16edo: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:~~Osmiorisbendi~~|~~Osmiorisbendi~~]] and made on <tt>2011-05-~~24 02~~:45:35 UTC</tt>.

: This revision was by author [[User:jdfreivald|jdfreivald]] and made on <tt>2011-05-31 19:53:30 UTC</tt>.

: The original revision id was <tt>~~231256160~~</tt>.

: The original revision id was <tt>233317148</tt>.

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

: The revision comment was: <tt>Added comma table.</tt>

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<h4>Original Wikitext content:</h4>

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8 octaves into 8 equal parts = 16 16 16 16 16 16 16 16 = Octave Progression

9 octaves into 8 equal parts = 18 18 18 18 18 18 18 18 = Ninth Progression

= =

=Commas=

16 EDO tempers out the following commas. (Note: This assumes val < 16 25 37 45 55 59 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||

|| 135/128 || | -7 3 1 > || 92.18 || Major Chroma || Major Limma || Pelogic Comma ||

|| 648/625 || | 3 4 -4 > || 62.57 || Major Diesis || Diminished Comma || ||

|| 3125/3072 || | -10 -1 5 > || 29.61 || Small Diesis || Magic Comma || ||

|| 1212717/1210381 || | 23 6 -14 > || 3.34 || Vishnuzma || Semisuper || ||

|| 36/35 || | 2 2 -1 -1 > || 48.77 || Septimal Quarter Tone || || ||

|| 525/512 || | -9 1 2 1 > || 43.41 || Avicennma || Avicenna's Enharmonic Diesis || ||

|| 50/49 || | 1 2 -2 > || 34.98 || Tritonic Diesis || Jubilisma || ||

|| 64827/64000 || | -9 3 -3 4 > || 22.23 || Squalentine || || ||

|| 3125/3087 || | -2 5 -3 > || 21.18 || Gariboh || || ||

|| 126/125 || | 1 2 -3 1 > || 13.79 || Septimal Semicomma || Starling Comma || ||

|| 1029/1024 || | -10 1 0 3 > || 8.43 || Gamelisma || || ||

|| 6144/6125 || | 11 1 -3 -2 > || 5.36 || Porwell || || ||

|| 121/120 || | -3 -1 -1 2 > || 14.37 || Biyatisma || || ||

|| 176/175 || | 4 -2 -1 1 > || 9.86 || Valinorsma || || ||

|| 385/384 || | -7 -1 1 1 1 > || 4.50 || Keenanisma || || ||

|| 441/440 || | -3 2 -1 2 -1 > || 3.93 || Werckisma || || ||

|| 3025/3024 || | -4 -3 2 -1 2 > || 0.57 || Lehmerisma || || ||

=Hexadecaphonic Notation:=

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[[@http://www.soundclick.com/bands/page_songInfo.cfm?bandID=660895&songID=7715803|Palestrina Morta, fantasia quasi una sonata]] by [[@http://fiale.tk|Fabrizio Fulvio Fausto Fiale]]</pre></div>

<h4>Original HTML content:</h4>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>16edo</title></head><body><a href="#Hexadecaphonic Octave Theory">Hexadecaphonic Octave Theory</a> | <a href="#~~toc1~~"> </a> | <a href="#Hexadecaphonic Notation:">Hexadecaphonic Notation:</a> | <a href="#Armodue theory">Armodue theory</a> | <a href="#External links">External links</a> | <a href="#Compositions">Compositions</a>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>16edo</title></head><body><a href="#Hexadecaphonic Octave Theory">Hexadecaphonic Octave Theory</a> | <a href="#Commas">Commas</a> | <a href="#Hexadecaphonic Notation:">Hexadecaphonic Notation:</a> | <a href="#Armodue theory">Armodue theory</a> | <a href="#External links">External links</a> | <a href="#Compositions">Compositions</a>

16-edo equal temperament is the division of the octave into sixteen narrow chromatic semitones each of 75 cents exactly. It is not especially good at representing most low-integer musical intervals, but it has a 7/4 which is six cents sharp, and a 5/4 which is eleven cents flat. Four steps of it gives the 300 cent minor third interval identical to that of 12-edo, giving it four diminished seventh chords exactly like those of 12-edo, and a diminished triad on each scale step.

