MOS scale: Difference between revisions

Revision as of 12:04, 29 November 2009

IMPORTED REVISION FROM WIKISPACES

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

This revision was by author guest and made on 2009-11-29 12:04:48 UTC.

The original revision id was 105842765.

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

Original Wikitext content:

=MOS scales= 

An important class of scales are MOS scales (MOS "Moment of symmetry"). These were invented by Erv Wilson. His original paper can be found here [[http://anaphoria.com/mos.PDF]] . There is also an introduction [[http://anaphoria.com/wilsonintroMOS.html]]
An MOS scale is a scale whose basic steps come in 2 different sizes. This is an interesting property because two basic scales of classical music theory have it: the diatonic scale (whole tone and semitone) and the pentatonic scale (minor third and whole tone).


=[[MOSDiagrams]]= 

==Classification of MOS== 
An obvious first rough classification of MOS scales is given by the number of elements of the scale - the number of large intervals (L) and the number of small intervals (s). E.g., the diatonic scale in 12-tone equal temperament could be described as [[5L 2s]] (5 large steps and 2 small steps).
Since numbers tend to be dry, Graham Breed has proposed a [[MOSNamingScheme|naming scheme for MOS scales]].

==MOS in equal temperaments== 
In the special case of an equal temperament, more concrete things about MOS can be stated.
In an equal temparement, all intervals - and hence also the intervals L and s - are integer multiples of a smallest unit. (Example: in case of the diatonic scale in 12EDO, L would be 2 and s 1.)
If we have an arbitrary MOS scale in an n-tone equal temperament, with a steps of size L and b steps of size s, there holds

a*L +b*s = n.

which is a linear diophantine equation! This means that given a, b and n, all possible MOS types can be calculated via the general solution of the corresponding linear diophantine equation.

Below is a list of MOS with number of elements from 5 to 10, in equal temperaments up to 36.
Not all mathematical possibilities are listed - solutions of the equation that would yield too "exotic" scale steps (too small/too big diffference between s and L) are excluded. (The concrete - sort of arbitrary - restrictions applied were: a solution appears if 7/6 < L/s < 5.)

|| [[PentatonicMOS|Pentatonic MOS]] ||   ||   ||   ||   ||   || [[1L 4s]] ||   || [[2L 3s]] ||   || [[3L 2s]] ||   || [[4L 1s]] ||   ||   ||   ||   ||   ||
|| [[HexatonicMOS|Hexatonic MOS]] ||   ||   ||   ||   || [[1L 5s]] ||   || [[2L 4s]] ||   || [[3L 3s]] ||   || [[4L 2s]] ||   || [[5L 1s]] ||   ||   ||   ||   ||
|| [[HeptatonicMOS|Heptatonic MOS]] ||   ||   ||   || [[1L 6s]] ||   || [[2L 5s]] ||   || [[3L 4s]] ||   || [[4L 3s]] ||   || [[5L 2s]] ||   || [[6L 1s]] ||   ||   ||   ||
|| [[OctatonicMOS|Octatonic MOS]] ||   ||   || [[1L 7s]] ||   || [[2L 6s]] ||   || [[3L 5s]] ||   || [[4L 4s]] ||   || [[5L 3s]] ||   || [[6L 2s]] ||   || [[7L 1s]] ||   ||   ||
|| [[NonatonicMOS|Nonatonic MOS]] ||   || [[1L 8s]] ||   || [[2L 7s]] ||   || [[3L 6s] ||   || [[4L 5s]] ||   || [[5L 4s]] ||   || [[6L 3s]] ||   || [[7L 2s]] ||   || [[8L 1s]] ||   ||
|| [[DecatonicMOS|Decatonic MOS]] || [[1L 9s]] ||   || [[2L 8s]] ||   || [[3L 7s]] ||   || [[4L 6s]] ||   || [[5L 5s]] ||   || [[6L 4s]] ||   || [[7L 3s]] ||   || [[8L 2s]] ||   || [[9L 1s]] ||
<span style="color: #0000ee;"> </span>
=MOS As Applied To Rhythms= 
David Canright was the first to suggest Fibonacci Rhythms in 1/1. This lead to Kraig Grady to be the first to apply MOS patterns to rhythms. Two papers on the subject can be found here [[http://anaphoria.com/hora.PDF]] and [[http://anaphoria.com/horo2.PDF]]
MOS structures and thinking can be applied to the design of rhythms as well. See [[MOS Rhythm Tutorial]]

Original HTML content:

