Generator ranges of MOS: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-05-~~11 23~~:49:44 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-05-12 01:14:29 UTC</tt>.<br>

: The original revision id was <tt>~~333901600~~</tt>.<br>

: The original revision id was <tt>333922658</tt>.<br>

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

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<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">Below are ranges of generators for various L-s patterns of MOS, with the number of steps in the scale from 2 to 22. The ranges are given in fractions of the interval of equivalence, which is normally an octave. The tables give the range of possible generators in the second column, normalized so that the lower end of the range is where L/s = 1 (Nedo.) The third column gives the range of propriety, where the proper MOS reside.

Suppose we have a scale of N steps to the interval of repetition, with Q steps to a period, so that there are P = N/Q periods to the repetition interval (usually, an octave.) If the generator range in question is u < g < v, then we may multiply by P to get C/Q < Pg < a/b, where C is the number of the [[Interval class|generic interval]] to which the generator g belongs, and both C/Q and a/b are reduced to lowest terms. We have normalized so that C/N < g, and hence C/Q < Pg, with C/N being the lower end of the range of possible generators, where L=s. If the range is C/Q < Pg < a/b, then when Pg = a/b, s has decreased to zero with increasing g, so s = (a - bg)/P. If ~~g = C/N so that L=s~~, then ~~L = a - Cb/N = (Na - Cb)/N~~ = 1~~/N, as expected, since C/Q = PC/N and a/b are adjacent ratios in the [[http://en.wikipedia.org/wiki/Farey_sequence|Farey sequence]]. The adjacent ratio on the other side is (C-a)/(Q-b)~~, so ~~that C/Q is the [[http://en.wikipedia.org/wiki/Mediant_(mathematics)|mediant]] of the two ratios. On that side, which brings us outside of the possible range since now~~ L~~<s, as Pg decreases to~~ (C-a)/~~(Q-b), L decreases to zero, so L~~ = ~~(Q-b)g~~/P ~~- (C~~-a).

Suppose we have a scale of N steps to the interval of repetition, with Q steps to a period, so that there are P = N/Q periods to the repetition interval (usually, an octave.) If the generator range in question is u < g < v, then we may multiply by P to get C/Q < Pg < a/b, where C is the number of the [[Interval class|generic interval]] to which the generator g belongs, and both C/Q and a/b are reduced to lowest terms. We have normalized so that C/N < g, and hence C/Q < Pg, with C/N being the lower end of the range of possible generators, where L=s. If the range is C/Q < Pg < a/b, then when Pg = a/b, s has decreased to zero with increasing g, so s = (a - bg)/P. If there are e large steps and f small steps, then eL + fs = 1, so L = (1 - fs)/e = 1/P - a + bg.

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<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Generator ranges of MOS</title></head><body>Below are ranges of generators for various L-s patterns of MOS, with the number of steps in the scale from 2 to 22. The ranges are given in fractions of the interval of equivalence, which is normally an octave. The tables give the range of possible generators in the second column, normalized so that the lower end of the range is where L/s = 1 (Nedo.) The third column gives the range of propriety, where the proper MOS reside.<br />

<br />

Suppose we have a scale of N steps to the interval of repetition, with Q steps to a period, so that there are P = N/Q periods to the repetition interval (usually, an octave.) If the generator range in question is u &lt; g &lt; v, then we may multiply by P to get C/Q &lt; Pg &lt; a/b, where C is the number of the <a class="wiki_link" href="/Interval%20class">generic interval</a> to which the generator g belongs, and both C/Q and a/b are reduced to lowest terms. We have normalized so that C/N &lt; g, and hence C/Q &lt; Pg, with C/N being the lower end of the range of possible generators, where L=s. If the range is C/Q &lt; Pg &lt; a/b, then when Pg = a/b, s has decreased to zero with increasing g, so s = (a - bg)/P. If ~~g = C/N so that L=s~~, then ~~L = a - Cb/N = (Na - Cb)/N~~ = 1/N, as expected, since C/Q = PC/N and a/b are adjacent ratios in the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Farey_sequence" rel="nofollow">Farey sequence</a>. The adjacent ratio on the other side is (C-a)/(Q-b), so ~~that C/Q is the <a class="wiki_link_ext" href~~=~~"http://en.wikipedia.org/wiki/Mediant_~~(~~mathematics)" rel="nofollow">mediant</a> of the two ratios. On that side, which brings us outside of the possible range since now L&lt;s, as Pg decreases to (C~~-a)/~~(Q-b), L decreases to zero, so L~~ = ~~(Q-b)g~~/P ~~- (C~~-a).<br />

Suppose we have a scale of N steps to the interval of repetition, with Q steps to a period, so that there are P = N/Q periods to the repetition interval (usually, an octave.) If the generator range in question is u &lt; g &lt; v, then we may multiply by P to get C/Q &lt; Pg &lt; a/b, where C is the number of the <a class="wiki_link" href="/Interval%20class">generic interval</a> to which the generator g belongs, and both C/Q and a/b are reduced to lowest terms. We have normalized so that C/N &lt; g, and hence C/Q &lt; Pg, with C/N being the lower end of the range of possible generators, where L=s. If the range is C/Q &lt; Pg &lt; a/b, then when Pg = a/b, s has decreased to zero with increasing g, so s = (a - bg)/P. If there are e large steps and f small steps, then eL + fs = 1, so L = (1 - fs)/e = 1/P - a + bg.<br />

