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:~~genewardsmith~~|~~genewardsmith~~]] and made on <tt>2014-02-~~22 09~~:40:13 UTC</tt>.<br>

: This revision was by author [[User:cookiemeows|cookiemeows]] and made on <tt>2014-03-17 14:25:36 UTC</tt>.<br>

: The original revision id was <tt>~~491203820~~</tt>.<br>

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</span>

[[toc]]

**16-EDO** is the [[equal division of the octave]] into sixteen narrow chromatic semitones each of 75 [[cent]]s exactly. It is not especially good at representing most low-integer musical intervals, but it has a [[7_4|7/4]] which is six cents sharp, and a [[5_4|5/4]] 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 [[12edo|12-EDO]], and a diminished triad on each scale step.

=Intervals=

|| Degree || Cents ||= Approximate

Ratios* || Interval Name ||

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*based on treating 16-EDO as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible.

==Intervals by patent val error==

|| Interval || Error ||

|| 12/11 || -0.637 ||

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and the "fourth" is the same as the distance between the "fourth" and the "fifth" (i.e. near a 12/11)...This mean(s) that

135/128 (the difference between 16/15 and 9/8) is tempered out...."

==Harmonizing Mavila/Armodue in 16 EDO==

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

However triadic harmony can be based on on sevenths rather than thirds. For instance, 16 edo approximates 7/4 well enough to use

it in place of the usual 3/2. A triad can then be constructed by adding another seventh on top producing two possibilities for asymmetric

sevenths triads. a small one: 0-975-1050 called hard and a large one: 0-1050-975 called soft. In addition two other symmetrical triads

0-975-975 and 0-1050-1050 are also obvious possible chords.

Their characteristic metallic sound has earned them the name "Metallic triads".

===MOS that support metallic harmony===

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

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

See Metallic Harmony

=Commas=

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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"><html><head><title>16edo</title></head><body><span style="display: block; text-align: right;">Other languages: <a class="wiki_link" href="http://xenharmonie.wikispaces.com/16edo">Deutsch</a><br />

</span><br />

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

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

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

<div style="margin-left: 2em;"><a href="#Intervals-Intervals by patent val error">Intervals by patent val 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: 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: 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="#Armodue Theory (4-line staff)">Armodue Theory (4-line staff)</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="#Books/Literature">Books/Literature</a></div>

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

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

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

</div>

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

~~<br />~~

<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="Intervals"></a>Intervals</h1>

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<br />

<h2 id="toc1"><a name="Intervals-Intervals by patent val error"></a>Intervals by patent val error</h2>

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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 />

<div class="objectEmbed"><a href="/file/view/16ed2-001.svg/480989378/16ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/16ed2-001.svg/480989378/16ed2-001.svg');"><img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="16ed2-001.svg" /></a><div><a href="/file/view/16ed2-001.svg/480989378/16ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/16ed2-001.svg/480989378/16ed2-001.svg');" class="filename" title="16ed2-001.svg">16ed2-001.svg</a><br /><ul><li><a href="/file/detail/16ed2-001.svg">Details</a></li><li><a href="/file/view/16ed2-001.svg/480989378/16ed2-001.svg">Download</a></li><li style="color: #666">18 KB</li></ul></div></div><br />

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<br />

<h1 id="toc2"><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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135/128 (the difference between 16/15 and 9/8) is tempered out....&quot;<br />

<br />

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

<h2 id="toc5"><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 />

However triadic harmony can be based on on sevenths rather than thirds. For instance, 16 edo approximates 7/4 well enough to use <br />

it in place of the usual 3/2. A triad can then be constructed by adding another seventh on top producing two possibilities for asymmetric<br />

sevenths triads. a small one: 0-975-1050 called hard and a large one: 0-1050-975 called soft. In addition two other symmetrical triads<br />

0-975-975 and 0-1050-1050 are also obvious possible chords. <br />

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

<br />

<h3 id="toc6"><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 />

<br />

See Metallic Harmony<br />

<br />

<h1 id="toc7"><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="~~toc6~~"><a name="Armodue Theory (4-line staff)"></a><strong>Armodue Theory (4-line staff)</strong></h1>

<h1 id="toc8"><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="~~toc7~~"><a name="Books/Literature"></a>Books/Literature</h1>

<h1 id="toc9"><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="~~toc8~~"><a name="Compositions"></a>Compositions</h1>

