15edo: Difference between revisions

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=15 tone equal temperament= 


15-edo is a [[7-limit]] tuning because of the amount of septimal intervals, but it does contain [[11-limit]] intervals.

from //wikipedia//:
""In music, 15 equal temperament, called 15-TET, 15-EDO, or 15-ET, is the tempered scale derived by dividing the [[octave]] into 15 equal steps. Each step represents a frequency ratio of 2 1/15, or 80 [[cent]]s. Because 15 factors into 3 times 5, it can be seen as being made up of three scales of [[5edo|5 equal divisions of the octave]] (or five scales of [[3edo]]). 15-ET matches the [[11_8|11:8]] and [[16_11|16:11]] ratios, and does not match the [[7_5|7:5]] or [[10_7|10:7]] tritone."

==Commas== 
15 EDO tempers out the following commas. (Note: This assumes the val < 15 24 35 42 52 56 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 ||< | 8 -5 > ||> 90.22 ||= Limma ||= Pythagorean Minor 2nd ||=   ||
||= 250/243 ||< | 1 -5 3 > ||> 49.17 ||= Maximal Diesis ||= Porcupine Comma ||=   ||
||= 128/125 ||< | 7 0 -3 > ||> 41.06 ||= Diesis ||= Augmented Comma ||=   ||
||= 15625/15552 ||< | -6 -5 6 > ||> 8.11 ||= Kleisma ||= Semicomma Majeur ||=   ||
||= 1029/1000 ||< | -3 1 -3 3 > ||> 49.49 ||= Keega ||=   ||=   ||
||= 49/48 ||< | -4 -1 2 > ||> 35.70 ||= Slendro Diesis ||=   ||=   ||
||= 64/63 ||< | 6 -2 0 -1 > ||> 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
||= 64827/64000 ||< | -9 3 -3 4 > ||> 22.23 ||= Squalentine ||=   ||=   ||
||= 875/864 ||< | -5 -3 3 1 > ||> 21.90 ||= Keema ||=   ||=   ||
||= 126/125 ||< | 1 2 -3 1 > ||> 13.79 ||= Septimal Semicomma ||= Starling Comma ||=   ||
||= 4000/3969 ||< | 5 -4 3 -2 > ||> 13.47 ||= Octagar ||=   ||=   ||
||= 1029/1024 ||< | -10 1 0 3 > ||> 8.43 ||= Gamelisma ||=   ||=   ||
||= 6144/6125 ||< | 11 1 -3 -2 > ||> 5.36 ||= Porwell ||=   ||=   ||
||= 250047/250000 ||< | -4 6 -6 3 > ||> 0.33 ||= Landscape Comma ||=   ||=   ||
||= 100/99 ||< | 2 -2 2 -1 > ||> 17.40 ||= Ptolemisma ||=   ||=   ||
||= 121/120 ||< | -3 -1 -1 2 > ||> 14.37 ||= Biyatisma ||=   ||=   ||
||= 176/175 ||< | 4 -2 -1 1 > ||> 9.86 ||= Valinorsma ||=   ||=   ||
||= 65536/65219 ||< | 16 -2 -3 > ||> 8.39 ||= Orgonisma ||=   ||=   ||
||= 385/384 ||< | -7 -1 1 1 1 > ||> 4.50 ||= Keenanisma ||=   ||=   ||
||= 441/440 ||< | -3 2 -1 2 -1 > ||> 3.93 ||= Werckisma ||=   ||=   ||
||= 4000/3993 ||< | 5 -1 3 -3 > ||> 3.03 ||= Wizardharry ||=   ||=   ||
||= 3025/3024 ||< | -4 -3 2 -1 2 > ||> 0.57 ||= Lehmerisma ||=   ||=   ||
||= 91/90 ||< | -1 -2 -1 1 1 > ||> 19.13 ||= Superleap ||=   ||=   ||
||= 676/675 ||< | 2 -3 -2 2 > ||> 2.56 ||= Parizeksma ||=   ||=   ||


==Additional notes== 

In the 15-edo system, major thirds cannot be divided perfectly into two, and coupled with the lack of a standard tritone, this tuning at first can be disorienting. However, because the guitar can be tuned symmetrically, from E to e (6th to 1st strings) unlike the 12-tone system on guitars, the learning curve is very manageable. All chords look the same modualted anywhere, and minor arpeggios are vertically stacked, making them very easy to play. 15-tone may be a promising start for anyone interested in superior harmony and xenharmony, a manageable number of tones, and the sonic fingerprint of multiples of 5-edo. .


