87edo: Difference between revisions

Line 1:

~~<h2>IMPORTED REVISION FROM WIKISPACES</h2>~~

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|cent]]s. It is solid as both a [[13-limit|13-limit]] (or 15 odd limit) and as a [[5-limit|5-limit]] system, and of course does well enough in any limit in between. It represents the [[13-limit|13-limit]] [[Tonality_diamond|tonality diamond]] both uniquely and [[consistent|consistent]]ly, and is the smallest equal temperament to do so.

~~This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>~~

~~: 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, <46 -29|, the misty comma, <26 -12 -3|, the kleisma, 15625/15552, 245/243, 1029/1024, 3136/3125, and 5120/5103.

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, <46 -29|, the misty comma, <26 -12 -3|, the kleisma, 15625/15552, 245/243, 1029/1024, 3136/3125, and 5120/5103.

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.

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.

=Rank two temperaments=

{| class="wikitable"

|-

! | Periods

~~=Rank two temperaments=~~

~~||~ Periods~~

per

octave ||~ Generator ||~ Cents ||~ Associated

ratio ||~ Temperament ||

octave

||~~>~~ 1 ||~~>~~ 4\87 ||~~>~~ 55.172 ||= 33/32 ||~~<~~ [[Sensa]] ||

! | Generator

||~~>~~ 1 ||~~>~~ 10\87 ||~~>~~ 137.931 ||= 13/12 || [[Quartemka]] ||

! | Cents

||~~>~~ 1 ||~~>~~ 14\87 ||~~>~~ 193.103 ||= 28/25 || [[Luna]]/[[Hemithirds|hemithirds]] ||

! | Associated

||~~>~~ 1 ||~~>~~ 17\87 ||~~>~~ 234.483 ||= 8/7 || [[Rodan]] ||

||~~>~~ 1 ||~~>~~ 23\87 ||~~>~~ 317.241 ||= 6/5 || [[Hanson]]/[[countercata]]/[[metakleismic]] ||

ratio

||~~>~~ 1 ||~~>~~ 32\87 ||~~>~~ 441.379 ||= 9/7 || [[Clyde]] ||

! | Temperament

||~~>~~ 1 ||~~>~~ 38\87 ||~~>~~ 524.138 ||= 65/48 ||~~<~~ [[Widefourth]] ||

|-

||~~>~~ 1 ||~~>~~ 40\87 ||~~>~~ 551.724 ||= 11/8 || [[Emkay]] ||

| style="text-align:right;" | 1

||~~>~~ 3 ||~~>~~ 23\87 ||~~>~~ 317.241 ||= 6/5 || [[Tritikleismic]] ||

| style="text-align:right;" | 4\87

||~~>~~ 29 ||~~>~~ 28\87 ||~~>~~ 386.207 ||= 5/4 || [[Mystery]] ||

| style="text-align:right;" | 55.172

| style="text-align:center;" | 33/32

| | [[Sensa|Sensa]]

|-

| style="text-align:right;" | 1

| style="text-align:right;" | 10\87

| style="text-align:right;" | 137.931

| style="text-align:center;" | 13/12

| | [[Quartemka|Quartemka]]

|-

| style="text-align:right;" | 1

| style="text-align:right;" | 14\87

| style="text-align:right;" | 193.103

| style="text-align:center;" | 28/25

| | [[Luna|Luna]]/[[Hemithirds|hemithirds]]

|-

| style="text-align:right;" | 1

| style="text-align:right;" | 17\87

| style="text-align:right;" | 234.483

| style="text-align:center;" | 8/7

| | [[Rodan|Rodan]]

|-

| style="text-align:right;" | 1

| style="text-align:right;" | 23\87

| style="text-align:right;" | 317.241

| style="text-align:center;" | 6/5

|-

| style="text-align:right;" | 1

| style="text-align:right;" | 32\87

| style="text-align:right;" | 441.379

| style="text-align:center;" | 9/7

| | [[Clyde|Clyde]]

