Fokker block: Difference between revisions

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==Product words and the fourth definition of a Fokker block==

Starting from our example 22 note per octave scale, we can produce a list of 22 steps: 33/32, 512/495, 33/32, 45/44, 25/24, 128/125, 25/24, 33/32, 128/125, 25/24, 45/44, 28/27, 36/35, 33/32, 512/495, 33/32, 80/77, 49/48, 33/32, 80/77, 77/75, 80/77. We can apply the four wedgies for pajara, magic, orwell and porcupine to these steps to obtain four abstract temperament MOS, each of which has two kinds of steps, expressed as vals. If a = -<10 16 23 28 34| and b = <12 19 28 34 42|, then pajara applied to the steps gives abababaabababababaabab. If c = -<3 5 7 9 10| and d = <19 30 44 53 66|, then magic gives cccdccccccdccccccdcccc. If e = <9 14 21 25 31| and f = -<13 21 30 37 45|, then orwell gives efeefefeefefeefefeefef. Finally, if g = <7 11 16 20 24| and h = -<15 24 35 42 52|, then porcupine gives ghggghgghgghgghgghgghg. By taking product words, we get not only the Fokker block itself, but also the various temperings in the associated temperaments. Here "product" means product in a quite literal sense, since these can be construed as wedge product words.

Starting from our example 22 note per octave scale, we can produce a list of 22 steps: steps[i] = 33/32, 512/495, 33/32, 45/44, 25/24, 128/125, 25/24, 33/32, 128/125, 25/24, 45/44, 28/27, 36/35, 33/32, 512/495, 33/32, 80/77, 49/48, 33/32, 80/77, 77/75, 80/77. We can apply the four wedgies for pajara, magic, orwell and porcupine to these steps to obtain four abstract temperament MOS, each of which has two kinds of steps, expressed as vals. If a = -<10 16 23 28 34| and b = <12 19 28 34 42|, then pajara applied to the steps gives abababaabababababaabab. If c = -<3 5 7 9 10| and d = <19 30 44 53 66|, then magic gives cccdccccccdccccccdcccc. If e = <9 14 21 25 31| and f = -<13 21 30 37 45|, then orwell gives efeefefeefefeefefeefef. Finally, if g = <7 11 16 20 24| and h = -<15 24 35 42 52|, then porcupine gives ghggghgghgghgghgghgghg. By taking product words, we get not only the Fokker block itself, but also the various temperings in the associated temperaments. Here "product" means product in a quite literal sense, since these can be construed as wedge product words.

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.

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<br />

<h2 id="toc9"><a name="Example-Product words and the fourth definition of a Fokker block"></a>Product words and the fourth definition of a Fokker block</h2>

Starting from our example 22 note per octave scale, we can produce a list of 22 steps: 33/32, 512/495, 33/32, 45/44, 25/24, 128/125, 25/24, 33/32, 128/125, 25/24, 45/44, 28/27, 36/35, 33/32, 512/495, 33/32, 80/77, 49/48, 33/32, 80/77, 77/75, 80/77. We can apply the four wedgies for pajara, magic, orwell and porcupine to these steps to obtain four abstract temperament MOS, each of which has two kinds of steps, expressed as vals. If a = -&lt;10 16 23 28 34| and b = &lt;12 19 28 34 42|, then pajara applied to the steps gives abababaabababababaabab. If c = -&lt;3 5 7 9 10| and d = &lt;19 30 44 53 66|, then magic gives cccdccccccdccccccdcccc. If e = &lt;9 14 21 25 31| and f = -&lt;13 21 30 37 45|, then orwell gives efeefefeefefeefefeefef. Finally, if g = &lt;7 11 16 20 24| and h = -&lt;15 24 35 42 52|, then porcupine gives ghggghgghgghgghgghgghg. By taking product words, we get not only the Fokker block itself, but also the various temperings in the associated temperaments. Here &quot;product&quot; means product in a quite literal sense, since these can be construed as wedge product words. <br />

