MOS scale: Difference between revisions

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==Theory of MOS==

Let us represent the period as 1. This would be the logarithm base 2 of 2 if the period is an octave, or in general we can measure intervals by the log base P when P is the period. Suppose the fractions a/b and c/d are a [[http://en.wikipedia.org/wiki/Farey_sequence#Farey_neighbours|Farey pair]], meaning that a/b < c/d and bc - ad = 1. If g = (1-t)(a/b) + t(c/d) for 0 <= t <= 1, then when t = 0, the scale generated by g will consist of an equal division of 1 (representing P) into steps of size 1/b, and when t = 1 into steps of size 1/d. In between, when t = b/(b + d), we obtain a generator equal to the [[http://en.wikipedia.org/wiki/Mediant_%28mathematics%29|mediant]] (a + c)/(b + d) and which will divide the period into b+d equal steps. For all other values a/b < g < c/d we obtain two different sizes of steps, the small steps s, and the large steps L, with the total number of steps b+d, and these scales are the MOS associated to the Farey pair.

Let us represent the period as 1. This would be the logarithm base 2 of 2 if the period is an octave, or in general we can measure intervals by the log base P when P is the period. Suppose the fractions a/b and c/d are a [[http://en.wikipedia.org/wiki/Farey_sequence#Farey_neighbours|Farey pair]], meaning that a/b < c/d and bc - ad = 1. If g = (1-t)(a/b) + t(c/d) for 0 <= t <= 1, then when t = 0, the scale generated by g will consist of an equal division of 1 (representing P) into steps of size 1/b, and when t = 1 into steps of size 1/d. In between, when t = b/(b + d), we obtain a generator equal to the [[http://en.wikipedia.org/wiki/Mediant_%28mathematics%29|mediant]] (a + c)/(b + d) and which will divide the period into b+d equal steps. For all other values a/b < g < c/d we obtain two different sizes of steps, the small steps s, and the large steps L, with the total number of steps b+d, and these scales are the MOS associated to the Farey pair. When g is between a/b and (a + c)/(b + d) there will be b large steps and d small steps, and when it is between (a + c)/(b + d) and c/d, d large steps and b small ones.

While all the scales constructed by generators g with a/b < g < c/d with the exception of the mediant which gives an equal temperament are MOS, not all the scales are [[http://en.wikipedia.org/wiki/Rothenberg_propriety|proper]] in the sense of Rothenberg. The //range of propriety// for MOS is (2a + c)/(2b + d) <= g <= (a + 2c)/(b + 2d), where MOS coming from a Farey pair (a/b, c/d) are proper when in this range, and improper when out of it. If (2a + c)/(2b + d) < g < (a + 2c)/(b + 2d), then the scales are strictly proper. Hence the diatonic scale in 12et, with generator 7/12, is proper but not strictly proper since starting from the pair (1/2, 3/5) we find the range of propriety for these seven-note MOS to be [5/9, 7/12].

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 succesive semiconvergents have the property they define a Farey pair with g contained in the pair, that is, a/b < g < c/d. Hence we may use b+d as the number of notes in a MOS, with b notes of one size and d of the other. Going back, we see we can also use b and d as sizes of scales, illustrating the general fact that denominators of semiconvergents tell us what MOS there are for a given generator.

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.

==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).

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<br />

<h2 id="toc1"><a name="MOS scales-Theory of MOS"></a>Theory of MOS</h2>

Let us represent the period as 1. This would be the logarithm base 2 of 2 if the period is an octave, or in general we can measure intervals by the log base P when P is the period. Suppose the fractions a/b and c/d are a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Farey_sequence#Farey_neighbours" rel="nofollow">Farey pair</a>, meaning that a/b &lt; c/d and bc - ad = 1. If g = (1-t)(a/b) + t(c/d) for 0 &lt;= t &lt;= 1, then when t = 0, the scale generated by g will consist of an equal division of 1 (representing P) into steps of size 1/b, and when t = 1 into steps of size 1/d. In between, when t = b/(b + d), we obtain a generator equal to the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Mediant_%28mathematics%29" rel="nofollow">mediant</a> (a + c)/(b + d) and which will divide the period into b+d equal steps. For all other values a/b &lt; g &lt; c/d we obtain two different sizes of steps, the small steps s, and the large steps L, with the total number of steps b+d, and these scales are the MOS associated to the Farey pair.<br />