[[image:http://www.swordguitars.com/Sword_15edoclassicalsm.JPG]](//15-tone Classical Guitar by [[Ron Sword]]// / //Sword Guitars//)

=Theory= 
[[http://sonic-arts.org/darreg/dar35.htm|The 15-Tone Scale System]] by [[Ivor Darreg]] [[http://www.webcitation.org/5xZyzKBEW|Permalink]]
[[http://www.inteas.com/Penta01.htm|The Pentadecaphonic System]]
[[http://home.comcast.net/%7Ebrentishere/15noteequaltempermenttutorial.html|15-EDO Tutorial]] by [[Brent Carson]] [[http://www.webcitation.org/5xeJYBsDg|Permalink]]

=Practical Theory / Books= 
[[image:http://ronsword.com/images/Pendecaphonic_coversm.jpg width="112" height="149" link="http://www.ronsword.com"]][[http://www.ronsword.com|Sword, Ronald. "Pendecaphonic Scales for Guitar" IAAA Press, UK-USA. First Ed: June 2009.]] - A large repository of all known scales and temperament families in the 15-edo system. 300+ examples /w chord-scale progressions
==**Compositions**== 
[[http://teamouse.googlepages.com/MizarianPorcupineOverture.mp3|Mizarian Porcupine Overture]] by [[Herman Miller]] ([[http://teamouse.googlepages.com/home|Herman Miller]]) ([[Regular Temperaments#porcupine|porcupine]] chord progressions)
[[http://www.microtonalmusic.net/audio/15edostudy.mp3|Study for Bells]] by [[Daniel Thompson]] ([[http://danielthompson.blogspot.com/|Daniel Thompson]]) (Jan. 2007)
Study for Kyle Gann by [[http://www.akjmusic.com/works.html|Aaron K. Johnson]] (12-out-of-15)
[[http://azuma-asobi.com/Music/index.html|Rick McGowan]]: [[http://azuma-asobi.com/Music/Music-FullWorks.html|Four Ballet Scenes]]
[[http://www.soundclick.com/bands/songInfo.cfm?bandID=145852&songID=2920478|Hyperimprovisation 3.3]] by [[Jacob Barton]] (2003)
[[http://www.h-pi.com/midi/15ET.mid|Elegy in 15ET]] by [[Aaron Andrew Hunt]]
[[http://www.h-pi.com/midi/15ETa3fugue2.mid|Fugue a3 in 15ET]] by Aaron Andrew Hunt

Original HTML content:

<html><head><title>15edo</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x15 tone equal temperament"></a><!-- ws:end:WikiTextHeadingRule:0 -->15 tone equal temperament</h1>
 <br />
<br />
15-edo is a <a class="wiki_link" href="/7-limit">7-limit</a> tuning because of the amount of septimal intervals, but it does contain <a class="wiki_link" href="/11-limit">11-limit</a> intervals.<br />
<br />
from <em>wikipedia</em>:<br />
&quot;&quot;In music, 15 equal temperament, called 15-TET, 15-EDO, or 15-ET, is the tempered scale derived by dividing the <a class="wiki_link" href="/octave">octave</a> into 15 equal steps. Each step represents a frequency ratio of 2 1/15, or 80 <a class="wiki_link" href="/cent">cent</a>s. Because 15 factors into 3 times 5, it can be seen as being made up of three scales of <a class="wiki_link" href="/5edo">5 equal divisions of the octave</a> (or five scales of <a class="wiki_link" href="/3edo">3edo</a>). 15-ET matches the <a class="wiki_link" href="/11_8">11:8</a> and <a class="wiki_link" href="/16_11">16:11</a> ratios, and does not match the <a class="wiki_link" href="/7_5">7:5</a> or <a class="wiki_link" href="/10_7">10:7</a> tritone.&quot;<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x15 tone equal temperament-Commas"></a><!-- ws:end:WikiTextHeadingRule:2 -->Commas</h2>
 15 EDO tempers out the following commas. (Note: This assumes the val &lt; 15 24 35 42 52 56 |.)<br />
<br />