Starting from our example 22 note per octave scale, we can produce a list of 22 steps: steps[i] = 33/32, 512/495, 33/32, 45/44, 25/24, 128/125, 25/24, 33/32, 128/125, 25/24, 45/44, 28/27, 36/35, 33/32, 512/495, 33/32, 80/77, 49/48, 33/32, 80/77, 77/75, 80/77. We can apply the four wedgies for pajara, magic, orwell and porcupine to these steps to obtain four abstract temperament MOS, each of which has two kinds of steps, expressed as vals. If a = -&lt;10 16 23 28 34| and b = &lt;12 19 28 34 42|, then pajara applied to the steps gives abababaabababababaabab. If c = -&lt;3 5 7 9 10| and d = &lt;19 30 44 53 66|, then magic gives cccdccccccdccccccdcccc. If e = &lt;9 14 21 25 31| and f = -&lt;13 21 30 37 45|, then orwell gives efeefefeefefeefefeefef. Finally, if g = &lt;7 11 16 20 24| and h = -&lt;15 24 35 42 52|, then porcupine gives ghggghgghgghgghgghgghg. By taking product words, we get not only the Fokker block itself, but also the various temperings in the associated temperaments. Here &quot;product&quot; means product in a quite literal sense, since these can be construed as wedge product words. <br />

