Fokker block: Difference between revisions

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<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>2012-03-02 23:49:41 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-03-03 02:24:36 UTC</tt>.<br>

: The original revision id was <tt>~~307335458~~</tt>.<br>

: The original revision id was <tt>307349344</tt>.<br>

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is the periodic scale with which we began this analysis.

==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.

==The tempered scales of a Fokker block==

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[[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>

<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>Fokker blocks</title></head><body><a href="#Preliminaries">Preliminaries</a> | <a href="#First definition of a Fokker block">First definition of a Fokker block</a> | <a href="#Second definition of a Fokker block">Second definition of a Fokker block</a> | <a href="#Third definition of a Fokker block">Third definition of a Fokker block</a> | <a href="#Fourth definition of a Fokker block">Fourth definition of a Fokker block</a> | <a href="#Determining if a scale is a Fokker block">Determining if a scale is a Fokker block</a> | <a href="#Example">Example</a>

<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>Fokker blocks</title></head><body><a href="#Preliminaries">Preliminaries</a> | <a href="#First definition of a Fokker block">First definition of a Fokker block</a> | <a href="#Second definition of a Fokker block">Second definition of a Fokker block</a> | <a href="#Third definition of a Fokker block">Third definition of a Fokker block</a> | <a href="#Fourth definition of a Fokker block">Fourth definition of a Fokker block</a> | <a href="#Determining if a scale is a Fokker block">Determining if a scale is a Fokker block</a> | <a href="#Example">Example</a>

<br />

<br />

The <strong>Fokker block</strong> is one of the most notable inventions of the physicist and music theorist <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Adriaan_Fokker" rel="nofollow">Adriaan Fokker</a>. While the idea generalizes easily to <a class="wiki_link" href="/just%20intonation%20subgroups">just intonation subgroups</a>, for ease of exposition we will suppose that we are in a <a class="wiki_link" href="/Harmonic%20Limit">p-limit</a> situation with n=pi(p) primes up to an including p.<br />

<br />

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is the periodic scale with which we began this analysis.<br />

<br />

<h2 id="toc9"><a name="Example-The tempered scales of a Fokker block"></a>The tempered scales of a Fokker block</h2>

<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. <br />

<br />

<h2 id="toc10"><a name="Example-The tempered scales of a Fokker block"></a>The tempered scales of a Fokker block</h2>

<br />

<img src="/file/view/1000px-Pajmagorpor22_temperament_support_lattice.svg.png/307264676/1000px-Pajmagorpor22_temperament_support_lattice.svg.png" alt="1000px-Pajmagorpor22_temperament_support_lattice.svg.png" title="1000px-Pajmagorpor22_temperament_support_lattice.svg.png" /><br />

<img src="/file/view/1000px-Pajmagorpor22_temperament_support_lattice.svg.png/307264676/1000px-Pajmagorpor22_temperament_support_lattice.svg.png" alt="1000px-Pajmagorpor22_temperament_support_lattice.svg.png" title="1000px-Pajmagorpor22_temperament_support_lattice.svg.png" /><br />