|-

| style="text-align:right;" | 1

| style="text-align:right;" | 38\87

| style="text-align:right;" | 524.138

| style="text-align:center;" | 65/48

| | [[Widefourth|Widefourth]]

|-

| style="text-align:right;" | 1

| style="text-align:right;" | 40\87

| style="text-align:right;" | 551.724

| style="text-align:center;" | 11/8

| | [[Emkay|Emkay]]

|-

| style="text-align:right;" | 3

| style="text-align:right;" | 23\87

| style="text-align:right;" | 317.241

| style="text-align:center;" | 6/5

| | [[Tritikleismic|Tritikleismic]]

|-

| style="text-align:right;" | 29

| style="text-align:right;" | 28\87

| style="text-align:right;" | 386.207

| style="text-align:center;" | 5/4

| | [[mystery|Mystery]]

|}

87 can serve as a MOS in these:

[[M&~~amp;N temperaments~~|270&87]] <<24 -9 -66 12 27 ~~... ||~~

[[M&N_temperaments|270&87]] <<24 -9 -66 12 27 ... ||

~~[[M&N temperaments|494&87]] <<51 -1 -133 11 32~~ ... ||

=13-limit detempering of 87et=

[[M&N_temperaments|494&87]] <<51 -1 -133 11 32 ... ||

=13-limit detempering of 87et=

[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]

=Music=

~~[[http://www.archive.org/details/Pianodactyl|Pianodactyl]] [[http://www.archive.org/download/Pianodactyl/pianodactyl.mp3|play]] by [[Gene Ward Smith]]</pre></div>~~

~~<h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>87edo</title></head><body>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 <a class="wiki_link" href="/cent">cent</a>s. It is solid as both a <a class="wiki_link" href="/13-limit">13-limit</a> (or 15 odd limit) and as a <a class="wiki_link" href="/5-limit">5-limit</a> system, and of course does well enough in any limit in between. It represents the <a class="wiki_link" href="/13-limit">13-limit</a> <a class="wiki_link" href="/tonality%20diamond">tonality diamond</a> both uniquely and <a class="wiki_link" href="/consistent">consistent</a>ly, and is the smallest equal temperament to do so.<br />

~~<br />~~

87et <a class="wiki_link" href="/tempering%20out">tempers out</a> 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.<br />

~~<br />~~

87et is a particularly good tuning for <a class="wiki_link" href="/Gamelismic%20clan">rodan temperament</a>. The 8/7 generator of 17\87 is a remarkable 0.00062 cents sharper than the 13-limit <a class="wiki_link" href="/POTE%20tuning">POTE</a> generator and is close to the 11-limit POTE generator also. Also, the 32\87 generator for <a class="wiki_link" href="/Kleismic%20family">clyde temperament</a> is 0.04455 cents sharp of the 7-limit POTE generator.<br />

~~<br />~~

<h1 id="toc0"><a name="Rank two temperaments"></a>Rank two temperaments</h1>

~~<table class="wiki_table">~~

~~<tr>~~

~~<th>Periods<br />~~

~~per<br />~~

~~octave<br />~~

~~</th>~~

~~<th>Generator<br />~~

~~</th>~~

~~<th>Cents<br />~~

~~</th>~~

~~<th>Associated<br />~~

~~ratio<br />~~

~~</th>~~

~~<th>Temperament<br />~~

~~</th>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: right;">1<br />~~

~~</td>~~

~~<td style="text-align: right;">4\87<br />~~

~~</td>~~

~~<td style="text-align: right;">55.172<br />~~

~~</td>~~

~~<td style="text-align: center;">33/32<br />~~

~~</td>~~

~~<td style="text-align: left;"><a class="wiki_link" href="/Sensa">Sensa</a><br />~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: right;">1<br />~~

~~</td>~~

~~<td style="text-align: right;">10\87<br />~~

~~</td>~~

~~<td style="text-align: right;">137.931<br />~~

~~</td>~~

~~<td style="text-align: center;">13/12<br />~~

~~</td>~~

~~<td><a class="wiki_link" href="/Quartemka">Quartemka</a><br />~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: right;">1<br />~~

