Fokker block: Difference between revisions
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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>2012-03-02 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-03-02 02:17:27 UTC</tt>.<br> | ||
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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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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. | 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. | ||
Graham complexity for S with respect to a wedgie W defines a complexity measure for the wedgies which makes the wedgies which determine if the scale S is a Fokker block precisely those f lowest complexity. However, for some purposed a quadratically (L2) defined complexity measure with similar properties is of use. We can define such a complexity measure for wedgies W by setting T[i] = (W∨S[i])(2), and then taking the sum ∑(T[i] - μ)^2 for i from 0 to P-1, where μ is the mean (∑T[i])/P. This can be analyzed in terms of the associated positive definite bilinear form on the linear combinations of basis elements giving W, and it is clear that past a certain range which can be determined the quadratic complexity measure will continue to increase, and that if needed one can in this way prove that a block is not Fokker.</pre></div> | Graham complexity for S with respect to a wedgie W defines a complexity measure for the wedgies which makes the wedgies which determine if the scale S is a Fokker block precisely those f lowest complexity. However, for some purposed a quadratically (L2) defined complexity measure with similar properties is of use. We can define such a complexity measure for wedgies W by setting T[i] = (W∨S[i])(2), and then taking the sum ∑(T[i] - μ)^2 for i from 0 to P-1, where μ is the mean (∑T[i])/P. This can be analyzed in terms of the associated positive definite bilinear form on the linear combinations of basis elements giving W, and it is clear that past a certain range which can be determined the quadratic complexity measure will continue to increase, and that if needed one can in this way prove that a block is not Fokker. | ||
=Example= | |||
Consider the periodic scale S[i] with quasiperiod P = 22 whose values for i from 0 to 22 are 1, 33/32, 16/15, 11/10, 9/8, 75/64, 6/5, 5/4, 165/128, 33/25, 11/8, 45/32, 35/24, 3/2, 99/64, 8/5, 33/20, 12/7, 7/4, 231/128, 15/8, 77/40, 2. By solving for the val, or simply testing to see if the patent val works, we quickly find that v = <22 35 51 62 76| sorts the scale in ascending order. A basis for the commas of this val is {50/49, 55/54, 64/63, 99/98}, and by taking three element subsets we find a basis for the wedgies to be {<<1 9 -2 -6 12 -6 -13 -30 -45 -10||, <<2 -4 -4 -12 -11 -12 -26 2 -14 -20||, <<6 10 10 8 2 -1 -8 -5 -16 -12||, <<2 -4 -4 10 -11 -12 9 2 37 42||}, which is to say, {suprapyth, pajara, hedgehog, pajarous}. Taking Z-linear (integer coefficient) combinations, we quickly find that there are four and only four wedgies which give a Graham complexity for the scale less than 22, </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:14:&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:14 --><!-- ws:start:WikiTextTocRule:15: --><a href="#Preliminaries">Preliminaries</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> | <a href="#First definition of a Fokker block">First definition of a Fokker block</a><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --> | <a href="#Second definition of a Fokker block">Second definition of a Fokker block</a><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --> | <a href="#Third definition of a Fokker block">Third definition of a Fokker block</a><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --> | <a href="#Fourth definition of a Fokker block">Fourth definition of a Fokker block</a><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --> | <a href="#Determining if a scale is a Fokker block">Determining if a scale is a Fokker block</a><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --> | <a href="#Example">Example</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --> | ||
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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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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.<br /> | 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.<br /> | ||
<br /> | <br /> | ||
Graham complexity for S with respect to a wedgie W defines a complexity measure for the wedgies which makes the wedgies which determine if the scale S is a Fokker block precisely those f lowest complexity. However, for some purposed a quadratically (L2) defined complexity measure with similar properties is of use. We can define such a complexity measure for wedgies W by setting T[i] = (W∨S[i])(2), and then taking the sum ∑(T[i] - μ)^2 for i from 0 to P-1, where μ is the mean (∑T[i])/P. This can be analyzed in terms of the associated positive definite bilinear form on the linear combinations of basis elements giving W, and it is clear that past a certain range which can be determined the quadratic complexity measure will continue to increase, and that if needed one can in this way prove that a block is not Fokker.</body></html></pre></div> | Graham complexity for S with respect to a wedgie W defines a complexity measure for the wedgies which makes the wedgies which determine if the scale S is a Fokker block precisely those f lowest complexity. However, for some purposed a quadratically (L2) defined complexity measure with similar properties is of use. We can define such a complexity measure for wedgies W by setting T[i] = (W∨S[i])(2), and then taking the sum ∑(T[i] - μ)^2 for i from 0 to P-1, where μ is the mean (∑T[i])/P. This can be analyzed in terms of the associated positive definite bilinear form on the linear combinations of basis elements giving W, and it is clear that past a certain range which can be determined the quadratic complexity measure will continue to increase, and that if needed one can in this way prove that a block is not Fokker.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Example"></a><!-- ws:end:WikiTextHeadingRule:12 -->Example</h1> | |||
Consider the periodic scale S[i] with quasiperiod P = 22 whose values for i from 0 to 22 are 1, 33/32, 16/15, 11/10, 9/8, 75/64, 6/5, 5/4, 165/128, 33/25, 11/8, 45/32, 35/24, 3/2, 99/64, 8/5, 33/20, 12/7, 7/4, 231/128, 15/8, 77/40, 2. By solving for the val, or simply testing to see if the patent val works, we quickly find that v = &lt;22 35 51 62 76| sorts the scale in ascending order. A basis for the commas of this val is {50/49, 55/54, 64/63, 99/98}, and by taking three element subsets we find a basis for the wedgies to be {&lt;&lt;1 9 -2 -6 12 -6 -13 -30 -45 -10||, &lt;&lt;2 -4 -4 -12 -11 -12 -26 2 -14 -20||, &lt;&lt;6 10 10 8 2 -1 -8 -5 -16 -12||, &lt;&lt;2 -4 -4 10 -11 -12 9 2 37 42||}, which is to say, {suprapyth, pajara, hedgehog, pajarous}. Taking Z-linear (integer coefficient) combinations, we quickly find that there are four and only four wedgies which give a Graham complexity for the scale less than 22,</body></html></pre></div> | |||