15edo: Difference between revisions

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<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">[[toc~~|flat~~]]

<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">[[toc]]

~~<span style~~=~~"font~~-~~size: 140%;">**15~~ tone equal temperament**</span>

=15-tone equal temperament=

From [[@http://en.wikipedia.org/wiki/15_equal_temperament|Wikipedia]]:

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There are a variety of ways to notate 15-edo, and the choice of notation depends heavily which rank-2 temperament or MOS scale one wishes to treat as being the "main focus" of 15-edo composition.

==**Blackwood~~/Blacksmith[10]~~ Notation**==

==Blackwood Notation==

* **Decimal Version:** Using the nominals 1-0 (with 0 representing "10"), one ~~chain~~ of 5-edo is represented by the odd numbers, the ~~other~~ by the even numbers~~. Accidentals are used to denote which of~~ the ~~three 5-edo chains within 15-edo are being used~~.

* **Decimal Version:** Using the nominals 1-0 (with 0 representing "10"), one of the three chains of 5-edo is represented by the odd numbers, the second by the even numbers, and the third by numbers with accidentals (either odd numbers with sharps, or even numbers with flats).

* **Guitar Version:** On a 15-edo guitar, because the "perfect fourth" comes from 5-edo, all of the open strings can be tuned a perfect fourth apart and still span exactly two octaves. If one starts the circle of fourths on B — B-E-A-D-G-(B) — then the open strings of the guitar can be notated as usual (E-A-D-G-B-E). However, because the circle of fourths closes at five, and does not continue to circulate through the other 10 notes of 15-edo, it is necessary to use accidentals to notate intervals on the other two chains of 5-edo. This notation is not particularly ideal as a basis for a staff notation (as it requires all non-5edo chords to be notated with accidentals). It is nevertheless useful because it reflects an intuitive approach to 15-edo on the guitar, since 5-edo provides a useful set of 3-limit landmarks (or "perfect fourths" and "perfect fifths") that can be used to navigate the fretboard. It's especially convenient for writing chord charts, where the funky accidental-laden spellings can be more or less ignored.

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<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>15edo</title></head><body><a href="#Intervals">Intervals</a> | <a href="#Notation">Notation</a> | <a href="#Rank two temperaments">Rank two temperaments</a> | <a href="#Commas">Commas</a> | <a href="#Theory">Theory</a> | <a href="#Practical Theory / Books">Practical Theory / Books</a> | <a href="#Compositions">Compositions</a>

<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>15edo</title></head><body><div id="toc"><h1 class="nopad">Table of Contents</h1><div style="margin-left: 1em;"><a href="#x15-tone equal temperament">15-tone equal temperament</a></div>

<~~span style="font~~-~~size~~: ~~140%~~;"><~~strong~~>15 tone equal temperament</~~strong~~><~~/span~~><br />

<div style="margin-left: 1em;"><a href="#Intervals">Intervals</a></div>

<div style="margin-left: 1em;"><a href="#Notation">Notation</a></div>

<div style="margin-left: 2em;"><a href="#Notation-Blackwood Notation">Blackwood Notation</a></div>

<div style="margin-left: 2em;"><a href="#Notation-Porcupine Notation">Porcupine Notation</a></div>

<div style="margin-left: 1em;"><a href="#Rank two temperaments">Rank two temperaments</a></div>

<div style="margin-left: 1em;"><a href="#Commas">Commas</a></div>

<div style="margin-left: 1em;"><a href="#Theory">Theory</a></div>

<div style="margin-left: 1em;"><a href="#Practical Theory / Books">Practical Theory / Books</a></div>

<div style="margin-left: 1em;"><a href="#Compositions">Compositions</a></div>

</div>

<h1 id="toc0"><a name="x15-tone equal temperament"></a>15-tone equal temperament</h1>

<br />

From <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/15_equal_temperament" rel="nofollow" target="_blank">Wikipedia</a>:<br />