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In 16-tone, because of the 25 cent difference in the steps from 100 in 12-tone, a western &quot;twelve tone ear&quot; hears dissonance with more complexity and less familiarity than even 24-tone, yet within a more manageable number of tones and a strange familiarity - the diminished family - making 16-edo a truly xenharmonic system.

If we take the 300-cent minor third as an approximation of the harmonic 19th (19/16, 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. 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;. Example on Goldsmith board: <img src="http://www.ronsword.com/161928%20copy.jpg" alt="external image 161928%20copy.jpg" title="external image 161928%20copy.jpg" style="height: 92px; width: 158px;" />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.

If we take the 300-cent minor third as an approximation of the harmonic 19th (19/16, 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. 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;. Example on Goldsmith board: <img src="http://www.ronsword.com/161928%20copy.jpg" alt="external image 161928%20copy.jpg" title="external image 161928%20copy.jpg" style="height: 92px; width: 158px;" />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.

<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;" />

<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;" />

<hr />

In 16-edo diatonic scales are dissonant because of the 25 cent raised superfourth in conjunction with the 25 cent subtracted fifth / poor 3/2 approximation. The septimal semi diminished fourth can be more desirable. Perhaps using Moment of Symmetry Scales an alternative temperament families like the &quot;Anti-Diatonic&quot; Mavila (which reverses step sizes of diatonic), Diminished, Happy, Rice, Grumpy, Mosh, Magic, Lemba, Cynder, and Decatonic can be more interesting and suitable:

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8 octaves into 8 equal parts = 16 16 16 16 16 16 16 16 = Octave Progression

9 octaves into 8 equal parts = 18 18 18 18 18 18 18 18 = Ninth Progression

<h1 id="toc1"> </h1>

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

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

16 EDO tempers out the following commas. (Note: This assumes val &lt; 16 25 37 45 55 59 |.)

<tr>

<th>Comma

</th>

<th>Monzo

</th>

<th>Value (Cents)

</th>

<th>Name 1

</th>

<th>Name 2

</th>

<th>Name 3

</th>

</tr>

<tr>

<td>135/128

</td>

<td>| -7 3 1 &gt;

</td>

<td>92.18

</td>

<td>Major Chroma

</td>

<td>Major Limma

</td>

<td>Pelogic Comma

</td>

</tr>

<tr>

<td>648/625

</td>

<td>| 3 4 -4 &gt;

</td>

<td>62.57

</td>

<td>Major Diesis

</td>

<td>Diminished Comma

</td>

<td>

</td>

</tr>

<tr>

<td>3125/3072

</td>

<td>| -10 -1 5 &gt;

</td>

<td>29.61

</td>

<td>Small Diesis

</td>

<td>Magic Comma

</td>

<td>

</td>

</tr>

<tr>

<td>1212717/1210381

</td>

<td>| 23 6 -14 &gt;

</td>

<td>3.34

</td>

<td>Vishnuzma

</td>

<td>Semisuper

</td>

<td>

</td>

</tr>

<tr>

<td>36/35

</td>

<td>| 2 2 -1 -1 &gt;

</td>

<td>48.77

</td>

<td>Septimal Quarter Tone

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>525/512

</td>

<td>| -9 1 2 1 &gt;

</td>

<td>43.41

</td>

<td>Avicennma

</td>

<td>Avicenna's Enharmonic Diesis

</td>

<td>

</td>

</tr>

<tr>

<td>50/49

</td>

<td>| 1 2 -2 &gt;

</td>

<td>34.98

</td>

<td>Tritonic Diesis

</td>

<td>Jubilisma

</td>

<td>

</td>

</tr>

<tr>

<td>64827/64000

</td>

<td>| -9 3 -3 4 &gt;

</td>

<td>22.23

</td>

<td>Squalentine

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>3125/3087

</td>

<td>| -2 5 -3 &gt;