<table class="wiki_table">
    <tr>
        <th>Comma<br />
</th>
        <th>Monzo<br />
</th>
        <th>Value (Cents)<br />
</th>
        <th>Name 1<br />
</th>
        <th>Name 2<br />
</th>
        <th>Name 3<br />
</th>
    </tr>
    <tr>
        <td style="text-align: center;">256/243<br />
</td>
        <td style="text-align: left;">| 8 -5 &gt;<br />
</td>
        <td style="text-align: right;">90.22<br />
</td>
        <td style="text-align: center;">Limma<br />
</td>
        <td style="text-align: center;">Pythagorean Minor 2nd<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">250/243<br />
</td>
        <td style="text-align: left;">| 1 -5 3 &gt;<br />
</td>
        <td style="text-align: right;">49.17<br />
</td>
        <td style="text-align: center;">Maximal Diesis<br />
</td>
        <td style="text-align: center;">Porcupine Comma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">128/125<br />
</td>
        <td style="text-align: left;">| 7 0 -3 &gt;<br />
</td>
        <td style="text-align: right;">41.06<br />
</td>
        <td style="text-align: center;">Diesis<br />
</td>
        <td style="text-align: center;">Augmented Comma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">15625/15552<br />
</td>
        <td style="text-align: left;">| -6 -5 6 &gt;<br />
</td>
        <td style="text-align: right;">8.11<br />
</td>
        <td style="text-align: center;">Kleisma<br />
</td>
        <td style="text-align: center;">Semicomma Majeur<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">1029/1000<br />
</td>
        <td style="text-align: left;">| -3 1 -3 3 &gt;<br />
</td>
        <td style="text-align: right;">49.49<br />
</td>
        <td style="text-align: center;">Keega<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">49/48<br />
</td>
        <td style="text-align: left;">| -4 -1 2 &gt;<br />
</td>
        <td style="text-align: right;">35.70<br />
</td>
        <td style="text-align: center;">Slendro Diesis<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">64/63<br />
</td>
        <td style="text-align: left;">| 6 -2 0 -1 &gt;<br />
</td>
        <td style="text-align: right;">27.26<br />
</td>
        <td style="text-align: center;">Septimal Comma<br />
</td>
        <td style="text-align: center;">Archytas' Comma<br />
</td>
        <td style="text-align: center;">Leipziger Komma<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">64827/64000<br />
</td>
        <td style="text-align: left;">| -9 3 -3 4 &gt;<br />
</td>
        <td style="text-align: right;">22.23<br />
</td>
        <td style="text-align: center;">Squalentine<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">875/864<br />
</td>
        <td style="text-align: left;">| -5 -3 3 1 &gt;<br />
</td>
        <td style="text-align: right;">21.90<br />
</td>
        <td style="text-align: center;">Keema<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">126/125<br />
</td>
        <td style="text-align: left;">| 1 2 -3 1 &gt;<br />
</td>
        <td style="text-align: right;">13.79<br />
</td>
        <td style="text-align: center;">Septimal Semicomma<br />
</td>
        <td style="text-align: center;">Starling Comma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">4000/3969<br />
</td>
        <td style="text-align: left;">| 5 -4 3 -2 &gt;<br />
</td>
        <td style="text-align: right;">13.47<br />
</td>
        <td style="text-align: center;">Octagar<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">1029/1024<br />
</td>
        <td style="text-align: left;">| -10 1 0 3 &gt;<br />
</td>
        <td style="text-align: right;">8.43<br />
</td>
        <td style="text-align: center;">Gamelisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">6144/6125<br />
</td>
        <td style="text-align: left;">| 11 1 -3 -2 &gt;<br />
</td>
        <td style="text-align: right;">5.36<br />
</td>
        <td style="text-align: center;">Porwell<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">250047/250000<br />
</td>
        <td style="text-align: left;">| -4 6 -6 3 &gt;<br />
</td>
        <td style="text-align: right;">0.33<br />
</td>
        <td style="text-align: center;">Landscape Comma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">100/99<br />
</td>
        <td style="text-align: left;">| 2 -2 2 -1 &gt;<br />
</td>
        <td style="text-align: right;">17.40<br />
</td>
        <td style="text-align: center;">Ptolemisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">121/120<br />
</td>
        <td style="text-align: left;">| -3 -1 -1 2 &gt;<br />
</td>
        <td style="text-align: right;">14.37<br />
</td>
        <td style="text-align: center;">Biyatisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">176/175<br />
</td>
        <td style="text-align: left;">| 4 -2 -1 1 &gt;<br />
</td>
        <td style="text-align: right;">9.86<br />
</td>
        <td style="text-align: center;">Valinorsma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">65536/65219<br />
</td>
        <td style="text-align: left;">| 16 -2 -3 &gt;<br />
</td>
        <td style="text-align: right;">8.39<br />
</td>
        <td style="text-align: center;">Orgonisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">385/384<br />
</td>
        <td style="text-align: left;">| -7 -1 1 1 1 &gt;<br />
</td>
        <td style="text-align: right;">4.50<br />
</td>
        <td style="text-align: center;">Keenanisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">441/440<br />
</td>
        <td style="text-align: left;">| -3 2 -1 2 -1 &gt;<br />
</td>
        <td style="text-align: right;">3.93<br />
</td>
        <td style="text-align: center;">Werckisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">4000/3993<br />
</td>
        <td style="text-align: left;">| 5 -1 3 -3 &gt;<br />
</td>
        <td style="text-align: right;">3.03<br />
</td>
        <td style="text-align: center;">Wizardharry<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">3025/3024<br />
</td>
        <td style="text-align: left;">| -4 -3 2 -1 2 &gt;<br />
</td>
        <td style="text-align: right;">0.57<br />
</td>
        <td style="text-align: center;">Lehmerisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">91/90<br />
</td>
        <td style="text-align: left;">| -1 -2 -1 1 1 &gt;<br />
</td>
        <td style="text-align: right;">19.13<br />
</td>
        <td style="text-align: center;">Superleap<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">676/675<br />
</td>
        <td style="text-align: left;">| 2 -3 -2 2 &gt;<br />
</td>
        <td style="text-align: right;">2.56<br />
</td>
        <td style="text-align: center;">Parizeksma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
</table>

