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:TallKite|TallKite]] and made on <tt>2016-12-~~25 20~~:33~~:06~~ UTC</tt>.<br>

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-12-26 06:37:33 UTC</tt>.<br>

: The original revision id was <tt>~~602809012~~</tt>.<br>

: The original revision id was <tt>602814266</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>

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||= Degree ||= Cents ||= Approximate

Ratios* ||||= ~~Interval~~ names,

Ratios* ||||= Melodic names,

major wider

major wider than minor ||||= Harmonic names,

than minor ||||= ~~Interval~~ names,

major narrower

than minor ||= Interval Names

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Septimal Minor 7th ||= Perf 7th ||

||= 14 ||= 1050 ||= 11/6, 64/35 ||= maj 7th ||= C ||= min 7th ||= C ||= Neutral 7th ||= Min 7th ||

||= 15 ||= 1125 ||= 27/14, 52/27 ||= aug 7th, dim 8ve ||= C# ||= dim 7th, aug 8ve ||= D# ||= Supermajor 7th ||= Maj 7th ||

||= 15 ||= 1125 ||= 27/14, 52/27 ||= aug 7th, dim 8ve ||= C#, Db ||= dim 7th, aug 8ve ||= Cb, D# ||= Supermajor 7th ||= Maj 7th ||

||= 16 ||= 1200 ||= 2/1 ||= 8ve ||= D ||= 8ve ||= D ||= Octave ||= Octave ||

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

=Chord names=

16edo chords can be named using us and downs. Using harmonic interval names, the names are easy to find, but they bear little relationship to the sound. 4:5:6 is a minor chord and 10:12:15 is a major chord! Using melodic names, the chord names will match the sound, but finding the name is much more complicated. First change sharps to flats, then find every interval from the root, then exchange major for minor and aug for dim, then name the chord from the intervals. (See [[xenharmonic/Ups and Downs Notation#Other%20EDOs|xenharmonic.wikispaces.com/Ups+and+Downs+Notation#Other%20EDOs]]).

||= chord ||= JI ratios ||||||= harmonic name ||||||= melodic name ||

||= 0-5-9 ||= 4:5:6 ||= D F A ||= Dm ||= D minor ||= D F A ||= D ||= D major ||

||= 0-4-9 ||= 10:12:15 ||= D F# A ||= D ||= D major ||= D Fb A ||= Dm ||= D minor ||

||= 0-4-8 ||= 5:6:7 ||= D F# A# ||= Daug ||= D augmented ||= D Fb Ab ||= Ddim ||= D diminished ||

||= 0-5-10 ||= ||= D F Ab ||= Ddim ||= D diminished ||= D F A# ||= Daug ||= D augmented ||

||= 0-5-9-13 ||= 4:5:6:7 ||= D F A C# ||= Dm(M7) ||= D minor-major ||= D F A Cb ||= D7 ||= D seven ||

||= 0-5-9-12 ||= ||= D F A Bb ||= Dm(b6) ||= D minor flat-six ||= D F A B# ||= D6 ||= D six ||

||= 0-5-9-14 ||= ||= D F A C ||= Dm7 ||= D minor seven ||= D F A C ||= DM7 ||= D major seven ||

||= 0-4-9-13 ||= ||= D F# A C# ||= DM7 ||= D major seven ||= D Fb A Cb ||= DM7 ||= D minor seven ||

==Selected just intervals by error==

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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: 1em;"><a href="#Chord names">Chord names</a></div>

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

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

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

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

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

<~~!-- ws:end:WikiTextTocRule:39 --~~><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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Ratios*<br />

</td>

<td colspan="2" style="text-align: center;">~~Interval~~ names,<br />

<td colspan="2" style="text-align: center;">Melodic names, <br />

major wider~~<br />~~

major wider than minor<br />

than minor<br />

</td>

<td colspan="2" style="text-align: center;">~~Interval~~ names,<br />

<td colspan="2" style="text-align: center;">Harmonic names,<br />

major narrower<br />

than minor<br />

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</td>

</td>

</td>

</td>

<td style="text-align: center;">Supermajor 7th<br />

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

<br />

<h2 id="~~toc2~~"><a name="~~Intervals~~-Selected just intervals by error"></a>Selected just intervals by error</h2>

