|
|
| (8 intermediate revisions by 5 users not shown) |
| Line 1: |
Line 1: |
| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | [[File:Wilson CPS names.png|400px|thumb|right|The names of all combination product sets up to 6 elements.]] |
| 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>
| |
| : 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> | |
| <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.
| | A '''combination product set''' ('''CPS''') is a [[scale|scale]] generated by the following means: |
| # 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. 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.
| | # 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]], 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 [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. |
|
| |
|
| </pre></div>
| | 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. |
| <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 />
| | CPS were invented by [[Erv Wilson]]. |
| <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 />
| | == See also == |
| 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 />
| | * [http://anaphoria.com/wilsoncps.html Wilson Archives - Combination Product Sets - CPS] |
| <br />
| | * [[Gallery of combination product sets]] |
| 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 />
| | [[Category:Combination product sets| ]] <!-- main article --> |
| 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>
| | [[Category:Erv Wilson]] |