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<br />

<h1 id="~~toc0~~"><a name="Intervals"></a>Intervals</h1>

<h1 id="toc1"><a name="Intervals"></a>Intervals</h1>

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<br />

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<div class="objectEmbed"><a href="/file/view/15ed2-001.svg/480716490/15ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/15ed2-001.svg/480716490/15ed2-001.svg');"><img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="15ed2-001.svg" /></a><div><a href="/file/view/15ed2-001.svg/480716490/15ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/15ed2-001.svg/480716490/15ed2-001.svg');" class="filename" title="15ed2-001.svg">15ed2-001.svg</a><br /><ul><li><a href="/file/detail/15ed2-001.svg">Details</a></li><li><a href="/file/view/15ed2-001.svg/480716490/15ed2-001.svg">Download</a></li><li style="color: #666">16 KB</li></ul></div></div><br />

<br />

<h1 id="~~toc1~~"><a name="Notation"></a>Notation</h1>

<h1 id="toc2"><a name="Notation"></a>Notation</h1>

<br />

There are a variety of ways to notate 15-edo, and the choice of notation depends heavily which rank-2 temperament or MOS scale one wishes to treat as being the &quot;main focus&quot; of 15-edo composition.<br />

<br />

<h2 id="~~toc2~~"><a name="Notation-Blackwood~~/Blacksmith[10]~~ Notation"></a><!-- ws:end:WikiTextHeadingRule:6 --~~><strong~~>Blackwood~~/Blacksmith[10]~~ Notation~~</strong>~~</h2>

<h2 id="toc3"><a name="Notation-Blackwood Notation"></a>Blackwood Notation</h2>

<ul><li><strong>Decimal Version:</strong> Using the nominals 1-0 (with 0 representing &quot;10&quot;), one ~~chain~~ of 5-edo is represented by the odd numbers, the ~~other~~ by the even numbers~~. Accidentals are used to denote which of~~ the ~~three 5-edo chains within 15-edo are being used~~.</li><li><strong>Guitar Version:</strong> On a 15-edo guitar, because the &quot;perfect fourth&quot; comes from 5-edo, all of the open strings can be tuned a perfect fourth apart and still span exactly two octaves. If one starts the circle of fourths on B — B-E-A-D-G-(B) — then the open strings of the guitar can be notated as usual (E-A-D-G-B-E). However, because the circle of fourths closes at five, and does not continue to circulate through the other 10 notes of 15-edo, it is necessary to use accidentals to notate intervals on the other two chains of 5-edo. This notation is not particularly ideal as a basis for a staff notation (as it requires all non-5edo chords to be notated with accidentals). It is nevertheless useful because it reflects an intuitive approach to 15-edo on the guitar, since 5-edo provides a useful set of 3-limit landmarks (or &quot;perfect fourths&quot; and &quot;perfect fifths&quot;) that can be used to navigate the fretboard. It's especially convenient for writing chord charts, where the funky accidental-laden spellings can be more or less ignored.</li></ul><br />

<ul><li><strong>Decimal Version:</strong> Using the nominals 1-0 (with 0 representing &quot;10&quot;), one of the three chains of 5-edo is represented by the odd numbers, the second by the even numbers, and the third by numbers with accidentals (either odd numbers with sharps, or even numbers with flats).</li><li><strong>Guitar Version:</strong> On a 15-edo guitar, because the &quot;perfect fourth&quot; comes from 5-edo, all of the open strings can be tuned a perfect fourth apart and still span exactly two octaves. If one starts the circle of fourths on B — B-E-A-D-G-(B) — then the open strings of the guitar can be notated as usual (E-A-D-G-B-E). However, because the circle of fourths closes at five, and does not continue to circulate through the other 10 notes of 15-edo, it is necessary to use accidentals to notate intervals on the other two chains of 5-edo. This notation is not particularly ideal as a basis for a staff notation (as it requires all non-5edo chords to be notated with accidentals). It is nevertheless useful because it reflects an intuitive approach to 15-edo on the guitar, since 5-edo provides a useful set of 3-limit landmarks (or &quot;perfect fourths&quot; and &quot;perfect fifths&quot;) that can be used to navigate the fretboard. It's especially convenient for writing chord charts, where the funky accidental-laden spellings can be more or less ignored.</li></ul><br />