~~</td>~~

~~<td style="text-align: right;">14\87<br />~~

~~</td>~~

~~<td style="text-align: right;">193.103<br />~~

~~</td>~~

~~<td style="text-align: center;">28/25<br />~~

~~</td>~~

~~<td><a class="wiki_link" href="/Luna">Luna</a>/<a class="wiki_link" href="/Hemithirds">hemithirds</a><br />~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: right;">1<br />~~

~~</td>~~

~~<td style="text-align: right;">17\87<br />~~

~~</td>~~

~~<td style="text-align: right;">234.483<br />~~

~~</td>~~

~~<td style="text-align: center;">8/7<br />~~

~~</td>~~

~~<td><a class="wiki_link" href="/Rodan">Rodan</a><br />~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: right;">1<br />~~

~~</td>~~

~~<td style="text-align: right;">23\87<br />~~

~~</td>~~

~~<td style="text-align: right;">317.241<br />~~

~~</td>~~

~~<td style="text-align: center;">6/5<br />~~

~~</td>~~

<td><a class="wiki_link" href="/Hanson">Hanson</a>/<a class="wiki_link" href="/countercata">countercata</a>/<a class="wiki_link" href="/metakleismic">metakleismic</a><br />

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: right;">1<br />~~

~~</td>~~

~~<td style="text-align: right;">32\87<br />~~

~~</td>~~

~~<td style="text-align: right;">441.379<br />~~

~~</td>~~

~~<td style="text-align: center;">9/7<br />~~

~~</td>~~

~~<td><a class="wiki_link" href="/Clyde">Clyde</a><br />~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: right;">1<br />~~

~~</td>~~

~~<td style="text-align: right;">38\87<br />~~

~~</td>~~

~~<td style="text-align: right;">524.138<br />~~

~~</td>~~

~~<td style="text-align: center;">65/48<br />~~

~~</td>~~

~~<td style="text-align: left;"><a class="wiki_link" href="/Widefourth">Widefourth</a><br />~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: right;">1<br />~~

~~</td>~~

~~<td style="text-align: right;">40\87<br />~~

~~</td>~~

~~<td style="text-align: right;">551.724<br />~~

~~</td>~~

~~<td style="text-align: center;">11/8<br />~~

~~</td>~~

~~<td><a class="wiki_link" href="/Emkay">Emkay</a><br />~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: right;">3<br />~~

~~</td>~~

~~<td style="text-align: right;">23\87<br />~~

~~</td>~~

~~<td style="text-align: right;">317.241<br />~~

~~</td>~~

~~<td style="text-align: center;">6/5<br />~~

~~</td>~~

~~<td><a class="wiki_link" href="/Tritikleismic">Tritikleismic</a><br />~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: right;">29<br />~~

~~</td>~~

~~<td style="text-align: right;">28\87<br />~~

~~</td>~~

~~<td style="text-align: right;">386.207<br />~~

~~</td>~~

~~<td style="text-align: center;">5/4<br />~~

~~</td>~~

~~<td><a class="wiki_link" href="/Mystery">Mystery</a><br />~~

~~</td>~~

~~</tr>~~

~~</table>~~

~~<br />~~

[http://www.archive.org/details/Pianodactyl Pianodactyl] [http://www.archive.org/download/Pianodactyl/pianodactyl.mp3 play] by [[Gene_Ward_Smith|Gene Ward Smith]] [[Category:87edo]]

~~87 can serve as a MOS in these:<br />~~

[[Category:clyde]]

~~<br />~~

[[Category:countercata]]

~~<a class="wiki_link" href="/M%26N%20temperaments">270&amp;87</a> &lt;&lt;24 -9 -66 12 27 ... ||<br />~~

[[Category:edo]]

~~<a class="wiki_link" href="/M%26N%20temperaments">494&amp;87</a> &lt;&lt;51 -1 -133 11 32 ... ||<br />~~

