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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> | ||
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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. | 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. | ||
Looking only at the number of independent generators of a tuning can obscure its real nature, at least as it is being applied. For instance, a 31et tuning of meantone temperament, with a meantone fifth of 18\31 octaves, is of rank one in the sense that all the intervals in the tuning are generated from 1\31; however, it is being used as a rank two tuning. This issue can be gotten around by means of [[ | Looking only at the number of independent generators of a tuning can obscure its real nature, at least as it is being applied. For instance, a 31et tuning of meantone temperament, with a meantone fifth of 18\31 octaves, is of rank one in the sense that all the intervals in the tuning are generated from 1\31; however, it is being used as a rank two tuning. This issue can be gotten around by means of [[@http://et3allem.blogspot.com|abstract regular temperaments]]; an abstract regular temperament is of rank r if it is defined by a [[Normal lists|normal val list]] of r vals, or equivalently by an r-multival. The abstractly characterized intervals of the abstract temperament can then be mapped to a tuning; if the mapping is to a rank one tuning such as 31et, that does not affect the rank of the temperament. | ||
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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;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 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.<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 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.<br /> | ||
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Looking only at the number of independent generators of a tuning can obscure its real nature, at least as it is being applied. For instance, a 31et tuning of meantone temperament, with a meantone fifth of 18\31 octaves, is of rank one in the sense that all the intervals in the tuning are generated from 1\31; however, it is being used as a rank two tuning. This issue can be gotten around by means of <a class=" | Looking only at the number of independent generators of a tuning can obscure its real nature, at least as it is being applied. For instance, a 31et tuning of meantone temperament, with a meantone fifth of 18\31 octaves, is of rank one in the sense that all the intervals in the tuning are generated from 1\31; however, it is being used as a rank two tuning. This issue can be gotten around by means of <a class="wiki_link_ext" href="http://et3allem.blogspot.com" rel="nofollow" target="_blank">abstract regular temperaments</a>; an abstract regular temperament is of rank r if it is defined by a <a class="wiki_link" href="/Normal%20lists">normal val list</a> of r vals, or equivalently by an r-multival. The abstractly characterized intervals of the abstract temperament can then be mapped to a tuning; if the mapping is to a rank one tuning such as 31et, that does not affect the rank of the temperament.<br /> | ||
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Revision as of 20:24, 31 October 2012
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author guest and made on 2012-10-31 20:24:48 UTC.
- The original revision id was 378163648.
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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 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. Looking only at the number of independent generators of a tuning can obscure its real nature, at least as it is being applied. For instance, a 31et tuning of meantone temperament, with a meantone fifth of 18\31 octaves, is of rank one in the sense that all the intervals in the tuning are generated from 1\31; however, it is being used as a rank two tuning. This issue can be gotten around by means of [[@http://et3allem.blogspot.com|abstract regular temperaments]]; an abstract regular temperament is of rank r if it is defined by a [[Normal lists|normal val list]] of r vals, or equivalently by an r-multival. The abstractly characterized intervals of the abstract temperament can then be mapped to a tuning; if the mapping is to a rank one tuning such as 31et, that does not affect the rank of the temperament. [[@http://et3allem.blogspot.com/|Click Here]] [[@http://et3allem.blogspot.com/|Click Here]] [[@http://et3allem.blogspot.com/|Click Here]] [[@http://et3allem.blogspot.com/|Click Here]] [[@http://et3allem.blogspot.com/|Click Here]] [[@http://et3allem.blogspot.com/|Click Here]] [[@http://et3allem.blogspot.com/|Click Here]] [[@http://et3allem.blogspot.com/|Click Here]] [[@http://et3allem.blogspot.com/|Click Here]] [[@http://et3allem.blogspot.com/|Click Here]] [[@http://et3allem.blogspot.com/|Click Here]] [[@http://et3allem.blogspot.com/|Click Here]] [[@http://et3allem.blogspot.com/|Click Here]] [[@http://et3allem.blogspot.com/|Click Here]] [[@http://et3allem.blogspot.com/|Click Here]] [[@http://et3allem.blogspot.com/|Click Here]]
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 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.<br /> <br /> Looking only at the number of independent generators of a tuning can obscure its real nature, at least as it is being applied. For instance, a 31et tuning of meantone temperament, with a meantone fifth of 18\31 octaves, is of rank one in the sense that all the intervals in the tuning are generated from 1\31; however, it is being used as a rank two tuning. This issue can be gotten around by means of <a class="wiki_link_ext" href="http://et3allem.blogspot.com" rel="nofollow" target="_blank">abstract regular temperaments</a>; an abstract regular temperament is of rank r if it is defined by a <a class="wiki_link" href="/Normal%20lists">normal val list</a> of r vals, or equivalently by an r-multival. The abstractly characterized intervals of the abstract temperament can then be mapped to a tuning; if the mapping is to a rank one tuning such as 31et, that does not affect the rank of the temperament.<br /> <br /> <a class="wiki_link_ext" href="http://et3allem.blogspot.com/" rel="nofollow" target="_blank">Click Here</a><br /> <a class="wiki_link_ext" href="http://et3allem.blogspot.com/" rel="nofollow" target="_blank">Click Here</a><br /> <a class="wiki_link_ext" href="http://et3allem.blogspot.com/" rel="nofollow" target="_blank">Click Here</a><br /> <a class="wiki_link_ext" href="http://et3allem.blogspot.com/" rel="nofollow" target="_blank">Click Here</a><br /> <a class="wiki_link_ext" href="http://et3allem.blogspot.com/" rel="nofollow" target="_blank">Click Here</a><br /> <a class="wiki_link_ext" href="http://et3allem.blogspot.com/" rel="nofollow" target="_blank">Click Here</a><br /> <a class="wiki_link_ext" href="http://et3allem.blogspot.com/" rel="nofollow" target="_blank">Click Here</a><br /> <a class="wiki_link_ext" href="http://et3allem.blogspot.com/" rel="nofollow" target="_blank">Click Here</a><br /> <a class="wiki_link_ext" href="http://et3allem.blogspot.com/" rel="nofollow" target="_blank">Click Here</a><br /> <a class="wiki_link_ext" href="http://et3allem.blogspot.com/" rel="nofollow" target="_blank">Click Here</a><br /> <a class="wiki_link_ext" href="http://et3allem.blogspot.com/" rel="nofollow" target="_blank">Click Here</a><br /> <a class="wiki_link_ext" href="http://et3allem.blogspot.com/" rel="nofollow" target="_blank">Click Here</a><br /> <a class="wiki_link_ext" href="http://et3allem.blogspot.com/" rel="nofollow" target="_blank">Click Here</a><br /> <a class="wiki_link_ext" href="http://et3allem.blogspot.com/" rel="nofollow" target="_blank">Click Here</a><br /> <br /> <a class="wiki_link_ext" href="http://et3allem.blogspot.com/" rel="nofollow" target="_blank">Click Here</a><br /> <a class="wiki_link_ext" href="http://et3allem.blogspot.com/" rel="nofollow" target="_blank">Click Here</a></body></html>