<h2 id="~~toc3~~"><a name="Notation-Porcupine Notation"></a>Porcupine Notation</h2>

<h2 id="toc4"><a name="Notation-Porcupine Notation"></a>Porcupine Notation</h2>

<ul><li>See the main <a class="wiki_link" href="/Porcupine%20Notation">porcupine notation</a> page.</li></ul><br />

<h1 id="~~toc4~~"><a name="Rank two temperaments"></a>Rank two temperaments</h1>

<h1 id="toc5"><a name="Rank two temperaments"></a>Rank two temperaments</h1>

<a class="wiki_link" href="/List%20of%2015et%20rank%20two%20temperaments%20by%20badness">List of 15et rank two temperaments by badness</a><br />

<a class="wiki_link" href="/List%20of%20edo-distinct%2015et%20rank%20two%20temperaments">List of edo-distinct 15et rank two temperaments</a><br />

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<br />

<h1 id="~~toc5~~"><a name="Commas"></a>Commas</h1>

<h1 id="toc6"><a name="Commas"></a>Commas</h1>

<br />

15 EDO <a class="wiki_link" href="/tempering%20out">tempers</a> out the following <a class="wiki_link" href="/comma">comma</a>s. (Note: This assumes the val &lt; 15 24 35 42 52 56 |.)<br />

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<br />

<h1 id="~~toc6~~"><a name="Theory"></a>Theory</h1>

<h1 id="toc7"><a name="Theory"></a>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 />

<h1 id="~~toc7~~"><a name="Practical Theory / Books"></a>Practical Theory / Books</h1>

<h1 id="toc8"><a name="Practical Theory / Books"></a>Practical Theory / Books</h1>

<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><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 />

<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><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 />

<br />

<h1 id="~~toc8~~"><a name="Compositions"></a><strong>Compositions</strong></h1>

<h1 id="toc9"><a name="Compositions"></a><strong>Compositions</strong></h1>

<br />

<strong><a class="wiki_link_ext" href="http://micro.soonlabel.com/15-ET/" rel="nofollow" target="_blank">XA 15-ET Directory</a></strong><br />

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<a class="wiki_link_ext" href="http://youtu.be/6TyQ9kDm2fk" rel="nofollow">Ode For Ada</a> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a> (<a class="wiki_link_ext" href="http://www.seraph.it/blog_files/5f07ebbdf8b4ba8ef067ed3d00a38357-158.html" rel="nofollow">blog entry</a>)<br />

Suite in 15-Note Equal Tuning, opus 33 by Easley Blackwood (as well as one of the Twelve Microtonal Etudes, opus 28)<br />

<div class="objectEmbed"><a href="/file/view/15edo-Chords.ogg/252472844/15edo-Chords.ogg" onclick="ws.common.trackFileLink('/file/view/15edo-Chords.ogg/252472844/15edo-Chords.ogg');"><img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="15edo-Chords.ogg" /></a><div><a href="/file/view/15edo-Chords.ogg/252472844/15edo-Chords.ogg" onclick="ws.common.trackFileLink('/file/view/15edo-Chords.ogg/252472844/15edo-Chords.ogg');" class="filename" title="15edo-Chords.ogg">15edo-Chords.ogg</a><br /><ul><li><a href="/file/detail/15edo-Chords.ogg">Details</a></li><li><a href="/file/view/15edo-Chords.ogg/252472844/15edo-Chords.ogg">Download</a></li><li style="color: #666">2 MB</li></ul></div></div> Some nice sounds I found in 15 EDO<br />

