Combination product set: Difference between revisions

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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.

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.

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CPS were invented by [[Erv Wilson]].

== ~~External links~~ ==

== See also ==

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

* [[Gallery of combination product sets]]

[[Category:Combination product sets| ]]

[[Category:Erv Wilson]]

@@ Line 10: / Line 10: @@
 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.
@@ Line 16: / Line 16: @@
 CPS were invented by [[Erv Wilson]].
-== External links ==
+== See also ==
 * [http://anaphoria.com/wilsoncps.html Wilson Archives - Combination Product Sets - CPS]
+* [[Gallery of combination product sets]]
 [[Category:Combination product sets| ]] <!-- main article -->
+[[Category:Erv Wilson]]