MOS scale
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=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]] ==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 [[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.) [[PentatonicMOS|Pentatonic MOS]] <More follow soon>
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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").<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 "moment of symmetry", 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:<h2> --><h2 id="toc1"><a name="MOS scales-Classification of MOS"></a><!-- ws:end:WikiTextHeadingRule:2 -->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:4:<h2> --><h2 id="toc2"><a name="MOS scales-MOS in equal temperaments"></a><!-- ws:end:WikiTextHeadingRule:4 -->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 "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.)<br /> <br /> <a class="wiki_link" href="/PentatonicMOS">Pentatonic MOS</a><br /> <More follow soon></body></html>