</td>

<td>21.18

</td>

<td>Gariboh

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>126/125

</td>

<td>| 1 2 -3 1 &gt;

</td>

<td>13.79

</td>

<td>Septimal Semicomma

</td>

<td>Starling Comma

</td>

<td>

</td>

</tr>

<tr>

<td>1029/1024

</td>

<td>| -10 1 0 3 &gt;

</td>

<td>8.43

</td>

<td>Gamelisma

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>6144/6125

</td>

<td>| 11 1 -3 -2 &gt;

</td>

<td>5.36

</td>

<td>Porwell

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>121/120

</td>

<td>| -3 -1 -1 2 &gt;

</td>

<td>14.37

</td>

<td>Biyatisma

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>176/175

</td>

<td>| 4 -2 -1 1 &gt;

</td>

<td>9.86

</td>

<td>Valinorsma

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>385/384

</td>

<td>| -7 -1 1 1 1 &gt;

</td>

<td>4.50

</td>

<td>Keenanisma

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>441/440

</td>

<td>| -3 2 -1 2 -1 &gt;

</td>

<td>3.93

</td>

<td>Werckisma

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>3025/3024

</td>

<td>| -4 -3 2 -1 2 &gt;

</td>

<td>0.57

</td>

<td>Lehmerisma

</td>

<td>

</td>

<td>

</td>

</tr>

</table>

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

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

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<h1 id="toc4"><a name="External links"></a>External links</h1>

<img src="http://ronsword.com/images/ESG_sm.jpg" alt="external image ESG_sm.jpg" title="external image ESG_sm.jpg" style="height: 161px; width: 120px;" />

<img src="http://ronsword.com/images/ESG_sm.jpg" alt="external image ESG_sm.jpg" title="external image ESG_sm.jpg" style="height: 161px; width: 120px;" />

