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>2010-05-20 02:39:36 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-20 05:28:45 UTC</tt>.<br>

: The original revision id was <tt>~~143392517~~</tt>.<br>

: The original revision id was <tt>143411365</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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Yet another way of classifying MOS is via [[http://en.wikipedia.org/wiki/Minkowski%27s_question_mark_function|Minkowski's ? function]]. Here ?(x) is a continuous increasing function from the real numbers to the real numbers which has some peculiar properties, one being that it sends rational numbers to [[http://en.wikipedia.org/wiki/Dyadic_rational|dyadic rationals]]. Hence if q is a rational number 0 < q < 1 in use in the mediant system of classifying MOS, r = ?(q) = A/2^n will be a dyadic rational number which can also be used. Note that the ? function is invertible, and it and its inverse function, the Box function, have code given for them in the algorithms section at the bottom of the article.

The integer n in the denominator of r (with A assumed to be odd) is the order (or n+1 is, according to some sources) of q in the [[http://en.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree|Stern-Brocot tree]]. The two neighboring numbers of order n+1, which ~~will~~ be Box(r - 2^(-n-1) and Box(r + 2^(-n-1)) ~~can also be expressed in terms of the ? and Box functions~~. If r represents a MOS, the range of possible values for a generator of the MOS will be Box(r) < g < Box(r + 2^(-n)). So, for example, the MOS denoted by 3/2048 will be between Box(3/2048) and Box(4/2048), which means that 2/11 < g < 1/10.

The integer n in the denominator of r (with A assumed to be odd) is the order (or n+1 is, according to some sources) of q in the [[http://en.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree|Stern-Brocot tree]]. The two neighboring numbers of order n+1, which define the range of propriety, can also be expressed in terms of the ? and Box functions as Box(r - 2^(-n-1) and Box(r + 2^(-n-1)). If r represents a MOS, the range of possible values for a generator of the MOS will be Box(r) < g < Box(r + 2^(-n)), and the proper generators will be Box(r) < g < Box(r + 2^(-n-1)). So, for example, the MOS denoted by 3/2048 will be between Box(3/2048) and Box(4/2048), which means that 2/21 < g < 1/10, and will be proper if 2/21 < g < 3/31. Hence 7/72, a generator for miracle temperament, will define a MOS but it will not be proper since 7/72 > 3/31 = Box(3/2048 + 1/4096)).

===Names for MOS===

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Yet another way of classifying MOS is via <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Minkowski%27s_question_mark_function" rel="nofollow">Minkowski's ? function</a>. Here ?(x) is a continuous increasing function from the real numbers to the real numbers which has some peculiar properties, one being that it sends rational numbers to <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Dyadic_rational" rel="nofollow">dyadic rationals</a>. Hence if q is a rational number 0 &lt; q &lt; 1 in use in the mediant system of classifying MOS, r = ?(q) = A/2^n will be a dyadic rational number which can also be used. Note that the ? function is invertible, and it and its inverse function, the Box function, have code given for them in the algorithms section at the bottom of the article.<br />

<br />

The integer n in the denominator of r (with A assumed to be odd) is the order (or n+1 is, according to some sources) of q in the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree" rel="nofollow">Stern-Brocot tree</a>. The two neighboring numbers of order n+1, which ~~will~~ be Box(r - 2^(-n-1) and Box(r + 2^(-n-1)) ~~can also be expressed in terms of the ? and Box functions~~. If r represents a MOS, the range of possible values for a generator of the MOS will be Box(r) &lt; g &lt; Box(r + 2^(-n)). So, for example, the MOS denoted by 3/2048 will be between Box(3/2048) and Box(4/2048), which means that 2/11 &lt; g &lt; 1/10.<br />

The integer n in the denominator of r (with A assumed to be odd) is the order (or n+1 is, according to some sources) of q in the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree" rel="nofollow">Stern-Brocot tree</a>. The two neighboring numbers of order n+1, which define the range of propriety, can also be expressed in terms of the ? and Box functions as Box(r - 2^(-n-1) and Box(r + 2^(-n-1)). If r represents a MOS, the range of possible values for a generator of the MOS will be Box(r) &lt; g &lt; Box(r + 2^(-n)), and the proper generators will be Box(r) &lt; g &lt; Box(r + 2^(-n-1)). So, for example, the MOS denoted by 3/2048 will be between Box(3/2048) and Box(4/2048), which means that 2/21 &lt; g &lt; 1/10, and will be proper if 2/21 &lt; g &lt; 3/31. Hence 7/72, a generator for miracle temperament, will define a MOS but it will not be proper since 7/72 &gt; 3/31 = Box(3/2048 + 1/4096)).<br />

