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:~~guest~~|~~guest~~]] and made on <tt>2010-07-06 20:17:49 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-07-06 21:41:19 UTC</tt>.<br>

: The original revision id was <tt>~~151747285~~</tt>.<br>

: The original revision id was <tt>151755221</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>

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===First definition of a Fokker block===

~~There exist positive real numbers t1, ... tn,~~ positive integers e1, ..., en, and ~~positive real numbers~~ a1, ...., an with e1+e2+...+en = N,

Now set P = v1(2) and choose two sets of n positive integers e1, ..., en, and a1, ...., an with e1+e2+...+en = P and ak < P; and let ti = log2(ci), choosing the e1 so that e1*t1+e2*t2+...+en*tn=1. Now define a function on the integers by

e1*t1 + ... + en*tn = ~~O, and ak <~~ 1 ~~so that~~

S[i] = floor((e1~~/N)~~*i + a1)*t1+...+floor((en/N)*i + an)*tn</pre></div>

S[i] = floor((e1*i + a1)/P)*t1 + ... + floor((en*i + an)/P)*tn

Here floor(x) is the [[http://en.wikipedia.org/wiki/Floor_and_ceiling_functions|floor function]], the [[http://en.wikipedia.org/wiki/Quasiperiodic_function|quasiperiodic function]] returning the largest integer less than or equal to x. When i=0, since ak < N each term is 0 and so S[0] = 0. Since for integer j, floor(x+j) = floor(x) + j, we have

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>

<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>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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<br />

<h3 id="toc1"><a name="x--First definition of a Fokker block"></a>First definition of a Fokker block</h3>

~~There exist positive real numbers t1, ... tn,~~ positive integers e1, ..., en, and ~~positive real numbers~~ a1, ...., an with e1+e2+...+en = N,<br />

Now set P = v1(2) and choose two sets of n positive integers e1, ..., en, and a1, ...., an with e1+e2+...+en = P and ak &lt; P; and let ti = log2(ci), choosing the e1 so that e1*t1+e2*t2+...+en*tn=1. Now define a function on the integers by<br />

e1*t1 + ... + en*tn = O, ~~and~~ ak &lt; 1 ~~so that~~<br />

<br />

S[i] = floor((e1*i + a1)/P)*t1 + ... + floor((en*i + an)/P)*tn<br />

<br />

Here floor(x) is the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Floor_and_ceiling_functions" rel="nofollow">floor function</a>, the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Quasiperiodic_function" rel="nofollow">quasiperiodic function</a> returning the largest integer less than or equal to x. When i=0, since ak &lt; N each term is 0 and so S[0] = 0. Since for integer j, floor(x+j) = floor(x) + j, we have<br />

<br />

S[i + P] = S[i] + e1*t1 + e2*t2 + ... + en*tn = S[i] + 1<br />

<br />

S~~[i]~~ = ~~floor((e1~~/~~N)*i + a1)*t1+.~~..~~+floor((en/N)*i + an)*tn~~</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.</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:guest|guest]] and made on <tt>2010-07-06 20:17:49 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-07-06 21:41:19 UTC</tt>.<br>
-: The original revision id was <tt>151747285</tt>.<br>
+: The original revision id was <tt>151755221</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 18: / Line 18: @@
 ===First definition of a Fokker block===
-There exist positive real numbers t1, ... tn, positive integers e1, ..., en, and positive real numbers a1, ...., an with e1+e2+...+en = N,
+Now set P = v1(2) and choose two sets of n positive integers e1, ..., en, and a1, ...., an with e1+e2+...+en = P and ak &lt; P; and let ti = log2(ci), choosing the e1 so that e1*t1+e2*t2+...+en*tn=1. Now define a function on the integers by
-e1*t1 + ... + en*tn = O, and ak &lt; 1 so that
-S[i] = floor((e1/N)*i + a1)*t1+...+floor((en/N)*i + an)*tn</pre></div>
+S[i] = floor((e1*i + a1)/P)*t1 + ... + floor((en*i + an)/P)*tn
+Here floor(x) is the [[http://en.wikipedia.org/wiki/Floor_and_ceiling_functions|floor function]], the [[http://en.wikipedia.org/wiki/Quasiperiodic_function|quasiperiodic function]] returning the largest integer less than or equal to x. When i=0, since ak &lt; N each term is 0 and so S[0] = 0. Since for integer j, floor(x+j) = floor(x) + j, we have
+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>
 <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;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;
@@ Line 35: / Line 40: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x--First definition of a Fokker block"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;First definition of a Fokker block&lt;/h3&gt;
-  There exist positive real numbers t1, ... tn, positive integers e1, ..., en, and positive real numbers a1, ...., an with e1+e2+...+en = N,&lt;br /&gt;
+  Now set P = v1(2) and choose two sets of n positive integers e1, ..., en, and a1, ...., an with e1+e2+...+en = P and ak &amp;lt; P; and let ti = log2(ci), choosing the e1 so that e1*t1+e2*t2+...+en*tn=1. Now define a function on the integers by&lt;br /&gt;
-e1*t1 + ... + en*tn = O, and ak &amp;lt; 1 so that&lt;br /&gt;
+&lt;br /&gt;
+S[i] = floor((e1*i + a1)/P)*t1 + ... + floor((en*i + an)/P)*tn&lt;br /&gt;
+&lt;br /&gt;
+Here floor(x) is the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Floor_and_ceiling_functions" rel="nofollow"&gt;floor function&lt;/a&gt;, the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Quasiperiodic_function" rel="nofollow"&gt;quasiperiodic function&lt;/a&gt; returning the largest integer less than or equal to x. When i=0, since ak &amp;lt; N each term is 0 and so S[0] = 0. Since for integer j, floor(x+j) = floor(x) + j, we have&lt;br /&gt;
+&lt;br /&gt;
+S[i + P] = S[i] + e1*t1 + e2*t2 + ... + en*tn = S[i] + 1&lt;br /&gt;
 &lt;br /&gt;
-S[i] = floor((e1/N)*i + a1)*t1+...+floor((en/N)*i + an)*tn&lt;/body&gt;&lt;/html&gt;</pre></div>
+Hence S satisfies the conditions for being a &lt;a class="wiki_link" href="/Periodic%20scale"&gt;periodic scale&lt;/a&gt;, 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.&lt;/body&gt;&lt;/html&gt;</pre></div>

Fokker block: Difference between revisions

Wikispaces>genewardsmith