Generator
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- This revision was by author genewardsmith and made on 2011-01-29 18:10:09 UTC.
- The original revision id was 197087796.
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Original Wikitext content:
A [[http://en.wikipedia.org/wiki/Generating_set_of_a_group|set of generators]] for a [[http://en.wikipedia.org/wiki/Group_%28mathematics%29|group]] is a subset of the elements of the group which is not contained in any [[http://en.wikipedia.org/wiki/Subgroup|proper subgroup]], which is to say, any subgroup which is not the whole group. If the set is a finite set, the group is called finitely generated. If it is also an [[http://en.wikipedia.org/wiki/Abelian_group|abelian group]], it is called a [[http://en.wikipedia.org/wiki/Finitely_generated_abelian_group|finitely generated abelian group]].
If the abelian group is written additively, then if {g1, g2, ... gk} is the generating set, every element g of the group can be written
g = n1g1 + n2g2 + ... + nkgk
where the ni are integers. If the group operation is multiplicative,
g = g1^n1 g2^n2 ... gk^nk
An important example is provided by [[Regular Temperaments|regular temperaments]], where if a particular tuning for the temperament is written additively, the generators can be taken as intervals expressed in terms of cents, and if multiplicatively, as intervals given as frequency ratios. Another example is provided by [[just intonation subgroups]], where the generators are a finite set of positive rational numbers.
These two example converge when we seek generators for the [[Abstract regular temperament|abstract temperament]] rather than any particular tuning of it. One way to obtain these is to use the [[http://mathworld.wolfram.com/InteriorProduct.html|interior product]] of a [[Wedgies and Multivals|wedgie]] for a p-limit temperament with the p-limit monzos. A less abstract approach is to define a [[transversal]] for the temperament by giving generators for a just intonation subgroup which when tempered becomes the notes of the temperament.
For example, for [[Gamelismic clan|miracle temperament]] [2, 15/14] defines a rank two 7-limit subgroup whose [[Normal lists|normal interval list]] is [2, 15/7]. We might also use [2, 16/15], with a normal interval list [2, 15]. When tempered by miracle, [2, 15/14] and [2, 16/15] lead to the same notes; hence we can use either for a transversal. Either pair may be considered a generating set for the abstract temperament.
Alternatively, using "v" to represent the interior product, we have mir = <<6 -7 -2 -25 -20 15|| for the wedgie of 7-limit miracle. Then the interior product mir v |1 0 0 0> is <0 -6 7 2|, with 15/14 we get mir v |-1 1 1 -1> is <1 1 3 3|, and with 16/15 we get mir v |4 -1 -1 0> is also <1 1 3 3|. Unlike the case for just intonation subgroups, the rank two group for the abstract temperament in terms of interior products is already exactly the same.Original HTML content:
<html><head><title>Generators</title></head><body>A <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Generating_set_of_a_group" rel="nofollow">set of generators</a> for a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Group_%28mathematics%29" rel="nofollow">group</a> is a subset of the elements of the group which is not contained in any <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Subgroup" rel="nofollow">proper subgroup</a>, which is to say, any subgroup which is not the whole group. If the set is a finite set, the group is called finitely generated. If it is also an <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Abelian_group" rel="nofollow">abelian group</a>, it is called a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Finitely_generated_abelian_group" rel="nofollow">finitely generated abelian group</a>.<br />
<br />
If the abelian group is written additively, then if {g1, g2, ... gk} is the generating set, every element g of the group can be written<br />
<br />
g = n1g1 + n2g2 + ... + nkgk<br />
<br />
where the ni are integers. If the group operation is multiplicative,<br />
<br />
g = g1^n1 g2^n2 ... gk^nk<br />
<br />
An important example is provided by <a class="wiki_link" href="/Regular%20Temperaments">regular temperaments</a>, where if a particular tuning for the temperament is written additively, the generators can be taken as intervals expressed in terms of cents, and if multiplicatively, as intervals given as frequency ratios. Another example is provided by <a class="wiki_link" href="/just%20intonation%20subgroups">just intonation subgroups</a>, where the generators are a finite set of positive rational numbers.<br />
<br />
These two example converge when we seek generators for the <a class="wiki_link" href="/Abstract%20regular%20temperament">abstract temperament</a> rather than any particular tuning of it. One way to obtain these is to use the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/InteriorProduct.html" rel="nofollow">interior product</a> of a <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a> for a p-limit temperament with the p-limit monzos. A less abstract approach is to define a <a class="wiki_link" href="/transversal">transversal</a> for the temperament by giving generators for a just intonation subgroup which when tempered becomes the notes of the temperament.<br />
<br />
For example, for <a class="wiki_link" href="/Gamelismic%20clan">miracle temperament</a> [2, 15/14] defines a rank two 7-limit subgroup whose <a class="wiki_link" href="/Normal%20lists">normal interval list</a> is [2, 15/7]. We might also use [2, 16/15], with a normal interval list [2, 15]. When tempered by miracle, [2, 15/14] and [2, 16/15] lead to the same notes; hence we can use either for a transversal. Either pair may be considered a generating set for the abstract temperament.<br />
<br />
Alternatively, using "v" to represent the interior product, we have mir = <<6 -7 -2 -25 -20 15|| for the wedgie of 7-limit miracle. Then the interior product mir v |1 0 0 0> is <0 -6 7 2|, with 15/14 we get mir v |-1 1 1 -1> is <1 1 3 3|, and with 16/15 we get mir v |4 -1 -1 0> is also <1 1 3 3|. Unlike the case for just intonation subgroups, the rank two group for the abstract temperament in terms of interior products is already exactly the same.</body></html>