Fokker block: Difference between revisions
Wikispaces>genewardsmith **Imported revision 306603170 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 306655448 - 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>2012-02 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-03-01 02:14:46 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>306655448</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> | ||
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=Fourth definition of a Fokker block= | =Fourth definition of a Fokker block= | ||
The n-1 abstract MOS scales discussed with the second definition of Fokker blocks can be put into some chosen order, and the [[product word]] taken. This entails that every Fokker block leads to a product word, and the process can be reversed, so that product words of n-1 abstract MOS scales lead to Fokker blocks. Given the n-1 vals obtained by taking the interior product with some interval q, q can be recovered either by wedging the vals together and taking the [[The dual|dual]], or by taking the determinant of the nxn matrix of vals whose first row consists of indeterminates, as in the Preliminaries section. | The n-1 abstract MOS scales discussed with the second definition of Fokker blocks can be put into some chosen order, and the [[product word]] taken. This entails that every Fokker block leads to a product word, and the process can be reversed, so that product words of n-1 abstract MOS scales lead to Fokker blocks. Given the n-1 vals obtained by taking the interior product with some interval q, q can be recovered either by wedging the vals together and taking the [[The dual|dual]], or by taking the determinant of the nxn matrix of vals whose first row consists of indeterminates, as in the Preliminaries section. | ||
</pre></div> | |||
=Determining if a scale is a Fokker block= | |||
The second definition of Fokker block can be used to determine if a given periodic JI scale is a Fokker block. The first step is to find if it is epimorphic; this can be done by starting with a val V with indeterminate coefficients, and finding if the linear equations V(S[i]) = i have a solution. [[Scala]] does this as a part of its "Show data" suite of scale analytics. Now we take note of the fact that if r is the rank of the group generated by the scale (which is therefore the minimal JI system it is defined in) the set of bivals associated to V (which may defined as all bivals W such that V∧W = 0) is a free abelian group of rank r-1. We will assume we are working in a full p-limit group, but nothing essential is changed in Fokker block theory in the case of subgroups. The free group, defined by addition of bivals, has a basis consisting of ∓Wk for some set of wedgies, and we may assume the sign is positive and the basis is a basis of wedgies. Using this basis, we may either find a basis of r-1 wedgies each of which gives a [[Graham complexity]] to the scale reduced to the octave; that is, to S = {S[i]| 0 ≤ i < P} which is less than P, in which case the scale is a Fokker block, or determine no such basis exists, in which case it is not Fokker.</pre></div> | |||
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<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:12:&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:12 --><!-- ws:start:WikiTextTocRule:13: --><a href="#Preliminaries">Preliminaries</a><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --> | <a href="#First definition of a Fokker block">First definition of a Fokker block</a><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --> | <a href="#Second definition of a Fokker block">Second definition of a Fokker block</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> | <a href="#Third definition of a Fokker block">Third definition of a Fokker block</a><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --> | <a href="#Fourth definition of a Fokker block">Fourth definition of a Fokker block</a><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --> | <a href="#Determining if a scale is a Fokker block">Determining if a scale is a Fokker block</a><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --> | ||
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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 /> | ||
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<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Fourth definition of a Fokker block"></a><!-- ws:end:WikiTextHeadingRule:8 -->Fourth definition of a Fokker block</h1> | <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Fourth definition of a Fokker block"></a><!-- ws:end:WikiTextHeadingRule:8 -->Fourth definition of a Fokker block</h1> | ||
The n-1 abstract MOS scales discussed with the second definition of Fokker blocks can be put into some chosen order, and the <a class="wiki_link" href="/product%20word">product word</a> taken. This entails that every Fokker block leads to a product word, and the process can be reversed, so that product words of n-1 abstract MOS scales lead to Fokker blocks. Given the n-1 vals obtained by taking the interior product with some interval q, q can be recovered either by wedging the vals together and taking the <a class="wiki_link" href="/The%20dual">dual</a>, or by taking the determinant of the nxn matrix of vals whose first row consists of indeterminates, as in the Preliminaries section.</body></html></pre></div> | The n-1 abstract MOS scales discussed with the second definition of Fokker blocks can be put into some chosen order, and the <a class="wiki_link" href="/product%20word">product word</a> taken. This entails that every Fokker block leads to a product word, and the process can be reversed, so that product words of n-1 abstract MOS scales lead to Fokker blocks. Given the n-1 vals obtained by taking the interior product with some interval q, q can be recovered either by wedging the vals together and taking the <a class="wiki_link" href="/The%20dual">dual</a>, or by taking the determinant of the nxn matrix of vals whose first row consists of indeterminates, as in the Preliminaries section.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Determining if a scale is a Fokker block"></a><!-- ws:end:WikiTextHeadingRule:10 -->Determining if a scale is a Fokker block</h1> | |||
The second definition of Fokker block can be used to determine if a given periodic JI scale is a Fokker block. The first step is to find if it is epimorphic; this can be done by starting with a val V with indeterminate coefficients, and finding if the linear equations V(S[i]) = i have a solution. <a class="wiki_link" href="/Scala">Scala</a> does this as a part of its &quot;Show data&quot; suite of scale analytics. Now we take note of the fact that if r is the rank of the group generated by the scale (which is therefore the minimal JI system it is defined in) the set of bivals associated to V (which may defined as all bivals W such that V∧W = 0) is a free abelian group of rank r-1. We will assume we are working in a full p-limit group, but nothing essential is changed in Fokker block theory in the case of subgroups. The free group, defined by addition of bivals, has a basis consisting of ∓Wk for some set of wedgies, and we may assume the sign is positive and the basis is a basis of wedgies. Using this basis, we may either find a basis of r-1 wedgies each of which gives a <a class="wiki_link" href="/Graham%20complexity">Graham complexity</a> to the scale reduced to the octave; that is, to S = {S[i]| 0 ≤ i &lt; P} which is less than P, in which case the scale is a Fokker block, or determine no such basis exists, in which case it is not Fokker.</body></html></pre></div> | |||