Permutation product set: Difference between revisions

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A '''permutation product set''' (PPS) is obtained from a [[chord]] C = {1,''a''_1,''a''_2,...,''a''_''n''} as follows:
A '''permutation product set''' ('''PPS''') is obtained from a [[chord]] C = {1, ''a''<sub>1</sub>, ''a''<sub>2</sub>, …, ''a''<sub>''n''</sub>} as follows:


Let ''b''_1,...,''b_''''n'' be the [[interval]]s between successive [[note]]s of the chord: ''b_i'' = ''a_i''''/a_''(''i''-1). These ''n'' intervals can be permuted in ''n''! ways, yielding ''n''! different chords:
Let ''b''<sub>1</sub>, …, ''b''<sub>''n''</sub> be the [[interval]]s between successive [[note]]s of the chord: ''b''<sub>''i''</sub> = ''a''<sub>''i''</sub>/''a''<sub>(''i'' - 1)</sub>. 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''}
{1, ''b''<sub>s(1)</sub>, ''b''<sub>s(1)</sub>*''b''<sub>s(2)</sub>, …} 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.
The union of these ''n'' chords is the PPS of C. PPSes may or may not be octave equivalent.
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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 equivalence|octave equivalent]] 6-limit HPPS is the union of the major and minor diatonic scales:
The [[octave equivalence|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
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The octave equivalent 16-limit HPPS has 1775 notes.
The octave equivalent 16-limit HPPS has 1775 notes.
[[Category:math]]
 
[[Category:Math]]
[[Category:Scale]]
[[Category:Scale]]
{{todo|link}}

Latest revision as of 07:13, 20 October 2024

A permutation product set (PPS) is obtained from a chord C = {1, a1, a2, …, an} as follows:

Let b1, …, bn be the intervals between successive notes of the chord: bi = ai/a(i - 1). These n intervals can be permuted in n! ways, yielding n! different chords:

{1, bs(1), bs(1)*bs(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.

The octave equivalent 16-limit HPPS has 1775 notes.

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