16edo: Difference between revisions

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<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:~~xenwolf~~|~~xenwolf~~]] and made on <tt>2016-10~~-30 17~~:33:41 UTC</tt>.<br>

: This revision was by author [[User:k9assassin|k9assassin]] and made on <tt>2016-11-10 09:15:35 UTC</tt>.<br>

: The original revision id was <tt>~~597503962~~</tt>.<br>

: The original revision id was <tt>599092924</tt>.<br>

: The revision comment was: <tt></tt><br>

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"Like the conventional 12-EDO diatonic and pentatonic (meantone) scales, these arise from tempering out a

unison vector from Fokker periodicity blocks. Only in 16-EDO, that unison vector is 135:128, instead of 81:80."

==16 Tone Piano Layout Based on the Anti-diatonic Scale==

[[image:16-EDO-PIano-Diagram.png width="748" height="293"]]

This Layout places Mavila[7] on the white keys and Mavila[9] on the white keys. As you can see, flats are higher than naturals and sharps are lower.

=Rank two temperaments=

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</span><br />

<br />

<div id="toc"><h1 class="nopad">Table of Contents</h1><div style="margin-left: 1em;"><a href="#Images">Images</a></div>

<div id="toc"><h1 class="nopad">Table of Contents</h1><div style="margin-left: 1em;"><a href="#Images">Images</a></div>

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

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

<div style="margin-left: 2em;"><a href="#Intervals-Selected just intervals by error">Selected just intervals by error</a></div>

<div style="margin-left: 2em;"><a href="#Intervals-Selected just intervals by error">Selected just intervals by error</a></div>

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

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

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

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

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

<div style="margin-left: 2em;"><a href="#Hexadecaphonic Notation-16 Tone Piano Layout Based on the Anti-diatonic Scale">16 Tone Piano Layout Based on the Anti-diatonic Scale</a></div>

<div style="margin-left: 2em;"><a href="#Rank two temperaments-Harmonizing Mavila/Armodue in 16 EDO">Harmonizing Mavila/Armodue in 16 EDO</a></div>

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

<div style="margin-left: 3em;"><a href="#Rank two temperaments-Harmonizing Mavila/Armodue in 16 EDO-MOS that support metallic harmony">MOS that support metallic harmony</a></div>

<div style="margin-left: 2em;"><a href="#Rank two temperaments-Harmonizing Mavila/Armodue in 16 EDO">Harmonizing Mavila/Armodue in 16 EDO</a></div>

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

<div style="margin-left: 3em;"><a href="#Rank two temperaments-Harmonizing Mavila/Armodue in 16 EDO-MOS that support metallic harmony">MOS that support metallic harmony</a></div>

<div style="margin-left: 1em;"><a href="#Armodue Theory (4-line staff)">Armodue Theory (4-line staff)</a></div>

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

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

<div style="margin-left: 1em;"><a href="#Armodue Theory (4-line staff)">Armodue Theory (4-line staff)</a></div>

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

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

</div>

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

<strong>16-EDO</strong> is the <a class="wiki_link" href="/equal%20division%20of%20the%20octave">equal division of the octave</a> into sixteen narrow chromatic semitones each of 75 <a class="wiki_link" href="/cent">cent</a>s exactly. It is not especially good at representing most low-integer musical intervals, but it has a <a class="wiki_link" href="/7_4">7/4</a> which is six cents sharp, and a <a class="wiki_link" href="/5_4">5/4</a> which is eleven cents flat. Four steps of it gives the 300 cent minor third interval, the same of that 12-EDO, giving it four diminished seventh chords exactly like those of <a class="wiki_link" href="/12edo">12-EDO</a>, and a diminished triad on each scale step.<br />