Let us represent the period as 1. This would be the logarithm base 2 of 2 if the period is an octave, or in general we can measure intervals by the log base P when P is the period. Suppose the fractions a/b and c/d are a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Farey_sequence#Farey_neighbours" rel="nofollow">Farey pair</a>, meaning that a/b &lt; c/d and bc - ad = 1. If g = (1-t)(a/b) + t(c/d) for 0 &lt;= t &lt;= 1, then when t = 0, the scale generated by g will consist of an equal division of 1 (representing P) into steps of size 1/b, and when t = 1 into steps of size 1/d. In between, when t = b/(b + d), we obtain a generator equal to the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Mediant_%28mathematics%29" rel="nofollow">mediant</a> (a + c)/(b + d) and which will divide the period into b+d equal steps. For all other values a/b &lt; g &lt; c/d we obtain two different sizes of steps, the small steps s, and the large steps L, with the total number of steps b+d, and these scales are the MOS associated to the Farey pair. When g is between a/b and (a + c)/(b + d) there will be b large steps and d small steps, and when it is between (a + c)/(b + d) and c/d, d large steps and b small ones.<br />

<br />

While all the scales constructed by generators g with a/b &lt; g &lt; c/d with the exception of the mediant which gives an equal temperament are MOS, not all the scales are <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rothenberg_propriety" rel="nofollow">proper</a> in the sense of Rothenberg. The <em>range of propriety</em> for MOS is (2a + c)/(2b + d) &lt;= g &lt;= (a + 2c)/(b + 2d), where MOS coming from a Farey pair (a/b, c/d) are proper when in this range, and improper when out of it. If (2a + c)/(2b + d) &lt; g &lt; (a + 2c)/(b + 2d), then the scales are strictly proper. Hence the diatonic scale in 12et, with generator 7/12, is proper but not strictly proper since starting from the pair (1/2, 3/5) we find the range of propriety for these seven-note MOS to be [5/9, 7/12].<br />

<br />

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 succesive semiconvergents have the property they define a Farey pair with g contained in the pair, that is, a/b &lt; g &lt; c/d. Hence we may use b+d as the number of notes in a MOS, with b notes of one size and d of the other. Going back, we see we can also use b and d as sizes of scales, illustrating the general fact that denominators of semiconvergents tell us what MOS there are for a given generator.<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 />

