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 [[Moments of Symmetry|http://anaphoria.com/mos.PDF]] . There is also an introduction h [[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]] [[HexatonicMOS|Hexatonic MOS]] [[HeptatonicMOS|Heptatonic MOS]] [[OctatonicMOS|Octatonic MOS]] [[NonatonicMOS|Nonatonic MOS]] [[DecatonicMOS|Decatonic MOS]] <span style="color: rgb(0, 0, 238);"> </span> =MOS As Applied To Rhythms= MOS structures and thinking can be applied to the design of rhythms as well. See [[MOS Rhythm Tutorial]]
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<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" href="/Moments%20of%20Symmetry">http://anaphoria.com/mos.PDF</a> . There is also an introduction h <a class="wiki_link" href="/introduction">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 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:<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 /> <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><br /> <span style="color: rgb(0, 0, 238);"><br /> </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> 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>