/Gamelismic%20clan"&gt;rodan temperament&lt;/a&gt;. The 8/7 generator of 17\87 is a remarkable 0.00062 cents sharper than the 13-limit &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE&lt;/a&gt; generator and is close to the 11-limit POTE generator also. Also, the 32\87 generator for &lt;a class="wiki_link" href="/Kleismic%20family"&gt;clyde temperament&lt;/a&gt; is 0.04455 cents sharp of the 7-limit POTE generator.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Rank two temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Rank two temperaments&lt;/h1&gt;


&lt;table class="wiki_table"&gt;
== Regular temperament properties ==
    &lt;tr&gt;
{| class="wikitable center-4 center-5 center-6"
        &lt;th&gt;Periods&lt;br /&gt;
|-
per&lt;br /&gt;
! rowspan="2" | [[Subgroup]]
octave&lt;br /&gt;
! rowspan="2" | [[Comma list]]
&lt;/th&gt;
! rowspan="2" | [[Mapping]]
        &lt;th&gt;Generator&lt;br /&gt;
! rowspan="2" | Optimal<br>8ve stretch (¢)
&lt;/th&gt;
! colspan="2" | Tuning error
        &lt;th&gt;Cents&lt;br /&gt;
|-
&lt;/th&gt;
! [[TE error|Absolute]] (¢)
        &lt;th&gt;Associated&lt;br /&gt;
! [[TE simple badness|Relative]] (%)
ratio&lt;br /&gt;
|-
&lt;/th&gt;
| 2.3.5
        &lt;th&gt;Temperament&lt;br /&gt;
| 15625/15552, 67108864/66430125
&lt;/th&gt;
| {{mapping| 87 138 202 }}
    &lt;/tr&gt;
| −0.299
    &lt;tr&gt;
| 0.455
        &lt;td style="text-align: right;"&gt;1&lt;br /&gt;
| 3.30
&lt;/td&gt;
|-
        &lt;td style="text-align: right;"&gt;4\87&lt;br /&gt;
| 2.3.5.7
&lt;/td&gt;
| 245/243, 1029/1024, 3136/3125
        &lt;td style="text-align: right;"&gt;55.172&lt;br /&gt;
| {{mapping| 87 138 202 244 }}
&lt;/td&gt;
| +0.070
        &lt;td style="text-align: center;"&gt;33/32&lt;br /&gt;
| 0.752
&lt;/td&gt;
| 5.45
        &lt;td style="text-align: left;"&gt;&lt;a class="wiki_link" href="/Sensa"&gt;Sensa&lt;/a&gt;&lt;br /&gt;
|-
&lt;/td&gt;
| 2.3.5.7.11
    &lt;/tr&gt;
| 245/243, 385/384, 441/440, 3136/3125
    &lt;tr&gt;
| {{mapping| 87 138 202 244 301 }}
        &lt;td style="text-align: right;"&gt;1&lt;br /&gt;
| +0.033
&lt;/td&gt;
| 0.676
        &lt;td style="text-align: right;"&gt;10\87&lt;br /&gt;
| 4.90
&lt;/td&gt;
|-
        &lt;td style="text-align: right;"&gt;137.931&lt;br /&gt;
| 2.3.5.7.11.13
&lt;/td&gt;
| 196/195, 245/243, 352/351, 364/363, 625/624
        &lt;td style="text-align: center;"&gt;13/12&lt;br /&gt;
| {{mapping| 87 138 202 244 301 322 }}
&lt;/td&gt;
| −0.011
        &lt;td&gt;&lt;a class="wiki_link" href="/Quartemka"&gt;Quartemka&lt;/a&gt;&lt;br /&gt;
| 0.625
&lt;/td&gt;
| 4.53
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 2.3.5.7.11.13.17
        &lt;td style="text-align: right;"&gt;1&lt;br /&gt;
| 154/153, 196/195, 245/243, 273/272, 364/363, 375/374
&lt;/td&gt;
| {{mapping| 87 138 202 244 301 322 356 }}
        &lt;td style="text-align: right;"&gt;14\87&lt;br /&gt;
| −0.198
&lt;/td&gt;
| 0.738
        &lt;td style="text-align: right;"&gt;193.103&lt;br /&gt;
| 5.35
&lt;/td&gt;
|-
        &lt;td style="text-align: center;"&gt;28/25&lt;br /&gt;
| 2.3.5.7.11.13.17.19
&lt;/td&gt;
| 154/153, 196/195, 210/209, 245/243, 273/272, 286/285, 364/363
        &lt;td&gt;&lt;a class="wiki_link" href="/Luna"&gt;Luna&lt;/a&gt;/&lt;a class="wiki_link" href="/Hemithirds"&gt;hemithirds&lt;/a&gt;&lt;br /&gt;
| {{mapping| 87 138 202 244 301 322 356 370 }}
&lt;/td&gt;
| −0.348
    &lt;/tr&gt;
| 0.796
    &lt;tr&gt;
| 5.77
        &lt;td style="text-align: right;"&gt;1&lt;br /&gt;
|}
