Fokker block: Difference between revisions
Wikispaces>genewardsmith **Imported revision 311257204 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 311585012 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<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>2012-03-15 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-03-15 21:44:41 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>311585012</tt>.<br> | ||
: The revision comment was: <tt></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> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
| Line 81: | Line 81: | ||
[[image:http://upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Pajmagorpor22_temperament_support_lattice.svg/2000px-Pajmagorpor22_temperament_support_lattice.svg.png height="170" width="714" link="http://upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Pajmagorpor22_temperament_support_lattice.svg/2000px-Pajmagorpor22_temperament_support_lattice.svg.png"]] | [[image:http://upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Pajmagorpor22_temperament_support_lattice.svg/2000px-Pajmagorpor22_temperament_support_lattice.svg.png height="170" width="714" link="http://upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Pajmagorpor22_temperament_support_lattice.svg/2000px-Pajmagorpor22_temperament_support_lattice.svg.png"]] | ||
One has first the [[pajmagorpor22|original JI scale]]. Then there are codimension one temperings of the scale, in each of the commas associated to the Fokker block; in our example these are [[pajmagorpor22_225|225/224]], [[pajmagorpor22_100|100/99]], [[pajmagorpor22_176|176/175]], and [[pajmagorpor22_385|385/384]]. The next level gives [[pajmagorpor22apollo|apollo]], [[pajmagorpor22minerva|minerva]], [[pajmagorpor22marvel|marvel]], [[pajmagorpor22ares|ares]], [[pajmagorpor22supermagic|supermagic]], and [[pajmagorpor22zeus|zeus]]. Next come pajara, magic, orwell and porcupine, with the range of generators already given, and then finally 22 equal. Exploring the changes wrought by the various scales in such a Fokker universe, not to mention all of the modes and domes, would certainly give the interested composer plenty to work with.</pre></div> | One has first the [[pajmagorpor22|original JI scale]]. Then there are codimension one temperings of the scale, in each of the commas associated to the Fokker block; in our example these are [[pajmagorpor22_225|225/224]], [[pajmagorpor22_100|100/99]], [[pajmagorpor22_176|176/175]], and [[pajmagorpor22_385|385/384]]. The next level gives [[pajmagorpor22apollo|apollo]], [[pajmagorpor22minerva|minerva]], [[pajmagorpor22marvel|marvel]], [[pajmagorpor22ares|ares]], [[pajmagorpor22supermagic|supermagic]], and [[pajmagorpor22zeus|zeus]]. Next come pajara, magic, orwell and porcupine, with the range of generators already given, and then finally 22 equal. Exploring the changes wrought by the various scales in such a Fokker universe, not to mention all of the modes and domes, would certainly give the interested composer plenty to work with. | ||
==Example of an abstract Fokker block== | |||
Let S be the abstract scale defined by, for scale steps from 1 to 22: | |||
[<<<1 -2 0 1 1 -2 9 5 -10 -4|||, <<<0 1 1 -1 -1 2 -4 -4 4 4|||, <<<1 -1 1 0 0 0 5 1 -6 0|||, | |||
<<<0 0 0 2 2 -4 3 3 -6 -8|||, <<<0 -2 -2 1 1 -2 6 6 -4 -4|||, <<<0 1 1 1 1 -2 -1 -1 -2 -4|||, | |||
<<<0 -1 -1 0 0 0 2 2 0 0|||, <<<1 -3 -1 1 1 -2 11 7 -10 -4|||, <<<1 0 2 1 1 -2 4 0 -8 -4|||, | |||
<<<1 -2 0 0 0 0 7 3 -6 0|||, <<<0 -1 -1 2 2 -4 5 5 -6 -8|||, <<<1 -1 1 -1 -1 2 3 -1 -2 4|||, | |||
<<<0 0 0 1 1 -2 1 1 -2 -4|||, <<<1 -2 0 2 2 -4 10 6 -12 -8|||, <<<0 1 1 0 0 0 -3 -3 2 0|||, | |||
<<<1 -1 1 1 1 -2 6 2 -8 -4|||, <<<-1 0 -2 1 1 -2 -2 2 4 -4|||, <<<1 0 2 0 0 0 2 -2 -4 0|||, | |||
<<<2 -2 2 1 1 -2 11 3 -14 -4|||, <<<0 -1 -1 1 1 -2 3 3 -2 -4|||, <<<2 -1 3 0 0 0 7 -1 -10 0|||, | |||
<<<0 0 0 0 0 0 -1 -1 2 0|||] | |||
By subtracting S[i+1]-S[i] we can form the scale steps, and by taking differences between scale steps we can find a finite set of commas of the scale--intervals belonging to the unison class. Solving for a val V such that V(c)=0 for all commas in the unison class gives the 11-limit 22edo patent val V = <22 35 51 62 76|. | |||
</pre></div> | |||
<h4>Original HTML content:</h4> | <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><!-- ws:start:WikiTextTocRule: | <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><!-- ws:start:WikiTextTocRule:26:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --><a href="#Preliminaries">Preliminaries</a><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --> | <a href="#First definition of a Fokker block">First definition of a Fokker block</a><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --> | <a href="#Second definition of a Fokker block">Second definition of a Fokker block</a><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --> | <a href="#Third definition of a Fokker block">Third definition of a Fokker block</a><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --> | <a href="#Fourth definition of a Fokker block">Fourth definition of a Fokker block</a><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --> | <a href="#Determining if a scale is a Fokker block">Determining if a scale is a Fokker block</a><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --> | <a href="#Expanding the definition">Expanding the definition</a><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --> | <a href="#Example">Example</a><!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --><!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --><!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --><!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --><!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --><!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:40 --><br /> | ||
