Fokker block: Difference between revisions
Wikispaces>genewardsmith **Imported revision 307420274 - Original comment: ** |
Wikispaces>clumma **Imported revision 307452870 - 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: | : This revision was by author [[User:clumma|clumma]] and made on <tt>2012-03-03 17:17:36 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>307452870</tt>.<br> | ||
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As noted above, pajara, magic, orwell and porcupine correspond (in a way which depends on reference to octaves) the commas 385/384, 176/175, 100/99 and 225/224. If we take for example 385/384 and 176/175, we get zeus temperament. Wedging the monzos for these two commas and taking [[The dual|dual]] we obtain the wedgie for zeus, which is <<<2 -3 1 -1 -1 2 11 3 -10 4|||. Taking the interior product of this with the steps of our scale gives wxwwyzywzywxwwxwyzwyzy, where w = <<1 -3 5 -1 -7 5 -5 20 8 -20||, x = <<-3 5 -9 1 15 -6 12 -35 -15 34||, y = <<4 2 -1 3 -6 -13 -9 -8 0 12||, and z = <<-6 0 -3 -3 14 12 16 -7 -7 2||. If we set Orw[i] = orwell∨steps[i] and Por[i] = porcupine∨steps[i], then Zeus[i] = Orw[i]∧Por[i], which exhibits the scale tempered in zeus as a product word of the orwell MOS with the porcupine MOS. This procedure is easily turned into a formal proof which generalizes a result of Marek Zabka. | As noted above, pajara, magic, orwell and porcupine correspond (in a way which depends on reference to octaves) the commas 385/384, 176/175, 100/99 and 225/224. If we take for example 385/384 and 176/175, we get zeus temperament. Wedging the monzos for these two commas and taking [[The dual|dual]] we obtain the wedgie for zeus, which is <<<2 -3 1 -1 -1 2 11 3 -10 4|||. Taking the interior product of this with the steps of our scale gives wxwwyzywzywxwwxwyzwyzy, where w = <<1 -3 5 -1 -7 5 -5 20 8 -20||, x = <<-3 5 -9 1 15 -6 12 -35 -15 34||, y = <<4 2 -1 3 -6 -13 -9 -8 0 12||, and z = <<-6 0 -3 -3 14 12 16 -7 -7 2||. If we set Orw[i] = orwell∨steps[i] and Por[i] = porcupine∨steps[i], then Zeus[i] = Orw[i]∧Por[i], which exhibits the scale tempered in zeus as a product word of the orwell MOS with the porcupine MOS. This procedure is easily turned into a formal proof which generalizes a result of Marek Zabka. | ||
==The tempered scales of a Fokker block== | ==The tempered scales of a Fokker block== | ||
[[image: | [[image:600px-Pajmagorpor22_temperament_support_lattice.svg.png]] | ||
[[http://upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Pajmagorpor22_temperament_support_lattice.svg/2000px-Pajmagorpor22_temperament_support_lattice.svg.png|Pajmagorpor22 lattice]] </pre></div> | [[http://upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Pajmagorpor22_temperament_support_lattice.svg/2000px-Pajmagorpor22_temperament_support_lattice.svg.png|Pajmagorpor22 lattice]] </pre></div> | ||
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As noted above, pajara, magic, orwell and porcupine correspond (in a way which depends on reference to octaves) the commas 385/384, 176/175, 100/99 and 225/224. If we take for example 385/384 and 176/175, we get zeus temperament. Wedging the monzos for these two commas and taking <a class="wiki_link" href="/The%20dual">dual</a> we obtain the wedgie for zeus, which is &lt;&lt;&lt;2 -3 1 -1 -1 2 11 3 -10 4|||. Taking the interior product of this with the steps of our scale gives wxwwyzywzywxwwxwyzwyzy, where w = &lt;&lt;1 -3 5 -1 -7 5 -5 20 8 -20||, x = &lt;&lt;-3 5 -9 1 15 -6 12 -35 -15 34||, y = &lt;&lt;4 2 -1 3 -6 -13 -9 -8 0 12||, and z = &lt;&lt;-6 0 -3 -3 14 12 16 -7 -7 2||. If we set Orw[i] = orwell∨steps[i] and Por[i] = porcupine∨steps[i], then Zeus[i] = Orw[i]∧Por[i], which exhibits the scale tempered in zeus as a product word of the orwell MOS with the porcupine MOS. This procedure is easily turned into a formal proof which generalizes a result of Marek Zabka.<br /> | As noted above, pajara, magic, orwell and porcupine correspond (in a way which depends on reference to octaves) the commas 385/384, 176/175, 100/99 and 225/224. If we take for example 385/384 and 176/175, we get zeus temperament. Wedging the monzos for these two commas and taking <a class="wiki_link" href="/The%20dual">dual</a> we obtain the wedgie for zeus, which is &lt;&lt;&lt;2 -3 1 -1 -1 2 11 3 -10 4|||. Taking the interior product of this with the steps of our scale gives wxwwyzywzywxwwxwyzwyzy, where w = &lt;&lt;1 -3 5 -1 -7 5 -5 20 8 -20||, x = &lt;&lt;-3 5 -9 1 15 -6 12 -35 -15 34||, y = &lt;&lt;4 2 -1 3 -6 -13 -9 -8 0 12||, and z = &lt;&lt;-6 0 -3 -3 14 12 16 -7 -7 2||. If we set Orw[i] = orwell∨steps[i] and Por[i] = porcupine∨steps[i], then Zeus[i] = Orw[i]∧Por[i], which exhibits the scale tempered in zeus as a product word of the orwell MOS with the porcupine MOS. This procedure is easily turned into a formal proof which generalizes a result of Marek Zabka.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="Example-The tempered scales of a Fokker block"></a><!-- ws:end:WikiTextHeadingRule:20 -->The tempered scales of a Fokker block</h2> | <!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="Example-The tempered scales of a Fokker block"></a><!-- ws:end:WikiTextHeadingRule:20 -->The tempered scales of a Fokker block</h2> | ||
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<!-- ws:start:WikiTextLocalImageRule:35:&lt;img src=&quot;/file/view/ | <!-- ws:start:WikiTextLocalImageRule:35:&lt;img src=&quot;/file/view/600px-Pajmagorpor22_temperament_support_lattice.svg.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/600px-Pajmagorpor22_temperament_support_lattice.svg.png" alt="600px-Pajmagorpor22_temperament_support_lattice.svg.png" title="600px-Pajmagorpor22_temperament_support_lattice.svg.png" /><!-- ws:end:WikiTextLocalImageRule:35 --><br /> | ||
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<a class="wiki_link_ext" href="http://upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Pajmagorpor22_temperament_support_lattice.svg/2000px-Pajmagorpor22_temperament_support_lattice.svg.png" rel="nofollow">Pajmagorpor22 lattice</a></body></html></pre></div> | <a class="wiki_link_ext" href="http://upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Pajmagorpor22_temperament_support_lattice.svg/2000px-Pajmagorpor22_temperament_support_lattice.svg.png" rel="nofollow">Pajmagorpor22 lattice</a></body></html></pre></div> | ||