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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>2012-05-11 23:49:44 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-05-12 01:14:29 UTC</tt>.<br>
-: The original revision id was <tt>333901600</tt>.<br>
+: The original revision id was <tt>333922658</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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 <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">Below are ranges of generators for various L-s patterns of MOS, with the number of steps in the scale from 2 to 22. The ranges are given in fractions of the interval of equivalence, which is normally an octave. The tables give the range of possible generators in the second column, normalized so that the lower end of the range is where L/s = 1 (Nedo.) The third column gives the range of propriety, where the proper MOS reside.
-Suppose we have a scale of N steps to the interval of repetition, with Q steps to a period, so that there are P = N/Q periods to the repetition interval (usually, an octave.) If the generator range in question is u &lt; g &lt; v, then we may multiply by P to get C/Q &lt; Pg &lt; a/b, where C is the number of the [[Interval class|generic interval]] to which the generator g belongs, and both C/Q and a/b are reduced to lowest terms. We have normalized so that C/N &lt; g, and hence C/Q &lt; Pg, with C/N being the lower end of the range of possible generators, where L=s. If the range is C/Q &lt; Pg &lt; a/b, then when Pg = a/b, s has decreased to zero with increasing g, so s = (a - bg)/P. If g = C/N so that L=s, then L = a - Cb/N = (Na - Cb)/N = 1/N, as expected, since C/Q = PC/N and a/b are adjacent ratios in the [[http://en.wikipedia.org/wiki/Farey_sequence|Farey sequence]]. The adjacent ratio on the other side is (C-a)/(Q-b), so that C/Q is the [[http://en.wikipedia.org/wiki/Mediant_(mathematics)|mediant]] of the two ratios. On that side, which brings us outside of the possible range since now L&lt;s, as Pg decreases to (C-a)/(Q-b), L decreases to zero, so L = (Q-b)g/P - (C-a).
+Suppose we have a scale of N steps to the interval of repetition, with Q steps to a period, so that there are P = N/Q periods to the repetition interval (usually, an octave.) If the generator range in question is u &lt; g &lt; v, then we may multiply by P to get C/Q &lt; Pg &lt; a/b, where C is the number of the [[Interval class|generic interval]] to which the generator g belongs, and both C/Q and a/b are reduced to lowest terms. We have normalized so that C/N &lt; g, and hence C/Q &lt; Pg, with C/N being the lower end of the range of possible generators, where L=s. If the range is C/Q &lt; Pg &lt; a/b, then when Pg = a/b, s has decreased to zero with increasing g, so s = (a - bg)/P. If there are e large steps and f small steps, then eL + fs = 1, so L = (1 - fs)/e = 1/P - a + bg.
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 <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Generator ranges of MOS&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Below are ranges of generators for various L-s patterns of MOS, with the number of steps in the scale from 2 to 22. The ranges are given in fractions of the interval of equivalence, which is normally an octave. The tables give the range of possible generators in the second column, normalized so that the lower end of the range is where L/s = 1 (Nedo.) The third column gives the range of propriety, where the proper MOS reside.&lt;br /&gt;
 &lt;br /&gt;
-Suppose we have a scale of N steps to the interval of repetition, with Q steps to a period, so that there are P = N/Q periods to the repetition interval (usually, an octave.) If the generator range in question is u &amp;lt; g &amp;lt; v, then we may multiply by P to get C/Q &amp;lt; Pg &amp;lt; a/b, where C is the number of the &lt;a class="wiki_link" href="/Interval%20class"&gt;generic interval&lt;/a&gt; to which the generator g belongs, and both C/Q and a/b are reduced to lowest terms. We have normalized so that C/N &amp;lt; g, and hence C/Q &amp;lt; Pg, with C/N being the lower end of the range of possible generators, where L=s. If the range is C/Q &amp;lt; Pg &amp;lt; a/b, then when Pg = a/b, s has decreased to zero with increasing g, so s = (a - bg)/P. If g = C/N so that L=s, then L = a - Cb/N = (Na - Cb)/N = 1/N, as expected, since C/Q = PC/N and a/b are adjacent ratios in the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Farey_sequence" rel="nofollow"&gt;Farey sequence&lt;/a&gt;. The adjacent ratio on the other side is (C-a)/(Q-b), so that C/Q is the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Mediant_(mathematics)" rel="nofollow"&gt;mediant&lt;/a&gt; of the two ratios. On that side, which brings us outside of the possible range since now L&amp;lt;s, as Pg decreases to (C-a)/(Q-b), L decreases to zero, so L = (Q-b)g/P - (C-a).&lt;br /&gt;
+Suppose we have a scale of N steps to the interval of repetition, with Q steps to a period, so that there are P = N/Q periods to the repetition interval (usually, an octave.) If the generator range in question is u &amp;lt; g &amp;lt; v, then we may multiply by P to get C/Q &amp;lt; Pg &amp;lt; a/b, where C is the number of the &lt;a class="wiki_link" href="/Interval%20class"&gt;generic interval&lt;/a&gt; to which the generator g belongs, and both C/Q and a/b are reduced to lowest terms. We have normalized so that C/N &amp;lt; g, and hence C/Q &amp;lt; Pg, with C/N being the lower end of the range of possible generators, where L=s. If the range is C/Q &amp;lt; Pg &amp;lt; a/b, then when Pg = a/b, s has decreased to zero with increasing g, so s = (a - bg)/P. If there are e large steps and f small steps, then eL + fs = 1, so L = (1 - fs)/e = 1/P - a + bg.&lt;br /&gt;
 &lt;br /&gt;
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