Step pattern product: Difference between revisions
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Wikispaces>guest **Imported revision 400665958 - Original comment: ** |
Wikispaces>mbattaglia1 **Imported revision 400669080 - Original comment: Reverted to Nov 15, 2011 4:00 pm: spam** |
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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: | : This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2013-01-23 03:39:38 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>400669080</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt>Reverted to Nov 15, 2011 4:00 pm: spam</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> | ||
<h4>Original Wikitext content:</h4> | <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 **word** is a sequence of letters from some finite alphabet. Words can be used to represent scales; for example "aabaaab" represents the ([[meantone]]) major scale if 'a' represents a whole step and 'b' a half step. | <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 **word** is a sequence of letters from some finite alphabet. Words can be used to represent scales; for example "aabaaab" represents the ([[meantone]]) major scale if 'a' represents a whole step and 'b' a half step. | ||
Given two words of the same length ("factors"), their **product word** is the word whose alphabet consists of ordered pairs of the alphabets of its factor words, and whose //n//th letter is the ordered pair of the //n//th letters of its factors. | Given two words of the same length ("factors"), their **product word** is the word whose alphabet consists of ordered pairs of the alphabets of its factor words, and whose //n//th letter is the ordered pair of the //n//th letters of its factors. | ||
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<h4>Original HTML content:</h4> | <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>Product word</title></head><body>A <strong>word</strong> is a sequence of letters from some finite alphabet. Words can be used to represent scales; for example &quot;aabaaab&quot; represents the (<a class="wiki_link" href="/meantone">meantone</a>) major scale if 'a' represents a whole step and 'b' a half step.<br /> | <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>Product word</title></head><body>A <strong>word</strong> is a sequence of letters from some finite alphabet. Words can be used to represent scales; for example &quot;aabaaab&quot; represents the (<a class="wiki_link" href="/meantone">meantone</a>) major scale if 'a' represents a whole step and 'b' a half step.<br /> | ||
<br /> | <br /> | ||
Given two words of the same length (&quot;factors&quot;), their <strong>product word</strong> is the word whose alphabet consists of ordered pairs of the alphabets of its factor words, and whose <em>n</em>th letter is the ordered pair of the <em>n</em>th letters of its factors.<br /> | Given two words of the same length (&quot;factors&quot;), their <strong>product word</strong> is the word whose alphabet consists of ordered pairs of the alphabets of its factor words, and whose <em>n</em>th letter is the ordered pair of the <em>n</em>th letters of its factors.<br /> | ||
<br /> | <br /> | ||
Revision as of 03:39, 23 January 2013
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author mbattaglia1 and made on 2013-01-23 03:39:38 UTC.
- The original revision id was 400669080.
- The revision comment was: Reverted to Nov 15, 2011 4:00 pm: spam
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
A **word** is a sequence of letters from some finite alphabet. Words can be used to represent scales; for example "aabaaab" represents the ([[meantone]]) major scale if 'a' represents a whole step and 'b' a half step.
Given two words of the same length ("factors"), their **product word** is the word whose alphabet consists of ordered pairs of the alphabets of its factor words, and whose //n//th letter is the ordered pair of the //n//th letters of its factors.
For example, the product word of "aabaaab" and "xxyxyxy" is "(a,x)(a,x)(b,y)(a,x)(a,y)(a,x)(b,y)". For brevity, we can substitute each ordered pair of letters by a new single letter and say this is equivalent to the word "rrsrtrs". This construction has an obvious generalization to the product of three or more words.
The importance of product words in music theory is that every [[Fokker blocks|Fokker block]] can be expressed as the product word of two or more [[distributional evenness|distributionally even]] scales in a unique way. Fokker blocks are therefore equivalent to product words of DE scales of the same size. If one or both of the DE scales are rotated (into different [[mode|modes]]), then the product Fokker block scale is not always a mode, but is often a [[dome]] instead.Original HTML content:
<html><head><title>Product word</title></head><body>A <strong>word</strong> is a sequence of letters from some finite alphabet. Words can be used to represent scales; for example "aabaaab" represents the (<a class="wiki_link" href="/meantone">meantone</a>) major scale if 'a' represents a whole step and 'b' a half step.<br /> <br /> Given two words of the same length ("factors"), their <strong>product word</strong> is the word whose alphabet consists of ordered pairs of the alphabets of its factor words, and whose <em>n</em>th letter is the ordered pair of the <em>n</em>th letters of its factors.<br /> <br /> For example, the product word of "aabaaab" and "xxyxyxy" is "(a,x)(a,x)(b,y)(a,x)(a,y)(a,x)(b,y)". For brevity, we can substitute each ordered pair of letters by a new single letter and say this is equivalent to the word "rrsrtrs". This construction has an obvious generalization to the product of three or more words.<br /> <br /> The importance of product words in music theory is that every <a class="wiki_link" href="/Fokker%20blocks">Fokker block</a> can be expressed as the product word of two or more <a class="wiki_link" href="/distributional%20evenness">distributionally even</a> scales in a unique way. Fokker blocks are therefore equivalent to product words of DE scales of the same size. If one or both of the DE scales are rotated (into different <a class="wiki_link" href="/mode">modes</a>), then the product Fokker block scale is not always a mode, but is often a <a class="wiki_link" href="/dome">dome</a> instead.</body></html>