</div>

<strong>16-EDO</strong> is the <a class="wiki_link" href="/equal%20division%20of%20the%20octave">equal division of the octave</a> into sixteen narrow chromatic semitones each of 75 <a class="wiki_link" href="/cent">cent</a>s exactly. It is not especially good at representing most low-integer musical intervals, but it has a <a class="wiki_link" href="/7_4">7/4</a> which is six cents sharp, and a <a class="wiki_link" href="/5_4">5/4</a> which is eleven cents flat. Four steps of it gives the 300 cent minor third interval, the same of that 12-EDO, giving it four diminished seventh chords exactly like those of <a class="wiki_link" href="/12edo">12-EDO</a>, and a diminished triad on each scale step.<br />

<br />

<h1 id="toc0"><a name="Images"></a>Images</h1>

<br />

<img src="/file/view/16edo%20wheel%2001.png/516947132/325x325/16edo%20wheel%2001.png" alt="16edo wheel 01.png" title="16edo wheel 01.png" style="height: 325px; width: 325px;" /><br />

<img src="/file/view/16edo%20wheel%2001.png/516947132/325x325/16edo%20wheel%2001.png" alt="16edo wheel 01.png" title="16edo wheel 01.png" style="height: 325px; width: 325px;" /><br />

<br />

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

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<object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/16ed2-001.svg">alt : Your browser has no SVG support.</object><br />

<br />

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<br />

<h1 id="toc3"><a name="Hexadecaphonic Octave Theory"></a>Hexadecaphonic Octave Theory</h1>

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<br />

Another interesting approach can include two interwoven <a class="wiki_link" href="/8edo">8-EDO</a> scales (narrow 12/11 neutral seconds). There are two major seventh intervals, a harmonic seventh at step 13\16, a 7/4 ratio approximation, sharp by 6.174 cents, followed by an undecimal 11/6 ratio or neutral seventh (which is mapped in 16's <strong>Mavila</strong> as a major seventh). If we take the 300-cent minor third as an approximation of the harmonic 19th (<a class="wiki_link" href="/19_16">19/16</a>, approximately 297.5 cents), that adds another overtone which can combine with the approximation of the harmonic seventh to form a 16:19:28 triad (pictured below).<br />

<img src="http://www.ronsword.com/161928%20copy.jpg" alt="external image 161928%20copy.jpg" title="external image 161928%20copy.jpg" style="height: 179px; width: 316px;" /><br />

<img src="http://www.ronsword.com/161928%20copy.jpg" alt="external image 161928%20copy.jpg" title="external image 161928%20copy.jpg" style="height: 179px; width: 316px;" /><br />

<br />

The interval between the 28th &amp; 19th overtones, 28:19, measures approximately 671.3 cents, which is 3.7 cents away from 16edo's &quot;narrow fifth&quot;. Another voicing for this chord is 14:16:19, which features 19:14 as the outer interval (528.7 cents just, 525.0 cents in 16edo). A perhaps more consonant open voicing is 7:16:19.<br />

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<br />

16-EDO notation can be easy utilizing Goldsmith's Circle of keys, nominals, and respective notation. The nominals for a 6 line staff can be switched for Wilson's Beta and Epsilon additions to A-G. Armodue of Italy uses a 4-line staff for 16-EDO.<br />

<img src="http://ronsword.com/DSgoldsmith_piece.jpg" alt="external image DSgoldsmith_piece.jpg" title="external image DSgoldsmith_piece.jpg" style="height: 342px; width: 1008px;" /><br />

<img src="http://ronsword.com/DSgoldsmith_piece.jpg" alt="external image DSgoldsmith_piece.jpg" title="external image DSgoldsmith_piece.jpg" style="height: 342px; width: 1008px;" /><br />

<hr />

In 16-EDO diatonic scales are dissonant and &quot;shimmery&quot; because of the 25 cent raised superfourth in conjunction with the 25 cent subtracted fifth / poor 3/2 approximation. Scales like the Harmonic Minor scale in 16-EDO require 4 step sizes.<br />

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unison vector from Fokker periodicity blocks. Only in 16-EDO, that unison vector is 135:128, instead of 81:80.&quot;<br />

<br />

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

<h2 id="toc5"><a name="Hexadecaphonic Notation-16 Tone Piano Layout Based on the Anti-diatonic Scale"></a>16 Tone Piano Layout Based on the Anti-diatonic Scale</h2>

