MOS scale: 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:hstraub|hstraub]] and made on <tt>2007-03-~~25 07~~:42:59 UTC</tt>.<br>

: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2007-03-27 11:09:08 UTC</tt>.<br>

: The original revision id was <tt>~~3452530~~</tt>.<br>

: The original revision id was <tt>3492831</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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==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]].</pre></div>

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 [[http://mathworld.wolfram.com/DiophantineEquation.html|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 from 5 to 36.

Not all mathematical possibilities are listed - solutions of the equation that would yield too "exotic" scale steps (too small/tto big diffference between s and L) are excluded. (The concrete restrictions applied were: a solution appears if 7/6 < L/s < 5.)</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>MOSScales</title></head><body><h1 id="toc0"><a name="MOS scales"></a>MOS scales</h1>

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<h2 id="toc1"><a name="MOS scales-Classification of MOS"></a>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>.</body></html></pre></div>

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 />

<h2 id="toc2"><a name="MOS scales-MOS in equal temperaments"></a>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 <a class="wiki_link_ext" href="http://mathworld.wolfram.com/DiophantineEquation.html" rel="nofollow">linear diophantine equation</a>! 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 from 5 to 36.<br />

Not all mathematical possibilities are listed - solutions of the equation that would yield too &quot;exotic&quot; scale steps (too small/tto big diffference between s and L) are excluded. (The concrete restrictions applied were: a solution appears if 7/6 &lt; L/s &lt; 5.)</body></html></pre></div>

@@ 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>2007-03-25 07:42:59 UTC</tt>.<br>
+: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2007-03-27 11:09:08 UTC</tt>.<br>
-: The original revision id was <tt>3452530</tt>.<br>
+: The original revision id was <tt>3492831</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 15: / Line 15: @@
 ==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]].</pre></div>
+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 [[http://mathworld.wolfram.com/DiophantineEquation.html|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 from 5 to 36.
+Not all mathematical possibilities are listed - solutions of the equation that would yield too "exotic" scale steps (too small/tto big diffference between s and L) are excluded. (The concrete restrictions applied were: a solution appears if 7/6 &lt; L/s &lt; 5.)</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;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;
@@ Line 26: / Line 38: @@
 &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;
 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;/body&gt;&lt;/html&gt;</pre></div>
+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;
+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;
+&lt;br /&gt;
+a*L +b*s = n.&lt;br /&gt;
+&lt;br /&gt;
+which is a &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/DiophantineEquation.html" rel="nofollow"&gt;linear diophantine equation&lt;/a&gt;! 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.&lt;br /&gt;
+&lt;br /&gt;
+Below is a list of MOS with number of elements from 5 to 10, in equal temperaments from 5 to 36.&lt;br /&gt;
+Not all mathematical possibilities are listed - solutions of the equation that would yield too &amp;quot;exotic&amp;quot; scale steps (too small/tto big diffference between s and L) are excluded. (The concrete restrictions applied were: a solution appears if 7/6 &amp;lt; L/s &amp;lt; 5.)&lt;/body&gt;&lt;/html&gt;</pre></div>