<div class="objectEmbed"><a href="/file/view/15edo-Chords.ogg/252472844/15edo-Chords.ogg" onclick="ws.common.trackFileLink('/file/view/15edo-Chords.ogg/252472844/15edo-Chords.ogg');"><img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="15edo-Chords.ogg" /></a><div><a href="/file/view/15edo-Chords.ogg/252472844/15edo-Chords.ogg" onclick="ws.common.trackFileLink('/file/view/15edo-Chords.ogg/252472844/15edo-Chords.ogg');" class="filename" title="15edo-Chords.ogg">15edo-Chords.ogg</a><br /><ul><li><a href="/file/detail/15edo-Chords.ogg">Details</a></li><li><a href="/file/view/15edo-Chords.ogg/252472844/15edo-Chords.ogg">Download</a></li><li style="color: #666">2 MB</li></ul></div></div> Some nice sounds I found in 15 EDO<br />

<div class="objectEmbed"><a href="/file/view/happenstance15.ogg/257380320/happenstance15.ogg" onclick="ws.common.trackFileLink('/file/view/happenstance15.ogg/257380320/happenstance15.ogg');"><img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="happenstance15.ogg" /></a><div><a href="/file/view/happenstance15.ogg/257380320/happenstance15.ogg" onclick="ws.common.trackFileLink('/file/view/happenstance15.ogg/257380320/happenstance15.ogg');" class="filename" title="happenstance15.ogg">happenstance15.ogg</a><br /><ul><li><a href="/file/detail/happenstance15.ogg">Details</a></li><li><a href="/file/view/happenstance15.ogg/257380320/happenstance15.ogg">Download</a></li><li style="color: #666">6 MB</li></ul></div></div>Sonic experiment in 15. Somewhat familiar tonality.<br />

<div class="objectEmbed"><a href="/file/view/happenstance15.ogg/257380320/happenstance15.ogg" onclick="ws.common.trackFileLink('/file/view/happenstance15.ogg/257380320/happenstance15.ogg');"><img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="happenstance15.ogg" /></a><div><a href="/file/view/happenstance15.ogg/257380320/happenstance15.ogg" onclick="ws.common.trackFileLink('/file/view/happenstance15.ogg/257380320/happenstance15.ogg');" class="filename" title="happenstance15.ogg">happenstance15.ogg</a><br /><ul><li><a href="/file/detail/happenstance15.ogg">Details</a></li><li><a href="/file/view/happenstance15.ogg/257380320/happenstance15.ogg">Download</a></li><li style="color: #666">6 MB</li></ul></div></div>Sonic experiment in 15. Somewhat familiar tonality.<br />

<span class="long-title"><a class="wiki_link_ext" href="http://www.youtube.com/watch?v=kQECU5ecCd4" rel="nofollow" target="_blank">Portrait of insects with 15-tone equal tempered guitar music </a></span><br />

<a class="wiki_link_ext" href="http://www.reverbnation.com/ffffiale/song/3073160-pentadecafonicoda-15et" rel="nofollow">PentadecafoniCoda (15et)</a> by <span class="wiki_link_ext">F.F.F. Fiale</span><br />

<script type="text/javascript" src="http://webplayer.yahooapis.com/player-beta.js">

