Fokker block: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-03-13 13:03:30 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-03-13 14:31:55 UTC</tt>.<br>

: The original revision id was <tt>~~310613022~~</tt>.<br>

: The original revision id was <tt>310655370</tt>.<br>

: The revision comment was: <tt></tt><br>

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=Third definition of a Fokker block=

The n-1 vals u2, u3, ..., un defined in the previous section gave us n-1 inequalities ak - P < uk(q) ≤ ak, which apply to any q in the Fokker block. If we restrict q to 1 ≤ q < 2, and regard it as representing a pitch class, then it is associated to a lattice point in an n-1 dimensional vector space, and in that space the n-1 inequalities define the boundries of a parallepiped. The Fokker blocks can be defined as the pitch classes lying within such a paralellepiped.

=Fourth definition of a Fokker block=

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=Expanding the definition=

A Fokker block as we have so far defined it is an epimorphic periodic scale S with period N repeating at the octave, with values in p-limit rational intonation, such that there exist pi(p)-1 = r-1 different rank-two wedgies {Wk} such that S has Graham complexity less than N for each Wk. If we unpack that definition we can extend it in several distinct ways.

A Fokker block as we have so far defined it is an epimorphic periodic scale S with period P repeating at the octave, with values in p-limit rational intonation, such that there exist pi(p)-1 = n-1 different rank-two wedgies {Wk} such that S has Graham complexity less than P for each Wk. If we unpack that definition we can extend it in several distinct ways.

Explicitly, S is a [[http://en.wikipedia.org/wiki/Quasiperiodic_function|quasiperiodic function]] from the integers to the p-limit rational numbers, such that S[0] = 1 and S[i + N] = 2S[i], for which there is a val V such that V(S[i]) = i. This entails that V(S[N]) = V(2) = N, so that V = <N ... |, with N a positive integer; in other words, V is ~~an N~~-edo val.

Explicitly, S is a [[http://en.wikipedia.org/wiki/Quasiperiodic_function|quasiperiodic function]] from the integers to the p-limit rational numbers, such that S[0] = 1 and S[i + P] = 2S[i], for which there is a val V such that V(S[i]) = i. This entails that V(S[P]) = V(2) = P, so that V = <P ... |, with P a positive integer; in other words, V is a P-edo val. For each of the n-1 wedgies Wk, we can form an abstract temperament periodic scale, meaning a periodic scale taking values in an [[abstract regular temperament]], by Tk[i] = Wk∨S[i]. The values Tk[i] are p-limit vals, and since Tk[P] = Wk∨S[i] = Wk∨2, Tk[P](2) = 0, and so Tk[i + P](2) = (Tk[i] + Tk[P])(2) = Tk[i](2). Hence Tk[i](2) takes on P or fewer values, with a ≤ Tk[i](2) ≤ b. The Graham complexity G(Wk) of S with respect to Wk is b-a, and if S is a Fokker block, for each Wk, G(Wk) < P.

One way to generalize this is to allow the [[Just intonation subgroups|group of the scale]] to be something other than the full p-limit group, adjusting the basis for vals, monzos and wedgies to correspond with a basis for this subgroup. We may also replace the interval of equivalence 2 with any rational number E which is not a power, so that S[i + P] = E S[i] and replacing Tk[i](2) with Tk[i](E).

Still another generalization is to consider regular temperament Fokker blocks. These are abstract temperament periodic scales S[i] with values in an abstract regular temperament belonging to some r-wedgie Y. It is Fokker if there are n-r (r+1)-wedgies with Graham complexity less than P.

