Combination product set - Revision history

BudjarnLambeth: /* External links */ link to orphaned page

2024-12-07T03:22:03Z

External links: link to orphaned page

← Older revision		Revision as of 03:22, 7 December 2024
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	CPS were invented by [[Erv Wilson]].		CPS were invented by [[Erv Wilson]].

	== ~~External links~~ ==		== See also ==
	* [http://anaphoria.com/wilsoncps.html Wilson Archives - Combination Product Sets - CPS]		* [http://anaphoria.com/wilsoncps.html Wilson Archives - Combination Product Sets - CPS]
			* [[Gallery of combination product sets]]

	[[Category:Combination product sets\| ]] <!-- main article -->		[[Category:Combination product sets\| ]] <!-- main article -->
	[[Category:Erv Wilson]]		[[Category:Erv Wilson]]

Jolohaga: Adding Erv Wilson category

2022-12-29T15:19:57Z

Adding Erv Wilson category

← Older revision		Revision as of 15:19, 29 December 2022
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	This is sometimes called a ''k'')''n'' CPS, where the ''n'' denotes the size of the set ''S''. There are special names for special cases: a 2)4 CPS is called a [[Hexany\|hexany]]; both 2)5 and 3)5 CPS are called [[Dekany\|dekanies]]; both 2)6 and 4)6 CPS are called [[Pentadekany\|pentadekanies]], a 3)6 CPS is called an [[Eikosany\|eikosany]], etc. 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.		This is sometimes called a ''k'')''n'' CPS, where the ''n'' denotes the size of the set ''S''. There are special names for special cases: a 2)4 CPS is called a [[Hexany\|hexany]]; both 2)5 and 3)5 CPS are called [[Dekany\|dekanies]]; both 2)6 and 4)6 CPS are called [[Pentadekany\|pentadekanies]], a 3)6 CPS is called an [[Eikosany\|eikosany]], etc. 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 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 multiplicities; 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 [https://en.wikipedia.org/wiki/Multiset multiset]. A multiset is like a set, but the elements have multiplicities; 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-Fokker genus\|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-Fokker genus\|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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	[[Category:Combination product sets\| ]] <!-- main article -->		[[Category:Combination product sets\| ]] <!-- main article -->
			[[Category:Erv Wilson]]

Fredg999: Recategorization

2021-08-07T22:52:13Z

Recategorization

← Older revision		Revision as of 22:52, 7 August 2021
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	[[Category:Combination product sets\| ]] <!-- main article -->		[[Category:Combination product sets\| ]] <!-- main article -->
	~~[[Category:Scale families]]~~

Fredg999: Added 1 figure (names of CPS), added 1 external link, misc. edits

2021-08-07T21:23:54Z

Added 1 figure (names of CPS), added 1 external link, misc. edits

← Older revision		Revision as of 21:23, 7 August 2021
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	~~A '''combination product set''' is a~~ [[~~scale~~\|~~scale~~]] ~~generated by the following means:~~		[[File:Wilson CPS names.png\|400px\|thumb\|right\|The names of all combination product sets up to 6 elements.]]

	~~<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 [[~~Periodic_scale~~\|~~periodic~~ scale]] (the ~~usual sort of scale, in other words) which we may call Cps(S, k).</li></ol>~~		A '''combination product set''' ('''CPS''') is a [[scale\|scale]] generated by the following means:

	~~This~~ is ~~sometimes called a~~ k~~)n cps, where the "n~~' ~~denotes the size~~ of the set S. ~~There~~ are ~~special names for special cases:~~ a ~~2)4 cps~~ is ~~called~~ a ~~[[Hexany\|hexany]]; both 2~~)~~5 and 3)5 cps~~ are ~~called~~ [[~~dekanies~~\|~~dekanies~~]]~~; both 2)6~~ and ~~4)6 cps are called [[pentadekanies\|pentadekanies]]~~, ~~and~~ a ~~3)6 cps an~~ [[~~eikosany~~\|~~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~~.		# 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 arbitrary; if a canonical choice is required, the smallest element could be used).
			# The resulting elements are [[Octave reduction\|octave-reduced]] and sorted in ascending order, resulting in an octave period of a [[Periodic_scale\|periodic scale]] (the usual sort of scale, in other words) which we may call CPS(''S'', ''k'').

	~~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~~.		This is sometimes called a ''k'')''n'' CPS, where the ''n'' denotes the size of the set ''S''. There are special names for special cases: a 2)4 CPS is called a [[Hexany\|hexany]]; both 2)5 and 3)5 CPS are called [[Dekany\|dekanies]]; both 2)6 and 4)6 CPS are called [[Pentadekany\|pentadekanies]], a 3)6 CPS is called an [[Eikosany\|eikosany]], etc. 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.

