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''1, ''a''2, …, ''a''''n''} 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''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:

{1,''b''_s(1),''b''_s(1)*''b''_s(2),~~...~~} where s is a permutation of {1,2,~~...~~,''n''}

{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.

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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

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The octave equivalent 16-limit HPPS has 1775 notes.

[[Category:~~math~~]]

[[Category:Math]]

[[Category:Scale]]

@@ Line 1: / Line 1: @@
-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.
@@ Line 9: / Line 9: @@
 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 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1
@@ Line 20: / Line 20: @@
 The octave equivalent 16-limit HPPS has 1775 notes.
-[[Category:math]]
+[[Category:Math]]
 [[Category:Scale]]