Generator complexity: Difference between revisions
Wikispaces>clumma **Imported revision 509978412 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 515964002 - 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: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-07-10 12:40:35 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>515964002</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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Now suppose X is a finite dimensional real normed vector space. A is then also a finite dimensional real normed vector space, inheriting its norm from X, and X/A is a finite dimensional real normed vector space, with a norm given by, for an equivalence class [x], ||[x]|| equals inf {||x + a||, a∈ A}. Algebraically X is (noncanonically) isomorphic to X*, but in general they are no longer isomorphic as normed spaces. Instead, we have the [[dual norm]] on X*, defined by setting, over all nonzero x ∈ X, ||f||* = sup <f|x>/||x||. Under the dual norm X* is also a finite dimensional normed vector space, A⁀ is isometrically isomorphic to (X/A)*, and X*/A⁀ is isometrically isomorphic to A*. | Now suppose X is a finite dimensional real normed vector space. A is then also a finite dimensional real normed vector space, inheriting its norm from X, and X/A is a finite dimensional real normed vector space, with a norm given by, for an equivalence class [x], ||[x]|| equals inf {||x + a||, a∈ A}. Algebraically X is (noncanonically) isomorphic to X*, but in general they are no longer isomorphic as normed spaces. Instead, we have the [[dual norm]] on X*, defined by setting, over all nonzero x ∈ X, ||f||* = sup <f|x>/||x||. Under the dual norm X* is also a finite dimensional normed vector space, A⁀ is isometrically isomorphic to (X/A)*, and X*/A⁀ is isometrically isomorphic to A*. | ||
In the situation which concerns us, X is the p-limit interval space of dimension n under a norm of one half times the L1 norm, A is a subspace of dimension n-1, whose coordinates sum to 0; hence A can be described as having the one-dimensional subspace A⁀ = {kJ}, where J is the JIP, as its annihilator. X has a norm of half the L1 norm, and hence X* has a norm of twice the L∞ norm. The norm on A* is defined by its isomorphism with X*/A⁀; the minimum defining inf {||f + kJ||} occurs for the value of k where the maximum of f + kJ and minus the minimum of f + kj are the same. In that case, 2||f + kJ||_L∞ = span(f), which is the generator complexity of f. Hence generator complexity is the dual norm for Kees expressibility as a norm on pitch classes.</pre></div> | In the situation which concerns us, X is the p-limit interval space of dimension n under a norm of one half times the L1 norm, A is a subspace of dimension n-1, whose coordinates sum to 0; hence A can be described as having the one-dimensional subspace A⁀ = {kJ}, where J is the [[JIP]], as its annihilator. X has a norm of half the L1 norm, and hence X* has a norm of twice the L∞ norm. The norm on A* is defined by its isomorphism with X*/A⁀; the minimum defining inf {||f + kJ||} occurs for the value of k where the maximum of f + kJ and minus the minimum of f + kj are the same. In that case, 2||f + kJ||_L∞ = span(f), which is the generator complexity of f. Hence generator complexity is the dual norm for Kees expressibility as a norm on pitch classes. | ||
=STD complexity= | |||
If B = <0 B₃ B₅ B₇ ... Bp| is the generator mapping val in weighted coordinates, and P is the period, then the //STD complexity// (a term due to Graham Breed) is P rms(B), where "rms" means the root mean square. If m is the mean of the coordinates of B (including the 0) and J is the [[JIP]] <1 1 1 ... 1|, rms(B) is √(C∙C/n) where C = B - mJ, n is the number of primes in the p-limit, and the dot denotes the dot product.</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>Generator complexity</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>Generator complexity</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="#Definition">Definition</a><!-- ws:end:WikiTextTocRule:7 --><!-- ws:start:WikiTextTocRule:8: --> | <a href="#Generator complexity and Kees expressibility">Generator complexity and Kees expressibility</a><!-- ws:end:WikiTextTocRule:8 --><!-- ws:start:WikiTextTocRule:9: --> | <a href="#STD complexity">STD complexity</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:10 --><br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definition</h1> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definition</h1> | ||
