15edo

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

<br />
<br />
Additional notes:<br />
<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:362:&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:362 -->(<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:4:&lt;h1&gt; --><h1 id="toc2"><a name="Theory"></a><!-- ws:end:WikiTextHeadingRule:4 -->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:6:&lt;h1&gt; --><h1 id="toc3"><a name="Practical Theory / Books"></a><!-- ws:end:WikiTextHeadingRule:6 -->Practical Theory / Books</h1>
 <!-- ws:start:WikiTextRemoteImageRule:364:&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:364 --><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:8:&lt;h2&gt; --><h2 id="toc4"><a name="Practical Theory / Books-Compositions"></a><!-- ws:end:WikiTextHeadingRule:8 --><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>