MOS scale
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=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:<h1> --><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 "Moment of symmetry"). 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:<h1> --><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:<h2> --><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:<h2> --><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 "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.)<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><a class="wiki_link" href="/3L%206s">3L 6s</a><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:<h1> --><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>