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;17\87&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;234.483&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;8/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/Rodan"&gt;Rodan&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: right;"&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;23\87&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;317.241&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;6/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/Hanson"&gt;Hanson&lt;/a&gt;/&lt;a class="wiki_link" href="/countercata"&gt;countercata&lt;/a&gt;/&lt;a class="wiki_link" href="/metakleismic"&gt;metakleismic&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: right;"&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;32\87&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;441.379&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;9/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/Clyde"&gt;Clyde&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: right;"&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;38\87&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;524.138&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;65/48&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;&lt;a class="wiki_link" href="/Widefourth"&gt;Widefourth&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: right;"&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;40\87&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;551.724&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;11/8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/Emkay"&gt;Emkay&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: right;"&gt;3&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;23\87&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;317.241&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;6/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/Tritikleismic"&gt;Tritikleismic&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: right;"&gt;29&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;28\87&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;386.207&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;5/4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/Mystery"&gt;Mystery&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
=== 13-limit detempering ===
87 can serve as a MOS in these:&lt;br /&gt;
{{Main|87edo/13-limit detempering}}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/M%26N%20temperaments"&gt;270&amp;amp;87&lt;/a&gt; &amp;lt;&amp;lt;24 -9 -66 12 27 ... ||&lt;br /&gt;
=== Rank-2 temperaments ===
&lt;a class="wiki_link" href="/M%26N%20temperaments"&gt;494&amp;amp;87&lt;/a&gt; &amp;lt;&amp;lt;51 -1 -133 11 32 ... ||&lt;br /&gt;
{| class="wikitable center-all left-5"
&lt;br /&gt;
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="x13-limit detempering of 87et"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;13-limit detempering of 87et&lt;/h1&gt;
|-
[91/90, 49/48, 40/39, 28/27, 25/24, 21/20, 35/33, 16/15, 14/13, 13/12, 12/11, 11/10, 10/9, 28/25, 9/8, 25/22, 8/7, 15/13, 7/6, 75/64, 13/11, 25/21, 6/5, 40/33, 11/9, 16/13, 26/21, 5/4, 44/35, 14/11, 32/25, 9/7, 13/10, 21/16, 33/25, 4/3, 35/26, 27/20, 15/11, 11/8, 18/13, 7/5, 45/32, 64/45, 10/7, 13/9, 16/11, 22/15, 40/27, 52/35, 3/2, 50/33, 32/21, 20/13, 14/9, 25/16, 11/7, 35/22, 8/5, 21/13, 13/8, 18/11, 33/20, 5/3, 42/25, 22/13, 75/44, 12/7, 26/15, 7/4, 44/25, 16/9, 25/14, 9/5, 20/11, 11/6, 24/13, 13/7, 15/8, 66/35, 21/11, 25/13, 27/14, 39/20, 55/28, 99/50, 2]&lt;br /&gt;
! Periods<br>per 8ve
&lt;br /&gt;
! Generator*
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Music"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Music&lt;/h1&gt;
! Cents*
&lt;br /&gt;
! Associated<br>ratio*
&lt;a class="wiki_link_ext" href="http://www.archive.org/details/Pianodactyl" rel="nofollow"&gt;Pianodactyl&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://www.archive.org/download/Pianodactyl/pianodactyl.mp3" rel="nofollow"&gt;play&lt;/a&gt; by &lt;a class="wiki_link" href="/Gene%20Ward%20Smith"&gt;Gene Ward Smith&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
! 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"