<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 />

@@ 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>2010-05-16 18:32:33 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-16 19:30:03 UTC</tt>.<br>
-: The original revision id was <tt>142359339</tt>.<br>
+: The original revision id was <tt>142367395</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 13: / Line 13: @@
 ==Theory of MOS==
-Let us represent the period as 1. This would be the logarithm base 2 of 2 if the period is an octave, or in general we can measure intervals by the log base P when P is the period. Suppose the fractions a/b and c/d are a [[http://en.wikipedia.org/wiki/Farey_sequence#Farey_neighbours|Farey pair]], meaning that a/b &lt; c/d and bc - ad = 1. If g = (1-t)(a/b) + t(c/d) for 0 &lt;= t &lt;= 1, then when t = 0, the scale generated by g will consist of an equal division of 1 (representing P) into steps of size 1/b, and when t = 1 into steps of size 1/d. In between, when t = b/(b + d), we obtain a generator equal to the [[http://en.wikipedia.org/wiki/Mediant_%28mathematics%29|mediant]] (a + c)/(b + d) and which will divide the period into b+d equal steps. For all other values a/b &lt; g &lt; c/d we obtain two different sizes of steps, the small steps s, and the large steps L, with the total number of steps b+d, and these scales are the MOS associated to the Farey pair.
+Let us represent the period as 1. This would be the logarithm base 2 of 2 if the period is an octave, or in general we can measure intervals by the log base P when P is the period. Suppose the fractions a/b and c/d are a [[http://en.wikipedia.org/wiki/Farey_sequence#Farey_neighbours|Farey pair]], meaning that a/b &lt; c/d and bc - ad = 1. If g = (1-t)(a/b) + t(c/d) for 0 &lt;= t &lt;= 1, then when t = 0, the scale generated by g will consist of an equal division of 1 (representing P) into steps of size 1/b, and when t = 1 into steps of size 1/d. In between, when t = b/(b + d), we obtain a generator equal to the [[http://en.wikipedia.org/wiki/Mediant_%28mathematics%29|mediant]] (a + c)/(b + d) and which will divide the period into b+d equal steps. For all other values a/b &lt; g &lt; c/d we obtain two different sizes of steps, the small steps s, and the large steps L, with the total number of steps b+d, and these scales are the MOS associated to the Farey pair. When g is between a/b and (a + c)/(b + d) there will be b large steps and d small steps, and when it is between (a + c)/(b + d) and c/d, d large steps and b small ones.
 While all the scales constructed by generators g with a/b &lt; g &lt; c/d with the exception of the mediant which gives an equal temperament are MOS, not all the scales are [[http://en.wikipedia.org/wiki/Rothenberg_propriety|proper]] in the sense of Rothenberg. The //range of propriety// for MOS is (2a + c)/(2b + d) &lt;= g &lt;= (a + 2c)/(b + 2d), where MOS coming from a Farey pair (a/b, c/d) are proper when in this range, and improper when out of it. If (2a + c)/(2b + d) &lt; g &lt; (a + 2c)/(b + 2d), then the scales are strictly proper. Hence the diatonic scale in 12et, with generator 7/12, is proper but not strictly proper since starting from the pair (1/2, 3/5) we find the range of propriety for these seven-note MOS to be [5/9, 7/12].
+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 succesive semiconvergents have the property they define a Farey pair with g contained in the pair, that is, a/b &lt; g &lt; c/d. Hence we may use b+d as the number of notes in a MOS, with b notes of one size and d of the other. Going back, we see we can also use b and d as sizes of scales, illustrating the general fact that denominators of semiconvergents tell us what MOS there are for a given generator.
+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.
 ==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).
@@ Line 52: / Line 57: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="MOS scales-Theory of MOS"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Theory of MOS&lt;/h2&gt;
-Let us represent the period as 1. This would be the logarithm base 2 of 2 if the period is an octave, or in general we can measure intervals by the log base P when P is the period. Suppose the fractions a/b and c/d are a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Farey_sequence#Farey_neighbours" rel="nofollow"&gt;Farey pair&lt;/a&gt;, meaning that a/b &amp;lt; c/d and bc - ad = 1. If g = (1-t)(a/b) + t(c/d) for 0 &amp;lt;= t &amp;lt;= 1, then when t = 0, the scale generated by g will consist of an equal division of 1 (representing P) into steps of size 1/b, and when t = 1 into steps of size 1/d. In between, when t = b/(b + d), we obtain a generator equal to the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Mediant_%28mathematics%29" rel="nofollow"&gt;mediant&lt;/a&gt; (a + c)/(b + d) and which will divide the period into b+d equal steps. For all other values a/b &amp;lt; g &amp;lt; c/d we obtain two different sizes of steps, the small steps s, and the large steps L, with the total number of steps b+d, and these scales are the MOS associated to the Farey pair.&lt;br /&gt;
+Let us represent the period as 1. This would be the logarithm base 2 of 2 if the period is an octave, or in general we can measure intervals by the log base P when P is the period. Suppose the fractions a/b and c/d are a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Farey_sequence#Farey_neighbours" rel="nofollow"&gt;Farey pair&lt;/a&gt;, meaning that a/b &amp;lt; c/d and bc - ad = 1. If g = (1-t)(a/b) + t(c/d) for 0 &amp;lt;= t &amp;lt;= 1, then when t = 0, the scale generated by g will consist of an equal division of 1 (representing P) into steps of size 1/b, and when t = 1 into steps of size 1/d. In between, when t = b/(b + d), we obtain a generator equal to the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Mediant_%28mathematics%29" rel="nofollow"&gt;mediant&lt;/a&gt; (a + c)/(b + d) and which will divide the period into b+d equal steps. For all other values a/b &amp;lt; g &amp;lt; c/d we obtain two different sizes of steps, the small steps s, and the large steps L, with the total number of steps b+d, and these scales are the MOS associated to the Farey pair. When g is between a/b and (a + c)/(b + d) there will be b large steps and d small steps, and when it is between (a + c)/(b + d) and c/d, d large steps and b small ones.&lt;br /&gt;
 &lt;br /&gt;
 While all the scales constructed by generators g with a/b &amp;lt; g &amp;lt; c/d with the exception of the mediant which gives an equal temperament are MOS, not all the scales are &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rothenberg_propriety" rel="nofollow"&gt;proper&lt;/a&gt; in the sense of Rothenberg. The &lt;em&gt;range of propriety&lt;/em&gt; for MOS is (2a + c)/(2b + d) &amp;lt;= g &amp;lt;= (a + 2c)/(b + 2d), where MOS coming from a Farey pair (a/b, c/d) are proper when in this range, and improper when out of it. If (2a + c)/(2b + d) &amp;lt; g &amp;lt; (a + 2c)/(b + 2d), then the scales are strictly proper. Hence the diatonic scale in 12et, with generator 7/12, is proper but not strictly proper since starting from the pair (1/2, 3/5) we find the range of propriety for these seven-note MOS to be [5/9, 7/12].&lt;br /&gt;
+&lt;br /&gt;
+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 succesive semiconvergents have the property they define a Farey pair with g contained in the pair, that is, a/b &amp;lt; g &amp;lt; c/d. Hence we may use b+d as the number of notes in a MOS, with b notes of one size and d of the other. Going back, we see we can also use b and d as sizes of scales, illustrating the general fact that denominators of semiconvergents tell us what MOS there are for a given generator.&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;
+&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;