<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x15 tone equal temperament-Additional notes"></a><!-- ws:end:WikiTextHeadingRule:4 -->Additional notes</h2>
 <br />
In the 15-edo system, major thirds cannot be divided perfectly into two, and coupled with the lack of a standard tritone, this tuning at first can be disorienting. However, because the guitar can be tuned symmetrically, from E to e (6th to 1st strings) unlike the 12-tone system on guitars, the learning curve is very manageable. All chords look the same modualted anywhere, and minor arpeggios are vertically stacked, making them very easy to play. 15-tone may be a promising start for anyone interested in superior harmony and xenharmony, a manageable number of tones, and the sonic fingerprint of multiples of 5-edo. .<br />
<br />
<br />
<!-- ws:start:WikiTextRemoteImageRule:364:&lt;img src=&quot;http://www.swordguitars.com/Sword_15edoclassicalsm.JPG&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="http://www.swordguitars.com/Sword_15edoclassicalsm.JPG" alt="external image Sword_15edoclassicalsm.JPG" title="external image Sword_15edoclassicalsm.JPG" /><!-- ws:end:WikiTextRemoteImageRule:364 -->(<em>15-tone Classical Guitar by <a class="wiki_link" href="/Ron%20Sword">Ron Sword</a></em> / <em>Sword Guitars</em>)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Theory"></a><!-- ws:end:WikiTextHeadingRule:6 -->Theory</h1>
 <a class="wiki_link_ext" href="http://sonic-arts.org/darreg/dar35.htm" rel="nofollow">The 15-Tone Scale System</a> by <a class="wiki_link" href="/Ivor%20Darreg">Ivor Darreg</a> <a class="wiki_link_ext" href="http://www.webcitation.org/5xZyzKBEW" rel="nofollow">Permalink</a><br />
<a class="wiki_link_ext" href="http://www.inteas.com/Penta01.htm" rel="nofollow">The Pentadecaphonic System</a><br />
<a class="wiki_link_ext" href="http://home.comcast.net/%7Ebrentishere/15noteequaltempermenttutorial.html" rel="nofollow">15-EDO Tutorial</a> by <a class="wiki_link" href="/Brent%20Carson">Brent Carson</a> <a class="wiki_link_ext" href="http://www.webcitation.org/5xeJYBsDg" rel="nofollow">Permalink</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Practical Theory / Books"></a><!-- ws:end:WikiTextHeadingRule:8 -->Practical Theory / Books</h1>
 <!-- ws:start:WikiTextRemoteImageRule:366:&lt;a href=&quot;http://www.ronsword.com&quot; rel=&quot;nofollow&quot;&gt;&lt;img src=&quot;http://ronsword.com/images/Pendecaphonic_coversm.jpg&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 149px; width: 112px;&quot; /&gt;&lt;/a&gt; --><a href="http://www.ronsword.com" rel="nofollow"><img src="http://ronsword.com/images/Pendecaphonic_coversm.jpg" alt="external image Pendecaphonic_coversm.jpg" title="external image Pendecaphonic_coversm.jpg" style="height: 149px; width: 112px;" /></a><!-- ws:end:WikiTextRemoteImageRule:366 --><a class="wiki_link_ext" href="http://www.ronsword.com" rel="nofollow">Sword, Ronald. &quot;Pendecaphonic Scales for Guitar&quot; IAAA Press, UK-USA. First Ed: June 2009.</a> - A large repository of all known scales and temperament families in the 15-edo system. 300+ examples /w chord-scale progressions<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Practical Theory / Books-Compositions"></a><!-- ws:end:WikiTextHeadingRule:10 --><strong>Compositions</strong></h2>
 <a class="wiki_link_ext" href="http://teamouse.googlepages.com/MizarianPorcupineOverture.mp3" rel="nofollow">Mizarian Porcupine Overture</a> by <a class="wiki_link" href="/Herman%20Miller">Herman Miller</a> (<a class="wiki_link_ext" href="http://teamouse.googlepages.com/home" rel="nofollow">Herman Miller</a>) (<a class="wiki_link" href="/Regular%20Temperaments#porcupine">porcupine</a> chord progressions)<br />
<a class="wiki_link_ext" href="http://www.microtonalmusic.net/audio/15edostudy.mp3" rel="nofollow">Study for Bells</a> by <a class="wiki_link" href="/Daniel%20Thompson">Daniel Thompson</a> (<a class="wiki_link_ext" href="http://danielthompson.blogspot.com/" rel="nofollow">Daniel Thompson</a>) (Jan. 2007)<br />
Study for Kyle Gann by <a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow">Aaron K. Johnson</a> (12-out-of-15)<br />
<a class="wiki_link_ext" href="http://azuma-asobi.com/Music/index.html" rel="nofollow">Rick McGowan</a>: <a class="wiki_link_ext" href="http://azuma-asobi.com/Music/Music-FullWorks.html" rel="nofollow">Four Ballet Scenes</a><br />
<a class="wiki_link_ext" href="http://www.soundclick.com/bands/songInfo.cfm?bandID=145852&amp;songID=2920478" rel="nofollow">Hyperimprovisation 3.3</a> by <a class="wiki_link" href="/Jacob%20Barton">Jacob Barton</a> (2003)<br />
<a class="wiki_link_ext" href="http://www.h-pi.com/midi/15ET.mid" rel="nofollow">Elegy in 15ET</a> by <a class="wiki_link" href="/Aaron%20Andrew%20Hunt">Aaron Andrew Hunt</a><br />
<a class="wiki_link_ext" href="http://www.h-pi.com/midi/15ETa3fugue2.mid" rel="nofollow">Fugue a3 in 15ET</a> by Aaron Andrew Hunt</body></html>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:jdfreivald|jdfreivald]] and made on <tt>2011-05-31 19:49:07 UTC</tt>.<br>
+: This revision was by author [[User:jdfreivald|jdfreivald]] and made on <tt>2011-05-31 19:50:31 UTC</tt>.<br>
-: The original revision id was <tt>233316194</tt>.<br>
+: The original revision id was <tt>233316530</tt>.<br>
 : The revision comment was: <tt>Added comma table.</tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 18: / Line 18: @@
 ||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
-|| 256/243 || | 8 -5 &gt; || 90.22 || Limma || Pythagorean Minor 2nd ||   ||
+||= 256/243 ||&lt; | 8 -5 &gt; ||&gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||=   ||
-|| 250/243 || | 1 -5 3 &gt; || 49.17 || Maximal Diesis || Porcupine Comma ||   ||
+||= 250/243 ||&lt; | 1 -5 3 &gt; ||&gt; 49.17 ||= Maximal Diesis ||= Porcupine Comma ||=   ||
-|| 128/125 || | 7 0 -3 &gt; || 41.06 || Diesis || Augmented Comma ||   ||
+||= 128/125 ||&lt; | 7 0 -3 &gt; ||&gt; 41.06 ||= Diesis ||= Augmented Comma ||=   ||
