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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
A '''permutation product set''' (PPS) is obtained from a chord C = {1,''a''_1,''a''_2,...,''a''_''n''} as follows:
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:Praimhin|Praimhin]] and made on <tt>2016-08-05 04:39:36 UTC</tt>.<br>
: The original revision id was <tt>588832490</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 **permutation product set** (PPS) is obtained from a chord C = {1,//a//_1,//a//_2,...,//a//_//n//} as follows:
Let //b//_1,...,//b_////n// be the intervals between successive notes of the chord: //b_i// = //a_i/////a_//(//i//-1). These //n// intervals can be permuted in //n//! ways, yielding //n//! different chords:
{1,//b//_s(1),//b//_s(1)*//b//_s(2),...} where s is a permutation of {1,2,...,//n//}


The union of these //n// chords is the PPS of C. PPSes may or may not be octave equivalent.
Let ''b''_1,...,''b_''''n'' be the intervals between successive notes of the chord: ''b_i'' = ''a_i''''/a_''(''i''-1). These ''n'' intervals can be permuted in ''n''! ways, yielding ''n''! different chords:
 
{1,''b''_s(1),''b''_s(1)*''b''_s(2),...} where s is a permutation of {1,2,...,''n''}
 
The union of these ''n'' chords is the PPS of C. PPSes may or may not be octave equivalent.


Permutation product sets were introduced by Marcel De Velde in 2009 to explain the diatonic scale.
Permutation product sets were introduced by Marcel De Velde in 2009 to explain the diatonic scale.


==Special cases==  
==Special cases==


If C is a harmonic series, {1/1,2/1,...,//n///1}, then the PPS of C is called the //n//-limit harmonic permutation product set (HPPS). //n// can be even.
If C is a harmonic series, {1/1,2/1,...,''n''/1}, then the PPS of C is called the ''n''-limit harmonic permutation product set (HPPS). ''n'' can be even.


The octave equivalent 6-limit HPPS is the union of the major and minor diatonic scales:
The octave equivalent 6-limit HPPS is the union of the major and minor diatonic scales:
1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1
1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1


The octave equivalent 8-limit HPPS has 33 notes.
The octave equivalent 8-limit HPPS has 33 notes.


The octave equivalent 16-limit HPPS has 1775 notes.</pre></div>
The octave equivalent 16-limit HPPS has 1775 notes.
<h4>Original HTML content:</h4>
[[Category:math]]
<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;permutation product set&lt;/title&gt;&lt;/head&gt;&lt;body&gt;A &lt;strong&gt;permutation product set&lt;/strong&gt; (PPS) is obtained from a chord C = {1,&lt;em&gt;a&lt;/em&gt;_1,&lt;em&gt;a&lt;/em&gt;_2,...,&lt;em&gt;a&lt;/em&gt;_&lt;em&gt;n&lt;/em&gt;} as follows:&lt;br /&gt;
[[Category:scale_theory]]
Let &lt;em&gt;b&lt;/em&gt;_1,...,&lt;em&gt;b_&lt;/em&gt;&lt;em&gt;n&lt;/em&gt; be the intervals between successive notes of the chord: &lt;em&gt;b_i&lt;/em&gt; = &lt;em&gt;a_i&lt;/em&gt;&lt;em&gt;/a_&lt;/em&gt;(&lt;em&gt;i&lt;/em&gt;-1). These &lt;em&gt;n&lt;/em&gt; intervals can be permuted in &lt;em&gt;n&lt;/em&gt;! ways, yielding &lt;em&gt;n&lt;/em&gt;! different chords:&lt;br /&gt;
[[Category:theory]]
{1,&lt;em&gt;b&lt;/em&gt;_s(1),&lt;em&gt;b&lt;/em&gt;_s(1)*&lt;em&gt;b&lt;/em&gt;_s(2),...} where s is a permutation of {1,2,...,&lt;em&gt;n&lt;/em&gt;}&lt;br /&gt;
&lt;br /&gt;
The union of these &lt;em&gt;n&lt;/em&gt; chords is the PPS of C. PPSes may or may not be octave equivalent.&lt;br /&gt;
&lt;br /&gt;
Permutation product sets were introduced by Marcel De Velde in 2009 to explain the diatonic scale.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Special cases"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Special cases&lt;/h2&gt;
&lt;br /&gt;
If C is a harmonic series, {1/1,2/1,...,&lt;em&gt;n&lt;/em&gt;/1}, then the PPS of C is called the &lt;em&gt;n&lt;/em&gt;-limit harmonic permutation product set (HPPS). &lt;em&gt;n&lt;/em&gt; can be even.&lt;br /&gt;
&lt;br /&gt;
The octave equivalent 6-limit HPPS is the union of the major and minor diatonic scales:&lt;br /&gt;
1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1&lt;br /&gt;
&lt;br /&gt;
The octave equivalent 8-limit HPPS has 33 notes.&lt;br /&gt;
&lt;br /&gt;
The octave equivalent 16-limit HPPS has 1775 notes.&lt;/body&gt;&lt;/html&gt;</pre></div>
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