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:genewardsmith|genewardsmith]] and made on <tt>2010-05-18 03:26:47 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-18 19:29:45 UTC</tt>.<br>

: The original revision id was <tt>~~142752091~~</tt>.<br>

: The original revision id was <tt>143005075</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]]~~.

A classification of MOS scales can also be given by the number of elements of the scale - the number of large intervals "L" and the number of small intervals "s", together with the period, assumed to be minimal (which entails that L and s are relatively prime.) E.g., the diatonic scale can be described as [5L 2s] (5 large steps and 2 small steps) or simply [5, 2] with period an octave. Alternatively, we could give a mediant for a Farey pair associated to the MOS, where this mediant is less than any generator for the MOS. In other words, we use the right hand part of the Farey pair interval, which means we must replace g with 1-g and use the complementary pair if g is in the left hand side.

The ~~analysis of MOS scales~~ in ~~terms of Farey pairs can be reverse engineered~~ starting from ~~this classification. If N = L + s is~~ the ~~number of notes in a period of~~ the ~~MOS, then we may take~~ the ~~two preceding~~ [[http://en.wikipedia.org/wiki/~~Continued_fraction#Theorem_1~~|~~convergents~~]] ~~to L/N. These will comprise a Farey pair with the mediant equal to L/N. Calling the smaller of the pair a/b and the larger c/d~~, ~~we have that L/N < g < c/d, since L is~~ the ~~number of large steps~~. ~~The MOS will~~ be ~~proper if L/N < g <= (L+c)/(N+d)~~, ~~and improper otherwise~~.

The two systems are equivalent; in the Algorithms section you will find code for routines starting from the mediant and going to the Ls pair (the "Ls" routine) and for starting from an Ls pair and going to the mediant (the "medi" routine.) The Ls routine uses [[http://en.wikipedia.org/wiki/Modular_multiplicative_inverse|modular inverses]], whereas the medi routine uses continued fractions.

Since numbers tend to be dry, Graham Breed has proposed a [[MOSNamingScheme|naming scheme for MOS scales]].

===Blackwood R constant===

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<br />

<h2 id="toc2"><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 <~~a class="wiki_link" href="~~/~~5L%202s"~~>~~5L 2s~~</a> (~~5 large steps~~ and ~~2 small steps~~)~~. Since numbers tend to be dry, Graham Breed has proposed a~~ <a class="~~wiki_link~~" href="/~~MOSNamingScheme~~">~~naming scheme for MOS scales~~</a>.<br />

A classification of MOS scales can also be given by the number of elements of the scale - the number of large intervals &quot;L&quot; and the number of small intervals &quot;s&quot;, together with the period, assumed to be minimal (which entails that L and s are relatively prime.) E.g., the diatonic scale can be described as [5L 2s] (5 large steps and 2 small steps) or simply [5, 2] with period an octave. Alternatively, we could give a mediant for a Farey pair associated to the MOS, where this mediant is less than any generator for the MOS. In other words, we use the right hand part of the Farey pair interval, which means we must replace g with 1-g and use the complementary pair if g is in the left hand side.<br />

<br />

The two systems are equivalent; in the Algorithms section you will find code for routines starting from the mediant and going to the Ls pair (the &quot;Ls&quot; routine) and for starting from an Ls pair and going to the mediant (the &quot;medi&quot; routine.) The Ls routine uses <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Modular_multiplicative_inverse" rel="nofollow">modular inverses</a>, whereas the medi routine uses continued fractions.<br />

<br />

~~The analysis of MOS scales in terms of Farey pairs can~~ be ~~reverse engineered starting from this classification. If N = L + s is the number of notes in~~ a ~~period of the MOS, then we may take the two preceding~~ <a class="~~wiki_link_ext~~" href="~~http:~~/~~/en.wikipedia.org/wiki/Continued_fraction#Theorem_1" rel="nofollow~~">~~convergents~~</a> to L/N. These will comprise a Farey pair with the mediant equal to L/N. Calling the smaller of the pair a/b and the larger c/d, we have that L/N &lt; g &lt; c/d, since L is the number of large steps. The MOS will be proper if L/N &lt; g &lt;= (L+c)/(N+d), and improper otherwise.<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 />