</script></body></html></pre></div>

@@ 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:spt3125|spt3125]] and made on <tt>2014-01-05 14:31:25 UTC</tt>.<br>
+: This revision was by author [[User:spt3125|spt3125]] and made on <tt>2014-01-05 14:48:58 UTC</tt>.<br>
-: The original revision id was <tt>480717160</tt>.<br>
+: The original revision id was <tt>480720476</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">[[toc|flat]]
+<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">[[toc]]
-&lt;span style="font-size: 140%;"&gt;**15 tone equal temperament**&lt;/span&gt;
+=15-tone equal temperament=
 From [[@http://en.wikipedia.org/wiki/15_equal_temperament|Wikipedia]]:
@@ Line 50: / Line 50: @@
 There are a variety of ways to notate 15-edo, and the choice of notation depends heavily which rank-2 temperament or MOS scale one wishes to treat as being the "main focus" of 15-edo composition.
-==**Blackwood/Blacksmith[10] Notation**==
+==Blackwood Notation==
-* **Decimal Version:** Using the nominals 1-0 (with 0 representing "10"), one chain of 5-edo is represented by the odd numbers, the other by the even numbers. Accidentals are used to denote which of the three 5-edo chains within 15-edo are being used.
+* **Decimal Version:** Using the nominals 1-0 (with 0 representing "10"), one of the three chains of 5-edo is represented by the odd numbers, the second by the even numbers, and the third by numbers with accidentals (either odd numbers with sharps, or even numbers with flats).
 * **Guitar Version:** On a 15-edo guitar, because the "perfect fourth" comes from 5-edo, all of the open strings can be tuned a perfect fourth apart and still span exactly two octaves. If one starts the circle of fourths on B — B-E-A-D-G-(B) — then the open strings of the guitar can be notated as usual (E-A-D-G-B-E). However, because the circle of fourths closes at five, and does not continue to circulate through the other 10 notes of 15-edo, it is necessary to use accidentals to notate intervals on the other two chains of 5-edo. This notation is not particularly ideal as a basis for a staff notation (as it requires all non-5edo chords to be notated with accidentals). It is nevertheless useful because it reflects an intuitive approach to 15-edo on the guitar, since 5-edo provides a useful set of 3-limit landmarks (or "perfect fourths" and "perfect fifths") that can be used to navigate the fretboard. It's especially convenient for writing chord charts, where the funky accidental-laden spellings can be more or less ignored.
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-<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;15edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:20:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt;&lt;a href="#Intervals"&gt;Intervals&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt; | &lt;a href="#Notation"&gt;Notation&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt;&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt;&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt; | &lt;a href="#Rank two temperaments"&gt;Rank two temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;!-- ws:start:WikiTextTocRule:26: --&gt; | &lt;a href="#Commas"&gt;Commas&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&gt; | &lt;a href="#Theory"&gt;Theory&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt; | &lt;a href="#Practical Theory / Books"&gt;Practical Theory / Books&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt; | &lt;a href="#Compositions"&gt;Compositions&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt;
+<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;15edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:22:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#x15-tone equal temperament"&gt;15-tone equal temperament&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;span style="font-size: 140%;"&gt;&lt;strong&gt;15 tone equal temperament&lt;/strong&gt;&lt;/span&gt;&lt;br /&gt;
+&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Intervals"&gt;Intervals&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Notation"&gt;Notation&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;!-- ws:start:WikiTextTocRule:26: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Notation-Blackwood Notation"&gt;Blackwood Notation&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Notation-Porcupine Notation"&gt;Porcupine Notation&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Rank two temperaments"&gt;Rank two temperaments&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Commas"&gt;Commas&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Theory"&gt;Theory&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Practical Theory / Books"&gt;Practical Theory / Books&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:31 --&gt;&lt;!-- ws:start:WikiTextTocRule:32: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Compositions"&gt;Compositions&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:32 --&gt;&lt;!-- ws:start:WikiTextTocRule:33: --&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:33 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x15-tone equal temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;15-tone equal temperament&lt;/h1&gt;
