Rank and codimension
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Original Wikitext content:
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.
Original HTML content:
<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 /> <br /> 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>