<h1 id="toc2"><a name="Chord names"></a>Chord names</h1>

<br />

16edo chords can be named using us and downs. Using harmonic interval names, the names are easy to find, but they bear little relationship to the sound. 4:5:6 is a minor chord and 10:12:15 is a major chord! Using melodic names, the chord names will match the sound, but finding the name is much more complicated. First change sharps to flats, then find every interval from the root, then exchange major for minor and aug for dim, then name the chord from the intervals. (See <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Other%20EDOs">xenharmonic.wikispaces.com/Ups+and+Downs+Notation#Other%20EDOs</a>).<br />

<br />

<tr>

<td style="text-align: center;">chord<br />

</td>

<td style="text-align: center;">JI ratios<br />

</td>

<td colspan="3" style="text-align: center;">harmonic name<br />

</td>

<td colspan="3" style="text-align: center;">melodic name<br />

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

<td style="text-align: center;">D minor<br />

</td>

</td>

</td>

<td style="text-align: center;">D major<br />

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

<td style="text-align: center;">D major<br />

</td>

</td>

</td>

<td style="text-align: center;">D minor<br />

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

<td style="text-align: center;">D augmented<br />

</td>

</td>

</td>

<td style="text-align: center;">D diminished<br />

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

<td style="text-align: center;">D diminished<br />

</td>

</td>

</td>

<td style="text-align: center;">D augmented<br />

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

<td style="text-align: center;">D minor-major<br />

</td>

</td>

</td>

<td style="text-align: center;">D seven<br />

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

<td style="text-align: center;">D minor flat-six<br />

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

<td style="text-align: center;">D minor seven<br />

</td>

</td>

</td>

<td style="text-align: center;">D major seven<br />

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

<td style="text-align: center;">D major seven<br />

</td>

</td>

</td>

<td style="text-align: center;">D minor seven<br />

</td>

</tr>

</table>

<br />

<h2 id="toc3"><a name="Chord names-Selected just intervals by error"></a>Selected just intervals by error</h2>

The following table shows how <a class="wiki_link" href="/Just-24">some prominent just intervals</a> are represented in 16-EDO (ordered by absolute error).<br />

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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 />

<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 />

<br />

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

<h1 id="toc4"><a name="Hexadecaphonic Octave Theory"></a>Hexadecaphonic Octave Theory</h1>

<br />

The scale supports the diminished temperament with its 1/4 octave period, though its generator size, equal to its step size of 75 cents, is smaller than ideal. Its very flat 3/2 of 675 cents supports Mavila temperament, where the mapping of major and minor is reversed. The temperament could be popular for its 150-cent &quot;3/4-tone&quot; equal division of the traditional 300-cent minor third.<br />

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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 />

<br />

<h1 id="~~toc4~~"><a name="Hexadecaphonic Notation"></a>Hexadecaphonic Notation</h1>

<h1 id="toc5"><a name="Hexadecaphonic Notation"></a>Hexadecaphonic Notation</h1>

<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 />

<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>

<h2 id="toc6"><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 />

<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 black 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>

<h1 id="toc7"><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="~~toc7~~"><a name="Rank two temperaments-Harmonizing Mavila/Armodue in 16 EDO"></a>Harmonizing Mavila/Armodue in 16 EDO</h2>

<h2 id="toc8"><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="~~toc8~~"><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="toc9"><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 introduces two more soft triads while the Wilson scale introduces two more hard triads.<br />

