Xenharmonic Wiki

Permutation product set: Difference between revisions

VisualWikitext
Wikispaces>Praimhin
**Imported revision 588832480 - Original comment: **
 
BudjarnLambeth (talk | contribs)
autopatrolled, emailconfirmed
18,215 edits
m Todo link
 
(6 intermediate revisions by 5 users not shown)
Line 1: Line 1:
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
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:
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>
: The original revision id was <tt>588832480</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''<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:


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


==Special cases==
The union of these ''n'' chords is the PPS of C. PPSes may or may not be octave equivalent.


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.
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 equivalence|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. 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.


The octave equivalent 16-limit HPPS has 1775 notes.</pre></div>
[[Category:Math]]
<h4>Original HTML content:</h4>
[[Category:Scale]]
<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;
{{todo|link}}
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;
{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. Coincidentally this scale can be obtained by arranging the notes in the &lt;a class="wiki_link" href="/Jekyll%20and%20Hyde%20diamonds"&gt;Jekyll or Hyde diamond&lt;/a&gt; in scalar order.&lt;br /&gt;
&lt;br /&gt;
The octave equivalent 16-limit HPPS has 1775 notes.&lt;/body&gt;&lt;/html&gt;</pre></div>
Retrieved from "https://en.xen.wiki/w/Permutation_product_set"