Suppose A = &lt;0 A₃ A₅ A₇ ... Ap| is the generator mapping val for a rank two temperament with P periods to the octave, and B = &lt;0 B₃ B₅ B₇ ... Bp| is the same val in weighted coordinates. For instance, &lt;0 1 -2 -2| is the generator mapping val for seven limit <a class="wiki_link" href="/pajara">pajara</a>, and &lt;0 1/log2(3) -2/log2(5) -2/log2(7)| ≅ &lt;0 0.631 -0.831 -0.712| is the val in weighted coordinates. For any vector V, let max(V) - min(V) = span(V). The <em>generator complexity</em> of the temperament is P span(B). In the case of pajara, which has two periods to the octave, this would be 2*(0.631 - (-0.861)) = 2.984. This can also be described in terms of the wedgie W of the temperament, as span(2∨W), which is the span of 0 followed by the first n-1 elements of W, where n is the number of primes in the p-limit. <br /> | Suppose A = &lt;0 A₃ A₅ A₇ ... Ap| is the generator mapping val for a rank two temperament with P periods to the octave, and B = &lt;0 B₃ B₅ B₇ ... Bp| is the same val in weighted coordinates. For instance, &lt;0 1 -2 -2| is the generator mapping val for seven limit <a class="wiki_link" href="/pajara">pajara</a>, and &lt;0 1/log2(3) -2/log2(5) -2/log2(7)| ≅ &lt;0 0.631 -0.831 -0.712| is the val in weighted coordinates. For any vector V, let max(V) - min(V) = span(V). The <em>generator complexity</em> of the temperament is P span(B). In the case of pajara, which has two periods to the octave, this would be 2*(0.631 - (-0.861)) = 2.984. This can also be described in terms of the wedgie W of the temperament, as span(2∨W), which is the span of 0 followed by the first n-1 elements of W, where n is the number of primes in the p-limit. <br /> | ||
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Now suppose X is a finite dimensional real normed vector space. A is then also a finite dimensional real normed vector space, inheriting its norm from X, and X/A is a finite dimensional real normed vector space, with a norm given by, for an equivalence class [x], ||[x]|| equals inf {||x + a||, a∈ A}. Algebraically X is (noncanonically) isomorphic to X*, but in general they are no longer isomorphic as normed spaces. Instead, we have the <a class="wiki_link" href="/dual%20norm">dual norm</a> on X*, defined by setting, over all nonzero x ∈ X, ||f||* = sup &lt;f|x&gt;/||x||. Under the dual norm X* is also a finite dimensional normed vector space, A⁀ is isometrically isomorphic to (X/A)*, and X*/A⁀ is isometrically isomorphic to A*.<br /> | Now suppose X is a finite dimensional real normed vector space. A is then also a finite dimensional real normed vector space, inheriting its norm from X, and X/A is a finite dimensional real normed vector space, with a norm given by, for an equivalence class [x], ||[x]|| equals inf {||x + a||, a∈ A}. Algebraically X is (noncanonically) isomorphic to X*, but in general they are no longer isomorphic as normed spaces. Instead, we have the <a class="wiki_link" href="/dual%20norm">dual norm</a> on X*, defined by setting, over all nonzero x ∈ X, ||f||* = sup &lt;f|x&gt;/||x||. Under the dual norm X* is also a finite dimensional normed vector space, A⁀ is isometrically isomorphic to (X/A)*, and X*/A⁀ is isometrically isomorphic to A*.<br /> | ||
<br /> | <br /> | ||
In the situation which concerns us, X is the p-limit interval space of dimension n under a norm of one half times the L1 norm, A is a subspace of dimension n-1, whose coordinates sum to 0; hence A can be described as having the one-dimensional subspace A⁀ = {kJ}, where J is the JIP, as its annihilator. X has a norm of half the L1 norm, and hence X* has a norm of twice the L∞ norm. The norm on A* is defined by its isomorphism with X*/A⁀; the minimum defining inf {||f + kJ||} occurs for the value of k where the maximum of f + kJ and minus the minimum of f + kj are the same. In that case, 2||f + kJ||_L∞ = span(f), which is the generator complexity of f. Hence generator complexity is the dual norm for Kees expressibility as a norm on pitch classes.</body></html></pre></div> | In the situation which concerns us, X is the p-limit interval space of dimension n under a norm of one half times the L1 norm, A is a subspace of dimension n-1, whose coordinates sum to 0; hence A can be described as having the one-dimensional subspace A⁀ = {kJ}, where J is the <a class="wiki_link" href="/JIP">JIP</a>, as its annihilator. X has a norm of half the L1 norm, and hence X* has a norm of twice the L∞ norm. The norm on A* is defined by its isomorphism with X*/A⁀; the minimum defining inf {||f + kJ||} occurs for the value of k where the maximum of f + kJ and minus the minimum of f + kj are the same. In that case, 2||f + kJ||_L∞ = span(f), which is the generator complexity of f. Hence generator complexity is the dual norm for Kees expressibility as a norm on pitch classes.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="STD complexity"></a><!-- ws:end:WikiTextHeadingRule:4 -->STD complexity</h1> | |||
If B = &lt;0 B₃ B₅ B₇ ... Bp| is the generator mapping val in weighted coordinates, and P is the period, then the <em>STD complexity</em> (a term due to Graham Breed) is P rms(B), where &quot;rms&quot; means the root mean square. If m is the mean of the coordinates of B (including the 0) and J is the <a class="wiki_link" href="/JIP">JIP</a> &lt;1 1 1 ... 1|, rms(B) is √(C∙C/n) where C = B - mJ, n is the number of primes in the p-limit, and the dot denotes the dot product.</body></html></pre></div> | |||