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<br />

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

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

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

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

<h1 id="toc13"><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:TallKite|TallKite]] and made on <tt>2016-12-25 20:33:06 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-12-26 06:37:33 UTC</tt>.<br>
-: The original revision id was <tt>602809012</tt>.<br>
+: The original revision id was <tt>602814266</tt>.<br>
 : The revision comment was: <tt></tt><br>
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@@ Line 23: / Line 23: @@
 ||= Degree ||= Cents ||= Approximate
-Ratios* ||||= Interval names,
+Ratios* ||||= Melodic names,
-major wider
+major wider than minor ||||= Harmonic names,
-than minor ||||= Interval names,
 major narrower
 than minor ||= Interval Names
@@ Line 51: / Line 50: @@
 Septimal Minor 7th ||= Perf 7th ||
 ||= 14 ||= 1050 ||= 11/6, 64/35 ||= maj 7th ||= C ||= min 7th ||= C ||= Neutral 7th ||= Min 7th ||
-||= 15 ||= 1125 ||= 27/14, 52/27 ||= aug 7th, dim 8ve ||= C# ||= dim 7th, aug 8ve ||= D# ||= Supermajor 7th ||= Maj 7th ||
+||= 15 ||= 1125 ||= 27/14, 52/27 ||= aug 7th, dim 8ve ||= C#, Db ||= dim 7th, aug 8ve ||= Cb, D# ||= Supermajor 7th ||= Maj 7th ||
 ||= 16 ||= 1200 ||= 2/1 ||= 8ve ||= D ||= 8ve ||= D ||= Octave ||= Octave ||
 *based on treating 16-EDO as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible.
+=Chord names=
+edo chords can be named using us and downs. Using harmonic interval names, the names are easy to find, but they bear little relationship to the sound. 4:5:6 is a minor chord and 10:12:15 is a major chord! Using melodic names, the chord names will match the sound, but finding the name is much more complicated. First change sharps to flats, then find every interval from the root, then exchange major for minor and aug for dim, then name the chord from the intervals. (See [[xenharmonic/Ups and Downs Notation#Other%20EDOs|xenharmonic.wikispaces.com/Ups+and+Downs+Notation#Other%20EDOs]]).
+||= chord ||= JI ratios ||||||= harmonic name ||||||= melodic name ||
+||= 0-5-9 ||= 4:5:6 ||= D F A ||= Dm ||= D minor ||= D F A ||= D ||= D major ||
+||= 0-4-9 ||= 10:12:15 ||= D F# A ||= D ||= D major ||= D Fb A ||= Dm ||= D minor ||
+||= 0-4-8 ||= 5:6:7 ||= D F# A# ||= Daug ||= D augmented ||= D Fb Ab ||= Ddim ||= D diminished ||
+||= 0-5-10 ||=   ||= D F Ab ||= Ddim ||= D diminished ||= D F A# ||= Daug ||= D augmented ||
+||= 0-5-9-13 ||= 4:5:6:7 ||= D F A C# ||= Dm(M7) ||= D minor-major ||= D F A Cb ||= D7 ||= D seven ||
+||= 0-5-9-12 ||=   ||= D F A Bb ||= Dm(b6) ||= D minor flat-six ||= D F A B# ||= D6 ||= D six ||
+||= 0-5-9-14 ||=   ||= D F A C ||= Dm7 ||= D minor seven ||= D F A C ||= DM7 ||= D major seven ||
+||= 0-4-9-13 ||=   ||= D F# A C# ||= DM7 ||= D major seven ||= D Fb A Cb ||= DM7 ||= D minor seven ||
 ==Selected just intervals by error==
@@ Line 220: / Line 234: @@
 &lt;/span&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextTocRule:29:&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:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Images"&gt;Images&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:start:WikiTextTocRule:31:&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:31 --&gt;&lt;!-- ws:start:WikiTextTocRule:32: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Images"&gt;Images&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="#Intervals"&gt;Intervals&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:32 --&gt;&lt;!-- ws:start:WikiTextTocRule:33: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Intervals"&gt;Intervals&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:31 --&gt;&lt;!-- ws:start:WikiTextTocRule:32: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Intervals-Selected just intervals by error"&gt;Selected just intervals by error&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:33 --&gt;&lt;!-- ws:start:WikiTextTocRule:34: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Chord names"&gt;Chord names&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:32 --&gt;&lt;!-- ws:start:WikiTextTocRule:33: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Hexadecaphonic Octave Theory"&gt;Hexadecaphonic Octave Theory&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:34 --&gt;&lt;!-- ws:start:WikiTextTocRule:35: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Chord names-Selected just intervals by error"&gt;Selected just intervals by error&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:33 --&gt;&lt;!-- ws:start:WikiTextTocRule:34: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Hexadecaphonic Notation"&gt;Hexadecaphonic Notation&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:35 --&gt;&lt;!-- ws:start:WikiTextTocRule:36: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Hexadecaphonic Octave Theory"&gt;Hexadecaphonic Octave Theory&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:34 --&gt;&lt;!-- ws:start:WikiTextTocRule:35: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Hexadecaphonic Notation-16 Tone Piano Layout Based on the Anti-diatonic Scale"&gt;16 Tone Piano Layout Based on the Anti-diatonic Scale&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:36 --&gt;&lt;!-- ws:start:WikiTextTocRule:37: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Hexadecaphonic Notation"&gt;Hexadecaphonic Notation&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:35 --&gt;&lt;!-- ws:start:WikiTextTocRule:36: --&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:37 --&gt;&lt;!-- ws:start:WikiTextTocRule:38: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Hexadecaphonic Notation-16 Tone Piano Layout Based on the Anti-diatonic Scale"&gt;16 Tone Piano Layout Based on the Anti-diatonic Scale&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:36 --&gt;&lt;!-- ws:start:WikiTextTocRule:37: --&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;
+&lt;!-- ws:end:WikiTextTocRule:38 --&gt;&lt;!-- ws:start:WikiTextTocRule:39: --&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:37 --&gt;&lt;!-- ws:start:WikiTextTocRule:38: --&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;
+&lt;!-- ws:end:WikiTextTocRule:39 --&gt;&lt;!-- ws:start:WikiTextTocRule:40: --&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;