<html><head><title>MOSScales</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="MOS scales"></a><!-- ws:end:WikiTextHeadingRule:0 -->MOS scales</h1>
 <br />
An important class of scales are MOS scales (MOS &quot;Moment of symmetry&quot;). These were invented by Erv Wilson. His original paper can be found here <a class="wiki_link_ext" href="http://anaphoria.com/mos.PDF" rel="nofollow">http://anaphoria.com/mos.PDF</a> . There is also an introduction <a class="wiki_link_ext" href="http://anaphoria.com/wilsonintroMOS.html" rel="nofollow">http://anaphoria.com/wilsonintroMOS.html</a><br />
An MOS scale is a scale whose basic steps come in 2 different sizes. This is an interesting property because two basic scales of classical music theory have it: the diatonic scale (whole tone and semitone) and the pentatonic scale (minor third and whole tone).<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="MOSDiagrams"></a><!-- ws:end:WikiTextHeadingRule:2 --><a class="wiki_link" href="/MOSDiagrams">MOSDiagrams</a></h1>
 <br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="MOSDiagrams-Classification of MOS"></a><!-- ws:end:WikiTextHeadingRule:4 -->Classification of MOS</h2>
 An obvious first rough classification of MOS scales is given by the number of elements of the scale - the number of large intervals (L) and the number of small intervals (s). E.g., the diatonic scale in 12-tone equal temperament could be described as <a class="wiki_link" href="/5L%202s">5L 2s</a> (5 large steps and 2 small steps).<br />
Since numbers tend to be dry, Graham Breed has proposed a <a class="wiki_link" href="/MOSNamingScheme">naming scheme for MOS scales</a>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="MOSDiagrams-MOS in equal temperaments"></a><!-- ws:end:WikiTextHeadingRule:6 -->MOS in equal temperaments</h2>
 In the special case of an equal temperament, more concrete things about MOS can be stated.<br />
In an equal temparement, all intervals - and hence also the intervals L and s - are integer multiples of a smallest unit. (Example: in case of the diatonic scale in 12EDO, L would be 2 and s 1.)<br />
If we have an arbitrary MOS scale in an n-tone equal temperament, with a steps of size L and b steps of size s, there holds<br />
<br />
a*L +b*s = n.<br />
<br />
which is a linear diophantine equation! This means that given a, b and n, all possible MOS types can be calculated via the general solution of the corresponding linear diophantine equation.<br />
<br />
Below is a list of MOS with number of elements from 5 to 10, in equal temperaments up to 36.<br />
Not all mathematical possibilities are listed - solutions of the equation that would yield too &quot;exotic&quot; scale steps (too small/too big diffference between s and L) are excluded. (The concrete - sort of arbitrary - restrictions applied were: a solution appears if 7/6 &lt; L/s &lt; 5.)<br />
<br />


<table class="wiki_table">
    <tr>
        <td><a class="wiki_link" href="/PentatonicMOS">Pentatonic MOS</a><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/1L%204s">1L 4s</a><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/2L%203s">2L 3s</a><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/3L%202s">3L 2s</a><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/4L%201s">4L 1s</a><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/HexatonicMOS">Hexatonic MOS</a><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/1L%205s">1L 5s</a><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/2L%204s">2L 4s</a><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/3L%203s">3L 3s</a><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/4L%202s">4L 2s</a><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/5L%201s">5L 1s</a><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/HeptatonicMOS">Heptatonic MOS</a><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/1L%206s">1L 6s</a><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/2L%205s">2L 5s</a><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/3L%204s">3L 4s</a><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/4L%203s">4L 3s</a><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/5L%202s">5L 2s</a><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/6L%201s">6L 1s</a><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/OctatonicMOS">Octatonic MOS</a><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/1L%207s">1L 7s</a><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/2L%206s">2L 6s</a><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/3L%205s">3L 5s</a><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/4L%204s">4L 4s</a><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/5L%203s">5L 3s</a><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/6L%202s">6L 2s</a><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/7L%201s">7L 1s</a><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/NonatonicMOS">Nonatonic MOS</a><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/1L%208s">1L 8s</a><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/2L%207s">2L 7s</a><br />
</td>
        <td><br />
</td>
        <td>[[3L 6s]<br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/4L%205s">4L 5s</a><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/5L%204s">5L 4s</a><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/6L%203s">6L 3s</a><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/7L%202s">7L 2s</a><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/8L%201s">8L 1s</a><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/DecatonicMOS">Decatonic MOS</a><br />
</td>
        <td><a class="wiki_link" href="/1L%209s">1L 9s</a><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/2L%208s">2L 8s</a><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/3L%207s">3L 7s</a><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/4L%206s">4L 6s</a><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/5L%205s">5L 5s</a><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/6L%204s">6L 4s</a><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/7L%203s">7L 3s</a><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/8L%202s">8L 2s</a><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/9L%201s">9L 1s</a><br />
</td>
    </tr>
</table>