=Example=

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<br />

<h1 id="toc3"><a name="Third definition of a Fokker block"></a>Third definition of a Fokker block</h1>

The n-1 vals u2, u3, ..., un defined in the previous section gave us n-1 inequalities ak - P &lt; uk(q) ≤ ak, which apply to any q in the Fokker block. If we restrict q to 1 ≤ q &lt; 2, and regard it as representing a pitch class, then it is associated to a lattice point in an n-1 dimensional vector space, and in that space the n-1 inequalities define the boundries of a parallepiped. The Fokker blocks can be defined as the pitch classes lying within such a paralellepiped.<br />

<br />

<h1 id="toc4"><a name="Fourth definition of a Fokker block"></a>Fourth definition of a Fokker block</h1>

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<br />

<h1 id="toc6"><a name="Expanding the definition"></a>Expanding the definition</h1>

A Fokker block as we have so far defined it is an epimorphic periodic scale S with period N repeating at the octave, with values in p-limit rational intonation, such that there exist pi(p)-1 = r-1 different rank-two wedgies {Wk} such that S has Graham complexity less than N for each Wk. If we unpack that definition we can extend it in several distinct ways.<br />

A Fokker block as we have so far defined it is an epimorphic periodic scale S with period P repeating at the octave, with values in p-limit rational intonation, such that there exist pi(p)-1 = n-1 different rank-two wedgies {Wk} such that S has Graham complexity less than P for each Wk. If we unpack that definition we can extend it in several distinct ways.<br />

<br />

Explicitly, S is a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Quasiperiodic_function" rel="nofollow">quasiperiodic function</a> from the integers to the p-limit rational numbers, such that S[0] = 1 and S[i + P] = 2S[i], for which there is a val V such that V(S[i]) = i. This entails that V(S[P]) = V(2) = P, so that V = &lt;P ... |, with P a positive integer; in other words, V is a P-edo val. For each of the n-1 wedgies Wk, we can form an abstract temperament periodic scale, meaning a periodic scale taking values in an <a class="wiki_link" href="/abstract%20regular%20temperament">abstract regular temperament</a>, by Tk[i] = Wk∨S[i]. The values Tk[i] are p-limit vals, and since Tk[P] = Wk∨S[i] = Wk∨2, Tk[P](2) = 0, and so Tk[i + P](2) = (Tk[i] + Tk[P])(2) = Tk[i](2). Hence Tk[i](2) takes on P or fewer values, with a ≤ Tk[i](2) ≤ b. The Graham complexity G(Wk) of S with respect to Wk is b-a, and if S is a Fokker block, for each Wk, G(Wk) &lt; P.<br />

<br />

One way to generalize this is to allow the <a class="wiki_link" href="/Just%20intonation%20subgroups">group of the scale</a> to be something other than the full p-limit group, adjusting the basis for vals, monzos and wedgies to correspond with a basis for this subgroup. We may also replace the interval of equivalence 2 with any rational number E which is not a power, so that S[i + P] = E S[i] and replacing Tk[i](2) with Tk[i](E). <br />

<br />

~~Explicitly, S~~ is ~~a <a class="wiki_link_ext" href="http://en.wikipedia~~.~~org/wiki/Quasiperiodic_function" rel="nofollow">quasiperiodic function</a> from the integers to the p-limit rational numbers, such that S[0] = 1 and~~ S[i ~~+ N~~] ~~= 2S[i], for which~~ there ~~is a val V such that V~~(~~S[i]~~) ~~= i. This entails that V(S[N]) = V(2) = N, so that V = &lt;N ... |,~~ with ~~N a positive integer; in other words, V is an N-edo val~~.<br />

Still another generalization is to consider regular temperament Fokker blocks. These are abstract temperament periodic scales S[i] with values in an abstract regular temperament belonging to some r-wedgie Y. It is Fokker if there are n-r (r+1)-wedgies with Graham complexity less than P.<br />

