Xenharmonic Wiki

87edo: Difference between revisions

← Older edit

Eliora (talk | contribs)

2 user groups
3,642 edits
VisualWikitext
Wikispaces>genewardsmith
**Imported revision 186512509 - Original comment: **
Eliora (talk | contribs)
autopatrolled, emailconfirmed
3,642 edits
 
(140 intermediate revisions by 30 users not shown)
Line 1: Line 1:
<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>2010-12-08 15:50:01 UTC</tt>.<br>
: The original revision id was <tt>186512509</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]], 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]].  


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. 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.</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;
[[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.
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;
 
&lt;br /&gt;
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.
Some 13-limit rank two temperaments supported by 87et are these:&lt;br /&gt;
 
&lt;br /&gt;
See also [[87th-octave temperaments]].
tritikleismic 72&amp;amp;87 &amp;lt;&amp;lt;18 15 -6 9 42 ... ||&lt;br /&gt;
 
rodan 41&amp;amp;87 &amp;lt;&amp;lt;3 17 -1 -13 -22 ... ||&lt;br /&gt;
== Intervals ==
mystery 29&amp;amp;58 &amp;lt;&amp;lt;0 29 29 29 29 ... ||&lt;br /&gt;
{| class="wikitable center-all right-2 left-3 left-4"
countercata 34&amp;amp;87 &amp;lt;&amp;lt;6 5 -31 32 14 ... ||&lt;br /&gt;
|-
luna 31&amp;amp;87 &amp;lt;&amp;lt;15 -2 -5 22 -23 ... ||&lt;br /&gt;
! rowspan="2" | #
&lt;br /&gt;
! rowspan="2" | Cents
87 can serve as a MOS in these:&lt;br /&gt;
! colspan="2" | Approximated ratios
&lt;br /&gt;
! colspan="2" rowspan="2" | [[Ups and downs notation]]
270&amp;amp;87 &amp;lt;&amp;lt;24 -9 -66 12 27 ... ||&lt;br /&gt;
|-
494&amp;amp;87 &amp;lt;&amp;lt;51 -1 -133 11 32 ... ||&lt;br /&gt;
! 13-limit
224&amp;amp;87 &amp;lt;&amp;lt;27 8 -67 -1 5 ...||&lt;br /&gt;
! 31-limit extension
&lt;br /&gt;
|-
87et is a particularly good tuning for rodan temperament. 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.&lt;/body&gt;&lt;/html&gt;</pre></div>
| 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
|-
| …
| …
| …
| …
| …
| …
|}
 
== 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 ==
{| 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 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:
 
* [[Avicenna (temperament)|Avicenna]] ([[Breed|87 & 270]])
* [[Breed|87 & 494]]
 
== Scales ==
=== Harmonic scales ===
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) ====
{| 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.
 
=== 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 ==
=== 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
 
[[Category:Listen]]
[[Category:Clyde]]
[[Category:Countercata]]
[[Category:Hemithirds]]
[[Category:Mystery]]
[[Category:Rodan]]
[[Category:Tritikleismic]]
Retrieved from "https://en.xen.wiki/w/87edo"