Fokker block: Difference between revisions
Wikispaces>genewardsmith **Imported revision 151755221 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 151756699 - 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>2010-07-06 21: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-07-06 21:55:32 UTC</tt>.<br> | ||
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===First definition of a Fokker block=== | ===First definition of a Fokker block=== | ||
Let us set ei = vi(2), and also P = e1 = v1(2), and choose n positive integers a1, ...., an with ak < P. Let ti = log2(ci), so that e1*t1+e2*t2+...+en*tn=1. Now define a function on the integers by | |||
S[i] = floor((e1*i + a1)/P)*t1 + ... + floor((en*i + an)/P)*tn | 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 < | 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 < P 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 | S[i + P] = S[i] + e1*t1 + e2*t2 + ... + en*tn = S[i] + 1 | ||
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<!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x--First definition of a Fokker block"></a><!-- ws:end:WikiTextHeadingRule:2 -->First definition of a Fokker block</h3> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x--First definition of a Fokker block"></a><!-- ws:end:WikiTextHeadingRule:2 -->First definition of a Fokker block</h3> | ||
Let us set ei = vi(2), and also P = e1 = v1(2), and choose n positive integers a1, ...., an with ak &lt; P. Let ti = log2(ci), so that e1*t1+e2*t2+...+en*tn=1. Now define a function on the integers by<br /> | |||
<br /> | <br /> | ||
S[i] = floor((e1*i + a1)/P)*t1 + ... + floor((en*i + an)/P)*tn<br /> | S[i] = floor((e1*i + a1)/P)*t1 + ... + floor((en*i + an)/P)*tn<br /> | ||
<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; | 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; P each term is 0 and so S[0] = 0. Since for integer j, floor(x+j) = floor(x) + j, we have<br /> | ||
<br /> | <br /> | ||
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.</body></html></pre></div> | ||