Combination product set: 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~~-06-~~28 23~~:43:42 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-12-18 22:30:37 UTC</tt>.

: The original revision id was <tt>~~150897057~~</tt>.

: The original revision id was <tt>393468168</tt>.

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

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# A set S of n positive real numbers is the starting point.

# All the combinations of k elements of the set are obtained, and their products taken.

# These are combined into a set, and then all of the elements of that set are divided by one of them (which one is ~~arbitary~~.)

# These are combined into a set, and then all of the elements of that set are divided by one of them (which one is arbitrary; if a canonical choice is required the smallest element could be used).

# The resulting elements are reduced to an octave and sorted in ascending order, resulting in an octave period of a [[periodic scale]] (the usual sort of scale, in other words) which we may call Cps(S, k).

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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>Combination product sets</title></head><body>A combination product set is a <a class="wiki_link" href="/scale">scale</a> generated by the following means:

<ol><li>A set S of n positive real numbers is the starting point.</li><li>All the combinations of k elements of the set are obtained, and their products taken.</li><li>These are combined into a set, and then all of the elements of that set are divided by one of them (which one is ~~arbitary~~.)</li><li>The resulting elements are reduced to an octave and sorted in ascending order, resulting in an octave period of a <a class="wiki_link" href="/periodic%20scale">periodic scale</a> (the usual sort of scale, in other words) which we may call Cps(S, k).</li></ol>

<ol><li>A set S of n positive real numbers is the starting point.</li><li>All the combinations of k elements of the set are obtained, and their products taken.</li><li>These are combined into a set, and then all of the elements of that set are divided by one of them (which one is arbitrary; if a canonical choice is required the smallest element could be used).</li><li>The resulting elements are reduced to an octave and sorted in ascending order, resulting in an octave period of a <a class="wiki_link" href="/periodic%20scale">periodic scale</a> (the usual sort of scale, in other words) which we may call Cps(S, k).</li></ol>

This is sometimes called a k)n cps. There are special names for special cases: a 2)4 cps is called a <a class="wiki_link" href="/hexany">hexany</a>; both 2)5 and 3)5 cps are called <a class="wiki_link" href="/dekanies">dekanies</a>; both 2)6 and 4)6 cps are called <a class="wiki_link" href="/pentadekanies">pentadekanies</a>, and a 3)6 cps an <a class="wiki_link" href="/eikosany">eikosany</a>. These are normally considered in connection with just intonation, so that the starting set is a set of positive rational numbers, but nothing prevents consideration of the more general case.

@@ 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>2010-06-28 23:43:42 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-12-18 22:30:37 UTC</tt>.<br>
-: The original revision id was <tt>150897057</tt>.<br>
+: The original revision id was <tt>393468168</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 10: / Line 10: @@
 # A set S of n positive real numbers is the starting point.
 # All the combinations of k elements of the set are obtained, and their products taken.
-# These are combined into a set, and then all of the elements of that set are divided by one of them (which one is arbitary.)
+# These are combined into a set, and then all of the elements of that set are divided by one of them (which one is arbitrary; if a canonical choice is required the smallest element could be used).
 # The resulting elements are reduced to an octave and sorted in ascending order, resulting in an octave period of a [[periodic scale]] (the usual sort of scale, in other words) which we may call Cps(S, k).
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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;Combination product sets&lt;/title&gt;&lt;/head&gt;&lt;body&gt;A &lt;strong&gt;combination product set&lt;/strong&gt; is a &lt;a class="wiki_link" href="/scale"&gt;scale&lt;/a&gt; generated by the following means:&lt;br /&gt;
 &lt;br /&gt;
-&lt;ol&gt;&lt;li&gt;A set S of n positive real numbers is the starting point.&lt;/li&gt;&lt;li&gt;All the combinations of k elements of the set are obtained, and their products taken.&lt;/li&gt;&lt;li&gt;These are combined into a set, and then all of the elements of that set are divided by one of them (which one is arbitary.)&lt;/li&gt;&lt;li&gt;The resulting elements are reduced to an octave and sorted in ascending order, resulting in an octave period of a &lt;a class="wiki_link" href="/periodic%20scale"&gt;periodic scale&lt;/a&gt; (the usual sort of scale, in other words) which we may call Cps(S, k).&lt;/li&gt;&lt;/ol&gt;&lt;br /&gt;
+&lt;ol&gt;&lt;li&gt;A set S of n positive real numbers is the starting point.&lt;/li&gt;&lt;li&gt;All the combinations of k elements of the set are obtained, and their products taken.&lt;/li&gt;&lt;li&gt;These are combined into a set, and then all of the elements of that set are divided by one of them (which one is arbitrary; if a canonical choice is required the smallest element could be used).&lt;/li&gt;&lt;li&gt;The resulting elements are reduced to an octave and sorted in ascending order, resulting in an octave period of a &lt;a class="wiki_link" href="/periodic%20scale"&gt;periodic scale&lt;/a&gt; (the usual sort of scale, in other words) which we may call Cps(S, k).&lt;/li&gt;&lt;/ol&gt;&lt;br /&gt;
 This is sometimes called a k)n cps. There are special names for special cases: a 2)4 cps is called a &lt;a class="wiki_link" href="/hexany"&gt;hexany&lt;/a&gt;; both 2)5 and 3)5 cps are called &lt;a class="wiki_link" href="/dekanies"&gt;dekanies&lt;/a&gt;; both 2)6 and 4)6 cps are called &lt;a class="wiki_link" href="/pentadekanies"&gt;pentadekanies&lt;/a&gt;, and a 3)6 cps an &lt;a class="wiki_link" href="/eikosany"&gt;eikosany&lt;/a&gt;. These are normally considered in connection with just intonation, so that the starting set is a set of positive rational numbers, but nothing prevents consideration of the more general case.&lt;br /&gt;
 &lt;br /&gt;