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>2010-05-17 02:49:27 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-17 06:34:11 UTC</tt>.<br>

: The original revision id was <tt>~~142436667~~</tt>.<br>

: The original revision id was <tt>142461149</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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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]].

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://~~Some useful theorems~~|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 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.

===Blackwood R constant===

In the context of the "recognizable diatonic" scales deriving from the Farey pair (1/2, 3/5) [[http://en.wikipedia.org/wiki/Easley_Blackwood,_Jr.|Easley Blackwood Jr.]] defined a characterizing constant R which we may generalize to any MOS as follows. If a/b < g < c/d is a generator with the given Farey pair, take the ratio of relative errors R = (bg - a)/(c - dg). Since this is a ratio of positive numbers, it is positive. As g tends towards a/b it tends to zero, and as g goes to c/d R goes to infinity. When g equals (a + c)/(b + d) it takes the value 1, and the range of propriety is 1/2 <= R <= 2.

When R is less than 1, it represents the ratio in (logarithmic) size between the smaller and the larger step. When it is greater than 1, it is larger/smaller. By replacing g with 1 - g if necessary, we can reduce always to the case where R>1 (or R<1 if we prefer.)

==MOS in equal temperaments==

In the special case of an equal temperament, more concrete things about MOS can be stated.

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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 />

<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://~~Some~~" rel="nofollow">~~http://Some~~</a>~~ useful theorems|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.<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 />

<br />

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

In the context of the &quot;recognizable diatonic&quot; scales deriving from the Farey pair (1/2, 3/5) <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Easley_Blackwood,_Jr." rel="nofollow">Easley Blackwood Jr.</a> defined a characterizing constant R which we may generalize to any MOS as follows. If a/b &lt; g &lt; c/d is a generator with the given Farey pair, take the ratio of relative errors R = (bg - a)/(c - dg). Since this is a ratio of positive numbers, it is positive. As g tends towards a/b it tends to zero, and as g goes to c/d R goes to infinity. When g equals (a + c)/(b + d) it takes the value 1, and the range of propriety is 1/2 &lt;= R &lt;= 2.<br />

<br />

<h2 id="~~toc3~~"><a name="MOS scales-MOS in equal temperaments"></a>MOS in equal temperaments</h2>

When R is less than 1, it represents the ratio in (logarithmic) size between the smaller and the larger step. When it is greater than 1, it is larger/smaller. By replacing g with 1 - g if necessary, we can reduce always to the case where R&gt;1 (or R&lt;1 if we prefer.)<br />

<h2 id="toc4"><a name="MOS scales-MOS in equal temperaments"></a>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 />

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<h1 id="~~toc4~~"><a name="MOS As Applied To Rhythms"></a>MOS As Applied To Rhythms</h1>

<h1 id="toc5"><a name="MOS As Applied To Rhythms"></a>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><br />

<br />

<h1 id="~~toc5~~"><a name="MOSDiagrams"></a><a class="wiki_link" href="/MOSDiagrams">MOSDiagrams</a></h1>

<h1 id="toc6"><a name="MOSDiagrams"></a><a class="wiki_link" href="/MOSDiagrams">MOSDiagrams</a></h1>

</body></html></pre></div>

@@ 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>2010-05-17 02:49:27 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-17 06:34:11 UTC</tt>.<br>
-: The original revision id was <tt>142436667</tt>.<br>
+: The original revision id was <tt>142461149</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 24: / Line 24: @@
 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]].
-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://Some useful theorems|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 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.
+===Blackwood R constant===
+In the context of the "recognizable diatonic" scales deriving from the Farey pair (1/2, 3/5) [[http://en.wikipedia.org/wiki/Easley_Blackwood,_Jr.|Easley Blackwood Jr.]] defined a characterizing constant R which we may generalize to any MOS as follows. If a/b &lt; g &lt; c/d is a generator with the given Farey pair, take the ratio of relative errors R = (bg - a)/(c - dg). Since this is a ratio of positive numbers, it is positive. As g tends towards a/b it tends to zero, and as g goes to c/d R goes to infinity. When g equals (a + c)/(b + d) it takes the value 1, and the range of propriety is 1/2 &lt;= R &lt;= 2.
+When R is less than 1, it represents the ratio in (logarithmic) size between the smaller and the larger step. When it is greater than 1, it is larger/smaller. By replacing g with 1 - g if necessary, we can reduce always to the case where R&gt;1 (or R&lt;1 if we prefer.)
 ==MOS in equal temperaments==
 In the special case of an equal temperament, more concrete things about MOS can be stated.
@@ Line 69: / Line 73: @@
   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;
 &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;!-- ws:start:WikiTextUrlRule:441:http://Some --&gt;&lt;a class="wiki_link_ext" href="http://Some" rel="nofollow"&gt;http://Some&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:441 --&gt; useful theorems|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 &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;
+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;
+&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;
+In the context of the &amp;quot;recognizable diatonic&amp;quot; scales deriving from the Farey pair (1/2, 3/5) &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Easley_Blackwood,_Jr." rel="nofollow"&gt;Easley Blackwood Jr.&lt;/a&gt; defined a characterizing constant R which we may generalize to any MOS as follows. If a/b &amp;lt; g &amp;lt; c/d is a generator with the given Farey pair, take the ratio of relative errors R = (bg - a)/(c - dg). Since this is a ratio of positive numbers, it is positive. As g tends towards a/b it tends to zero, and as g goes to c/d R goes to infinity. When g equals (a + c)/(b + d) it takes the value 1, and the range of propriety is 1/2 &amp;lt;= R &amp;lt;= 2.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="MOS scales-MOS in equal temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;MOS in equal temperaments&lt;/h2&gt;
+When R is less than 1, it represents the ratio in (logarithmic) size between the smaller and the larger step. When it is greater than 1, it is larger/smaller. By replacing g with 1 - g if necessary, we can reduce always to the case where R&amp;gt;1 (or R&amp;lt;1 if we prefer.)&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="MOS scales-MOS in equal temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;MOS in equal temperaments&lt;/h2&gt;
   In the special case of an equal temperament, more concrete things about MOS can be stated.&lt;br /&gt;
 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.)&lt;br /&gt;
@@ Line 317: / Line 325: @@
 &lt;span style="color: #0000ee;"&gt; &lt;/span&gt;&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="MOS As Applied To Rhythms"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;MOS As Applied To Rhythms&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="MOS As Applied To Rhythms"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;MOS As Applied To Rhythms&lt;/h1&gt;
   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 &lt;a class="wiki_link_ext" href="http://anaphoria.com/hora.PDF" rel="nofollow"&gt;http://anaphoria.com/hora.PDF&lt;/a&gt; and &lt;a class="wiki_link_ext" href="http://anaphoria.com/horo2.PDF" rel="nofollow"&gt;http://anaphoria.com/horo2.PDF&lt;/a&gt;&lt;br /&gt;
 MOS structures and thinking can be applied to the design of rhythms as well. See &lt;a class="wiki_link" href="/MOS%20Rhythm%20Tutorial"&gt;MOS Rhythm Tutorial&lt;/a&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="MOSDiagrams"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;&lt;a class="wiki_link" href="/MOSDiagrams"&gt;MOSDiagrams&lt;/a&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="MOSDiagrams"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;&lt;a class="wiki_link" href="/MOSDiagrams"&gt;MOSDiagrams&lt;/a&gt;&lt;/h1&gt;
 &lt;/body&gt;&lt;/html&gt;</pre></div>