Gencom: Difference between revisions
Wikispaces>genewardsmith **Imported revision 345412104 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 345428762 - 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>2012-06-14 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-06-14 15:51:15 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>345428762</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
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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The rows of the extended 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 fractional vals always return an integer value, which gives the number of times the corresponding generator or comma appears in the expression of the interval in terms of the gencom. 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. | The rows of the extended 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 fractional vals always return an integer value, which gives the number of times the corresponding generator or comma appears in the expression of the interval in terms of the gencom. 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. | ||
Converting a gencom to a [[Normal lists#x-Normal interval lists|normal interval list]] gives a canonical form for the subgroup, which is an invariant of the temperament. Doing the same to just the commas produces another invariant, and together these determine the temperament: it is the unique temperament on the given group tempering out the given commas. The normal list defined by the generators alone is not an invariant of the temperament, since the generators give only a transversal for the tempered intervals of the temperament, not the full set of intervals being tempered. Hence, for instance, [2 40/27; 81/80] and [2 3/2; 81/80] both define 5-limit meantone, but the normal list for [2 40/27] is 2.27/5 and for [2 3/2] is 2.3.</pre></div> | Converting a gencom to a [[Normal lists#x-Normal interval lists|normal interval list]] gives a canonical form for the subgroup, which is an invariant of the temperament. Doing the same to just the commas produces another invariant, and together these determine the temperament: it is the unique temperament on the given group tempering out the given commas. The normal list defined by the generators alone is not an invariant of the temperament, since the generators give only a transversal for the tempered intervals of the temperament, not the full set of intervals being tempered. Hence, for instance, [2 40/27; 81/80] and [2 3/2; 81/80] both define 5-limit meantone, but the normal list for [2 40/27] is 2.27/5 and for [2 3/2] is 2.3. However, the extended gencom mapping can be used to determine if an interval q is in the group of the temperament. Suppose [c1 c2 ... cn] is a gencom and [v1 v2 ... vn] is the corresponding extended mapping. Then each of v1(q), v2(q) ... vn(q) must be an integer, and moreover we must have q = c1^v1(q) * c2^v2(q) ... cn^vn(q). This provides sufficient coditions as well as necessary ones.</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>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 /> | <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>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 /> | ||
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The rows of the extended 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 fractional vals always return an integer value, which gives the number of times the corresponding generator or comma appears in the expression of the interval in terms of the gencom. 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.<br /> | The rows of the extended 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 fractional vals always return an integer value, which gives the number of times the corresponding generator or comma appears in the expression of the interval in terms of the gencom. 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.<br /> | ||
<br /> | <br /> | ||
Converting a gencom to a <a class="wiki_link" href="/Normal%20lists#x-Normal interval lists">normal interval list</a> gives a canonical form for the subgroup, which is an invariant of the temperament. Doing the same to just the commas produces another invariant, and together these determine the temperament: it is the unique temperament on the given group tempering out the given commas. The normal list defined by the generators alone is not an invariant of the temperament, since the generators give only a transversal for the tempered intervals of the temperament, not the full set of intervals being tempered. Hence, for instance, [2 40/27; 81/80] and [2 3/2; 81/80] both define 5-limit meantone, but the normal list for [2 40/27] is 2.27/5 and for [2 3/2] is 2.3.</body></html></pre></div> | Converting a gencom to a <a class="wiki_link" href="/Normal%20lists#x-Normal interval lists">normal interval list</a> gives a canonical form for the subgroup, which is an invariant of the temperament. Doing the same to just the commas produces another invariant, and together these determine the temperament: it is the unique temperament on the given group tempering out the given commas. The normal list defined by the generators alone is not an invariant of the temperament, since the generators give only a transversal for the tempered intervals of the temperament, not the full set of intervals being tempered. Hence, for instance, [2 40/27; 81/80] and [2 3/2; 81/80] both define 5-limit meantone, but the normal list for [2 40/27] is 2.27/5 and for [2 3/2] is 2.3. However, the extended gencom mapping can be used to determine if an interval q is in the group of the temperament. Suppose [c1 c2 ... cn] is a gencom and [v1 v2 ... vn] is the corresponding extended mapping. Then each of v1(q), v2(q) ... vn(q) must be an integer, and moreover we must have q = c1^v1(q) * c2^v2(q) ... cn^vn(q). This provides sufficient coditions as well as necessary ones.</body></html></pre></div> | ||