 &lt;br /&gt;
 From &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/15_equal_temperament" rel="nofollow" target="_blank"&gt;Wikipedia&lt;/a&gt;:&lt;br /&gt;
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+&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Notation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Notation&lt;/h1&gt;
   &lt;br /&gt;
 There are a variety of ways to notate 15-edo, and the choice of notation depends heavily which rank-2 temperament or MOS scale one wishes to treat as being the &amp;quot;main focus&amp;quot; of 15-edo composition.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Notation-Blackwood/Blacksmith[10] Notation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;&lt;strong&gt;Blackwood/Blacksmith[10] Notation&lt;/strong&gt;&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Notation-Blackwood Notation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Blackwood Notation&lt;/h2&gt;
-  &lt;ul&gt;&lt;li&gt;&lt;strong&gt;Decimal Version:&lt;/strong&gt; Using the nominals 1-0 (with 0 representing &amp;quot;10&amp;quot;), one chain of 5-edo is represented by the odd numbers, the other by the even numbers. Accidentals are used to denote which of the three 5-edo chains within 15-edo are being used.&lt;/li&gt;&lt;li&gt;&lt;strong&gt;Guitar Version:&lt;/strong&gt; On a 15-edo guitar, because the &amp;quot;perfect fourth&amp;quot; comes from 5-edo, all of the open strings can be tuned a perfect fourth apart and still span exactly two octaves. If one starts the circle of fourths on B — B-E-A-D-G-(B) — then the open strings of the guitar can be notated as usual (E-A-D-G-B-E). However, because the circle of fourths closes at five, and does not continue to circulate through the other 10 notes of 15-edo, it is necessary to use accidentals to notate intervals on the other two chains of 5-edo. This notation is not particularly ideal as a basis for a staff notation (as it requires all non-5edo chords to be notated with accidentals). It is nevertheless useful because it reflects an intuitive approach to 15-edo on the guitar, since 5-edo provides a useful set of 3-limit landmarks (or &amp;quot;perfect fourths&amp;quot; and &amp;quot;perfect fifths&amp;quot;) that can be used to navigate the fretboard. It's especially convenient for writing chord charts, where the funky accidental-laden spellings can be more or less ignored.&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
+  &lt;ul&gt;&lt;li&gt;&lt;strong&gt;Decimal Version:&lt;/strong&gt; Using the nominals 1-0 (with 0 representing &amp;quot;10&amp;quot;), one of the three chains of 5-edo is represented by the odd numbers, the second by the even numbers, and the third by numbers with accidentals (either odd numbers with sharps, or even numbers with flats).&lt;/li&gt;&lt;li&gt;&lt;strong&gt;Guitar Version:&lt;/strong&gt; On a 15-edo guitar, because the &amp;quot;perfect fourth&amp;quot; comes from 5-edo, all of the open strings can be tuned a perfect fourth apart and still span exactly two octaves. If one starts the circle of fourths on B — B-E-A-D-G-(B) — then the open strings of the guitar can be notated as usual (E-A-D-G-B-E). However, because the circle of fourths closes at five, and does not continue to circulate through the other 10 notes of 15-edo, it is necessary to use accidentals to notate intervals on the other two chains of 5-edo. This notation is not particularly ideal as a basis for a staff notation (as it requires all non-5edo chords to be notated with accidentals). It is nevertheless useful because it reflects an intuitive approach to 15-edo on the guitar, since 5-edo provides a useful set of 3-limit landmarks (or &amp;quot;perfect fourths&amp;quot; and &amp;quot;perfect fifths&amp;quot;) that can be used to navigate the fretboard. It's especially convenient for writing chord charts, where the funky accidental-laden spellings can be more or less ignored.&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Notation-Porcupine Notation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Porcupine Notation&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Notation-Porcupine Notation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Porcupine Notation&lt;/h2&gt;
   &lt;ul&gt;&lt;li&gt;See the main &lt;a class="wiki_link" href="/Porcupine%20Notation"&gt;porcupine notation&lt;/a&gt; page.&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Rank two temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Rank two temperaments&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Rank two temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Rank two temperaments&lt;/h1&gt;