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

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

@@ 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:Osmiorisbendi|Osmiorisbendi]] and made on <tt>2011-05-24 02:45:35 UTC</tt>.<br>
+: This revision was by author [[User:jdfreivald|jdfreivald]] and made on <tt>2011-05-31 19:53:30 UTC</tt>.<br>
-: The original revision id was <tt>231256160</tt>.<br>
+: The original revision id was <tt>233317148</tt>.<br>
-: The revision comment was: <tt></tt><br>
+: The revision comment was: <tt>Added comma table.</tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
 <h4>Original Wikitext content:</h4>
@@ Line 66: / Line 66: @@
 octaves into 8 equal parts = 16 16 16 16 16 16 16 16 = Octave Progression
 octaves into 8 equal parts = 18 18 18 18 18 18 18 18 = Ninth Progression
-= =
+=Commas=
+EDO tempers out the following commas. (Note: This assumes val &lt; 16 25 37 45 55 59 |.)
+||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
+|| 135/128 || | -7 3 1 &gt; || 92.18 || Major Chroma || Major Limma || Pelogic Comma ||
+|| 648/625 || | 3 4 -4 &gt; || 62.57 || Major Diesis || Diminished Comma ||   ||
+|| 3125/3072 || | -10 -1 5 &gt; || 29.61 || Small Diesis || Magic Comma ||   ||
+|| 1212717/1210381 || | 23 6 -14 &gt; || 3.34 || Vishnuzma || Semisuper ||   ||
+|| 36/35 || | 2 2 -1 -1 &gt; || 48.77 || Septimal Quarter Tone ||   ||   ||
+|| 525/512 || | -9 1 2 1 &gt; || 43.41 || Avicennma || Avicenna's Enharmonic Diesis ||   ||
+|| 50/49 || | 1 2 -2 &gt; || 34.98 || Tritonic Diesis || Jubilisma ||   ||
+|| 64827/64000 || | -9 3 -3 4 &gt; || 22.23 || Squalentine ||   ||   ||
+|| 3125/3087 || | -2 5 -3 &gt; || 21.18 || Gariboh ||   ||   ||
+|| 126/125 || | 1 2 -3 1 &gt; || 13.79 || Septimal Semicomma || Starling Comma ||   ||
+|| 1029/1024 || | -10 1 0 3 &gt; || 8.43 || Gamelisma ||   ||   ||
+|| 6144/6125 || | 11 1 -3 -2 &gt; || 5.36 || Porwell ||   ||   ||
+|| 121/120 || | -3 -1 -1 2 &gt; || 14.37 || Biyatisma ||   ||   ||
+|| 176/175 || | 4 -2 -1 1 &gt; || 9.86 || Valinorsma ||   ||   ||
+|| 385/384 || | -7 -1 1 1 1 &gt; || 4.50 || Keenanisma ||   ||   ||
+|| 441/440 || | -3 2 -1 2 -1 &gt; || 3.93 || Werckisma ||   ||   ||
+|| 3025/3024 || | -4 -3 2 -1 2 &gt; || 0.57 || Lehmerisma ||   ||   ||
 =Hexadecaphonic Notation:=
@@ Line 94: / Line 113: @@
 [[@http://www.soundclick.com/bands/page_songInfo.cfm?bandID=660895&amp;songID=7715803|Palestrina Morta, fantasia quasi una sonata]] by [[@http://fiale.tk|Fabrizio Fulvio Fausto Fiale]]</pre></div>
 <h4>Original HTML content:</h4>
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;16edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:12:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;&lt;a href="#Hexadecaphonic Octave Theory"&gt;Hexadecaphonic Octave Theory&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextTocRule:14: --&gt; | &lt;a href="#toc1"&gt; &lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt; | &lt;a href="#Hexadecaphonic Notation:"&gt;Hexadecaphonic Notation:&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt; | &lt;a href="#Armodue theory"&gt;Armodue theory&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt; | &lt;a href="#External links"&gt;External links&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt; | &lt;a href="#Compositions"&gt;Compositions&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;
+<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;16edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:12:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;&lt;a href="#Hexadecaphonic Octave Theory"&gt;Hexadecaphonic Octave Theory&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextTocRule:14: --&gt; | &lt;a href="#Commas"&gt;Commas&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt; | &lt;a href="#Hexadecaphonic Notation:"&gt;Hexadecaphonic Notation:&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt; | &lt;a href="#Armodue theory"&gt;Armodue theory&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt; | &lt;a href="#External links"&gt;External links&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt; | &lt;a href="#Compositions"&gt;Compositions&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;
 &lt;!-- ws:end:WikiTextTocRule:19 --&gt;16-edo equal temperament is the division of the octave into sixteen narrow chromatic semitones each of 75 cents exactly. It is not especially good at representing most low-integer musical intervals, but it has a 7/4 which is six cents sharp, and a 5/4 which is eleven cents flat. Four steps of it gives the 300 cent minor third interval identical to that of 12-edo, giving it four diminished seventh chords exactly like those of 12-edo, and a diminished triad on each scale step.&lt;br /&gt;
 &lt;br /&gt;
@@ Line 107: / Line 126: @@
 In 16-tone, because of the 25 cent difference in the steps from 100 in 12-tone, a western &amp;quot;twelve tone ear&amp;quot; hears dissonance with more complexity and less familiarity than even 24-tone, yet within a more manageable number of tones and a strange familiarity - the diminished family - making 16-edo a truly xenharmonic system.&lt;br /&gt;
 &lt;br /&gt;
-If we take the 300-cent minor third as an approximation of the harmonic 19th (19/16, 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. 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;. Example on Goldsmith board: &lt;!-- ws:start:WikiTextRemoteImageRule:21:&amp;lt;img src=&amp;quot;http://www.ronsword.com/161928%20copy.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 92px; width: 158px;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.ronsword.com/161928%20copy.jpg" alt="external image 161928%20copy.jpg" title="external image 161928%20copy.jpg" style="height: 92px; width: 158px;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:21 --&gt;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;
+If we take the 300-cent minor third as an approximation of the harmonic 19th (19/16, 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. 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;. Example on Goldsmith board: &lt;!-- ws:start:WikiTextRemoteImageRule:275:&amp;lt;img src=&amp;quot;http://www.ronsword.com/161928%20copy.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 92px; width: 158px;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.ronsword.com/161928%20copy.jpg" alt="external image 161928%20copy.jpg" title="external image 161928%20copy.jpg" style="height: 92px; width: 158px;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:275 --&gt;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;!-- ws:start:WikiTextRemoteImageRule:22:&amp;lt;img src=&amp;quot;http://ronsword.com/DSgoldsmith_piece.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 342px; width: 1008px;&amp;quot; /&amp;gt; --&gt;&lt;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;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:22 --&gt;&lt;br /&gt;
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 &lt;hr /&gt;