-|| 15625/15552 || | -6 -5 6 &gt; || 8.11 || Kleisma || Semicomma Majeur ||   ||
+||= 15625/15552 ||&lt; | -6 -5 6 &gt; ||&gt; 8.11 ||= Kleisma ||= Semicomma Majeur ||=   ||
-|| 1029/1000 || | -3 1 -3 3 &gt; || 49.49 || Keega ||   ||   ||
+||= 1029/1000 ||&lt; | -3 1 -3 3 &gt; ||&gt; 49.49 ||= Keega ||=   ||=   ||
-|| 49/48 || | -4 -1 2 &gt; || 35.70 || Slendro Diesis ||   ||   ||
+||= 49/48 ||&lt; | -4 -1 2 &gt; ||&gt; 35.70 ||= Slendro Diesis ||=   ||=   ||
-|| 64/63 || | 6 -2 0 -1 &gt; || 27.26 || Septimal Comma || Archytas' Comma || Leipziger Komma ||
+||= 64/63 ||&lt; | 6 -2 0 -1 &gt; ||&gt; 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
-|| 64827/64000 || | -9 3 -3 4 &gt; || 22.23 || Squalentine ||   ||   ||
+||= 64827/64000 ||&lt; | -9 3 -3 4 &gt; ||&gt; 22.23 ||= Squalentine ||=   ||=   ||
-|| 875/864 || | -5 -3 3 1 &gt; || 21.90 || Keema ||   ||   ||
+||= 875/864 ||&lt; | -5 -3 3 1 &gt; ||&gt; 21.90 ||= Keema ||=   ||=   ||
-|| 126/125 || | 1 2 -3 1 &gt; || 13.79 || Septimal Semicomma || Starling Comma ||   ||
+||= 126/125 ||&lt; | 1 2 -3 1 &gt; ||&gt; 13.79 ||= Septimal Semicomma ||= Starling Comma ||=   ||
-|| 4000/3969 || | 5 -4 3 -2 &gt; || 13.47 || Octagar ||   ||   ||
+||= 4000/3969 ||&lt; | 5 -4 3 -2 &gt; ||&gt; 13.47 ||= Octagar ||=   ||=   ||
-|| 1029/1024 || | -10 1 0 3 &gt; || 8.43 || Gamelisma ||   ||   ||
+||= 1029/1024 ||&lt; | -10 1 0 3 &gt; ||&gt; 8.43 ||= Gamelisma ||=   ||=   ||
-|| 6144/6125 || | 11 1 -3 -2 &gt; || 5.36 || Porwell ||   ||   ||
+||= 6144/6125 ||&lt; | 11 1 -3 -2 &gt; ||&gt; 5.36 ||= Porwell ||=   ||=   ||
-|| 250047/250000 || | -4 6 -6 3 &gt; || 0.33 || Landscape Comma ||   ||   ||
+||= 250047/250000 ||&lt; | -4 6 -6 3 &gt; ||&gt; 0.33 ||= Landscape Comma ||=   ||=   ||
-|| 100/99 || | 2 -2 2 -1 &gt; || 17.40 || Ptolemisma ||   ||   ||
+||= 100/99 ||&lt; | 2 -2 2 -1 &gt; ||&gt; 17.40 ||= Ptolemisma ||=   ||=   ||
-|| 121/120 || | -3 -1 -1 2 &gt; || 14.37 || Biyatisma ||   ||   ||
+||= 121/120 ||&lt; | -3 -1 -1 2 &gt; ||&gt; 14.37 ||= Biyatisma ||=   ||=   ||
-|| 176/175 || | 4 -2 -1 1 &gt; || 9.86 || Valinorsma ||   ||   ||
+||= 176/175 ||&lt; | 4 -2 -1 1 &gt; ||&gt; 9.86 ||= Valinorsma ||=   ||=   ||
-|| 65536/65219 || | 16 -2 -3 &gt; || 8.39 || Orgonisma ||   ||   ||
+||= 65536/65219 ||&lt; | 16 -2 -3 &gt; ||&gt; 8.39 ||= Orgonisma ||=   ||=   ||
-|| 385/384 || | -7 -1 1 1 1 &gt; || 4.50 || Keenanisma ||   ||   ||
+||= 385/384 ||&lt; | -7 -1 1 1 1 &gt; ||&gt; 4.50 ||= Keenanisma ||=   ||=   ||
-|| 441/440 || | -3 2 -1 2 -1 &gt; || 3.93 || Werckisma ||   ||   ||
+||= 441/440 ||&lt; | -3 2 -1 2 -1 &gt; ||&gt; 3.93 ||= Werckisma ||=   ||=   ||
-|| 4000/3993 || | 5 -1 3 -3 &gt; || 3.03 || Wizardharry ||   ||   ||
+||= 4000/3993 ||&lt; | 5 -1 3 -3 &gt; ||&gt; 3.03 ||= Wizardharry ||=   ||=   ||
-|| 3025/3024 || | -4 -3 2 -1 2 &gt; || 0.57 || Lehmerisma ||   ||   ||
+||= 3025/3024 ||&lt; | -4 -3 2 -1 2 &gt; ||&gt; 0.57 ||= Lehmerisma ||=   ||=   ||
-|| 91/90 || | -1 -2 -1 1 1 &gt; || 19.13 || Superleap ||   ||   ||
+||= 91/90 ||&lt; | -1 -2 -1 1 1 &gt; ||&gt; 19.13 ||= Superleap ||=   ||=   ||
-|| 676/675 || | 2 -3 -2 2 &gt; || 2.56 || Parizeksma ||   ||   ||
+||= 676/675 ||&lt; | 2 -3 -2 2 &gt; ||&gt; 2.56 ||= Parizeksma ||=   ||=   ||
-Additional notes:
+==Additional notes==
 In the 15-edo system, major thirds cannot be divided perfectly into two, and coupled with the lack of a standard tritone, this tuning at first can be disorienting. However, because the guitar can be tuned symmetrically, from E to e (6th to 1st strings) unlike the 12-tone system on guitars, the learning curve is very manageable. All chords look the same modualted anywhere, and minor arpeggios are vertically stacked, making them very easy to play. 15-tone may be a promising start for anyone interested in superior harmony and xenharmony, a manageable number of tones, and the sonic fingerprint of multiples of 5-edo. .
@@ Line 96: / Line 96: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;256/243&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;256/243&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;| 8 -5 &amp;gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;| 8 -5 &amp;gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;90.22&lt;br /&gt;
+         &lt;td style="text-align: right;"&gt;90.22&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Limma&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Limma&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Pythagorean Minor 2nd&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Pythagorean Minor 2nd&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;250/243&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;250/243&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;| 1 -5 3 &amp;gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;| 1 -5 3 &amp;gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;49.17&lt;br /&gt;
+         &lt;td style="text-align: right;"&gt;49.17&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Maximal Diesis&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Maximal Diesis&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Porcupine Comma&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Porcupine Comma&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;128/125&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;128/125&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;| 7 0 -3 &amp;gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;| 7 0 -3 &amp;gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;41.06&lt;br /&gt;
+         &lt;td style="text-align: right;"&gt;41.06&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Diesis&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Diesis&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Augmented Comma&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Augmented Comma&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;15625/15552&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;15625/15552&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;| -6 -5 6 &amp;gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;| -6 -5 6 &amp;gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;8.11&lt;br /&gt;
