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:

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-~~16 19~~:30:03 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-17 01:29:42 UTC</tt>.

: The original revision id was <tt>~~142367395~~</tt>.

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

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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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~~.

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.

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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 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 <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~~.

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.

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.

@@ 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 19:30:03 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-17 01:29:42 UTC</tt>.<br>
-: The original revision id was <tt>142367395</tt>.<br>
+: The original revision id was <tt>142429537</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 17: / Line 17: @@
 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.
+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.
@@ Line 61: / Line 61: @@
 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;
+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;