<h3 id="toc3"><a name="MOS scales-Classification of MOS-Blackwood R constant"></a>Blackwood R constant</h3>

@@ 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:genewardsmith|genewardsmith]] and made on <tt>2010-05-18 03:26:47 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-18 19:29:45 UTC</tt>.<br>
-: The original revision id was <tt>142752091</tt>.<br>
+: The original revision id was <tt>143005075</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 22: / Line 22: @@
 ==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]].
+A classification of MOS scales can also be given by the number of elements of the scale - the number of large intervals "L" and the number of small intervals "s", together with the period, assumed to be minimal (which entails that L and s are relatively prime.) E.g., the diatonic scale can be described as [5L 2s] (5 large steps and 2 small steps) or simply [5, 2] with period an octave. Alternatively, we could give a mediant for a Farey pair associated to the MOS, where this mediant is less than any generator for the MOS. In other words, we use the right hand part of the Farey pair interval, which means we must replace g with 1-g and use the complementary pair if g is in the left hand side.
-The analysis of MOS scales in terms of Farey pairs can be reverse engineered starting from this classification. If N = L + s is the number of notes in a period of the MOS, then we may take the two preceding [[http://en.wikipedia.org/wiki/Continued_fraction#Theorem_1|convergents]]  to L/N. These will comprise a Farey pair with the mediant equal to L/N. Calling the smaller of the pair a/b and the larger c/d, we have that L/N &lt; g &lt; c/d, since L is the number of large steps. The MOS will be proper if L/N &lt; g &lt;= (L+c)/(N+d), and improper otherwise.
+The two systems are equivalent; in the Algorithms section you will find code for routines starting from the mediant and going to the Ls pair (the "Ls" routine) and for starting from an Ls pair and going to the mediant (the "medi" routine.) The Ls routine uses [[http://en.wikipedia.org/wiki/Modular_multiplicative_inverse|modular inverses]], whereas the medi routine uses continued fractions.
+Since numbers tend to be dry, Graham Breed has proposed a [[MOSNamingScheme|naming scheme for MOS scales]].
 ===Blackwood R constant===
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 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="MOS scales-Classification of MOS"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&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 &lt;a class="wiki_link" href="/5L%202s"&gt;5L 2s&lt;/a&gt; (5 large steps and 2 small steps). 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;
+  A classification of MOS scales can also be given by the number of elements of the scale - the number of large intervals &amp;quot;L&amp;quot; and the number of small intervals &amp;quot;s&amp;quot;, together with the period, assumed to be minimal (which entails that L and s are relatively prime.) E.g., the diatonic scale can be described as [5L 2s] (5 large steps and 2 small steps) or simply [5, 2] with period an octave. Alternatively, we could give a mediant for a Farey pair associated to the MOS, where this mediant is less than any generator for the MOS. In other words, we use the right hand part of the Farey pair interval, which means we must replace g with 1-g and use the complementary pair if g is in the left hand side.&lt;br /&gt;
+&lt;br /&gt;
+The two systems are equivalent; in the Algorithms section you will find code for routines starting from the mediant and going to the Ls pair (the &amp;quot;Ls&amp;quot; routine) and for starting from an Ls pair and going to the mediant (the &amp;quot;medi&amp;quot; routine.) The Ls routine uses &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Modular_multiplicative_inverse" rel="nofollow"&gt;modular inverses&lt;/a&gt;, whereas the medi routine uses continued fractions.&lt;br /&gt;
 &lt;br /&gt;
-The analysis of MOS scales in terms of Farey pairs can be reverse engineered starting from this classification. If N = L + s is the number of notes in a period of the MOS, then we may take the two preceding &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Continued_fraction#Theorem_1" rel="nofollow"&gt;convergents&lt;/a&gt;  to L/N. These will comprise a Farey pair with the mediant equal to L/N. Calling the smaller of the pair a/b and the larger c/d, we have that L/N &amp;lt; g &amp;lt; c/d, since L is the number of large steps. The MOS will be proper if L/N &amp;lt; g &amp;lt;= (L+c)/(N+d), and improper otherwise.&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;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="MOS scales-Classification of MOS-Blackwood R constant"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Blackwood R constant&lt;/h3&gt;