MOS scale: Difference between revisions

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-04-06 15:29:58 UTC</tt>.<br>

: The original revision id was <tt>217830218</tt>.<br>

<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">An important class of scales are MOS scales (the acronym **MOS** coming from **"Moment Of Symmetry"**). These are derived by iterating an interval g, called the generator, inside a larger interval, called the period, and reducing to the period when the iterates become larger than the period. Usually the period is an octave or an nth root of 2, but it can in theory be any positive number. The resulting scale is called a MOS, when, as a [[periodic scale]], it has [[http://en.wikipedia.org/wiki/Myhill's_property|Myhill's propery]]: every generic interval aside from the initial unison interval has two specific intervals.  

Another characterization of when a generated scale is a MOS is that the number of scale steps is the denominator of a [[http://en.wikipedia.org/wiki/Continued_fraction|convergent or semiconvergent]] of the ratio g/P of the generator and the period. These conditions entail that the generated scale has exactly two sizes of steps when sorted into ascending order of size, and usually that latter condition suffices to define a MOS. However, when the generator is a rational fraction of the period and the number of steps is more than half of the total possible, a generated scale can have only two sizes of steps and the pseudo-Myhill property, meaning that not all non-unison classes have only two specific intervals.

Note that an octave scale with a period a 1/N fraction of an octave will be a MOS if it is a MOS, identically, within each period.

The term and scale construction method were invented by Erv Wilson in 1975. His original paper can be found here [[http://anaphoria.com/mos.PDF]]. There is also an introduction [[http://anaphoria.com/wilsonintroMOS.html]].In academic music theory, MOS are known as //well-formed scales// and the introduction of the concept is attributed to a 1989 paper by Norman Carey and David Clampitt. A great deal of interesting work has been done on scales in academic circles extending these ideas.

=The Definition of a Moment of Symmetry=

<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;An important class of scales are MOS scales (the acronym &lt;strong&gt;MOS&lt;/strong&gt; coming from &lt;strong&gt;&amp;quot;Moment Of Symmetry&amp;quot;&lt;/strong&gt;). These are derived by iterating an interval g, called the generator, inside a larger interval, called the period, and reducing to the period when the iterates become larger than the period. Usually the period is an octave or an nth root of 2, but it can in theory be any positive number. The resulting scale is called a MOS, when, as a &lt;a class="wiki_link" href="/periodic%20scale"&gt;periodic scale&lt;/a&gt;, it has &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Myhill's_property" rel="nofollow"&gt;Myhill's propery&lt;/a&gt;: every generic interval aside from the initial unison interval has two specific intervals. &lt;br /&gt;

Another characterization of when a generated scale is a MOS is that the number of scale steps is the denominator of a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Continued_fraction" rel="nofollow"&gt;convergent or semiconvergent&lt;/a&gt; of the ratio g/P of the generator and the period. These conditions entail that the generated scale has exactly two sizes of steps when sorted into ascending order of size, and usually that latter condition suffices to define a MOS. However, when the generator is a rational fraction of the period and the number of steps is more than half of the total possible, a generated scale can have only two sizes of steps and the pseudo-Myhill property, meaning that not all non-unison classes have only two specific intervals.&lt;br /&gt;