+         &lt;td style="text-align: right;"&gt;8.11&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Kleisma&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Kleisma&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Semicomma Majeur&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Semicomma Majeur&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;1029/1000&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;1029/1000&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;| -3 1 -3 3 &amp;gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;| -3 1 -3 3 &amp;gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;49.49&lt;br /&gt;
+         &lt;td style="text-align: right;"&gt;49.49&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Keega&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Keega&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;49/48&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;49/48&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;| -4 -1 2 &amp;gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;| -4 -1 2 &amp;gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;35.70&lt;br /&gt;
+         &lt;td style="text-align: right;"&gt;35.70&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Slendro Diesis&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Slendro Diesis&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;64/63&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;64/63&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;| 6 -2 0 -1 &amp;gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;| 6 -2 0 -1 &amp;gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;27.26&lt;br /&gt;
+         &lt;td style="text-align: right;"&gt;27.26&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Septimal Comma&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Septimal Comma&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Archytas' Comma&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Archytas' Comma&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Leipziger Komma&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Leipziger Komma&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;64827/64000&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;64827/64000&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;| -9 3 -3 4 &amp;gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;| -9 3 -3 4 &amp;gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;22.23&lt;br /&gt;
+         &lt;td style="text-align: right;"&gt;22.23&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Squalentine&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Squalentine&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;875/864&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;875/864&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;| -5 -3 3 1 &amp;gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;| -5 -3 3 1 &amp;gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;21.90&lt;br /&gt;
+         &lt;td style="text-align: right;"&gt;21.90&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Keema&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Keema&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;126/125&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;126/125&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;| 1 2 -3 1 &amp;gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;| 1 2 -3 1 &amp;gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;13.79&lt;br /&gt;
+         &lt;td style="text-align: right;"&gt;13.79&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Septimal Semicomma&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Septimal Semicomma&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Starling Comma&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Starling Comma&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;4000/3969&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;4000/3969&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;| 5 -4 3 -2 &amp;gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;| 5 -4 3 -2 &amp;gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;13.47&lt;br /&gt;
+         &lt;td style="text-align: right;"&gt;13.47&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Octagar&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Octagar&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;1029/1024&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;1029/1024&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;| -10 1 0 3 &amp;gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;| -10 1 0 3 &amp;gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;8.43&lt;br /&gt;
+         &lt;td style="text-align: right;"&gt;8.43&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Gamelisma&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Gamelisma&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;6144/6125&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;6144/6125&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;| 11 1 -3 -2 &amp;gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;| 11 1 -3 -2 &amp;gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;5.36&lt;br /&gt;
+         &lt;td style="text-align: right;"&gt;5.36&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Porwell&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Porwell&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;250047/250000&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;250047/250000&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;| -4 6 -6 3 &amp;gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;| -4 6 -6 3 &amp;gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;0.33&lt;br /&gt;