<img src="/file/view/16-EDO-PIano-Diagram.png/599092760/748x293/16-EDO-PIano-Diagram.png" alt="16-EDO-PIano-Diagram.png" title="16-EDO-PIano-Diagram.png" style="height: 293px; width: 748px;" /><br />

This Layout places Mavila[7] on the white keys and Mavila[9] on the white keys. As you can see, flats are higher than naturals and sharps are lower. <br />

<br />

<h1 id="toc6"><a name="Rank two temperaments"></a>Rank two temperaments</h1>

<a class="wiki_link" href="/List%20of%2016et%20rank%20two%20temperaments%20by%20badness">List of 16et rank two temperaments by badness</a><br />

<br />

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135/128 (the difference between 16/15 and 9/8) is tempered out....&quot;<br />

<br />

<h2 id="~~toc6~~"><a name="Rank two temperaments-Harmonizing Mavila/Armodue in 16 EDO"></a>Harmonizing Mavila/Armodue in 16 EDO</h2>

<h2 id="toc7"><a name="Rank two temperaments-Harmonizing Mavila/Armodue in 16 EDO"></a>Harmonizing Mavila/Armodue in 16 EDO</h2>

<br />

Because 16 edo doesn't approximate 3/2 well at all, triadic harmony based on thirds isn't a good option.<br />

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Their characteristic metallic sound has earned them the name &quot;Metallic triads&quot;.<br />

<br />

<h3 id="~~toc7~~"><a name="Rank two temperaments-Harmonizing Mavila/Armodue in 16 EDO-MOS that support metallic harmony"></a>MOS that support metallic harmony</h3>

<h3 id="toc8"><a name="Rank two temperaments-Harmonizing Mavila/Armodue in 16 EDO-MOS that support metallic harmony"></a>MOS that support metallic harmony</h3>

Mavila 7 contains two hard triads on degrees 1 and 4 and two soft triads on degrees 2 and 6. The other three chords<br />

are wide symmetrical triads 0-1050-1050. Mavila 9 introduced two more soft triads while the Wilson scale introduces two more hard triads.<br />

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<br />

<h1 id="~~toc8~~"><a name="Commas"></a>Commas</h1>

<h1 id="toc9"><a name="Commas"></a>Commas</h1>

16 EDO <a class="wiki_link" href="/tempering%20out">tempers out</a> the following <a class="wiki_link" href="/comma">comma</a>s. (Note: This assumes <a class="wiki_link" href="/val">val</a> &lt; 16 25 37 45 55 59 |.)<br />

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<br />

<h1 id="~~toc9~~"><a name="Armodue Theory (4-line staff)"></a><strong>Armodue Theory (4-line staff)</strong></h1>

<h1 id="toc10"><a name="Armodue Theory (4-line staff)"></a><strong>Armodue Theory (4-line staff)</strong></h1>

<a class="wiki_link_ext" href="http://www.armodue.com/ricerche.htm" rel="nofollow">Armodue</a>: Italian pages of theory for 16-tone (esadekaphonic) system, including compositions.<br />

Translations of parts of the Armodue pages can be found <a class="wiki_link" href="/Armodue">here</a> on this wiki.<br />

<br />

<h1 id="~~toc10~~"><a name="Books/Literature"></a>Books/Literature</h1>

<h1 id="toc11"><a name="Books/Literature"></a>Books/Literature</h1>

Sword, Ronald. &quot;Thesaurus of Melodic Patterns and Intervals for 16-Tones&quot; IAAA Press, USA. First Ed: August, 2011<br />

Sword, Ronald. &quot;Hexadecaphonic Scales for Guitar.&quot; IAAA Press, UK-USA. First Ed: Feb, 2010. (superfourth tuning)<br />

Sword, Ronald. &quot;Esadekaphonic Scales for Guitar.&quot; IAAA Press, UK-USA. First Ed: April, 2009. (semi-diminished fourth tuning)<br />