<br />

<h3 id="toc4"><a name="MOS scales-Classification of MOS-Names for MOS"></a>Names for MOS</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>2010-05-20 02:39:36 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-20 05:28:45 UTC</tt>.<br>
-: The original revision id was <tt>143392517</tt>.<br>
+: The original revision id was <tt>143411365</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 31: / Line 31: @@
 Yet another way of classifying MOS is via [[http://en.wikipedia.org/wiki/Minkowski%27s_question_mark_function|Minkowski's ? function]]. Here ?(x) is a continuous increasing function from the real numbers to the real numbers which has some peculiar properties, one being that it sends rational numbers to [[http://en.wikipedia.org/wiki/Dyadic_rational|dyadic rationals]]. Hence if q is a rational number 0 &lt; q &lt; 1 in use in the mediant system of classifying MOS, r = ?(q) = A/2^n will be a dyadic rational number which can also be used. Note that the ? function is invertible, and it and its inverse function, the Box function, have code given for them in the algorithms section at the bottom of the article.
-The integer n in the denominator of r (with A assumed to be odd) is the order (or n+1 is, according to some sources) of q in the [[http://en.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree|Stern-Brocot tree]]. The two neighboring numbers of order n+1, which will be Box(r - 2^(-n-1) and Box(r + 2^(-n-1)) can also be expressed in terms of the ? and Box functions. If r represents a MOS, the range of possible values for a generator of the MOS will be Box(r) &lt; g &lt; Box(r + 2^(-n)). So, for example, the MOS denoted by 3/2048 will be between Box(3/2048) and Box(4/2048), which means that 2/11 &lt; g &lt; 1/10.
+The integer n in the denominator of r (with A assumed to be odd) is the order (or n+1 is, according to some sources) of q in the [[http://en.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree|Stern-Brocot tree]]. The two neighboring numbers of order n+1, which define the range of propriety, can also be expressed in terms of the ? and Box functions as Box(r - 2^(-n-1) and Box(r + 2^(-n-1)). If r represents a MOS, the range of possible values for a generator of the MOS will be Box(r) &lt; g &lt; Box(r + 2^(-n)), and the proper generators will be Box(r) &lt; g &lt; Box(r + 2^(-n-1)). So, for example, the MOS denoted by 3/2048 will be between Box(3/2048) and Box(4/2048), which means that 2/21 &lt; g &lt; 1/10, and will be proper if 2/21 &lt; g &lt; 3/31. Hence 7/72, a generator for miracle temperament, will define a MOS but it will not be proper since 7/72 &gt; 3/31 = Box(3/2048 + 1/4096)).
 ===Names for MOS===
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 Yet another way of classifying MOS is via &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Minkowski%27s_question_mark_function" rel="nofollow"&gt;Minkowski's ? function&lt;/a&gt;. Here ?(x) is a continuous increasing function from the real numbers to the real numbers which has some peculiar properties, one being that it sends rational numbers to &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Dyadic_rational" rel="nofollow"&gt;dyadic rationals&lt;/a&gt;. Hence if q is a rational number 0 &amp;lt; q &amp;lt; 1 in use in the mediant system of classifying MOS, r = ?(q) = A/2^n will be a dyadic rational number which can also be used. Note that the ? function is invertible, and it and its inverse function, the Box function, have code given for them in the algorithms section at the bottom of the article.&lt;br /&gt;
 &lt;br /&gt;
-The integer n in the denominator of r (with A assumed to be odd) is the order (or n+1 is, according to some sources) of q in the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree" rel="nofollow"&gt;Stern-Brocot tree&lt;/a&gt;. The two neighboring numbers of order n+1, which will be Box(r - 2^(-n-1) and Box(r + 2^(-n-1)) can also be expressed in terms of the ? and Box functions. If r represents a MOS, the range of possible values for a generator of the MOS will be Box(r) &amp;lt; g &amp;lt; Box(r + 2^(-n)). So, for example, the MOS denoted by 3/2048 will be between Box(3/2048) and Box(4/2048), which means that 2/11 &amp;lt; g &amp;lt; 1/10.&lt;br /&gt;
+The integer n in the denominator of r (with A assumed to be odd) is the order (or n+1 is, according to some sources) of q in the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree" rel="nofollow"&gt;Stern-Brocot tree&lt;/a&gt;. The two neighboring numbers of order n+1, which define the range of propriety, can also be expressed in terms of the ? and Box functions as Box(r - 2^(-n-1) and Box(r + 2^(-n-1)). If r represents a MOS, the range of possible values for a generator of the MOS will be Box(r) &amp;lt; g &amp;lt; Box(r + 2^(-n)), and the proper generators will be Box(r) &amp;lt; g &amp;lt; Box(r + 2^(-n-1)). So, for example, the MOS denoted by 3/2048 will be between Box(3/2048) and Box(4/2048), which means that 2/21 &amp;lt; g &amp;lt; 1/10, and will be proper if 2/21 &amp;lt; g &amp;lt; 3/31. Hence 7/72, a generator for miracle temperament, will define a MOS but it will not be proper since 7/72 &amp;gt; 3/31 = Box(3/2048 + 1/4096)).&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc4"&gt;&lt;a name="MOS scales-Classification of MOS-Names for MOS"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Names for MOS&lt;/h3&gt;