<h1 id="toc10"><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 />

@@ 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:genewardsmith|genewardsmith]] and made on <tt>2014-02-22 09:40:13 UTC</tt>.<br>
+: This revision was by author [[User:cookiemeows|cookiemeows]] and made on <tt>2014-03-17 14:25:36 UTC</tt>.<br>
-: The original revision id was <tt>491203820</tt>.<br>
+: The original revision id was <tt>496394488</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 9: / Line 9: @@
 &lt;/span&gt;
 [[toc]]
 **16-EDO** is the [[equal division of the octave]] into sixteen narrow chromatic semitones each of 75 [[cent]]s exactly. It is not especially good at representing most low-integer musical intervals, but it has a [[7_4|7/4]] which is six cents sharp, and a [[5_4|5/4]] 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 [[12edo|12-EDO]], and a diminished triad on each scale step.
 =Intervals=
 || Degree || Cents ||= Approximate
 Ratios* || Interval Name ||
@@ Line 38: / Line 37: @@
 *based on treating 16-EDO as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible.
 ==Intervals by patent val error==
 || Interval || Error ||
 || 12/11 || -0.637 ||
@@ Line 132: / Line 131: @@
 and the "fourth" is the same as the distance between the "fourth" and the "fifth" (i.e. near a 12/11)...This mean(s) that
 /128 (the difference between 16/15 and 9/8) is tempered out...."
+==Harmonizing Mavila/Armodue in 16 EDO==
+Because 16 edo doesn't approximate 3/2 well at all, triadic harmony based on thirds isn't a good option.
+However triadic harmony can be based on on sevenths rather than thirds. For instance, 16 edo approximates 7/4 well enough to use
+it in place of the usual 3/2. A triad can then be constructed by adding another seventh on top producing two possibilities for asymmetric
+sevenths triads. a small one: 0-975-1050 called hard and a large one: 0-1050-975 called soft. In addition two other symmetrical triads
+-975-975 and 0-1050-1050 are also obvious possible chords.
+Their characteristic metallic sound has earned them the name "Metallic triads".
+===MOS that support metallic harmony===
+Mavila 7 contains two hard triads on degrees 1 and 4 and two soft triads on degrees 2 and 6. The other three chords
+are wide symmetrical triads 0-1050-1050. Mavila 9 introduced two more soft triads while the Wilson scale introduces two more hard triads.
+See Metallic Harmony
 =Commas=
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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;16edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;span style="display: block; text-align: right;"&gt;Other languages: &lt;a class="wiki_link" href="http://xenharmonie.wikispaces.com/16edo"&gt;Deutsch&lt;/a&gt;&lt;br /&gt;
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+&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Hexadecaphonic Notation"&gt;Hexadecaphonic Notation&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Rank two temperaments"&gt;Rank two temperaments&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Rank two temperaments-Harmonizing Mavila/Armodue in 16 EDO"&gt;Harmonizing Mavila/Armodue in 16 EDO&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:31 --&gt;&lt;!-- ws:start:WikiTextTocRule:32: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Rank two temperaments-Harmonizing Mavila/Armodue in 16 EDO-MOS that support metallic harmony"&gt;MOS that support metallic harmony&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:32 --&gt;&lt;!-- ws:start:WikiTextTocRule:33: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Commas"&gt;Commas&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:33 --&gt;&lt;!-- ws:start:WikiTextTocRule:34: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Armodue Theory (4-line staff)"&gt;Armodue Theory (4-line staff)&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:34 --&gt;&lt;!-- ws:start:WikiTextTocRule:35: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Books/Literature"&gt;Books/Literature&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:31 --&gt;&lt;br /&gt;
+&lt;!-- ws:end:WikiTextTocRule:35 --&gt;&lt;!-- ws:start:WikiTextTocRule:36: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Compositions"&gt;Compositions&lt;/a&gt;&lt;/div&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:36 --&gt;&lt;!-- ws:start:WikiTextTocRule:37: --&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:37 --&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="Intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:3 --&gt;Intervals&lt;/h1&gt;
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 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:7:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Hexadecaphonic Octave Theory"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:7 --&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 563: / Line 579: @@
   &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 703: / Line 719: @@
 /128 (the difference between 16/15 and 9/8) is tempered out....&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="Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:13 --&gt;Commas&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:13:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Rank two temperaments-Harmonizing Mavila/Armodue in 16 EDO"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:13 --&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;
+However triadic harmony can be based on on sevenths rather than thirds. For instance, 16 edo approximates 7/4 well enough to use &lt;br /&gt;
+it in place of the usual 3/2. A triad can then be constructed by adding another seventh on top producing two possibilities for asymmetric&lt;br /&gt;
+sevenths triads. a small one: 0-975-1050 called hard and a large one: 0-1050-975 called soft. In addition two other symmetrical triads&lt;br /&gt;
+-975-975 and 0-1050-1050 are also obvious possible chords. &lt;br /&gt;
+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:15:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc6"&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:15 --&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;
+&lt;br /&gt;
+See Metallic Harmony&lt;br /&gt;
+&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:17:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:17 --&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 963: / Line 995: @@
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:15:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Armodue Theory (4-line staff)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:15 --&gt;&lt;strong&gt;Armodue Theory (4-line staff)&lt;/strong&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:19:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Armodue Theory (4-line staff)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:19 --&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:17:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Books/Literature"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:17 --&gt;Books/Literature&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:21:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc9"&gt;&lt;a name="Books/Literature"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:21 --&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:19:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Compositions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:19 --&gt;Compositions&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:23:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc10"&gt;&lt;a name="Compositions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:23 --&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;