Combination product set: Difference between revisions
Wikispaces>genewardsmith **Imported revision 150729241 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 150759127 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | 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- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-06-28 03:49:07 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>150759127</tt>.<br> | ||
: The revision comment was: <tt></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> | 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">A combination product set is a scale generated by the following means: | <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">A combination product set is a scale generated by the following means: | ||
# A set of n positive | # A set of n positive real numbers is the starting point. | ||
# All the combinations of k elements of the set are obtained, and their | # 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 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 arbitary.) | ||
# 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.) | # 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.) | ||
This is sometimes called an k)n cps. There are special names for special cases: a 2)4 cps is called a [[hexany]]; both 2)5 and 3)5 cps are called dekanies; both 2)6 and 4)6 cps are called pentadekanies, and a 3)6 cps an eikosany. | This is sometimes called an k)n cps. There are special names for special cases: a 2)4 cps is called a [[hexany]]; both 2)5 and 3)5 cps are called dekanies; both 2)6 and 4)6 cps are called pentadekanies, and a 3)6 cps an eikosany. 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. | ||
The idea can be generalized so that the thing we start from is not a set but a [[http://en.wikipedia.org/wiki/Multiset|multiset]]. A multiset is like a set, but the elements have multiplicites; there can be two, three or any number of a given kind of element. Submultisets of a multiset are also multisets, and the product over a multiset takes account of multiplicity. Hence the 2)4 hexany resulting from the multiset [1,1,3,5] first generates the two-element submultisets, which are [1, 1], [1, 3], [1, 5], [3, 5], and we obtain products 1, 3, 5, and 15. Reducing to an octave gives 5/4, 3/2, 15/8, 2 which can be transposed to 5/4, 4/3, 5/3, 2. In spite of being called a hexany it has only four notes. | The idea can be further generalized so that the thing we start from is not a set but a [[http://en.wikipedia.org/wiki/Multiset|multiset]]. A multiset is like a set, but the elements have multiplicites; there can be two, three or any number of a given kind of element. Submultisets of a multiset are also multisets, and the product over a multiset takes account of multiplicity. Hence the 2)4 hexany resulting from the multiset [1,1,3,5] first generates the two-element submultisets, which are [1, 1], [1, 3], [1, 5], [3, 5], and we obtain products 1, 3, 5, and 15. Reducing to an octave gives 5/4, 3/2, 15/8, 2 which can be transposed to 5/4, 4/3, 5/3, 2. In spite of being called a hexany it has only four notes. | ||
Cps are closely related to [[Euler genera]], since if we combine 0)n, 1)n, 2)n ... n)n before reducing to an octave, and then reduce, we get an Euler genus. | Cps are closely related to [[Euler genera]], since if we combine 0)n, 1)n, 2)n ... n)n before reducing to an octave, and then reduce, we get an Euler genus. | ||
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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 scale generated by the following means:<br /> | <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 scale generated by the following means:<br /> | ||
<br /> | <br /> | ||
<ol><li>A set of n positive | <ol><li>A set 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.)</li></ol><br /> | ||
This is sometimes called an 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 dekanies; both 2)6 and 4)6 cps are called pentadekanies, and a 3)6 cps an eikosany.<br /> | This is sometimes called an 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 dekanies; both 2)6 and 4)6 cps are called pentadekanies, and a 3)6 cps an eikosany. 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.<br /> | ||
<br /> | <br /> | ||
The idea can be generalized so that the thing we start from is not a set but a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Multiset" rel="nofollow">multiset</a>. A multiset is like a set, but the elements have multiplicites; there can be two, three or any number of a given kind of element. Submultisets of a multiset are also multisets, and the product over a multiset takes account of multiplicity. Hence the 2)4 hexany resulting from the multiset [1,1,3,5] first generates the two-element submultisets, which are [1, 1], [1, 3], [1, 5], [3, 5], and we obtain products 1, 3, 5, and 15. Reducing to an octave gives 5/4, 3/2, 15/8, 2 which can be transposed to 5/4, 4/3, 5/3, 2. In spite of being called a hexany it has only four notes.<br /> | The idea can be further generalized so that the thing we start from is not a set but a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Multiset" rel="nofollow">multiset</a>. A multiset is like a set, but the elements have multiplicites; there can be two, three or any number of a given kind of element. Submultisets of a multiset are also multisets, and the product over a multiset takes account of multiplicity. Hence the 2)4 hexany resulting from the multiset [1,1,3,5] first generates the two-element submultisets, which are [1, 1], [1, 3], [1, 5], [3, 5], and we obtain products 1, 3, 5, and 15. Reducing to an octave gives 5/4, 3/2, 15/8, 2 which can be transposed to 5/4, 4/3, 5/3, 2. In spite of being called a hexany it has only four notes.<br /> | ||
<br /> | <br /> | ||
Cps are closely related to <a class="wiki_link" href="/Euler%20genera">Euler genera</a>, since if we combine 0)n, 1)n, 2)n ... n)n before reducing to an octave, and then reduce, we get an Euler genus.</body></html></pre></div> | Cps are closely related to <a class="wiki_link" href="/Euler%20genera">Euler genera</a>, since if we combine 0)n, 1)n, 2)n ... n)n before reducing to an octave, and then reduce, we get an Euler genus.</body></html></pre></div> | ||