-&lt;!-- ws:end:WikiTextTocRule:38 --&gt;&lt;!-- ws:start:WikiTextTocRule:39: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Commas"&gt;Commas&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:40 --&gt;&lt;!-- ws:start:WikiTextTocRule:41: --&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;
-&lt;!-- ws:end:WikiTextTocRule:39 --&gt;&lt;!-- ws:start:WikiTextTocRule:40: --&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;
+&lt;!-- ws:end:WikiTextTocRule:41 --&gt;&lt;!-- ws:start:WikiTextTocRule:42: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Commas"&gt;Commas&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:40 --&gt;&lt;!-- ws:start:WikiTextTocRule:41: --&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:42 --&gt;&lt;!-- ws:start:WikiTextTocRule:43: --&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;
-&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:43 --&gt;&lt;!-- ws:start:WikiTextTocRule:44: --&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:42 --&gt;&lt;!-- ws:start:WikiTextTocRule:43: --&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:44 --&gt;&lt;!-- ws:start:WikiTextTocRule:45: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Compositions"&gt;Compositions&lt;/a&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;!-- ws:end:WikiTextTocRule:45 --&gt;&lt;!-- ws:start:WikiTextTocRule:46: --&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:46 --&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;
-&lt;!-- ws:start:WikiTextLocalImageRule:909:&amp;lt;img src=&amp;quot;/file/view/16edo%20wheel%2001.png/516947132/325x325/16edo%20wheel%2001.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 325px; width: 325px;&amp;quot; /&amp;gt; --&gt;&lt;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;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:909 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:1068:&amp;lt;img src=&amp;quot;/file/view/16edo%20wheel%2001.png/516947132/325x325/16edo%20wheel%2001.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 325px; width: 325px;&amp;quot; /&amp;gt; --&gt;&lt;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;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:1068 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 &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;
@@ Line 257: / Line 272: @@
 Ratios*&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td colspan="2" style="text-align: center;"&gt;Interval names,&lt;br /&gt;
+         &lt;td colspan="2" style="text-align: center;"&gt;Melodic names, &lt;br /&gt;
-major wider&lt;br /&gt;
+major wider than minor&lt;br /&gt;
-than minor&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td colspan="2" style="text-align: center;"&gt;Interval names,&lt;br /&gt;
+         &lt;td colspan="2" style="text-align: center;"&gt;Harmonic names,&lt;br /&gt;
 major narrower&lt;br /&gt;
 than minor&lt;br /&gt;
@@ Line 587: / Line 601: @@
          &lt;td style="text-align: center;"&gt;aug 7th, dim 8ve&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C#, Db&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;dim 7th, aug 8ve&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;D#&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Cb, D#&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Supermajor 7th&lt;br /&gt;
@@ Line 622: / Line 636: @@
 *based on treating 16-EDO as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:7:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Intervals-Selected just intervals by error"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:7 --&gt;Selected just intervals by error&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:7:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Chord names"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:7 --&gt;Chord names&lt;/h1&gt;
+ &lt;br /&gt;
+edo chords can be named using us and downs. Using harmonic interval names, the names are easy to find, but they bear little relationship to the sound. 4:5:6 is a minor chord and 10:12:15 is a major chord! Using melodic names, the chord names will match the sound, but finding the name is much more complicated. First change sharps to flats, then find every interval from the root, then exchange major for minor and aug for dim, then name the chord from the intervals. (See &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Other%20EDOs"&gt;xenharmonic.wikispaces.com/Ups+and+Downs+Notation#Other%20EDOs&lt;/a&gt;).&lt;br /&gt;
+&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;chord&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;JI ratios&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td colspan="3" style="text-align: center;"&gt;harmonic name&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td colspan="3" style="text-align: center;"&gt;melodic name&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;0-5-9&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;4:5:6&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D F A&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Dm&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D minor&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D F A&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D major&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;0-4-9&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;10:12:15&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D F# A&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D major&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D Fb A&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Dm&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D minor&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;0-4-8&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;5:6:7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D F# A#&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Daug&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D augmented&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D Fb Ab&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Ddim&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D diminished&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;0-5-10&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D F Ab&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Ddim&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D