Permutation product set - Revision history

BudjarnLambeth: Todo link

2024-10-20T07:13:09Z

Todo link

← Older revision		Revision as of 07:13, 20 October 2024
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	[[Category:Math]]		[[Category:Math]]
	[[Category:Scale]]		[[Category:Scale]]
			{{todo\|link}}

FloraC: Improve readability

2024-01-09T11:08:23Z

Improve readability

← Older revision		Revision as of 11:08, 9 January 2024
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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]]

BudjarnLambeth: Added links to dead end pg

2024-01-09T06:25:10Z

Added links to dead end pg

← Older revision		Revision as of 06:25, 9 January 2024
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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 ~~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:		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''}
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	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.

	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 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

Fredg999 category edits: Removing from Category:Theory using Cat-a-lot

2023-03-05T05:32:47Z

Removing from Category:Theory using Cat-a-lot

← Older revision		Revision as of 05:32, 5 March 2023
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	[[Category:math]]		[[Category:math]]
	[[Category:Scale]]		[[Category:Scale]]
	~~[[Category:theory]]~~

Fredg999 category edits: Moving from Category:Scale theory to Category:Scale using Cat-a-lot

2022-02-26T16:32:52Z

Moving from Category:Scale theory to Category:Scale using Cat-a-lot

← Older revision		Revision as of 16:32, 26 February 2022
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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_theory~~]]		[[Category:Scale]]
	[[Category:theory]]		[[Category:theory]]

Wikispaces>FREEZE at 00:00, 17 July 2018

2018-07-17T00:00:00Z

← Older revision		Revision as of 00:00, 17 July 2018
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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"><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 />		[[Category:scale_theory]]
	~~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 />		[[Category:theory]]
	~~{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:&lt;h2&gt; --><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.<br />~~
	~~<br />~~
	~~The octave equivalent 16-limit HPPS has 1775 notes.</body></html></pre></div>~~

Wikispaces>Praimhin: Imported revision 588832490 - Original comment:

2016-08-05T04:39:36Z

**Imported revision 588832490 - Original comment: **

Wikispaces>Praimhin: Imported revision 588832480 - Original comment:

2016-08-05T04:38:48Z

**Imported revision 588832480 - Original comment: **

New page

<h2>IMPORTED REVISION FROM WIKISPACES</h2>
This is an imported revision from Wikispaces. The revision metadata is included below for reference: 
: This revision was by author [[User:Praimhin|Praimhin]] and made on <tt>2016-08-05 04:38:48 UTC</tt>. 
: The original revision id was <tt>588832480</tt>. 
: The revision comment was: <tt></tt> 
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it. 
<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.

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.</pre></div>
<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>permutation product set</title></head><body>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. 
 
<h2 id="toc0"><a name="x-Special cases"></a>Special cases</h2>
 
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 <a class="wiki_link" href="/Jekyll%20and%20Hyde%20diamonds">Jekyll or Hyde diamond</a> in scalar order. 
 
The octave equivalent 16-limit HPPS has 1775 notes.</body></html></pre></div>

← Older revision		Revision as of 04:39, 5 August 2016
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	<h2>IMPORTED REVISION FROM WIKISPACES</h2>		<h2>IMPORTED REVISION FROM WIKISPACES</h2>
	This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>		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 04:38:48 UTC</tt>.<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>~~588832480~~</tt>.<br>		: The original revision id was <tt>588832490</tt>.<br>
	: The revision comment was: <tt></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>		The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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	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~~. 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 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.</pre></div>
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	1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1<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 />		<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 />		The octave equivalent 8-limit HPPS has 33 notes.<br />
	<br />		<br />
	The octave equivalent 16-limit HPPS has 1775 notes.</body></html></pre></div>		The octave equivalent 16-limit HPPS has 1775 notes.</body></html></pre></div>