<br />

<h1 id="toc7"><a name="Example"></a>Example</h1>

@@ 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:genewardsmith|genewardsmith]] and made on <tt>2012-03-13 13:03:30 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-03-13 14:31:55 UTC</tt>.<br>
-: The original revision id was <tt>310613022</tt>.<br>
+: The original revision id was <tt>310655370</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 39: / Line 39: @@
 =Third definition of a Fokker block=
-The n-1 vals u2, u3, ..., un defined in the previous section gave us n-1 inequalities ak - P &lt; uk(q) ≤  ak, which apply to any q in the Fokker block. If we restrict q to 1 ≤ q &lt; 2, and regard it as representing a pitch class, then it is associated to a lattice point in an n-1 dimensional vector space, and in that space the n-1 inequalities define the boundries of a parallepiped. The Fokker blocks can be defined as the pitch classes lying within such a paralellepiped.
+The n-1 vals u2, u3, ..., un defined in the previous section gave us n-1 inequalities ak - P &lt; uk(q) ≤ ak, which apply to any q in the Fokker block. If we restrict q to 1 ≤ q &lt; 2, and regard it as representing a pitch class, then it is associated to a lattice point in an n-1 dimensional vector space, and in that space the n-1 inequalities define the boundries of a parallepiped. The Fokker blocks can be defined as the pitch classes lying within such a paralellepiped.
 =Fourth definition of a Fokker block=
@@ Line 50: / Line 50: @@
 =Expanding the definition=
-A Fokker block as we have so far defined it is an epimorphic periodic scale S with period N repeating at the octave, with values in p-limit rational intonation, such that there exist pi(p)-1 = r-1 different rank-two wedgies {Wk} such that S has Graham complexity less than N for each Wk. If we unpack that definition we can extend it in several distinct ways.
+A Fokker block as we have so far defined it is an epimorphic periodic scale S with period P repeating at the octave, with values in p-limit rational intonation, such that there exist pi(p)-1 = n-1 different rank-two wedgies {Wk} such that S has Graham complexity less than P for each Wk. If we unpack that definition we can extend it in several distinct ways.
-Explicitly, S is a [[http://en.wikipedia.org/wiki/Quasiperiodic_function|quasiperiodic function]] from the integers to the p-limit rational numbers, such that S[0] = 1 and S[i + N] = 2S[i], for which there is a val V such that V(S[i]) = i. This entails that V(S[N]) = V(2) = N, so that V = &lt;N ... |, with N a positive integer; in other words, V is an N-edo val.
+Explicitly, S is a [[http://en.wikipedia.org/wiki/Quasiperiodic_function|quasiperiodic function]] from the integers to the p-limit rational numbers, such that S[0] = 1 and S[i + P] = 2S[i], for which there is a val V such that V(S[i]) = i. This entails that V(S[P]) = V(2) = P, so that V = &lt;P ... |, with P a positive integer; in other words, V is a P-edo val. For each of the n-1 wedgies Wk, we can form an abstract temperament periodic scale, meaning a periodic scale taking values in an [[abstract regular temperament]], by Tk[i] = Wk∨S[i]. The values Tk[i] are p-limit vals, and since Tk[P] = Wk∨S[i] = Wk∨2, Tk[P](2) = 0, and so Tk[i + P](2) = (Tk[i] + Tk[P])(2) = Tk[i](2). Hence Tk[i](2) takes on P or fewer values, with a ≤ Tk[i](2) ≤ b. The Graham complexity G(Wk) of S with respect to Wk is b-a, and if S is a Fokker block, for each Wk, G(Wk) &lt; P.
+One way to generalize this is to allow the [[Just intonation subgroups|group of the scale]] to be something other than the full p-limit group, adjusting the basis for vals, monzos and wedgies to correspond with a basis for this subgroup. We may also replace the interval of equivalence 2 with any rational number E which is not a power, so that S[i + P] = E S[i] and replacing Tk[i](2) with Tk[i](E).
+Still another generalization is to consider regular temperament Fokker blocks. These are abstract temperament periodic scales S[i] with values in an abstract regular temperament belonging to some r-wedgie Y. It is Fokker if there are n-r (r+1)-wedgies with Graham complexity less than P.
 =Example=
@@ Line 112: / Line 116: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Third definition of a Fokker block"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Third definition of a Fokker block&lt;/h1&gt;