[[Category:hemithirds]]

~~<br />~~

[[Category:listen]]

<h1 id="toc1"><a name="x13-limit detempering of 87et"></a>13-limit detempering of 87et</h1>

[[Category:mystery]]

[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]<br />

[[Category:rodan]]

~~<br />~~

[[Category:theory]]

~~<h1 id="toc2"><a name="Music"></a>Music</h1>~~

[[Category:tritikleismic]]

~~<br />~~

~~<a class="wiki_link_ext" href="~~http://www.archive.org/details/Pianodactyl~~" rel="nofollow">~~Pianodactyl~~</a> <a class="wiki_link_ext" href="~~http://www.archive.org/download/Pianodactyl/pianodactyl.mp3~~" rel="nofollow">~~play~~</a>~~ by ~~<a class="wiki_link" href="/Gene%20Ward%20Smith">~~Gene Ward Smith~~</a></body></html></pre></div>~~

@@ Line 1: / Line 1: @@
-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+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|cent]]s. It is solid as both a [[13-limit|13-limit]] (or 15 odd limit) and as a [[5-limit|5-limit]] system, and of course does well enough in any limit in between. It represents the [[13-limit|13-limit]] [[Tonality_diamond|tonality diamond]] both uniquely and [[consistent|consistent]]ly, and is the smallest equal temperament to do so.
-This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: 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.
-et [[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.
+et [[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.
-et 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.
+et 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.
+=Rank two temperaments=
+{| class="wikitable"
+|-
+! | Periods
-=Rank two temperaments=
-||~ Periods
 per
-octave ||~ Generator ||~ Cents ||~ Associated
-ratio ||~ Temperament ||
+octave
-||&gt; 1 ||&gt; 4\87 ||&gt; 55.172 ||= 33/32 ||&lt; [[Sensa]] ||
+! | Generator
-||&gt; 1 ||&gt; 10\87 ||&gt; 137.931 ||= 13/12 || [[Quartemka]] ||
+! | Cents
-||&gt; 1 ||&gt; 14\87 ||&gt; 193.103 ||= 28/25 || [[Luna]]/[[Hemithirds|hemithirds]] ||
+! | Associated
-||&gt; 1 ||&gt; 17\87 ||&gt; 234.483 ||= 8/7 || [[Rodan]] ||
-||&gt; 1 ||&gt; 23\87 ||&gt; 317.241 ||= 6/5 || [[Hanson]]/[[countercata]]/[[metakleismic]] ||
+ratio
-||&gt; 1 ||&gt; 32\87 ||&gt; 441.379 ||= 9/7 || [[Clyde]] ||
+! | Temperament
-||&gt; 1 ||&gt; 38\87 ||&gt; 524.138 ||= 65/48 ||&lt; [[Widefourth]] ||
+|-
-||&gt; 1 ||&gt; 40\87 ||&gt; 551.724 ||= 11/8 || [[Emkay]] ||
+| style="text-align:right;" | 1
-||&gt; 3 ||&gt; 23\87 ||&gt; 317.241 ||= 6/5 || [[Tritikleismic]] ||
+| style="text-align:right;" | 4\87
-||&gt; 29 ||&gt; 28\87 ||&gt; 386.207 ||= 5/4 || [[Mystery]] ||
+| style="text-align:right;" | 55.172
+| style="text-align:center;" | 33/32
+| | [[Sensa|Sensa]]
+|-
+| style="text-align:right;" | 1
+| style="text-align:right;" | 10\87
+| style="text-align:right;" | 137.931
+| style="text-align:center;" | 13/12
+| | [[Quartemka|Quartemka]]
+|-
+| style="text-align:right;" | 1
+| style="text-align:right;" | 14\87
+| style="text-align:right;" | 193.103
+| style="text-align:center;" | 28/25
+| | [[Luna|Luna]]/[[Hemithirds|hemithirds]]
+|-
+| style="text-align:right;" | 1
+| style="text-align:right;" | 17\87
+| style="text-align:right;" | 234.483
+| style="text-align:center;" | 8/7
+| | [[Rodan|Rodan]]
+|-
+| style="text-align:right;" | 1
+| style="text-align:right;" | 23\87
+| style="text-align:right;" | 317.241
+| style="text-align:center;" | 6/5
+| | [[Hanson|Hanson]]/[[Countercata|countercata]]/[[metakleismic|metakleismic]]
+|-
+| style="text-align:right;" | 1
+| style="text-align:right;" | 32\87
+| style="text-align:right;" | 441.379
+| style="text-align:center;" | 9/7
+| | [[Clyde|Clyde]]
+|-
+| style="text-align:right;" | 1