	~~Cps~~ are ~~closely related to~~ [[~~Euler_genera\|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~~.		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 multiplicities; 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.
	~~[[Category:math]]~~

	~~CPSs~~ were invented by [[~~Erv Wilson\|~~Erv Wilson]].		CPS are closely related to [[Euler-Fokker genus\|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 were invented by [[Erv Wilson]].

	== External links ==		== External links ==
	* [http://anaphoria.com/wilsoncps.html Wilson Archives - Combination Product Sets - CPS]		* [http://anaphoria.com/wilsoncps.html Wilson Archives - Combination Product Sets - CPS]

			[[Category:Combination product sets\| ]] <!-- main article -->
			[[Category:Scale families]]

Fredg999: Added external link (archive of various papers and images from Erv Wilson)

2021-08-07T20:45:56Z

Added external link (archive of various papers and images from Erv Wilson)

← Older revision		Revision as of 20:45, 7 August 2021
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	CPSs were invented by [[Erv Wilson\|Erv Wilson]].		CPSs were invented by [[Erv Wilson\|Erv Wilson]].

			== External links ==
			* [http://anaphoria.com/wilsoncps.html Wilson Archives - Combination Product Sets - CPS]

Fredg999: Fredg999 moved page Combination product sets to Combination product set: Singular title by convention

2021-08-07T17:01:19Z

Fredg999 moved page Combination product sets to Combination product set: Singular title by convention

← Older revision	Revision as of 17:01, 7 August 2021
(No difference)

Cmloegcmluin at 21:23, 11 February 2020

2020-02-11T21:23:42Z

← Older revision		Revision as of 21:23, 11 February 2020
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	Cps are closely related to [[Euler_genera\|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\|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.
	[[Category:math]]		[[Category:math]]

			CPSs were invented by [[Erv Wilson\|Erv Wilson]].

Wikispaces>FREEZE at 00:00, 17 July 2018

2018-07-17T00:00:00Z

← Older revision		Revision as of 00:00, 17 July 2018
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	~~<h2>IMPORTED REVISION FROM WIKISPACES</h2>~~		A '''combination product set''' is a [[scale\|scale]] generated by the following means:
	~~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>2014-04-23 10:27:51 UTC</tt>.<br>~~
	~~: The original revision id was <tt>504046960</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>~~
	~~<h4>Original Wikitext content:</h4>~~
	<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 S of n positive real numbers is the starting point.		<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 [[Periodic_scale\|periodic scale]] (the usual sort of scale, in other words) which we may call Cps(S, k).</li></ol>
	# 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 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).

	This is sometimes called a k)n cps, where the "n' denotes the size of the set S. 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.		This is sometimes called a k)n cps, where the "n' denotes the size of the set S. There are special names for special cases: a 2)4 cps is called a [[Hexany\|hexany]]; both 2)5 and 3)5 cps are called [[dekanies\|dekanies]]; both 2)6 and 4)6 cps are called [[pentadekanies\|pentadekanies]], and a 3)6 cps an [[eikosany\|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 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.		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\|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.
			[[Category:math]]
	~~</pre></div>~~
	~~<h4>Original HTML content:</h4>~~
	<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 <strong>combination product set</strong> is a <a class="wiki_link" href="/scale">scale</a> generated by the following means:<br />
	~~<br />~~
	<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><br />
	This is sometimes called a k)n cps, where the &quot;n' denotes the size of the set S. 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.<br />
	~~<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 />~~
	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>

Wikispaces>genewardsmith: Imported revision 504046960 - Original comment:

2014-04-23T10:27:51Z

**Imported revision 504046960 - Original comment: **

Wikispaces>genewardsmith: Imported revision 393468168 - Original comment:

2012-12-18T22:30:37Z

**Imported revision 393468168 - Original comment: **

← Older revision		Revision as of 10:27, 23 April 2014
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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>~~2012~~-12-~~18 22~~:30:37 UTC</tt>.<br>		: This revision was by author [[User:genewardsmith\|genewardsmith]] and made on <tt>2014-04-23 10:27:51 UTC</tt>.<br>
	: The original revision id was <tt>~~393468168~~</tt>.<br>		: The original revision id was <tt>504046960</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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	# 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).		# 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).

	This is sometimes called a 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.		This is sometimes called a k)n cps, where the "n' denotes the size of the set S. 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 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.		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.
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	<br />		<br />
	<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><br />		<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><br />
	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.<br />		This is sometimes called a k)n cps, where the &quot;n' denotes the size of the set S. 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.<br />
	<br />		<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 />		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>

← Older revision		Revision as of 22:30, 18 December 2012
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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-~~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 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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	# A set S of n positive real numbers is the starting point.		# 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.		# 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).		# 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 <strong>combination product set</strong> is a <a class="wiki_link" href="/scale">scale</a> 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 <strong>combination product set</strong> is a <a class="wiki_link" href="/scale">scale</a> generated by the following means:<br />
	<br />		<br />
	<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><br />		<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><br />
	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.<br />		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.<br />
	<br />		<br />