87edo
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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 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= ||~ Periods per octave ||~ Generator ||~ Cents ||~ Associated ratio ||~ Temperament || ||> 1 ||> 4\87 ||> 55.172 ||= 33/32 ||< [[Sensa]] || ||> 1 ||> 10\87 ||> 137.931 ||= 13/12 || [[Quartemka]] || ||> 1 ||> 14\87 ||> 193.103 ||= 28/25 || [[Luna]]/[[Hemithirds|hemithirds]] || ||> 1 ||> 17\87 ||> 234.483 ||= 8/7 || [[Rodan]] || ||> 1 ||> 23\87 ||> 317.241 ||= 6/5 || [[Hanson]]/[[countercata]]/[[metakleismic]] || ||> 1 ||> 32\87 ||> 441.379 ||= 9/7 || [[Clyde]] || ||> 1 ||> 38\87 ||> 524.138 ||= 65/48 ||< [[Widefourth]] || ||> 1 ||> 40\87 ||> 551.724 ||= 11/8 || [[Emkay]] || ||> 3 ||> 23\87 ||> 317.241 ||= 6/5 || [[Tritikleismic]] || ||> 29 ||> 28\87 ||> 386.207 ||= 5/4 || [[Mystery]] || 87 can serve as a MOS in these: 270&87 <<24 -9 -66 12 27 ... || 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]]
Original HTML content:
<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, <46 -29|, the misty comma, <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 />
<!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Rank two temperaments"></a><!-- ws:end:WikiTextHeadingRule:0 -->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 />
87 can serve as a MOS in these:<br />
<br />
270&87 <<24 -9 -66 12 27 ... ||<br />
494&87 <<51 -1 -133 11 32 ... ||<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="x13-limit detempering of 87et"></a><!-- ws:end:WikiTextHeadingRule:2 -->13-limit detempering of 87et</h1>
[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 />
<br />
<!-- ws:start:WikiTextHeadingRule:4:<h1> --><h1 id="toc2"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:4 -->Music</h1>
<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>