<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 /> | 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 /> | ||
| Line 156: | Line 171: | ||
A Fokker block is not just a scale, but a little scale universe of tempered versions of that scale which identify various steps of the scale, as depicted below.<br /> | A Fokker block is not just a scale, but a little scale universe of tempered versions of that scale which identify various steps of the scale, as depicted below.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextRemoteImageRule: | <!-- ws:start:WikiTextRemoteImageRule:42:&lt;a href=&quot;http://upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Pajmagorpor22_temperament_support_lattice.svg/2000px-Pajmagorpor22_temperament_support_lattice.svg.png&quot; rel=&quot;nofollow&quot;&gt;&lt;img src=&quot;http://upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Pajmagorpor22_temperament_support_lattice.svg/2000px-Pajmagorpor22_temperament_support_lattice.svg.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 170px; width: 714px;&quot; /&gt;&lt;/a&gt; --><a href="http://upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Pajmagorpor22_temperament_support_lattice.svg/2000px-Pajmagorpor22_temperament_support_lattice.svg.png" rel="nofollow"><img src="http://upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Pajmagorpor22_temperament_support_lattice.svg/2000px-Pajmagorpor22_temperament_support_lattice.svg.png" alt="external image 2000px-Pajmagorpor22_temperament_support_lattice.svg.png" title="external image 2000px-Pajmagorpor22_temperament_support_lattice.svg.png" style="height: 170px; width: 714px;" /></a><!-- ws:end:WikiTextRemoteImageRule:42 --><br /> | ||
<br /> | |||
One has first the <a class="wiki_link" href="/pajmagorpor22">original JI scale</a>. Then there are codimension one temperings of the scale, in each of the commas associated to the Fokker block; in our example these are <a class="wiki_link" href="/pajmagorpor22_225">225/224</a>, <a class="wiki_link" href="/pajmagorpor22_100">100/99</a>, <a class="wiki_link" href="/pajmagorpor22_176">176/175</a>, and <a class="wiki_link" href="/pajmagorpor22_385">385/384</a>. The next level gives <a class="wiki_link" href="/pajmagorpor22apollo">apollo</a>, <a class="wiki_link" href="/pajmagorpor22minerva">minerva</a>, <a class="wiki_link" href="/pajmagorpor22marvel">marvel</a>, <a class="wiki_link" href="/pajmagorpor22ares">ares</a>, <a class="wiki_link" href="/pajmagorpor22supermagic">supermagic</a>, and <a class="wiki_link" href="/pajmagorpor22zeus">zeus</a>. Next come pajara, magic, orwell and porcupine, with the range of generators already given, and then finally 22 equal. Exploring the changes wrought by the various scales in such a Fokker universe, not to mention all of the modes and domes, would certainly give the interested composer plenty to work with.<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:24:&lt;h2&gt; --><h2 id="toc12"><a name="Example-Example of an abstract Fokker block"></a><!-- ws:end:WikiTextHeadingRule:24 -->Example of an abstract Fokker block</h2> | |||
Let S be the abstract scale defined by, for scale steps from 1 to 22:<br /> | |||
[&lt;&lt;&lt;1 -2 0 1 1 -2 9 5 -10 -4|||, &lt;&lt;&lt;0 1 1 -1 -1 2 -4 -4 4 4|||, &lt;&lt;&lt;1 -1 1 0 0 0 5 1 -6 0|||, <br /> | |||
&lt;&lt;&lt;0 0 0 2 2 -4 3 3 -6 -8|||, &lt;&lt;&lt;0 -2 -2 1 1 -2 6 6 -4 -4|||, &lt;&lt;&lt;0 1 1 1 1 -2 -1 -1 -2 -4|||, <br /> | |||
&lt;&lt;&lt;0 -1 -1 0 0 0 2 2 0 0|||, &lt;&lt;&lt;1 -3 -1 1 1 -2 11 7 -10 -4|||, &lt;&lt;&lt;1 0 2 1 1 -2 4 0 -8 -4|||, <br /> | |||
&lt;&lt;&lt;1 -2 0 0 0 0 7 3 -6 0|||, &lt;&lt;&lt;0 -1 -1 2 2 -4 5 5 -6 -8|||, &lt;&lt;&lt;1 -1 1 -1 -1 2 3 -1 -2 4|||, <br /> | |||
&lt;&lt;&lt;0 0 0 1 1 -2 1 1 -2 -4|||, &lt;&lt;&lt;1 -2 0 2 2 -4 10 6 -12 -8|||, &lt;&lt;&lt;0 1 1 0 0 0 -3 -3 2 0|||, <br /> | |||
&lt;&lt;&lt;1 -1 1 1 1 -2 6 2 -8 -4|||, &lt;&lt;&lt;-1 0 -2 1 1 -2 -2 2 4 -4|||, &lt;&lt;&lt;1 0 2 0 0 0 2 -2 -4 0|||, <br /> | |||
&lt;&lt;&lt;2 -2 2 1 1 -2 11 3 -14 -4|||, &lt;&lt;&lt;0 -1 -1 1 1 -2 3 3 -2 -4|||, &lt;&lt;&lt;2 -1 3 0 0 0 7 -1 -10 0|||, <br /> | |||
&lt;&lt;&lt;0 0 0 0 0 0 -1 -1 2 0|||]<br /> | |||
<br /> | <br /> | ||
By subtracting S[i+1]-S[i] we can form the scale steps, and by taking differences between scale steps we can find a finite set of commas of the scale--intervals belonging to the unison class. Solving for a val V such that V(c)=0 for all commas in the unison class gives the 11-limit 22edo patent val V = &lt;22 35 51 62 76|.</body></html></pre></div> | |||