+| style="text-align:right;" | 38\87
+| style="text-align:right;" | 524.138
+| style="text-align:center;" | 65/48
+| | [[Widefourth|Widefourth]]
+|-
+| style="text-align:right;" | 1
+| style="text-align:right;" | 40\87
+| style="text-align:right;" | 551.724
+| style="text-align:center;" | 11/8
+| | [[Emkay|Emkay]]
+|-
+| style="text-align:right;" | 3
+| style="text-align:right;" | 23\87
+| style="text-align:right;" | 317.241
+| style="text-align:center;" | 6/5
+| | [[Tritikleismic|Tritikleismic]]
+|-
+| style="text-align:right;" | 29
+| style="text-align:right;" | 28\87
+| style="text-align:right;" | 386.207
+| style="text-align:center;" | 5/4
+| | [[mystery|Mystery]]
+|}
 can serve as a MOS in these:
-[[M&amp;N temperaments|270&amp;87]] &lt;&lt;24 -9 -66 12 27 ... ||
+[[M&N_temperaments|270&amp;87]] &lt;&lt;24 -9 -66 12 27 ... ||
-[[M&amp;N temperaments|494&amp;87]] &lt;&lt;51 -1 -133 11 32 ... ||
-=13-limit detempering of 87et=
+[[M&N_temperaments|494&amp;87]] &lt;&lt;51 -1 -133 11 32 ... ||
+=13-limit detempering of 87et=
 [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]
 =Music=
-[[http://www.archive.org/details/Pianodactyl|Pianodactyl]] [[http://www.archive.org/download/Pianodactyl/pianodactyl.mp3|play]] by [[Gene Ward Smith]]</pre></div>
-<h4>Original HTML 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;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;
-&lt;br /&gt;
-et &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;
-et 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;
-    &lt;tr&gt;
-        &lt;th&gt;Periods&lt;br /&gt;
-per&lt;br /&gt;
-octave&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;Generator&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;Cents&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;Associated&lt;br /&gt;
-ratio&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;Temperament&lt;br /&gt;
-&lt;/th&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;4\87&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;55.172&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;33/32&lt;br /&gt;
-&lt;/td&gt;
-        &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;
-    &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;10\87&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;137.931&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;13/12&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;a class="wiki_link" href="/Quartemka"&gt;Quartemka&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;14\87&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;193.103&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;28/25&lt;br /&gt;
-&lt;/td&gt;
-        &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;
-&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;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;
+[http://www.archive.org/details/Pianodactyl Pianodactyl] [http://www.archive.org/download/Pianodactyl/pianodactyl.mp3 play] by [[Gene_Ward_Smith|Gene Ward Smith]]      [[Category:87edo]]
-can serve as a MOS in these:&lt;br /&gt;
+[[Category:clyde]]
-&lt;br /&gt;
+[[Category:countercata]]
-&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;
+[[Category:edo]]
-&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;
+[[Category:hemithirds]]
-&lt;br /&gt;
+[[Category:listen]]
-&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;
+[[Category:mystery]]
- [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;
+[[Category:rodan]]
-&lt;br /&gt;
+[[Category:theory]]
-&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;
+[[Category:tritikleismic]]
- &lt;br /&gt;
-&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>