Rank and codimension: Difference between revisions
Wikispaces>genewardsmith **Imported revision 297580356 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 297580644 - Original comment: ** |
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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:genewardsmith|genewardsmith]] and made on <tt>2012-02-01 17: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-02-01 17:57:15 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>297580644</tt>.<br> | ||
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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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<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">The //rank// of a [[Regular Temperaments|regular temperament]] is the number of independent intervals, called //generators//, which can be combined together to obtain any interval of the temperament. For instance, every interval of [[meantone]] can be obtained as a combination of a certain number of octaves up or down, plus a certain number of flattened meantone fifths up or down. The terminology comes from group theory; in the parlance of group theory, the intervals of a regular temperament comprise a [[http://en.wikipedia.org/wiki/Free_abelian_group#Rank|finitely generated free abelian group]] with a rank equal to the number of generators. | <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">The //rank// of a [[Regular Temperaments|regular temperament]] is the number of independent intervals, called //generators//, which can be combined together to obtain any interval of the temperament. For instance, every interval of [[meantone]] can be obtained as a combination of a certain number of octaves up or down, plus a certain number of flattened meantone fifths up or down. The terminology comes from group theory; in the parlance of group theory, the intervals of a regular temperament comprise a [[http://en.wikipedia.org/wiki/Free_abelian_group#Rank|finitely generated free abelian group]] with a rank equal to the number of generators. | ||
The [[http://en.wikipedia.org/wiki/Codimension|codimension]] or [[http://en.wikipedia.org/wiki/Free_abelian_group#Rank|co-rank]] of a temperament is the number of [[Comma|commas]] needed to completely define the temperament. If the temperament tempers the [[Harmonic Limit|p-limit]] just intonation group generated by the first n primes, then if it tempers out n-r independent commas, it will be of rank r. The terminology can also be applied to [[just intonation subgroups]]. In all cases care must be taken to specify the exact just intonation group which is being tempered by the tempering out of a set of commas.</pre></div> | The [[http://en.wikipedia.org/wiki/Codimension|codimension]] or [[http://en.wikipedia.org/wiki/Free_abelian_group#Rank|co-rank]] of a temperament is the number of [[Comma|commas]] needed to completely define the temperament. If the temperament tempers the [[Harmonic Limit|p-limit]] just intonation group generated by the first n primes, then if it tempers out n-r independent commas, it will be of rank r and codimension n-r. The terminology can also be applied to [[just intonation subgroups]]. In all cases care must be taken to specify the exact just intonation group which is being tempered by the tempering out of a set of commas.</pre></div> | ||
<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>Rank and codimension</title></head><body>The <em>rank</em> of a <a class="wiki_link" href="/Regular%20Temperaments">regular temperament</a> is the number of independent intervals, called <em>generators</em>, which can be combined together to obtain any interval of the temperament. For instance, every interval of <a class="wiki_link" href="/meantone">meantone</a> can be obtained as a combination of a certain number of octaves up or down, plus a certain number of flattened meantone fifths up or down. The terminology comes from group theory; in the parlance of group theory, the intervals of a regular temperament comprise a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Free_abelian_group#Rank" rel="nofollow">finitely generated free abelian group</a> with a rank equal to the number of generators.<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>Rank and codimension</title></head><body>The <em>rank</em> of a <a class="wiki_link" href="/Regular%20Temperaments">regular temperament</a> is the number of independent intervals, called <em>generators</em>, which can be combined together to obtain any interval of the temperament. For instance, every interval of <a class="wiki_link" href="/meantone">meantone</a> can be obtained as a combination of a certain number of octaves up or down, plus a certain number of flattened meantone fifths up or down. The terminology comes from group theory; in the parlance of group theory, the intervals of a regular temperament comprise a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Free_abelian_group#Rank" rel="nofollow">finitely generated free abelian group</a> with a rank equal to the number of generators.<br /> | ||
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The <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Codimension" rel="nofollow">codimension</a> or <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Free_abelian_group#Rank" rel="nofollow">co-rank</a> of a temperament is the number of <a class="wiki_link" href="/Comma">commas</a> needed to completely define the temperament. If the temperament tempers the <a class="wiki_link" href="/Harmonic%20Limit">p-limit</a> just intonation group generated by the first n primes, then if it tempers out n-r independent commas, it will be of rank r. The terminology can also be applied to <a class="wiki_link" href="/just%20intonation%20subgroups">just intonation subgroups</a>. In all cases care must be taken to specify the exact just intonation group which is being tempered by the tempering out of a set of commas.</body></html></pre></div> | The <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Codimension" rel="nofollow">codimension</a> or <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Free_abelian_group#Rank" rel="nofollow">co-rank</a> of a temperament is the number of <a class="wiki_link" href="/Comma">commas</a> needed to completely define the temperament. If the temperament tempers the <a class="wiki_link" href="/Harmonic%20Limit">p-limit</a> just intonation group generated by the first n primes, then if it tempers out n-r independent commas, it will be of rank r and codimension n-r. The terminology can also be applied to <a class="wiki_link" href="/just%20intonation%20subgroups">just intonation subgroups</a>. In all cases care must be taken to specify the exact just intonation group which is being tempered by the tempering out of a set of commas.</body></html></pre></div> | ||