<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 />

<br />

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2012-03-03 12:22:16 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-03-03 13:35:10 UTC</tt>.<br>
-: The original revision id was <tt>307410518</tt>.<br>
+: The original revision id was <tt>307420274</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>
@@ Line 62: / Line 62: @@
 ==Product words and the fourth definition of a Fokker block==
-Starting from our example 22 note per octave scale, we can produce a list of 22 steps: 33/32, 512/495, 33/32, 45/44, 25/24, 128/125, 25/24, 33/32, 128/125, 25/24, 45/44, 28/27, 36/35, 33/32, 512/495, 33/32, 80/77, 49/48, 33/32, 80/77, 77/75, 80/77. We can apply the four wedgies for pajara, magic, orwell and porcupine to these steps to obtain four abstract temperament MOS, each of which has two kinds of steps, expressed as vals. If a = -&lt;10 16 23 28 34| and b = &lt;12 19 28 34 42|, then pajara applied to the steps gives abababaabababababaabab. If c = -&lt;3 5 7 9 10| and d = &lt;19 30 44 53 66|, then magic gives cccdccccccdccccccdcccc. If e = &lt;9 14 21 25 31| and f = -&lt;13 21 30 37 45|, then orwell gives efeefefeefefeefefeefef. Finally, if g = &lt;7 11 16 20 24| and h = -&lt;15 24 35 42 52|, then porcupine gives ghggghgghgghgghgghgghg. By taking product words, we get not only the Fokker block itself, but also the various temperings in the associated temperaments. Here "product" means product in a quite literal sense, since these can be construed as wedge product words.
+Starting from our example 22 note per octave scale, we can produce a list of 22 steps: steps[i] = 33/32, 512/495, 33/32, 45/44, 25/24, 128/125, 25/24, 33/32, 128/125, 25/24, 45/44, 28/27, 36/35, 33/32, 512/495, 33/32, 80/77, 49/48, 33/32, 80/77, 77/75, 80/77. We can apply the four wedgies for pajara, magic, orwell and porcupine to these steps to obtain four abstract temperament MOS, each of which has two kinds of steps, expressed as vals. If a = -&lt;10 16 23 28 34| and b = &lt;12 19 28 34 42|, then pajara applied to the steps gives abababaabababababaabab. If c = -&lt;3 5 7 9 10| and d = &lt;19 30 44 53 66|, then magic gives cccdccccccdccccccdcccc. If e = &lt;9 14 21 25 31| and f = -&lt;13 21 30 37 45|, then orwell gives efeefefeefefeefefeefef. Finally, if g = &lt;7 11 16 20 24| and h = -&lt;15 24 35 42 52|, then porcupine gives ghggghgghgghgghgghgghg. By taking product words, we get not only the Fokker block itself, but also the various temperings in the associated temperaments. Here "product" means product in a quite literal sense, since these can be construed as wedge product words.
+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 &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.
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 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Example-Product words and the fourth definition of a Fokker block"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Product words and the fourth definition of a Fokker block&lt;/h2&gt;
-Starting from our example 22 note per octave scale, we can produce a list of 22 steps: 33/32, 512/495, 33/32, 45/44, 25/24, 128/125, 25/24, 33/32, 128/125, 25/24, 45/44, 28/27, 36/35, 33/32, 512/495, 33/32, 80/77, 49/48, 33/32, 80/77, 77/75, 80/77. We can apply the four wedgies for pajara, magic, orwell and porcupine to these steps to obtain four abstract temperament MOS, each of which has two kinds of steps, expressed as vals. If a = -&amp;lt;10 16 23 28 34| and b = &amp;lt;12 19 28 34 42|, then pajara applied to the steps gives abababaabababababaabab. If c = -&amp;lt;3 5 7 9 10| and d = &amp;lt;19 30 44 53 66|, then magic gives cccdccccccdccccccdcccc. If e = &amp;lt;9 14 21 25 31| and f = -&amp;lt;13 21 30 37 45|, then orwell gives efeefefeefefeefefeefef. Finally, if g = &amp;lt;7 11 16 20 24| and h = -&amp;lt;15 24 35 42 52|, then porcupine gives ghggghgghgghgghgghgghg. By taking product words, we get not only the Fokker block itself, but also the various temperings in the associated temperaments. Here &amp;quot;product&amp;quot; means product in a quite literal sense, since these can be construed as wedge product words. &lt;br /&gt;
+Starting from our example 22 note per octave scale, we can produce a list of 22 steps: steps[i] = 33/32, 512/495, 33/32, 45/44, 25/24, 128/125, 25/24, 33/32, 128/125, 25/24, 45/44, 28/27, 36/35, 33/32, 512/495, 33/32, 80/77, 49/48, 33/32, 80/77, 77/75, 80/77. We can apply the four wedgies for pajara, magic, orwell and porcupine to these steps to obtain four abstract temperament MOS, each of which has two kinds of steps, expressed as vals. If a = -&amp;lt;10 16 23 28 34| and b = &amp;lt;12 19 28 34 42|, then pajara applied to the steps gives abababaabababababaabab. If c = -&amp;lt;3 5 7 9 10| and d = &amp;lt;19 30 44 53 66|, then magic gives cccdccccccdccccccdcccc. If e = &amp;lt;9 14 21 25 31| and f = -&amp;lt;13 21 30 37 45|, then orwell gives efeefefeefefeefefeefef. Finally, if g = &amp;lt;7 11 16 20 24| and h = -&amp;lt;15 24 35 42 52|, then porcupine gives ghggghgghgghgghgghgghg. By taking product words, we get not only the Fokker block itself, but also the various temperings in the associated temperaments. Here &amp;quot;product&amp;quot; means product in a quite literal sense, since these can be construed as wedge product words. &lt;br /&gt;
+&lt;br /&gt;
+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 &lt;a class="wiki_link" href="/The%20dual"&gt;dual&lt;/a&gt; we obtain the wedgie for zeus, which is &amp;lt;&amp;lt;&amp;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 = &amp;lt;&amp;lt;1 -3 5 -1 -7 5 -5 20 8 -20||, x = &amp;lt;&amp;lt;-3 5 -9 1 15 -6 12 -35 -15 34||, y = &amp;lt;&amp;lt;4 2 -1 3 -6 -13 -9 -8 0 12||, and z = &amp;lt;&amp;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.&lt;br /&gt;
 &lt;br /&gt;
 &lt;br /&gt;