<br />

<h1 id="~~toc11~~"><a name="Compositions"></a>Compositions</h1>

<h1 id="toc12"><a name="Compositions"></a>Compositions</h1>

<a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=660895&amp;songID=12370649&amp;showPlayer=true" rel="nofollow" target="_blank">Prenestyna Highway</a> by <a class="wiki_link_ext" href="http://fiale.tk" rel="nofollow" target="_blank">Fabrizio Fulvio Fausto Fiale</a><br />

<a class="wiki_link_ext" href="http://lastsacrament.bandcamp.com/album/enantiodromia" rel="nofollow" target="_blank">Enantiodromia (album)</a> by Last Sacrament<br />

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 <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:xenwolf|xenwolf]] and made on <tt>2016-10-30 17:33:41 UTC</tt>.<br>
+: This revision was by author [[User:k9assassin|k9assassin]] and made on <tt>2016-11-10 09:15:35 UTC</tt>.<br>
-: The original revision id was <tt>597503962</tt>.<br>
+: The original revision id was <tt>599092924</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>
@@ Line 117: / Line 117: @@
 "Like the conventional 12-EDO diatonic and pentatonic (meantone) scales, these arise from tempering out a
 unison vector from Fokker periodicity blocks. Only in 16-EDO, that unison vector is 135:128, instead of 81:80."
+==16 Tone Piano Layout Based on the Anti-diatonic Scale==
+[[image:16-EDO-PIano-Diagram.png width="748" height="293"]]
+This Layout places Mavila[7] on the white keys and Mavila[9] on the white keys. As you can see, flats are higher than naturals and sharps are lower.
 =Rank two temperaments=
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+&lt;!-- ws:end:WikiTextTocRule:41 --&gt;&lt;!-- ws:start:WikiTextTocRule:42: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Compositions"&gt;Compositions&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:40 --&gt;&lt;strong&gt;16-EDO&lt;/strong&gt; is the &lt;a class="wiki_link" href="/equal%20division%20of%20the%20octave"&gt;equal division of the octave&lt;/a&gt; into sixteen narrow chromatic semitones each of 75 &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s exactly. It is not especially good at representing most low-integer musical intervals, but it has a &lt;a class="wiki_link" href="/7_4"&gt;7/4&lt;/a&gt; which is six cents sharp, and a &lt;a class="wiki_link" href="/5_4"&gt;5/4&lt;/a&gt; which is eleven cents flat. Four steps of it gives the 300 cent minor third interval, the same of that 12-EDO, giving it four diminished seventh chords exactly like those of &lt;a class="wiki_link" href="/12edo"&gt;12-EDO&lt;/a&gt;, and a diminished triad on each scale step.&lt;br /&gt;
+&lt;!-- ws:end:WikiTextTocRule:42 --&gt;&lt;!-- ws:start:WikiTextTocRule:43: --&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:43 --&gt;&lt;strong&gt;16-EDO&lt;/strong&gt; is the &lt;a class="wiki_link" href="/equal%20division%20of%20the%20octave"&gt;equal division of the octave&lt;/a&gt; into sixteen narrow chromatic semitones each of 75 &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s exactly. It is not especially good at representing most low-integer musical intervals, but it has a &lt;a class="wiki_link" href="/7_4"&gt;7/4&lt;/a&gt; which is six cents sharp, and a &lt;a class="wiki_link" href="/5_4"&gt;5/4&lt;/a&gt; which is eleven cents flat. Four steps of it gives the 300 cent minor third interval, the same of that 12-EDO, giving it four diminished seventh chords exactly like those of &lt;a class="wiki_link" href="/12edo"&gt;12-EDO&lt;/a&gt;, and a diminished triad on each scale step.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:3:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Images"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:3 --&gt;Images&lt;/h1&gt;
   &lt;br /&gt;
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 &lt;!-- ws:start:WikiTextHeadingRule:5:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:5 --&gt;Intervals&lt;/h1&gt;
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 &lt;br /&gt;
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 &lt;!-- ws:start:WikiTextHeadingRule:9:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Hexadecaphonic Octave Theory"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:9 --&gt;Hexadecaphonic Octave Theory&lt;/h1&gt;
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 &lt;br /&gt;
