Fokker block: Difference between revisions
Wikispaces>genewardsmith **Imported revision 310655370 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 311257204 - 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>2012-03- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-03-15 04:11:52 UTC</tt>.<br> | ||
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One way to generalize this is to allow the [[Just intonation subgroups|group of the scale]] to be something other than the full p-limit group, adjusting the basis for vals, monzos and wedgies to correspond with a basis for this subgroup. We may also replace the interval of equivalence 2 with any rational number E which is not a power, so that S[i + P] = E S[i] and replacing Tk[i](2) with Tk[i](E). | One way to generalize this is to allow the [[Just intonation subgroups|group of the scale]] to be something other than the full p-limit group, adjusting the basis for vals, monzos and wedgies to correspond with a basis for this subgroup. We may also replace the interval of equivalence 2 with any rational number E which is not a power, so that S[i + P] = E S[i] and replacing Tk[i](2) with Tk[i](E). | ||
Still another generalization is to consider regular temperament Fokker blocks. These are abstract temperament periodic scales S[i] with values in an abstract regular temperament belonging to some r-wedgie Y. It | Still another generalization is to consider regular temperament Fokker blocks. These are abstract temperament periodic scales S[i] with values in an abstract regular temperament belonging to some r-wedgie Y. It can be determined if these scales are Fokker by a variant of the method used for JI Fokker blocks. First, taking differences between steps gives intervals belonging to the zero class. Finding commas for these, we can recover the val V as the val tempering them all out, and we can recover the defining wedgie W for the values by the condition that wedging it with the scale steps leads to zero. Now by taking interior products of the scale steps with multimonzos of rank r-2 (modulo commatic ones which lead to zero), we can define Graham complexity for abstract temperament scales and thus determine if the scale is an abstract Fokker block. | ||
=Example= | =Example= | ||
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One way to generalize this is to allow the <a class="wiki_link" href="/Just%20intonation%20subgroups">group of the scale</a> to be something other than the full p-limit group, adjusting the basis for vals, monzos and wedgies to correspond with a basis for this subgroup. We may also replace the interval of equivalence 2 with any rational number E which is not a power, so that S[i + P] = E S[i] and replacing Tk[i](2) with Tk[i](E). <br /> | One way to generalize this is to allow the <a class="wiki_link" href="/Just%20intonation%20subgroups">group of the scale</a> to be something other than the full p-limit group, adjusting the basis for vals, monzos and wedgies to correspond with a basis for this subgroup. We may also replace the interval of equivalence 2 with any rational number E which is not a power, so that S[i + P] = E S[i] and replacing Tk[i](2) with Tk[i](E). <br /> | ||
<br /> | <br /> | ||
Still another generalization is to consider regular temperament Fokker blocks. These are abstract temperament periodic scales S[i] with values in an abstract regular temperament belonging to some r-wedgie Y. It | Still another generalization is to consider regular temperament Fokker blocks. These are abstract temperament periodic scales S[i] with values in an abstract regular temperament belonging to some r-wedgie Y. It can be determined if these scales are Fokker by a variant of the method used for JI Fokker blocks. First, taking differences between steps gives intervals belonging to the zero class. Finding commas for these, we can recover the val V as the val tempering them all out, and we can recover the defining wedgie W for the values by the condition that wedging it with the scale steps leads to zero. Now by taking interior products of the scale steps with multimonzos of rank r-2 (modulo commatic ones which lead to zero), we can define Graham complexity for abstract temperament scales and thus determine if the scale is an abstract Fokker block.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc7"><a name="Example"></a><!-- ws:end:WikiTextHeadingRule:14 -->Example</h1> | <!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc7"><a name="Example"></a><!-- ws:end:WikiTextHeadingRule:14 -->Example</h1> | ||