MOS scale: Difference between revisions

Revision as of 20:59, 31 May 2007

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This revision was by author xenjacob and made on 2007-05-31 20:59:34 UTC.

The original revision id was 4722523.

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Original Wikitext content:

=MOS scales= 

An important class of scales are MOS scales (MOS "Moment of symmetry").
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).
For more information of the background and why it is called "moment of symmetry", see
[[http://tonalsoft.com/enc/m/mos.aspx|Joe Monzo's encyclopedia of microtonal music theory]]

=[[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]]
[[HexatonicMOS|Hexatonic MOS]]
[[HeptatonicMOS|Heptatonic MOS]]
[[OctatonicMOS|Octatonic MOS]]
[[NonatonicMOS|Nonatonic MOS]]
[[DecatonicMOS|Decatonic MOS]]

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;).<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 />
For more information of the background and why it is called &quot;moment of symmetry&quot;, see<br />
<a class="wiki_link_ext" href="http://tonalsoft.com/enc/m/mos.aspx" rel="nofollow">Joe Monzo's encyclopedia of microtonal music theory</a><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 5L 2s (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 />
<a class="wiki_link" href="/PentatonicMOS">Pentatonic MOS</a><br />
<a class="wiki_link" href="/HexatonicMOS">Hexatonic MOS</a><br />
<a class="wiki_link" href="/HeptatonicMOS">Heptatonic MOS</a><br />
<a class="wiki_link" href="/OctatonicMOS">Octatonic MOS</a><br />
<a class="wiki_link" href="/NonatonicMOS">Nonatonic MOS</a><br />
<a class="wiki_link" href="/DecatonicMOS">Decatonic MOS</a></body></html>

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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:hstraub|hstraub]] and made on <tt>2007-03-27 13:26:10 UTC</tt>.<br>
+: This revision was by author [[User:xenjacob|xenjacob]] and made on <tt>2007-05-31 20:59:34 UTC</tt>.<br>
-: The original revision id was <tt>3495576</tt>.<br>
+: The original revision id was <tt>4722523</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>
 <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;white-space: pre-wrap ! important" class="old-revision-html">=MOS scales=
 An important class of scales are MOS scales (MOS "Moment of symmetry").
@@ Line 13: / Line 13: @@
 [[http://tonalsoft.com/enc/m/mos.aspx|Joe Monzo's encyclopedia of microtonal music theory]]
-==Classification of MOS==
+=[[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.)
@@ Line 37: / Line 39: @@
 <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;MOSScales&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="MOS scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;MOS scales&lt;/h1&gt;
-&lt;br /&gt;
+ &lt;br /&gt;
 An important class of scales are MOS scales (MOS &amp;quot;Moment of symmetry&amp;quot;).&lt;br /&gt;
 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).&lt;br /&gt;
@@ Line 43: / Line 45: @@
 &lt;a class="wiki_link_ext" href="http://tonalsoft.com/enc/m/mos.aspx" rel="nofollow"&gt;Joe Monzo's encyclopedia of microtonal music theory&lt;/a&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="MOS scales-Classification of MOS"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Classification of MOS&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="MOSDiagrams"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;&lt;a class="wiki_link" href="/MOSDiagrams"&gt;MOSDiagrams&lt;/a&gt;&lt;/h1&gt;
-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).&lt;br /&gt;
+ &lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="MOSDiagrams-Classification of MOS"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Classification of MOS&lt;/h2&gt;
+ 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).&lt;br /&gt;
 Since numbers tend to be dry, Graham Breed has proposed a &lt;a class="wiki_link" href="/MOSNamingScheme"&gt;naming scheme for MOS scales&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="MOS scales-MOS in equal temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;MOS in equal temperaments&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="MOSDiagrams-MOS in equal temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;MOS in equal temperaments&lt;/h2&gt;
-In the special case of an equal temperament, more concrete things about MOS can be stated.&lt;br /&gt;
+ In the special case of an equal temperament, more concrete things about MOS can be stated.&lt;br /&gt;
 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.)&lt;br /&gt;
 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&lt;br /&gt;

MOS scale: Difference between revisions

Revision as of 20:59, 31 May 2007

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