+         &lt;td style="text-align: right;"&gt;0.33&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Landscape Comma&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Landscape Comma&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;100/99&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;100/99&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;| 2 -2 2 -1 &amp;gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;| 2 -2 2 -1 &amp;gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;17.40&lt;br /&gt;
+         &lt;td style="text-align: right;"&gt;17.40&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Ptolemisma&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Ptolemisma&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;121/120&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;121/120&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;| -3 -1 -1 2 &amp;gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;| -3 -1 -1 2 &amp;gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;14.37&lt;br /&gt;
+         &lt;td style="text-align: right;"&gt;14.37&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Biyatisma&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Biyatisma&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;176/175&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;176/175&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;| 4 -2 -1 1 &amp;gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;| 4 -2 -1 1 &amp;gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;9.86&lt;br /&gt;
+         &lt;td style="text-align: right;"&gt;9.86&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Valinorsma&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Valinorsma&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;65536/65219&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;65536/65219&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;| 16 -2 -3 &amp;gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;| 16 -2 -3 &amp;gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;8.39&lt;br /&gt;
+         &lt;td style="text-align: right;"&gt;8.39&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Orgonisma&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Orgonisma&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;385/384&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;385/384&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;| -7 -1 1 1 1 &amp;gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;| -7 -1 1 1 1 &amp;gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;4.50&lt;br /&gt;
+         &lt;td style="text-align: right;"&gt;4.50&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Keenanisma&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Keenanisma&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;441/440&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;441/440&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;| -3 2 -1 2 -1 &amp;gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;| -3 2 -1 2 -1 &amp;gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;3.93&lt;br /&gt;
+         &lt;td style="text-align: right;"&gt;3.93&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Werckisma&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Werckisma&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;4000/3993&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;4000/3993&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;| 5 -1 3 -3 &amp;gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;| 5 -1 3 -3 &amp;gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;3.03&lt;br /&gt;
+         &lt;td style="text-align: right;"&gt;3.03&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Wizardharry&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Wizardharry&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;3025/3024&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;3025/3024&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;| -4 -3 2 -1 2 &amp;gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;| -4 -3 2 -1 2 &amp;gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;0.57&lt;br /&gt;
+         &lt;td style="text-align: right;"&gt;0.57&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Lehmerisma&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Lehmerisma&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;91/90&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;91/90&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;| -1 -2 -1 1 1 &amp;gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;| -1 -2 -1 1 1 &amp;gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;19.13&lt;br /&gt;
+         &lt;td style="text-align: right;"&gt;19.13&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Superleap&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Superleap&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;676/675&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;676/675&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;| 2 -3 -2 2 &amp;gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;| 2 -3 -2 2 &amp;gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;2.56&lt;br /&gt;
+         &lt;td style="text-align: right;"&gt;2.56&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Parizeksma&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Parizeksma&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 435: / Line 435: @@
 &lt;br /&gt;
 &lt;br /&gt;
-Additional notes:&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x15 tone equal temperament-Additional notes"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Additional notes&lt;/h2&gt;
-&lt;br /&gt;
+ &lt;br /&gt;
 In the 15-edo system, major thirds cannot be divided perfectly into two, and coupled with the lack of a standard tritone, this tuning at first can be disorienting. However, because the guitar can be tuned symmetrically, from E to e (6th to 1st strings) unlike the 12-tone system on guitars, the learning curve is very manageable. All chords look the same modualted anywhere, and minor arpeggios are vertically stacked, making them very easy to play. 15-tone may be a promising start for anyone interested in superior harmony and xenharmony, a manageable number of tones, and the sonic fingerprint of multiples of 5-edo. .&lt;br /&gt;
