Fokker block: Difference between revisions

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:clumma|clumma]] and made on <tt>2012-03-03 17:30:33 UTC</tt>.<br>

: This revision was by author [[User:clumma|clumma]] and made on <tt>2012-03-03 17:31:41 UTC</tt>.<br>

: The original revision id was <tt>~~307454932~~</tt>.<br>

: The original revision id was <tt>307455132</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 47:

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 purposes 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. Like Graham complexity, this gives a slightly lower value to a MOS with more than one period to the octave. WE can make them exactly the same by modifying things slightly so that T[i] is (W∨S[i])(2) in the first period of the octave, (W∨S[i])(2) + 1 for the second period, and so forth. This makes all MOS to result in P contiguous values, so that the resulting quadratic form returns P(P^2-1)/12 in all cases when the wedgie results in a MOS of P notes per octave, and more otherwise.

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 of lowest complexity. However, for some purposes 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. Like Graham complexity, this gives a slightly lower value to a MOS with more than one period to the octave. WE can make them exactly the same by modifying things slightly so that T[i] is (W∨S[i])(2) in the first period of the octave, (W∨S[i])(2) + 1 for the second period, and so forth. This makes all MOS to result in P contiguous values, so that the resulting quadratic form returns P(P^2-1)/12 in all cases when the wedgie results in a MOS of P notes per octave, and more otherwise.

=Example=

Line 112:

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 />

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 purposes 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. Like Graham complexity, this gives a slightly lower value to a MOS with more than one period to the octave. WE can make them exactly the same by modifying things slightly so that T[i] is (W∨S[i])(2) in the first period of the octave, (W∨S[i])(2) + 1 for the second period, and so forth. This makes all MOS to result in P contiguous values, so that the resulting quadratic form returns P(P^2-1)/12 in all cases when the wedgie results in a MOS of P notes per octave, and more otherwise. <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 of lowest complexity. However, for some purposes 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. Like Graham complexity, this gives a slightly lower value to a MOS with more than one period to the octave. WE can make them exactly the same by modifying things slightly so that T[i] is (W∨S[i])(2) in the first period of the octave, (W∨S[i])(2) + 1 for the second period, and so forth. This makes all MOS to result in P contiguous values, so that the resulting quadratic form returns P(P^2-1)/12 in all cases when the wedgie results in a MOS of P notes per octave, and more otherwise. <br />

<br />

<h1 id="toc6"><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:clumma|clumma]] and made on <tt>2012-03-03 17:30:33 UTC</tt>.<br>
+: This revision was by author [[User:clumma|clumma]] and made on <tt>2012-03-03 17:31:41 UTC</tt>.<br>
-: The original revision id was <tt>307454932</tt>.<br>
+: The original revision id was <tt>307455132</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 47: / Line 47: @@
 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 &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.
-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 purposes 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.  Like Graham complexity, this gives a slightly lower value to a MOS with more than one period to the octave. WE can make them exactly the same by modifying things slightly so that T[i] is (W∨S[i])(2) in the first period of the octave, (W∨S[i])(2) + 1 for the second period, and so forth. This makes all MOS to result in P contiguous values, so that the resulting quadratic form returns P(P^2-1)/12 in all cases when the wedgie results in a MOS of P notes per octave, and more otherwise.
+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 of lowest complexity. However, for some purposes 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.  Like Graham complexity, this gives a slightly lower value to a MOS with more than one period to the octave. WE can make them exactly the same by modifying things slightly so that T[i] is (W∨S[i])(2) in the first period of the octave, (W∨S[i])(2) + 1 for the second period, and so forth. This makes all MOS to result in P contiguous values, so that the resulting quadratic form returns P(P^2-1)/12 in all cases when the wedgie results in a MOS of P notes per octave, and more otherwise.
 =Example=
@@ Line 112: / Line 112: @@
 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. &lt;a class="wiki_link" href="/Scala"&gt;Scala&lt;/a&gt; does this as a part of its &amp;quot;Show data&amp;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 &lt;a class="wiki_link" href="/Graham%20complexity"&gt;Graham complexity&lt;/a&gt; to the scale reduced to the octave; that is, to S = {S[i]| 0 ≤ i &amp;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.&lt;br /&gt;
 &lt;br /&gt;
-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 purposes 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.  Like Graham complexity, this gives a slightly lower value to a MOS with more than one period to the octave. WE can make them exactly the same by modifying things slightly so that T[i] is (W∨S[i])(2) in the first period of the octave, (W∨S[i])(2) + 1 for the second period, and so forth. This makes all MOS to result in P contiguous values, so that the resulting quadratic form returns P(P^2-1)/12 in all cases when the wedgie results in a MOS of P notes per octave, and more otherwise. &lt;br /&gt;
+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 of lowest complexity. However, for some purposes 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.  Like Graham complexity, this gives a slightly lower value to a MOS with more than one period to the octave. WE can make them exactly the same by modifying things slightly so that T[i] is (W∨S[i])(2) in the first period of the octave, (W∨S[i])(2) + 1 for the second period, and so forth. This makes all MOS to result in P contiguous values, so that the resulting quadratic form returns P(P^2-1)/12 in all cases when the wedgie results in a MOS of P notes per octave, and more otherwise. &lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Example"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Example&lt;/h1&gt;