   &lt;a class="wiki_link" href="/List%20of%2015et%20rank%20two%20temperaments%20by%20badness"&gt;List of 15et rank two temperaments by badness&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/List%20of%20edo-distinct%2015et%20rank%20two%20temperaments"&gt;List of edo-distinct 15et rank two temperaments&lt;/a&gt;&lt;br /&gt;
@@ Line 524: / Line 534: @@
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-&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Commas&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Commas&lt;/h1&gt;
   &lt;br /&gt;
 EDO &lt;a class="wiki_link" href="/tempering%20out"&gt;tempers&lt;/a&gt; out the following &lt;a class="wiki_link" href="/comma"&gt;comma&lt;/a&gt;s. (Note: This assumes the val &amp;lt; 15 24 35 42 52 56 |.)&lt;br /&gt;
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-&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Theory"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Theory&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Theory"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&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:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Practical Theory / Books"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Practical Theory / Books&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Practical Theory / Books"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Practical Theory / Books&lt;/h1&gt;
-  &lt;!-- ws:start:WikiTextRemoteImageRule:734:&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:734 --&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:737:&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:737 --&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;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Compositions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;&lt;strong&gt;Compositions&lt;/strong&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc9"&gt;&lt;a name="Compositions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;&lt;strong&gt;Compositions&lt;/strong&gt;&lt;/h1&gt;
   &lt;br /&gt;
 &lt;strong&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/15-ET/" rel="nofollow" target="_blank"&gt;XA 15-ET Directory&lt;/a&gt;&lt;/strong&gt;&lt;br /&gt;
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 &lt;a class="wiki_link_ext" href="http://youtu.be/6TyQ9kDm2fk" rel="nofollow"&gt;Ode For Ada&lt;/a&gt; by &lt;a class="wiki_link" href="/Carlo%20Serafini"&gt;Carlo Serafini&lt;/a&gt; (&lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/5f07ebbdf8b4ba8ef067ed3d00a38357-158.html" rel="nofollow"&gt;blog entry&lt;/a&gt;)&lt;br /&gt;
 Suite in 15-Note Equal Tuning, opus 33 by Easley Blackwood (as well as one of the Twelve Microtonal Etudes, opus 28)&lt;br /&gt;
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-&lt;!-- ws:start:WikiTextFileRule:737:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/happenstance15.ogg?h=52&amp;amp;w=320&amp;quot; class=&amp;quot;WikiFile&amp;quot; id=&amp;quot;wikitext@@file@@happenstance15.ogg&amp;quot; title=&amp;quot;File: happenstance15.ogg&amp;quot; width=&amp;quot;320&amp;quot; height=&amp;quot;52&amp;quot; /&amp;gt; --&gt;&lt;div class="objectEmbed"&gt;&lt;a href="/file/view/happenstance15.ogg/257380320/happenstance15.ogg" onclick="ws.common.trackFileLink('/file/view/happenstance15.ogg/257380320/happenstance15.ogg');"&gt;&lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="happenstance15.ogg" /&gt;&lt;/a&gt;&lt;div&gt;&lt;a href="/file/view/happenstance15.ogg/257380320/happenstance15.ogg" onclick="ws.common.trackFileLink('/file/view/happenstance15.ogg/257380320/happenstance15.ogg');" class="filename" title="happenstance15.ogg"&gt;happenstance15.ogg&lt;/a&gt;&lt;br /&gt;&lt;ul&gt;&lt;li&gt;&lt;a href="/file/detail/happenstance15.ogg"&gt;Details&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a href="/file/view/happenstance15.ogg/257380320/happenstance15.ogg"&gt;Download&lt;/a&gt;&lt;/li&gt;&lt;li style="color: #666"&gt;6 MB&lt;/li&gt;&lt;/ul&gt;&lt;/div&gt;&lt;/div&gt;&lt;!-- ws:end:WikiTextFileRule:737 --&gt;Sonic experiment in 15. Somewhat familiar tonality.&lt;br /&gt;
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 &lt;span class="long-title"&gt;&lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=kQECU5ecCd4" rel="nofollow" target="_blank"&gt;Portrait of insects with 15-tone equal tempered guitar music &lt;/a&gt;&lt;/span&gt;&lt;br /&gt;
 &lt;a class="wiki_link_ext" href="http://www.reverbnation.com/ffffiale/song/3073160-pentadecafonicoda-15et" rel="nofollow"&gt;PentadecafoniCoda (15et)&lt;/a&gt; by &lt;span class="wiki_link_ext"&gt;F.F.F. Fiale&lt;/span&gt;&lt;br /&gt;
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