<span style="color: #0000ee;"> </span><br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="MOS As Applied To Rhythms"></a><!-- ws:end:WikiTextHeadingRule:8 -->MOS As Applied To Rhythms</h1>
 David Canright was the first to suggest Fibonacci Rhythms in 1/1. This lead to Kraig Grady to be the first to apply MOS patterns to rhythms. Two papers on the subject can be found here <a class="wiki_link_ext" href="http://anaphoria.com/hora.PDF" rel="nofollow">http://anaphoria.com/hora.PDF</a> and <a class="wiki_link_ext" href="http://anaphoria.com/horo2.PDF" rel="nofollow">http://anaphoria.com/horo2.PDF</a><br />
MOS structures and thinking can be applied to the design of rhythms as well. See <a class="wiki_link" href="/MOS%20Rhythm%20Tutorial">MOS Rhythm Tutorial</a></body></html>

@@ 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:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2009-11-11 17:15:48 UTC</tt>.<br>
+: This revision was by author [[User:guest|guest]] and made on <tt>2009-11-29 12:04:48 UTC</tt>.<br>
-: The original revision id was <tt>101990339</tt>.<br>
+: The original revision id was <tt>105842765</tt>.<br>
-: The revision comment was: <tt>added internal link</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>
 <h4>Original Wikitext content:</h4>
@@ Line 30: / Line 30: @@
 Not all mathematical possibilities are listed - solutions of the equation that would yield too "exotic" scale steps (too small/too big diffference between s and L) are excluded. (The concrete - sort of arbitrary - restrictions applied were: a solution appears if 7/6 &lt; L/s &lt; 5.)
-[[PentatonicMOS|Pentatonic MOS]]
+|| [[PentatonicMOS|Pentatonic MOS]] ||   ||   ||   ||   ||   || [[1L 4s]] ||   || [[2L 3s]] ||   || [[3L 2s]] ||   || [[4L 1s]] ||   ||   ||   ||   ||   ||
-[[HexatonicMOS|Hexatonic MOS]]
+|| [[HexatonicMOS|Hexatonic MOS]] ||   ||   ||   ||   || [[1L 5s]] ||   || [[2L 4s]] ||   || [[3L 3s]] ||   || [[4L 2s]] ||   || [[5L 1s]] ||   ||   ||   ||   ||
-[[HeptatonicMOS|Heptatonic MOS]]
+|| [[HeptatonicMOS|Heptatonic MOS]] ||   ||   ||   || [[1L 6s]] ||   || [[2L 5s]] ||   || [[3L 4s]] ||   || [[4L 3s]] ||   || [[5L 2s]] ||   || [[6L 1s]] ||   ||   ||   ||
-[[OctatonicMOS|Octatonic MOS]]
+|| [[OctatonicMOS|Octatonic MOS]] ||   ||   || [[1L 7s]] ||   || [[2L 6s]] ||   || [[3L 5s]] ||   || [[4L 4s]] ||   || [[5L 3s]] ||   || [[6L 2s]] ||   || [[7L 1s]] ||   ||   ||
-[[NonatonicMOS|Nonatonic MOS]]
+|| [[NonatonicMOS|Nonatonic MOS]] ||   || [[1L 8s]] ||   || [[2L 7s]] ||   || [[3L 6s] ||   || [[4L 5s]] ||   || [[5L 4s]] ||   || [[6L 3s]] ||   || [[7L 2s]] ||   || [[8L 1s]] ||   ||
-[[DecatonicMOS|Decatonic MOS]]
+|| [[DecatonicMOS|Decatonic MOS]] || [[1L 9s]] ||   || [[2L 8s]] ||   || [[3L 7s]] ||   || [[4L 6s]] ||   || [[5L 5s]] ||   || [[6L 4s]] ||   || [[7L 3s]] ||   || [[8L 2s]] ||   || [[9L 1s]] ||
-&lt;span style="color: #0000ee;"&gt;
+&lt;span style="color: #0000ee;"&gt; &lt;/span&gt;
-&lt;/span&gt;
 =MOS As Applied To Rhythms=
 David Canright was the first to suggest Fibonacci Rhythms in 1/1. This lead to Kraig Grady to be the first to apply MOS patterns to rhythms. Two papers on the subject can be found here [[http://anaphoria.com/hora.PDF]] and [[http://anaphoria.com/horo2.PDF]]
@@ Line 66: / Line 65: @@
 Not all mathematical possibilities are listed - solutions of the equation that would yield too &amp;quot;exotic&amp;quot; scale steps (too small/too big diffference between s and L) are excluded. (The concrete - sort of arbitrary - restrictions applied were: a solution appears if 7/6 &amp;lt; L/s &amp;lt; 5.)&lt;br /&gt;
 &lt;br /&gt;
-&lt;a class="wiki_link" href="/PentatonicMOS"&gt;Pentatonic MOS&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link" href="/HexatonicMOS"&gt;Hexatonic MOS&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link" href="/HeptatonicMOS"&gt;Heptatonic MOS&lt;/a&gt;&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
-&lt;a class="wiki_link" href="/OctatonicMOS"&gt;Octatonic MOS&lt;/a&gt;&lt;br /&gt;
+    &lt;tr&gt;
-&lt;a class="wiki_link" href="/NonatonicMOS"&gt;Nonatonic MOS&lt;/a&gt;&lt;br /&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/PentatonicMOS"&gt;Pentatonic MOS&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link" href="/DecatonicMOS"&gt;Decatonic MOS&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
-&lt;span style="color: #0000ee;"&gt;&lt;br /&gt;
+        &lt;td&gt;&lt;br /&gt;
