Fokker block: Difference between revisions
Wikispaces>genewardsmith **Imported revision 311585012 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 312095674 - 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>2012-03-18 11:58:38 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 can be determined if these scales are Fokker by a variant of the method used for JI Fokker blocks. First, | 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, a [[transversal]] for the abstract scale is obtained by truncating the p-limit multivals to the the q-limit which makes them [[Rank and codimension|codimension]] one, and then taking [[the dual]] of each multival to obtain a monzo; the monzos then defining a scale in the ordinary way. If the transversal is a Fokker block, the original abstract scale is an abstract Fokker block. This procedure is illustrated below. | ||
=Example= | =Example= | ||
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<<<0 0 0 0 0 0 -1 -1 2 0|||] | <<<0 0 0 0 0 0 -1 -1 2 0|||] | ||
This represents an abstract scale defined in terms of 11-limit trivals derived from taking interior products of an unknown scale with an unknown 11-limit rank four temperament. Working with it directly is more difficult than dealing with the [[transversal]] we may obtain by [[The wedgie#Truncation of wedgies|truncation]]. If we truncate each scale step to the 7-limit, we obtain a list of 7-limit trivals. Each of these is [[The dual|dual]] to a monzo, which we may express in terms of a 7-limit rational number, leading to the following scale, from 1 to 22: 525/512, 16/15, 35/32, 9/8, 75/64, 6/5, 5/4, 2625/2048, 21/16, 175/128, 45/32, 35/24, 3/2, 1575/1024, 8/5, 105/64, 12/7, 7/4, 3675/2048, 15/8, 245/128, 2. This we may now test for Fokker properties in the usual way. | |||
The first order of business is to determine if the scale is epimorphic, which it is, with 22 patent val <22 35 51 62|. Using a basis for the Fokker group, for instance the one listed [[Minkowski reduced bases for Fokker groups of certain vals|here]], pajara-magic-porcupine, we find that pajara, porcupine and orwell all temper it to a MOS, so that the scale is a Fokker block. This is enough to prove the original scale is an abstract Fokker block; however, we might want a result in terms of the original 11-limit problem. By solving for the condition that the interior product with each scale step is zero, we find that 176/175 is the unique comma tempered out by the rank-four temperament which tempered to the abstract scale. Adding 176/175 to the commas of pajara, porcupine and orwell leads to the 11-limit versions of each of these. Taking the interior product of the dual scale of bimonzos with each of these 11-limit wedgies leads to the conclusion that each of these temper the abstract scale to a MOS. | |||
</pre></div> | </pre></div> | ||
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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 can be determined if these scales are Fokker by a variant of the method used for JI Fokker blocks. First, | 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, a <a class="wiki_link" href="/transversal">transversal</a> for the abstract scale is obtained by truncating the p-limit multivals to the the q-limit which makes them <a class="wiki_link" href="/Rank%20and%20codimension">codimension</a> one, and then taking <a class="wiki_link" href="/the%20dual">the dual</a> of each multival to obtain a monzo; the monzos then defining a scale in the ordinary way. If the transversal is a Fokker block, the original abstract scale is an abstract Fokker block. This procedure is illustrated below.<br /> | ||
<br /> | <br /> | ||
<!-- 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> | ||
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&lt;&lt;&lt;0 0 0 0 0 0 -1 -1 2 0|||]<br /> | &lt;&lt;&lt;0 0 0 0 0 0 -1 -1 2 0|||]<br /> | ||
<br /> | <br /> | ||
This represents an abstract scale defined in terms of 11-limit trivals derived from taking interior products of an unknown scale with an unknown 11-limit rank four temperament. Working with it directly is more difficult than dealing with the <a class="wiki_link" href="/transversal">transversal</a> we may obtain by <a class="wiki_link" href="/The%20wedgie#Truncation of wedgies">truncation</a>. If we truncate each scale step to the 7-limit, we obtain a list of 7-limit trivals. Each of these is <a class="wiki_link" href="/The%20dual">dual</a> to a monzo, which we may express in terms of a 7-limit rational number, leading to the following scale, from 1 to 22: 525/512, 16/15, 35/32, 9/8, 75/64, 6/5, 5/4, 2625/2048, 21/16, 175/128, 45/32, 35/24, 3/2, 1575/1024, 8/5, 105/64, 12/7, 7/4, 3675/2048, 15/8, 245/128, 2. This we may now test for Fokker properties in the usual way.<br /> | |||
<br /> | |||
The first order of business is to determine if the scale is epimorphic, which it is, with 22 patent val &lt;22 35 51 62|. Using a basis for the Fokker group, for instance the one listed <a class="wiki_link" href="/Minkowski%20reduced%20bases%20for%20Fokker%20groups%20of%20certain%20vals">here</a>, pajara-magic-porcupine, we find that pajara, porcupine and orwell all temper it to a MOS, so that the scale is a Fokker block. This is enough to prove the original scale is an abstract Fokker block; however, we might want a result in terms of the original 11-limit problem. By solving for the condition that the interior product with each scale step is zero, we find that 176/175 is the unique comma tempered out by the rank-four temperament which tempered to the abstract scale. Adding 176/175 to the commas of pajara, porcupine and orwell leads to the 11-limit versions of each of these. Taking the interior product of the dual scale of bimonzos with each of these 11-limit wedgies leads to the conclusion that each of these temper the abstract scale to a MOS.</body></html></pre></div> | |||