 Another interesting approach can include two interwoven &lt;a class="wiki_link" href="/8edo"&gt;8-EDO&lt;/a&gt; scales (narrow 12/11 neutral seconds). There are two major seventh intervals, a harmonic seventh at step 13\16, a 7/4 ratio approximation, sharp by 6.174 cents, followed by an undecimal 11/6 ratio or neutral seventh (which is mapped in 16's &lt;strong&gt;Mavila&lt;/strong&gt; as a major seventh). If we take the 300-cent minor third as an approximation of the harmonic 19th (&lt;a class="wiki_link" href="/19_16"&gt;19/16&lt;/a&gt;, approximately 297.5 cents), that adds another overtone which can combine with the approximation of the harmonic seventh to form a 16:19:28 triad (pictured below).&lt;br /&gt;
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 &lt;br /&gt;
 The interval between the 28th &amp;amp; 19th overtones, 28:19, measures approximately 671.3 cents, which is 3.7 cents away from 16edo's &amp;quot;narrow fifth&amp;quot;. Another voicing for this chord is 14:16:19, which features 19:14 as the outer interval (528.7 cents just, 525.0 cents in 16edo). A perhaps more consonant open voicing is 7:16:19.&lt;br /&gt;
@@ Line 629: / Line 634: @@
   &lt;br /&gt;
 -EDO notation can be easy utilizing Goldsmith's Circle of keys, nominals, and respective notation. The nominals for a 6 line staff can be switched for Wilson's Beta and Epsilon additions to A-G. Armodue of Italy uses a 4-line staff for 16-EDO.&lt;br /&gt;
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 &lt;hr /&gt;
 In 16-EDO diatonic scales are dissonant and &amp;quot;shimmery&amp;quot; because of the 25 cent raised superfourth in conjunction with the 25 cent subtracted fifth / poor 3/2 approximation. Scales like the Harmonic Minor scale in 16-EDO require 4 step sizes.&lt;br /&gt;
@@ Line 638: / Line 643: @@
 unison vector from Fokker periodicity blocks. Only in 16-EDO, that unison vector is 135:128, instead of 81:80.&amp;quot;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:13:&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:13 --&gt;Rank two temperaments&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:13:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Hexadecaphonic Notation-16 Tone Piano Layout Based on the Anti-diatonic Scale"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:13 --&gt;16 Tone Piano Layout Based on the Anti-diatonic Scale&lt;/h2&gt;
+ &lt;!-- ws:start:WikiTextLocalImageRule:734:&amp;lt;img src=&amp;quot;/file/view/16-EDO-PIano-Diagram.png/599092760/748x293/16-EDO-PIano-Diagram.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 293px; width: 748px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/16-EDO-PIano-Diagram.png/599092760/748x293/16-EDO-PIano-Diagram.png" alt="16-EDO-PIano-Diagram.png" title="16-EDO-PIano-Diagram.png" style="height: 293px; width: 748px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:734 --&gt;&lt;br /&gt;
+This Layout places Mavila[7] on the white keys and Mavila[9] on the white keys. As you can see, flats are higher than naturals and sharps are lower. &lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:15:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Rank two temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:15 --&gt;Rank two temperaments&lt;/h1&gt;
   &lt;a class="wiki_link" href="/List%20of%2016et%20rank%20two%20temperaments%20by%20badness"&gt;List of 16et rank two temperaments by badness&lt;/a&gt;&lt;br /&gt;
 &lt;br /&gt;
@@ Line 769: / Line 778: @@
 /128 (the difference between 16/15 and 9/8) is tempered out....&amp;quot;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:15:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Rank two temperaments-Harmonizing Mavila/Armodue in 16 EDO"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:15 --&gt;Harmonizing Mavila/Armodue in 16 EDO&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:17:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Rank two temperaments-Harmonizing Mavila/Armodue in 16 EDO"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:17 --&gt;Harmonizing Mavila/Armodue in 16 EDO&lt;/h2&gt;
   &lt;br /&gt;
 Because 16 edo doesn't approximate 3/2 well at all, triadic harmony based on thirds isn't a good option.&lt;br /&gt;
@@ Line 778: / Line 787: @@