diminished&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D F A#&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Daug&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D augmented&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;0-5-9-13&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;4:5:6:7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D F A C#&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Dm(M7)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D minor-major&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D F A Cb&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D seven&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;0-5-9-12&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D F A Bb&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Dm(b6)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D minor flat-six&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D F A B#&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D6&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D six&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;0-5-9-14&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D F A C&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Dm7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D minor seven&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D F A C&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;DM7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D major seven&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;0-4-9-13&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D F# A C#&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;DM7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D major seven&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D Fb A Cb&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;DM7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D minor seven&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
+&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:9:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Chord names-Selected just intervals by error"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:9 --&gt;Selected just intervals by error&lt;/h2&gt;
   The following table shows how &lt;a class="wiki_link" href="/Just-24"&gt;some prominent just intervals&lt;/a&gt; are represented in 16-EDO (ordered by absolute error).&lt;br /&gt;
@@ Line 782: / Line 961: @@
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 &lt;br /&gt;
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+&lt;!-- ws:start:WikiTextHeadingRule:11:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Hexadecaphonic Octave Theory"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:11 --&gt;Hexadecaphonic Octave Theory&lt;/h1&gt;
   &lt;br /&gt;
 The scale supports the diminished temperament with its 1/4 octave period, though its generator size, equal to its step size of 75 cents, is smaller than ideal. Its very flat 3/2 of 675 cents supports Mavila temperament, where the mapping of major and minor is reversed. The temperament could be popular for its 150-cent &amp;quot;3/4-tone&amp;quot; equal division of the traditional 300-cent minor third.&lt;br /&gt;
@@ Line 794: / Line 973: @@
 &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;
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:11:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Hexadecaphonic Notation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:11 --&gt;Hexadecaphonic Notation&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:13:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Hexadecaphonic Notation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:13 --&gt;Hexadecaphonic Notation&lt;/h1&gt;
   &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 811: / Line 990: @@
 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;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;
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 This Layout places Mavila[7] on the black 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;!-- ws:start:WikiTextHeadingRule:17:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Rank two temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:17 --&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 946: / Line 1,125: @@
 /128 (the difference between 16/15 and 9/8) is tempered out....&amp;quot;&lt;br /&gt;
 &lt;br /&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;!-- ws:start:WikiTextHeadingRule:19:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Rank two temperaments-Harmonizing Mavila/Armodue in 16 EDO"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:19 --&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 955: / Line 1,134: @@
 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: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;
+&lt;!-- ws:start:WikiTextHeadingRule:21:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc9"&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:21 --&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 introduces two more soft triads while the Wilson scale introduces two more hard triads.&lt;br /&gt;
@@ Line 962: / Line 1,141: @@
 &lt;br /&gt;
 &lt;br /&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;
+&lt;!-- ws:start:WikiTextHeadingRule:23:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc10"&gt;&lt;a name="Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:23 --&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 1,222: / Line 1,401: @@
 &lt;br /&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;!-- ws:start:WikiTextHeadingRule:25:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc11"&gt;&lt;a name="Armodue Theory (4-line staff)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:25 --&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: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;
+&lt;!-- ws:start:WikiTextHeadingRule:27:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc12"&gt;&lt;a name="Books/Literature"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:27 --&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: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;!-- ws:start:WikiTextHeadingRule:29:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc13"&gt;&lt;a name="Compositions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:29 --&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;