 In 16-edo diatonic scales are dissonant because of the 25 cent raised superfourth in conjunction with the 25 cent subtracted fifth / poor 3/2 approximation. The septimal semi diminished fourth can be more desirable. Perhaps using Moment of Symmetry Scales an alternative temperament families like the &amp;quot;Anti-Diatonic&amp;quot; Mavila (which reverses step sizes of diatonic), Diminished, Happy, Rice, Grumpy, Mosh, Magic, Lemba, Cynder, and Decatonic can be more interesting and suitable:&lt;br /&gt;
@@ Line 154: / Line 173: @@
 octaves into 8 equal parts = 16 16 16 16 16 16 16 16 = Octave Progression&lt;br /&gt;
 octaves into 8 equal parts = 18 18 18 18 18 18 18 18 = Ninth Progression&lt;br /&gt;
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+&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Commas&lt;/h1&gt;
-  &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Hexadecaphonic Notation:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Hexadecaphonic Notation:&lt;/h1&gt;
+EDO tempers out the following commas. (Note: This assumes val &amp;lt; 16 25 37 45 55 59 |.)&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;th&gt;Comma&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;Monzo&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;Value (Cents)&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;Name 1&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;Name 2&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;Name 3&lt;br /&gt;
+&lt;/th&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;135/128&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -7 3 1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;92.18&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Major Chroma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Major Limma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Pelogic Comma&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;648/625&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| 3 4 -4 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;62.57&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Major Diesis&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Diminished Comma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;3125/3072&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -10 -1 5 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;29.61&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Small Diesis&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Magic Comma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;1212717/1210381&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| 23 6 -14 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;3.34&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Vishnuzma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Semisuper&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;36/35&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| 2 2 -1 -1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;48.77&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Septimal Quarter Tone&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;525/512&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -9 1 2 1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;43.41&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Avicennma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Avicenna's Enharmonic Diesis&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;50/49&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| 1 2 -2 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;34.98&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Tritonic Diesis&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Jubilisma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;64827/64000&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -9 3 -3 4 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;22.23&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Squalentine&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;3125/3087&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -2 5 -3 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;21.18&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Gariboh&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;126/125&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| 1 2 -3 1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;13.79&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Septimal Semicomma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Starling Comma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;1029/1024&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -10 1 0 3 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;8.43&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Gamelisma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;6144/6125&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| 11 1 -3 -2 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;5.36&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Porwell&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;121/120&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -3 -1 -1 2 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;14.37&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Biyatisma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;176/175&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| 4 -2 -1 1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;9.86&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Valinorsma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;385/384&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -7 -1 1 1 1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;4.50&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Keenanisma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;441/440&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -3 2 -1 2 -1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;3.93&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Werckisma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;3025/3024&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -4 -3 2 -1 2 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;0.57&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Lehmerisma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Hexadecaphonic Notation:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&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&lt;br /&gt;
@@ Line 171: / Line 448: @@
 &lt;br /&gt;
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+  &lt;!-- ws:start:WikiTextRemoteImageRule:277:&amp;lt;img src=&amp;quot;http://ronsword.com/images/ESG_sm.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 161px; width: 120px;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://ronsword.com/images/ESG_sm.jpg" alt="external image ESG_sm.jpg" title="external image ESG_sm.jpg" style="height: 161px; width: 120px;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:277 --&gt;&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;