 Their characteristic metallic sound has earned them the name &amp;quot;Metallic triads&amp;quot;.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:17:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc7"&gt;&lt;a name="Rank two temperaments-Harmonizing Mavila/Armodue in 16 EDO-MOS that support metallic harmony"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:17 --&gt;MOS that support metallic harmony&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:19:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc8"&gt;&lt;a name="Rank two temperaments-Harmonizing Mavila/Armodue in 16 EDO-MOS that support metallic harmony"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:19 --&gt;MOS that support metallic harmony&lt;/h3&gt;
   Mavila 7 contains two hard triads on degrees 1 and 4 and two soft triads on degrees 2 and 6. The other three chords&lt;br /&gt;
 are wide symmetrical triads 0-1050-1050. Mavila 9 introduced two more soft triads while the Wilson scale introduces two more hard triads.&lt;br /&gt;
@@ Line 785: / Line 794: @@
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:19:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:19 --&gt;Commas&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:21:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc9"&gt;&lt;a name="Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:21 --&gt;Commas&lt;/h1&gt;
 EDO &lt;a class="wiki_link" href="/tempering%20out"&gt;tempers out&lt;/a&gt; the following &lt;a class="wiki_link" href="/comma"&gt;comma&lt;/a&gt;s. (Note: This assumes &lt;a class="wiki_link" href="/val"&gt;val&lt;/a&gt; &amp;lt; 16 25 37 45 55 59 |.)&lt;br /&gt;
@@ Line 1,045: / Line 1,054: @@
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:21:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc9"&gt;&lt;a name="Armodue Theory (4-line staff)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:21 --&gt;&lt;strong&gt;Armodue Theory (4-line staff)&lt;/strong&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:23:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc10"&gt;&lt;a name="Armodue Theory (4-line staff)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:23 --&gt;&lt;strong&gt;Armodue Theory (4-line staff)&lt;/strong&gt;&lt;/h1&gt;
   &lt;a class="wiki_link_ext" href="http://www.armodue.com/ricerche.htm" rel="nofollow"&gt;Armodue&lt;/a&gt;: Italian pages of theory for 16-tone (esadekaphonic) system, including compositions.&lt;br /&gt;
 Translations of parts of the Armodue pages can be found &lt;a class="wiki_link" href="/Armodue"&gt;here&lt;/a&gt; on this wiki.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:23:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc10"&gt;&lt;a name="Books/Literature"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:23 --&gt;Books/Literature&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:25:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc11"&gt;&lt;a name="Books/Literature"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:25 --&gt;Books/Literature&lt;/h1&gt;
   Sword, Ronald. &amp;quot;Thesaurus of Melodic Patterns and Intervals for 16-Tones&amp;quot; IAAA Press, USA. First Ed: August, 2011&lt;br /&gt;
 Sword, Ronald. &amp;quot;Hexadecaphonic Scales for Guitar.&amp;quot; IAAA Press, UK-USA. First Ed: Feb, 2010. (superfourth tuning)&lt;br /&gt;
 Sword, Ronald. &amp;quot;Esadekaphonic Scales for Guitar.&amp;quot; IAAA Press, UK-USA. First Ed: April, 2009. (semi-diminished fourth tuning)&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:25:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc11"&gt;&lt;a name="Compositions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:25 --&gt;Compositions&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:27:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc12"&gt;&lt;a name="Compositions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:27 --&gt;Compositions&lt;/h1&gt;
   &lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=660895&amp;amp;songID=12370649&amp;amp;showPlayer=true" rel="nofollow" target="_blank"&gt;Prenestyna Highway&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://fiale.tk" rel="nofollow" target="_blank"&gt;Fabrizio Fulvio Fausto Fiale&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link_ext" href="http://lastsacrament.bandcamp.com/album/enantiodromia" rel="nofollow" target="_blank"&gt;Enantiodromia (album)&lt;/a&gt; by Last Sacrament&lt;br /&gt;