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextRemoteImageRule:362:&amp;lt;img src=&amp;quot;http://www.swordguitars.com/Sword_15edoclassicalsm.JPG&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.swordguitars.com/Sword_15edoclassicalsm.JPG" alt="external image Sword_15edoclassicalsm.JPG" title="external image Sword_15edoclassicalsm.JPG" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:362 --&gt;(&lt;em&gt;15-tone Classical Guitar by &lt;a class="wiki_link" href="/Ron%20Sword"&gt;Ron Sword&lt;/a&gt;&lt;/em&gt; / &lt;em&gt;Sword Guitars&lt;/em&gt;)&lt;br /&gt;
+&lt;!-- ws:start:WikiTextRemoteImageRule:364:&amp;lt;img src=&amp;quot;http://www.swordguitars.com/Sword_15edoclassicalsm.JPG&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.swordguitars.com/Sword_15edoclassicalsm.JPG" alt="external image Sword_15edoclassicalsm.JPG" title="external image Sword_15edoclassicalsm.JPG" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:364 --&gt;(&lt;em&gt;15-tone Classical Guitar by &lt;a class="wiki_link" href="/Ron%20Sword"&gt;Ron Sword&lt;/a&gt;&lt;/em&gt; / &lt;em&gt;Sword Guitars&lt;/em&gt;)&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Theory"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Theory&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Theory"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Theory&lt;/h1&gt;
   &lt;a class="wiki_link_ext" href="http://sonic-arts.org/darreg/dar35.htm" rel="nofollow"&gt;The 15-Tone Scale System&lt;/a&gt; by &lt;a class="wiki_link" href="/Ivor%20Darreg"&gt;Ivor Darreg&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://www.webcitation.org/5xZyzKBEW" rel="nofollow"&gt;Permalink&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link_ext" href="http://www.inteas.com/Penta01.htm" rel="nofollow"&gt;The Pentadecaphonic System&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link_ext" href="http://home.comcast.net/%7Ebrentishere/15noteequaltempermenttutorial.html" rel="nofollow"&gt;15-EDO Tutorial&lt;/a&gt; by &lt;a class="wiki_link" href="/Brent%20Carson"&gt;Brent Carson&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://www.webcitation.org/5xeJYBsDg" rel="nofollow"&gt;Permalink&lt;/a&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Practical Theory / Books"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Practical Theory / Books&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Practical Theory / Books"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Practical Theory / Books&lt;/h1&gt;
-  &lt;!-- ws:start:WikiTextRemoteImageRule:364:&amp;lt;a href=&amp;quot;http://www.ronsword.com&amp;quot; rel=&amp;quot;nofollow&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;http://ronsword.com/images/Pendecaphonic_coversm.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 149px; width: 112px;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="http://www.ronsword.com" rel="nofollow"&gt;&lt;img src="http://ronsword.com/images/Pendecaphonic_coversm.jpg" alt="external image Pendecaphonic_coversm.jpg" title="external image Pendecaphonic_coversm.jpg" style="height: 149px; width: 112px;" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:364 --&gt;&lt;a class="wiki_link_ext" href="http://www.ronsword.com" rel="nofollow"&gt;Sword, Ronald. &amp;quot;Pendecaphonic Scales for Guitar&amp;quot; IAAA Press, UK-USA. First Ed: June 2009.&lt;/a&gt; - A large repository of all known scales and temperament families in the 15-edo system. 300+ examples /w chord-scale progressions&lt;br /&gt;
+  &lt;!-- ws:start:WikiTextRemoteImageRule:366:&amp;lt;a href=&amp;quot;http://www.ronsword.com&amp;quot; rel=&amp;quot;nofollow&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;http://ronsword.com/images/Pendecaphonic_coversm.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 149px; width: 112px;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="http://www.ronsword.com" rel="nofollow"&gt;&lt;img src="http://ronsword.com/images/Pendecaphonic_coversm.jpg" alt="external image Pendecaphonic_coversm.jpg" title="external image Pendecaphonic_coversm.jpg" style="height: 149px; width: 112px;" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:366 --&gt;&lt;a class="wiki_link_ext" href="http://www.ronsword.com" rel="nofollow"&gt;Sword, Ronald. &amp;quot;Pendecaphonic Scales for Guitar&amp;quot; IAAA Press, UK-USA. First Ed: June 2009.&lt;/a&gt; - A large repository of all known scales and temperament families in the 15-edo system. 300+ examples /w chord-scale progressions&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Practical Theory / Books-Compositions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;&lt;strong&gt;Compositions&lt;/strong&gt;&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Practical Theory / Books-Compositions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;&lt;strong&gt;Compositions&lt;/strong&gt;&lt;/h2&gt;
   &lt;a class="wiki_link_ext" href="http://teamouse.googlepages.com/MizarianPorcupineOverture.mp3" rel="nofollow"&gt;Mizarian Porcupine Overture&lt;/a&gt; by &lt;a class="wiki_link" href="/Herman%20Miller"&gt;Herman Miller&lt;/a&gt; (&lt;a class="wiki_link_ext" href="http://teamouse.googlepages.com/home" rel="nofollow"&gt;Herman Miller&lt;/a&gt;) (&lt;a class="wiki_link" href="/Regular%20Temperaments#porcupine"&gt;porcupine&lt;/a&gt; chord progressions)&lt;br /&gt;
 &lt;a class="wiki_link_ext" href="http://www.microtonalmusic.net/audio/15edostudy.mp3" rel="nofollow"&gt;Study for Bells&lt;/a&gt; by &lt;a class="wiki_link" href="/Daniel%20Thompson"&gt;Daniel Thompson&lt;/a&gt; (&lt;a class="wiki_link_ext" href="http://danielthompson.blogspot.com/" rel="nofollow"&gt;Daniel Thompson&lt;/a&gt;) (Jan. 2007)&lt;br /&gt;

15edo: Difference between revisions

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