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 01:29:42 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-17 02:49:27 UTC</tt>.<br>

: The original revision id was <tt>~~142429537~~</tt>.<br>

: The original revision id was <tt>142436667</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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Given a generator g, we can find MOS for g with period 1 by means of the [[http://en.wikipedia.org/wiki/Continued_fraction#Semiconvergents|semiconvergents]] to g. A pair of successive semiconvergents have the property that they define a Farey pair, and when g is contained in the pair, that is, a/b < g < c/d, we have defined a MOS for g with b+d as the number of notes in the MOS, with b notes of one size and d of the other.

For example, suppose we want MOS for 1/4-comma meantone. The generator will then be log2(5)/4, which has semiconvergents 1/2, 2/3, 3/5, 4/7, 7/12, 11/19, 18/31, 29/50, 47/81, 65/112... If we settle on 31 as a good size for our MOS, we see it is the mediant between the Farey pair 11/19 and 7/12, for which the range of strict propriety is 29/50 < x < 25/43. Since g is in that range, we will get a strictly proper MOS.

For example, suppose we want MOS for 1/4-comma meantone. The generator will then be log2(5)/4, which has semiconvergents 1/2, 2/3, 3/5, 4/7, 7/12, 11/19, 18/31, 29/50, 47/81, 65/112... If we settle on 31 as a good size for our MOS, we see 18/31 is the mediant between the Farey pair 11/19 and 7/12, for which the range of strict propriety is 29/50 < x < 25/43. Since g is in that range and not equal to 18/31, we will get a strictly proper MOS.

==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).

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]].

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.

==MOS in equal temperaments==

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Given a generator g, we can find MOS for g with period 1 by means of the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Continued_fraction#Semiconvergents" rel="nofollow">semiconvergents</a> to g. A pair of successive semiconvergents have the property that they define a Farey pair, and when g is contained in the pair, that is, a/b &lt; g &lt; c/d, we have defined a MOS for g with b+d as the number of notes in the MOS, with b notes of one size and d of the other.<br />

<br />

For example, suppose we want MOS for 1/4-comma meantone. The generator will then be log2(5)/4, which has semiconvergents 1/2, 2/3, 3/5, 4/7, 7/12, 11/19, 18/31, 29/50, 47/81, 65/112... If we settle on 31 as a good size for our MOS, we see it is the mediant between the Farey pair 11/19 and 7/12, for which the range of strict propriety is 29/50 &lt; x &lt; 25/43. Since g is in that range, we will get a strictly proper MOS. <br />

For example, suppose we want MOS for 1/4-comma meantone. The generator will then be log2(5)/4, which has semiconvergents 1/2, 2/3, 3/5, 4/7, 7/12, 11/19, 18/31, 29/50, 47/81, 65/112... If we settle on 31 as a good size for our MOS, we see 18/31 is the mediant between the Farey pair 11/19 and 7/12, for which the range of strict propriety is 29/50 &lt; x &lt; 25/43. Since g is in that range and not equal to 18/31, we will get a strictly proper MOS. <br />

<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).~~<br />~~

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 />

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 />

<br />

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

@@ 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 01:29:42 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-17 02:49:27 UTC</tt>.<br>
-: The original revision id was <tt>142429537</tt>.<br>
+: The original revision id was <tt>142436667</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 19: / Line 19: @@
 Given a generator g, we can find MOS for g with period 1 by means of the [[http://en.wikipedia.org/wiki/Continued_fraction#Semiconvergents|semiconvergents]] to g. A pair of successive semiconvergents have the property that they define a Farey pair, and when g is contained in the pair, that is, a/b &lt; g &lt; c/d, we have defined a MOS for g with b+d as the number of notes in the MOS, with b notes of one size and d of the other.
-For example, suppose we want MOS for 1/4-comma meantone. The generator will then be log2(5)/4, which has semiconvergents 1/2, 2/3, 3/5, 4/7, 7/12, 11/19, 18/31, 29/50, 47/81, 65/112... If we settle on 31 as a good size for our MOS, we see it is the mediant between the Farey pair 11/19  and 7/12, for which the range of strict propriety is 29/50 &lt; x &lt; 25/43. Since g is in that range, we will get a strictly proper MOS.
+For example, suppose we want MOS for 1/4-comma meantone. The generator will then be log2(5)/4, which has semiconvergents 1/2, 2/3, 3/5, 4/7, 7/12, 11/19, 18/31, 29/50, 47/81, 65/112... If we settle on 31 as a good size for our MOS, we see 18/31 is the mediant between the Farey pair 11/19  and 7/12, for which the range of strict propriety is 29/50 &lt; x &lt; 25/43. Since g is in that range and not equal to 18/31, we will get a strictly proper MOS.
 ==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).
+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]].
-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.
 ==MOS in equal temperaments==
@@ Line 63: / Line 64: @@
 Given a generator g, we can find MOS for g with period 1 by means of the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Continued_fraction#Semiconvergents" rel="nofollow"&gt;semiconvergents&lt;/a&gt; to g. A pair of successive semiconvergents have the property that they define a Farey pair, and when g is contained in the pair, that is, a/b &amp;lt; g &amp;lt; c/d, we have defined a MOS for g with b+d as the number of notes in the MOS, with b notes of one size and d of the other.&lt;br /&gt;
 &lt;br /&gt;
-For example, suppose we want MOS for 1/4-comma meantone. The generator will then be log2(5)/4, which has semiconvergents 1/2, 2/3, 3/5, 4/7, 7/12, 11/19, 18/31, 29/50, 47/81, 65/112... If we settle on 31 as a good size for our MOS, we see it is the mediant between the Farey pair 11/19  and 7/12, for which the range of strict propriety is 29/50 &amp;lt; x &amp;lt; 25/43. Since g is in that range, we will get a strictly proper MOS. &lt;br /&gt;
+For example, suppose we want MOS for 1/4-comma meantone. The generator will then be log2(5)/4, which has semiconvergents 1/2, 2/3, 3/5, 4/7, 7/12, 11/19, 18/31, 29/50, 47/81, 65/112... If we settle on 31 as a good size for our MOS, we see 18/31 is the mediant between the Farey pair 11/19  and 7/12, for which the range of strict propriety is 29/50 &amp;lt; x &amp;lt; 25/43. Since g is in that range and not equal to 18/31, we will get a strictly proper MOS. &lt;br /&gt;
 &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).&lt;br /&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;
-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;
 &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;