<br />

<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>

@@ 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>2012-03-02 23:49:41 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-03-03 02:24:36 UTC</tt>.<br>
-: The original revision id was <tt>307335458</tt>.<br>
+: The original revision id was <tt>307349344</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 58: / Line 58: @@
 is the periodic scale with which we began this analysis.
+==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.
 ==The tempered scales of a Fokker block==
@@ Line 65: / Line 73: @@
 [[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>
 <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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Fokker blocks&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:20:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt;&lt;a href="#Preliminaries"&gt;Preliminaries&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt; | &lt;a href="#First definition of a Fokker block"&gt;First definition of a Fokker block&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt; | &lt;a href="#Second definition of a Fokker block"&gt;Second definition of a Fokker block&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt; | &lt;a href="#Third definition of a Fokker block"&gt;Third definition of a Fokker block&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt; | &lt;a href="#Fourth definition of a Fokker block"&gt;Fourth definition of a Fokker block&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;!-- ws:start:WikiTextTocRule:26: --&gt; | &lt;a href="#Determining if a scale is a Fokker block"&gt;Determining if a scale is a Fokker block&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&gt; | &lt;a href="#Example"&gt;Example&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt;&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt;&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt;&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt;
+<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Fokker blocks&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:22:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt;&lt;a href="#Preliminaries"&gt;Preliminaries&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt; | &lt;a href="#First definition of a Fokker block"&gt;First definition of a Fokker block&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt; | &lt;a href="#Second definition of a Fokker block"&gt;Second definition of a Fokker block&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;!-- ws:start:WikiTextTocRule:26: --&gt; | &lt;a href="#Third definition of a Fokker block"&gt;Third definition of a Fokker block&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&gt; | &lt;a href="#Fourth definition of a Fokker block"&gt;Fourth definition of a Fokker block&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt; | &lt;a href="#Determining if a scale is a Fokker block"&gt;Determining if a scale is a Fokker block&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt; | &lt;a href="#Example"&gt;Example&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt;&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt;&lt;!-- ws:end:WikiTextTocRule:31 --&gt;&lt;!-- ws:start:WikiTextTocRule:32: --&gt;&lt;!-- ws:end:WikiTextTocRule:32 --&gt;&lt;!-- ws:start:WikiTextTocRule:33: --&gt;&lt;!-- ws:end:WikiTextTocRule:33 --&gt;&lt;!-- ws:start:WikiTextTocRule:34: --&gt;
-&lt;!-- ws:end:WikiTextTocRule:31 --&gt;&lt;br /&gt;
+&lt;!-- ws:end:WikiTextTocRule:34 --&gt;&lt;br /&gt;
 The &lt;strong&gt;Fokker block&lt;/strong&gt; is one of the most notable inventions of the physicist and music theorist &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Adriaan_Fokker" rel="nofollow"&gt;Adriaan Fokker&lt;/a&gt;. While the idea generalizes easily to &lt;a class="wiki_link" href="/just%20intonation%20subgroups"&gt;just intonation subgroups&lt;/a&gt;, for ease of exposition we will suppose that we are in a &lt;a class="wiki_link" href="/Harmonic%20Limit"&gt;p-limit&lt;/a&gt; situation with n=pi(p) primes up to an including p.&lt;br /&gt;
 &lt;br /&gt;
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 is the periodic scale with which we began this analysis.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Example-The tempered scales of a Fokker block"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;The tempered scales of a Fokker block&lt;/h2&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. &lt;br /&gt;
+&lt;br /&gt;
+&lt;br /&gt;
+&lt;br /&gt;
+&lt;br /&gt;
+&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Example-The tempered scales of a Fokker block"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;The tempered scales of a Fokker block&lt;/h2&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:32:&amp;lt;img src=&amp;quot;/file/view/1000px-Pajmagorpor22_temperament_support_lattice.svg.png/307264676/1000px-Pajmagorpor22_temperament_support_lattice.svg.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/1000px-Pajmagorpor22_temperament_support_lattice.svg.png/307264676/1000px-Pajmagorpor22_temperament_support_lattice.svg.png" alt="1000px-Pajmagorpor22_temperament_support_lattice.svg.png" title="1000px-Pajmagorpor22_temperament_support_lattice.svg.png" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:32 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:35:&amp;lt;img src=&amp;quot;/file/view/1000px-Pajmagorpor22_temperament_support_lattice.svg.png/307264676/1000px-Pajmagorpor22_temperament_support_lattice.svg.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/1000px-Pajmagorpor22_temperament_support_lattice.svg.png/307264676/1000px-Pajmagorpor22_temperament_support_lattice.svg.png" alt="1000px-Pajmagorpor22_temperament_support_lattice.svg.png" title="1000px-Pajmagorpor22_temperament_support_lattice.svg.png" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:35 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 &lt;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"&gt;Pajmagorpor22 lattice&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

Fokker block: Difference between revisions

Wikispaces>genewardsmith