Permutation product set
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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.
Permutation product sets were introduced by Marcel De Velde in 2009 to explain the diatonic scale.
==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.
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
The octave equivalent 8-limit HPPS has 33 notes. Coincidentally this scale can be obtained by arranging the notes in the [[Jekyll and Hyde diamonds|Jekyll or Hyde diamond]] in scalar order.
The octave equivalent 16-limit HPPS has 1775 notes.Original HTML content:
<html><head><title>permutation product set</title></head><body>A <strong>permutation product set</strong> (PPS) is obtained from a chord C = {1,<em>a</em>_1,<em>a</em>_2,...,<em>a</em>_<em>n</em>} as follows:<br />
Let <em>b</em>_1,...,<em>b_</em><em>n</em> be the intervals between successive notes of the chord: <em>b_i</em> = <em>a_i</em><em>/a_</em>(<em>i</em>-1). These <em>n</em> intervals can be permuted in <em>n</em>! ways, yielding <em>n</em>! different chords:<br />
{1,<em>b</em>_s(1),<em>b</em>_s(1)*<em>b</em>_s(2),...} where s is a permutation of {1,2,...,<em>n</em>}<br />
<br />
The union of these <em>n</em> chords is the PPS of C. PPSes may or may not be octave equivalent.<br />
<br />
Permutation product sets were introduced by Marcel De Velde in 2009 to explain the diatonic scale.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:<h2> --><h2 id="toc0"><a name="x-Special cases"></a><!-- ws:end:WikiTextHeadingRule:0 -->Special cases</h2>
<br />
If C is a harmonic series, {1/1,2/1,...,<em>n</em>/1}, then the PPS of C is called the <em>n</em>-limit harmonic permutation product set (HPPS). <em>n</em> can be even.<br />
<br />
The octave equivalent 6-limit HPPS is the union of the major and minor diatonic scales:<br />
1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1<br />
<br />
The octave equivalent 8-limit HPPS has 33 notes. Coincidentally this scale can be obtained by arranging the notes in the <a class="wiki_link" href="/Jekyll%20and%20Hyde%20diamonds">Jekyll or Hyde diamond</a> in scalar order.<br />
<br />
The octave equivalent 16-limit HPPS has 1775 notes.</body></html>