-The n-1 vals u2, u3, ..., un defined in the previous section gave us n-1 inequalities ak - P &amp;lt; uk(q) ≤  ak, which apply to any q in the Fokker block. If we restrict q to 1 ≤ q &amp;lt; 2, and regard it as representing a pitch class, then it is associated to a lattice point in an n-1 dimensional vector space, and in that space the n-1 inequalities define the boundries of a parallepiped. The Fokker blocks can be defined as the pitch classes lying within such a paralellepiped.&lt;br /&gt;
+The n-1 vals u2, u3, ..., un defined in the previous section gave us n-1 inequalities ak - P &amp;lt; uk(q) ≤ ak, which apply to any q in the Fokker block. If we restrict q to 1 ≤ q &amp;lt; 2, and regard it as representing a pitch class, then it is associated to a lattice point in an n-1 dimensional vector space, and in that space the n-1 inequalities define the boundries of a parallepiped. The Fokker blocks can be defined as the pitch classes lying within such a paralellepiped.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Fourth definition of a Fokker block"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Fourth definition of a Fokker block&lt;/h1&gt;
@@ Line 123: / Line 127: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Expanding the definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Expanding the definition&lt;/h1&gt;
-A Fokker block as we have so far defined it is an epimorphic periodic scale S with period N repeating at the octave, with values in p-limit rational intonation, such that there exist pi(p)-1 = r-1 different rank-two wedgies {Wk} such that S has Graham complexity less than N for each Wk. If we unpack that definition we can extend it in several distinct ways.&lt;br /&gt;
+A Fokker block as we have so far defined it is an epimorphic periodic scale S with period P repeating at the octave, with values in p-limit rational intonation, such that there exist pi(p)-1 = n-1 different rank-two wedgies {Wk} such that S has Graham complexity less than P for each Wk. If we unpack that definition we can extend it in several distinct ways.&lt;br /&gt;
+&lt;br /&gt;
+Explicitly, S is a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Quasiperiodic_function" rel="nofollow"&gt;quasiperiodic function&lt;/a&gt; from the integers to the p-limit rational numbers, such that S[0] = 1 and S[i + P] = 2S[i], for which there is a val V such that V(S[i]) = i. This entails that V(S[P]) = V(2) = P, so that V = &amp;lt;P ... |, with P a positive integer; in other words, V is a P-edo val. For each of the n-1 wedgies Wk, we can form an abstract temperament periodic scale, meaning a periodic scale taking values in an &lt;a class="wiki_link" href="/abstract%20regular%20temperament"&gt;abstract regular temperament&lt;/a&gt;, by Tk[i] = Wk∨S[i]. The values Tk[i] are p-limit vals, and since Tk[P] = Wk∨S[i] = Wk∨2, Tk[P](2) = 0, and so Tk[i + P](2) = (Tk[i] + Tk[P])(2) = Tk[i](2). Hence Tk[i](2) takes on P or fewer values, with a ≤ Tk[i](2) ≤ b. The Graham complexity G(Wk) of S with respect to Wk is b-a, and if S is a Fokker block, for each Wk, G(Wk) &amp;lt; P.&lt;br /&gt;
+&lt;br /&gt;
+One way to generalize this is to allow the &lt;a class="wiki_link" href="/Just%20intonation%20subgroups"&gt;group of the scale&lt;/a&gt; to be something other than the full p-limit group, adjusting the basis for vals, monzos and wedgies to correspond with a basis for this subgroup. We may also replace the interval of equivalence 2 with any rational number E which is not a power, so that S[i + P] = E S[i] and replacing Tk[i](2) with Tk[i](E). &lt;br /&gt;
 &lt;br /&gt;
-Explicitly, S is a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Quasiperiodic_function" rel="nofollow"&gt;quasiperiodic function&lt;/a&gt; from the integers to the p-limit rational numbers, such that S[0] = 1 and S[i + N] = 2S[i], for which there is a val V such that V(S[i]) = i. This entails that V(S[N]) = V(2) = N, so that V = &amp;lt;N ... |, with N a positive integer; in other words, V is an N-edo val.&lt;br /&gt;
+Still another generalization is to consider regular temperament Fokker blocks. These are abstract temperament periodic scales S[i] with values in an abstract regular temperament belonging to some r-wedgie Y. It is Fokker if there are n-r (r+1)-wedgies with Graham complexity less than P.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Example"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Example&lt;/h1&gt;