Gencom
IMPORTED REVISION FROM WIKISPACES
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- This revision was by author genewardsmith and made on 2012-06-14 02:20:22 UTC.
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Original Wikitext content:
A //gencom// is a list of generators for a temperament followed by commas for the temperament, in a specific order. The generators are [[transversal generators]], meaning rational intervals belonging to the JI group the temperament tempers, which it tempers to generators for the temperment. The gencom is denoted [generator list; comma list[, with a semicolon between the generators and the commas. For instance, [16/15, 25/24; 81/80] is a gencom for 5-limit meantone. On the other hand, the exact same intervals with a different placement of the semicolon is a gencom for 5-limit JI: [16/15, 25/24, 81/80;], and another, [16/15; 25/24, 81/80] gives 5-limit 7-equal. The reason for putting the generators together with the commas is that notating the gencom as a list of [[monzos]] allows it to be treated as a matrix. The group of intervals generated by the gencom is the same no matter how we place the semicolon, and so is the matrix. When this group is a full p-limit group, as in the example above, the matrix is a [[http://en.wikipedia.org/wiki/Unimodular_matrix|unimodular matrix]]. Inverting and transposing gives a matrix whose rows are vals; if r is the rank of the temperament then the first r rows are the mapping matrix correspondign to the generator transversal. More interesting is the case where the gencom generates a [[Just intonation subgroups|JI subgroup]] of some p-limit. In all cases the transpose of the [[Tenney-Euclidean Tuning#The pseudoinverse|pseudoinverse]] of the matrix of monzos gives a matrix of vals whose first r rows we call the gencom mapping, and which in its entirety we call the extended gencom mapping. The extended gencom mapping is only a unimodular matrix, and the inversion ordinary matrix inversion, in the case of the full p-limit. However in all cases the transpose pesudoinverse of the gencom matrix is the extended gencom mapping, and the transpose pseudoinverse of the extended mapping is the gencom matrix. The rows of the gencom mapping are in general fractional vals, meaning the coefficients are allowed to be rational numbers. When applied to elements of the subgroup generated by the gencom, these always return an integer value. However, the converse is not the case: if the gencom mapping returns integer values, it does not mean the interval must belong to the gencom subgroup.
Original HTML content:
<html><head><title>Gencom</title></head><body>A <em>gencom</em> is a list of generators for a temperament followed by commas for the temperament, in a specific order. The generators are <a class="wiki_link" href="/transversal%20generators">transversal generators</a>, meaning rational intervals belonging to the JI group the temperament tempers, which it tempers to generators for the temperment. The gencom is denoted [generator list; comma list[, with a semicolon between the generators and the commas. For instance, [16/15, 25/24; 81/80] is a gencom for 5-limit meantone. On the other hand, the exact same intervals with a different placement of the semicolon is a gencom for 5-limit JI: [16/15, 25/24, 81/80;], and another, [16/15; 25/24, 81/80] gives 5-limit 7-equal.<br /> <br /> The reason for putting the generators together with the commas is that notating the gencom as a list of <a class="wiki_link" href="/monzos">monzos</a> allows it to be treated as a matrix. The group of intervals generated by the gencom is the same no matter how we place the semicolon, and so is the matrix. When this group is a full p-limit group, as in the example above, the matrix is a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Unimodular_matrix" rel="nofollow">unimodular matrix</a>. Inverting and transposing gives a matrix whose rows are vals; if r is the rank of the temperament then the first r rows are the mapping matrix correspondign to the generator transversal. More interesting is the case where the gencom generates a <a class="wiki_link" href="/Just%20intonation%20subgroups">JI subgroup</a> of some p-limit. In all cases the transpose of the <a class="wiki_link" href="/Tenney-Euclidean%20Tuning#The pseudoinverse">pseudoinverse</a> of the matrix of monzos gives a matrix of vals whose first r rows we call the gencom mapping, and which in its entirety we call the extended gencom mapping. The extended gencom mapping is only a unimodular matrix, and the inversion ordinary matrix inversion, in the case of the full p-limit. However in all cases the transpose pesudoinverse of the gencom matrix is the extended gencom mapping, and the transpose pseudoinverse of the extended mapping is the gencom matrix.<br /> <br /> The rows of the gencom mapping are in general fractional vals, meaning the coefficients are allowed to be rational numbers. When applied to elements of the subgroup generated by the gencom, these always return an integer value. However, the converse is not the case: if the gencom mapping returns integer values, it does not mean the interval must belong to the gencom subgroup.</body></html>