Fokker block: Difference between revisions
Wikispaces>genewardsmith **Imported revision 306112932 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 306531666 - 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>2012-02-29 16:39:34 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>306531666</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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S[i + P] = S[i] + e1*t1 + e2*t2 + ... + en*tn = S[i] + 1 | S[i + P] = S[i] + e1*t1 + e2*t2 + ... + en*tn = S[i] + 1 | ||
Hence S satisfies the conditions for being a [[Periodic scale|periodic scale]], and since our unit of measurement is the octave, ie we are using log base two to define intervals, the repetition interval 1 represents an octave. This gives us our first definition of Fokker block.</pre></div> | Hence S satisfies the conditions for being a [[Periodic scale|periodic scale]], and since our unit of measurement is the octave, ie we are using log base two to define intervals, the repetition interval 1 represents an octave. This gives us our first definition of Fokker block. | ||
=Second definition of a Fokker block= | |||
Let is define a new set of vals by uk = P*vk - vk(2)*v1. To apply these vals to S[i], note first that floor((e1*i+a1)/P) = floor(i+a1/P) = i, so that v1(S[i]) = i. Hence for k>1, uk(S[i]) = P*vk(S[i]) - vk(2)*i. Since x-1 < floor(x) ≤ x, we have (ek*i + ak)/P-1 < floor((ek*i + ak)/P) ≤ (ek*i + ak)/P, so that | |||
ek*i + ak - P < P*vk(S[i]) ≤ ek*i + ak. Since ek = vk(2), this gives us ak - P < uk(S[i]) ≤ ak. This means that for each of the vals uk, the scale is mapped to a set of P integers.</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:6:&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:6 --><!-- ws:start:WikiTextTocRule:7: --><a href="#Preliminaries">Preliminaries</a><!-- ws:end:WikiTextTocRule:7 --><!-- ws:start:WikiTextTocRule:8: --> | <a href="#First definition of a Fokker block">First definition of a Fokker block</a><!-- ws:end:WikiTextTocRule:8 --><!-- ws:start:WikiTextTocRule:9: --> | <a href="#Second definition of a Fokker block">Second definition of a Fokker block</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:10 --><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 /> | ||
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S[i + P] = S[i] + e1*t1 + e2*t2 + ... + en*tn = S[i] + 1<br /> | S[i + P] = S[i] + e1*t1 + e2*t2 + ... + en*tn = S[i] + 1<br /> | ||
<br /> | <br /> | ||
Hence S satisfies the conditions for being a <a class="wiki_link" href="/Periodic%20scale">periodic scale</a>, and since our unit of measurement is the octave, ie we are using log base two to define intervals, the repetition interval 1 represents an octave. This gives us our first definition of Fokker block.</body></html></pre></div> | Hence S satisfies the conditions for being a <a class="wiki_link" href="/Periodic%20scale">periodic scale</a>, and since our unit of measurement is the octave, ie we are using log base two to define intervals, the repetition interval 1 represents an octave. This gives us our first definition of Fokker block.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Second definition of a Fokker block"></a><!-- ws:end:WikiTextHeadingRule:4 -->Second definition of a Fokker block</h1> | |||
Let is define a new set of vals by uk = P*vk - vk(2)*v1. To apply these vals to S[i], note first that floor((e1*i+a1)/P) = floor(i+a1/P) = i, so that v1(S[i]) = i. Hence for k&gt;1, uk(S[i]) = P*vk(S[i]) - vk(2)*i. Since x-1 &lt; floor(x) ≤ x, we have (ek*i + ak)/P-1 &lt; floor((ek*i + ak)/P) ≤ (ek*i + ak)/P, so that<br /> | |||
ek*i + ak - P &lt; P*vk(S[i]) ≤ ek*i + ak. Since ek = vk(2), this gives us ak - P &lt; uk(S[i]) ≤ ak. This means that for each of the vals uk, the scale is mapped to a set of P integers.</body></html></pre></div> | |||