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:

: This revision was by author [[User:~~hstraub~~|~~hstraub~~]] and made on <tt>2011-07-~~18 03~~:09:53 UTC</tt>.

: This revision was by author [[User:igliashon|igliashon]] and made on <tt>2011-07-21 00:47:25 UTC</tt>.

: The original revision id was <tt>~~241737375~~</tt>.

: The original revision id was <tt>242219447</tt>.

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

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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**16edo** 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 identical to that of 12-edo, giving it four diminished seventh chords exactly like those of [[12edo]], and a diminished triad on each scale step.

|| Degree || Cents || Approximate

Ratios* || Interval Name ||

|| 0 || 0 || 1/1 || ||

|| 1 || 75 || 28/27, 27/26 || ||

|| 2 || 150 || 35/32 || ||

|| 3 || 225 || 8/7 || ||

|| 4 || 300 || 19/16, 32/27 || ||

|| 5 || 375 || 5/4, 16/13 || ||

|| 6 || 450 || 13/10 || ||

|| 7 || 525 || 27/20, 70/52, 26/19 || ||

|| 8 || 600 || 7/5, 10/7 || ||

|| 9 || 675 || 40/27, 52/35, 19/13 || ||

|| 10 || 750 || 20/13 || ||

|| 11 || 825 || 8/5, 13/8 || ||

|| 12 || 900 || 27/16, 32/19 || ||

|| 13 || 975 || 7/4 || ||

|| 14 || 1050 || 64/35 || ||

|| 15 || 1125 || 27/14, 52/27 || ||

|| 16 || 1200 || 2/1 || ||

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

=Hexadecaphonic Octave Theory=

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16edo 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 identical to that of 12-edo, giving it four diminished seventh chords exactly like those of <a class="wiki_link" href="/12edo">12edo</a>, and a diminished triad on each scale step.

<tr>

<td>Degree

</td>

<td>Cents

</td>

<td>Approximate

Ratios*

</td>

<td>Interval Name

</td>

</tr>

<tr>

<td>0

</td>

<td>0

</td>

<td>1/1

</td>

<td>

</td>

</tr>

<tr>

<td>1

</td>

<td>75

</td>

<td>28/27, 27/26

</td>

<td>

</td>

</tr>

<tr>

<td>2

</td>

<td>150

</td>

<td>35/32

</td>

<td>

</td>

</tr>

<tr>

<td>3

</td>

<td>225

</td>

<td>8/7

</td>

<td>

</td>

</tr>

<tr>

<td>4

</td>

<td>300

</td>

<td>19/16, 32/27

</td>

<td>

</td>

</tr>

<tr>

<td>5

</td>

<td>375

</td>

<td>5/4, 16/13

</td>

<td>

</td>

</tr>

<tr>

<td>6

</td>

<td>450

</td>

<td>13/10

</td>

<td>

</td>

</tr>

<tr>

<td>7

</td>

<td>525

</td>

<td>27/20, 70/52, 26/19

</td>

<td>

</td>

</tr>

<tr>

<td>8

</td>

<td>600

</td>

<td>7/5, 10/7

</td>

<td>

</td>

</tr>

<tr>

<td>9

</td>

<td>675

</td>

<td>40/27, 52/35, 19/13

</td>

<td>

</td>

</tr>

<tr>

<td>10

</td>

<td>750

</td>

<td>20/13

</td>

<td>

</td>

</tr>

<tr>

<td>11

</td>

<td>825

</td>

<td>8/5, 13/8

</td>

<td>

</td>

</tr>

<tr>

<td>12

</td>

<td>900

</td>

<td>27/16, 32/19

</td>

<td>

</td>

</tr>

<tr>

<td>13

</td>

<td>975

</td>

<td>7/4

</td>

<td>

</td>

</tr>

<tr>

<td>14

</td>

<td>1050

</td>

<td>64/35

</td>

<td>

</td>

</tr>

<tr>

<td>15

</td>

<td>1125

</td>

<td>27/14, 52/27

</td>

<td>

</td>

</tr>

<tr>

<td>16

</td>

<td>1200

</td>

<td>2/1

</td>

<td>

</td>

</tr>

</table>

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

<h1 id="toc0"><a name="Hexadecaphonic Octave Theory"></a>Hexadecaphonic Octave Theory</h1>

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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 <a class="wiki_link" href="/24edo">24-tone</a>, 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 (<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. 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 (<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. 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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<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:hstraub|hstraub]] and made on <tt>2011-07-18 03:09:53 UTC</tt>.<br>
+: This revision was by author [[User:igliashon|igliashon]] and made on <tt>2011-07-21 00:47:25 UTC</tt>.<br>
-: The original revision id was <tt>241737375</tt>.<br>
+: The original revision id was <tt>242219447</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 11: / Line 11: @@
 **16edo** 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 identical to that of 12-edo, giving it four diminished seventh chords exactly like those of [[12edo]], and a diminished triad on each scale step.
+|| Degree || Cents || Approximate
+Ratios* || Interval Name ||
+|| 0 || 0 || 1/1 ||   ||
+|| 1 || 75 || 28/27, 27/26 ||   ||
+|| 2 || 150 || 35/32 ||   ||
+|| 3 || 225 || 8/7 ||   ||
+|| 4 || 300 || 19/16, 32/27 ||   ||
+|| 5 || 375 || 5/4, 16/13 ||   ||
+|| 6 || 450 || 13/10 ||   ||
+|| 7 || 525 || 27/20, 70/52, 26/19 ||   ||
+|| 8 || 600 || 7/5, 10/7 ||   ||
+|| 9 || 675 || 40/27, 52/35, 19/13 ||   ||
+|| 10 || 750 || 20/13 ||   ||
+|| 11 || 825 || 8/5, 13/8 ||   ||
+|| 12 || 900 || 27/16, 32/19 ||   ||
+|| 13 || 975 || 7/4 ||   ||
+|| 14 || 1050 || 64/35 ||   ||
+|| 15 || 1125 || 27/14, 52/27 ||   ||
+|| 16 || 1200 || 2/1 ||   ||
+*based on treating 16-EDO as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible
 =Hexadecaphonic Octave Theory=
@@ Line 116: / Line 136: @@
 &lt;strong&gt;16edo&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 identical to that of 12-edo, giving it four diminished seventh chords exactly like those of &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, and a diminished triad on each scale step.&lt;br /&gt;
 &lt;br /&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;td&gt;Degree&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Cents&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Approximate&lt;br /&gt;
+Ratios*&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Interval Name&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;1/1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;75&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;28/27, 27/26&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;150&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;35/32&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;225&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;8/7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;300&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;19/16, 32/27&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;375&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;5/4, 16/13&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;6&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;450&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;13/10&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;525&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;27/20, 70/52, 26/19&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;8&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;600&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;7/5, 10/7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;9&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;675&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;40/27, 52/35, 19/13&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;10&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;750&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;20/13&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;11&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;825&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;8/5, 13/8&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;12&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;900&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;27/16, 32/19&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;13&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;975&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;7/4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;14&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;1050&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;64/35&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;15&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;1125&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;27/14, 52/27&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;16&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;1200&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;2/1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
+*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:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Hexadecaphonic Octave Theory"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Hexadecaphonic Octave Theory&lt;/h1&gt;
@@ Line 126: / Line 333: @@
 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 &lt;a class="wiki_link" href="/24edo"&gt;24-tone&lt;/a&gt;, 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 (&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. 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:276:&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:276 --&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 (&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. 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:458:&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:458 --&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;
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 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;
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 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;