-&lt;/span&gt;&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;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/1L%204s"&gt;1L 4s&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/2L%203s"&gt;2L 3s&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/3L%202s"&gt;3L 2s&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/4L%201s"&gt;4L 1s&lt;/a&gt;&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;td&gt;&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;&lt;a class="wiki_link" href="/HexatonicMOS"&gt;Hexatonic MOS&lt;/a&gt;&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;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/1L%205s"&gt;1L 5s&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/2L%204s"&gt;2L 4s&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/3L%203s"&gt;3L 3s&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/4L%202s"&gt;4L 2s&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/5L%201s"&gt;5L 1s&lt;/a&gt;&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;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;&lt;a class="wiki_link" href="/HeptatonicMOS"&gt;Heptatonic MOS&lt;/a&gt;&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;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/1L%206s"&gt;1L 6s&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/2L%205s"&gt;2L 5s&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/3L%204s"&gt;3L 4s&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/4L%203s"&gt;4L 3s&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/5L%202s"&gt;5L 2s&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/6L%201s"&gt;6L 1s&lt;/a&gt;&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;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/OctatonicMOS"&gt;Octatonic MOS&lt;/a&gt;&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;td&gt;&lt;a class="wiki_link" href="/1L%207s"&gt;1L 7s&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/2L%206s"&gt;2L 6s&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/3L%205s"&gt;3L 5s&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/4L%204s"&gt;4L 4s&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/5L%203s"&gt;5L 3s&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/6L%202s"&gt;6L 2s&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/7L%201s"&gt;7L 1s&lt;/a&gt;&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;&lt;a class="wiki_link" href="/NonatonicMOS"&gt;Nonatonic MOS&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/1L%208s"&gt;1L 8s&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/2L%207s"&gt;2L 7s&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;[[3L 6s]&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/4L%205s"&gt;4L 5s&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/5L%204s"&gt;5L 4s&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/6L%203s"&gt;6L 3s&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/7L%202s"&gt;7L 2s&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/8L%201s"&gt;8L 1s&lt;/a&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;&lt;a class="wiki_link" href="/DecatonicMOS"&gt;Decatonic MOS&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/1L%209s"&gt;1L 9s&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/2L%208s"&gt;2L 8s&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/3L%207s"&gt;3L 7s&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/4L%206s"&gt;4L 6s&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/5L%205s"&gt;5L 5s&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/6L%204s"&gt;6L 4s&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/7L%203s"&gt;7L 3s&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/8L%202s"&gt;8L 2s&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/9L%201s"&gt;9L 1s&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
+&lt;span style="color: #0000ee;"&gt; &lt;/span&gt;&lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="MOS As Applied To Rhythms"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;MOS As Applied To Rhythms&lt;/h1&gt;
   David Canright was the first to suggest Fibonacci Rhythms in 1/1. This lead to Kraig Grady to be the first to apply MOS patterns to rhythms. Two papers on the subject can be found here &lt;a class="wiki_link_ext" href="http://anaphoria.com/hora.PDF" rel="nofollow"&gt;http://anaphoria.com/hora.PDF&lt;/a&gt; and &lt;a class="wiki_link_ext" href="http://anaphoria.com/horo2.PDF" rel="nofollow"&gt;http://anaphoria.com/horo2.PDF&lt;/a&gt;&lt;br /&gt;
 MOS structures and thinking can be applied to the design of rhythms as well. See &lt;a class="wiki_link" href="/MOS%20Rhythm%20Tutorial"&gt;MOS Rhythm Tutorial&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

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