Generalized Tenney dual norms and Tp tuning space: Difference between revisions
Wikispaces>mbattaglia1 **Imported revision 356537966 - Original comment: ** |
Wikispaces>mbattaglia1 **Imported revision 356540846 - 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:mbattaglia1|mbattaglia1]] and made on <tt>2012-08-06 12: | : This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2012-08-06 12:41:40 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>356540846</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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[[math]] | [[math]] | ||
||f||_\mathbf{Tq*} = \text{sup}\left \{\frac{|f(\vec{v})|}{||\vec{v}||_\mathbf{Tp}}: \vec{v} \in \textbf{Lp}\right \} | |||
[[math]] | [[math]] | ||
for all f in **Tq<span style="font-size: 10px; vertical-align: super;">G</span>***. This normed space, for which the group of vals on **G** comprise the lattice of covectors with integer coefficients, is called **Tq* Tuning Space.** | for all f in **Tq<span style="font-size: 10px; vertical-align: super;">G</span>***. This normed space, for which the group of vals on **G** comprise the lattice of covectors with integer coefficients, is called **Tq* Tuning Space**. Other vectors in this space may be interpreted as tuning maps that send intervals in **G** to a certain number of cents (or other logarithmic units), although only tuning maps lying near the **JIP** will be of much musical relevance. | ||
Note that this norm enables us to define something like a complexity metric on vals, where vals that are closer to the origin (such as <7 11 16|) are rated less complex than vals which are further from the origin (such as <171 271 397|). Additionally, if this metric is used on tuning maps, we can evaluate the average error for any tuning map **t** and the **JIP** by looking at the quantity ||**t** - **JIP**||. As per the definition of dual norm above, the Tq* norm of this vector gives us the maximum Tp-weighted mapping for **t - JIP** over all intervals, and hence also gives us the maximum error for **t** over all intervals. | |||
==Prime Power Interval Groups== | ==Prime Power Interval Groups== | ||
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[[math]] | [[math]] | ||
\left \| \vec{v} \right \|_{\textbf{Tp}}^\textbf{G} = \left \| \mathbf{W_G} \cdot \vec{v} \right \|_\textbf{p} | |||
[[math]] | [[math]] | ||
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[[math]] | [[math]] | ||
\left \|f \right \|_{\textbf{Tq*}}^\textbf{G} = \left \| f \cdot \mathbf{W}_\mathbf{G}^{-1} \right \|_\textbf{q} | |||
[[math]] | [[math]] | ||
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[[math]] | [[math]] | ||
\left \| \vec{v} \right \|_{\textbf{T1}}^\textbf{G} = \left \| \mathbf{W_L} \cdot \mathbf{V_\textbf{G}} \cdot \vec{v} \right \|_\textbf{1} | |||
[[math]] | [[math]] | ||
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[[math]] | [[math]] | ||
\left \|f \right \|_{\textbf{Tq*}}^\textbf{G} = \inf_{n \in \text{ker}(\mathbf{V_G})} \left \{ \left \| (f-n) \cdot \mathbf{W}_\mathbf{L}^{-1} \right \|_\textbf{Tq*}^\mathbf{L} \right \} | |||
[[math]] | [[math]] | ||
Note that this is the quotient norm induced on the space **Tq<span style="font-size: 10px; vertical-align: super;">L</span>***/ker(**V<span style="font-size: 10px; vertical-align: sub;">G</span>**), where ker(**V<span style="font-size: 10px; vertical-align: sub;">G</span>**) is the set of vals in the V-map that are restricted away. This result is due to a corollary of the [[@http://www.math.unl.edu/~s-bbockel1/928/node25.html|Hahn-Banach theorem]], which demonstrates that the dual space M* to any subspace M of a Banach space V must be isometrically isomorphic to the quotient space V*/ker(M), where ker(M) is the set of all f in V* such that f(M) = 0. Since our vector space **Tp<span style="font-size: 10px; vertical-align: super;">L</span>** and our subspace is **Tp<span style="font-size: 10px; vertical-align: super;">G</span>**, this proves that our dual space **Tq<span style="font-size: 10px; vertical-align: super;">G</span>*** must be isometrically isomorphic to **Tq<span style="font-size: 10px; vertical-align: super;">L</span>***/ker(**V<span style="font-size: 10px; vertical-align: sub;">G</span>**). </pre></div> | Note that this is the quotient norm induced on the space **Tq<span style="font-size: 10px; vertical-align: super;">L</span>***/ker(**V<span style="font-size: 10px; vertical-align: sub;">G</span>**), where ker(**V<span style="font-size: 10px; vertical-align: sub;">G</span>**) is the set of vals in the V-map that are restricted away. This result is due to a corollary of the [[@http://www.math.unl.edu/~s-bbockel1/928/node25.html|Hahn-Banach theorem]], which demonstrates that the dual space M* to any subspace M of a Banach space V must be isometrically isomorphic to the quotient space V*/ker(M), where ker(M) is the set of all f in V* such that f(M) = 0. Since our vector space **Tp<span style="font-size: 10px; vertical-align: super;">L</span>** and our subspace is **Tp<span style="font-size: 10px; vertical-align: super;">G</span>**, this proves that our dual space **Tq<span style="font-size: 10px; vertical-align: super;">G</span>*** must be isometrically isomorphic to **Tq<span style="font-size: 10px; vertical-align: super;">L</span>***/ker(**V<span style="font-size: 10px; vertical-align: sub;">G</span>**).</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>Generalized Tenney Dual Norms and Tp Tuning Space</title></head><body><!-- ws:start:WikiTextHeadingRule:5:&lt;h1&gt; --><h1 id="toc0"><a name="Dual Norms"></a><!-- ws:end:WikiTextHeadingRule:5 -->Dual Norms</h1> | <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>Generalized Tenney Dual Norms and Tp Tuning Space</title></head><body><!-- ws:start:WikiTextHeadingRule:5:&lt;h1&gt; --><h1 id="toc0"><a name="Dual Norms"></a><!-- ws:end:WikiTextHeadingRule:5 -->Dual Norms</h1> | ||
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[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
||f||_\mathbf{Tq*} = \text{sup}\left \{\frac{|f(\vec{v})|}{||\vec{v}||_\mathbf{Tp}}: \vec{v} \in \textbf{Lp}\right \}&lt;br/&gt;[[math]] | |||
--><script type="math/tex"> ||f||_\mathbf{Tq*} = \text{sup}\left \{\frac{|f(\vec{v})|}{||\vec{v}||_\mathbf{Tp}}: \vec{v} \in \textbf{Lp}\right \}</script><!-- ws:end:WikiTextMathRule:0 --><br /> | --><script type="math/tex">||f||_\mathbf{Tq*} = \text{sup}\left \{\frac{|f(\vec{v})|}{||\vec{v}||_\mathbf{Tp}}: \vec{v} \in \textbf{Lp}\right \}</script><!-- ws:end:WikiTextMathRule:0 --><br /> | ||
<br /> | |||
for all f in <strong>Tq<span style="font-size: 10px; vertical-align: super;">G</span></strong>*. This normed space, for which the group of vals on <strong>G</strong> comprise the lattice of covectors with integer coefficients, is called <strong>Tq* Tuning Space</strong>. Other vectors in this space may be interpreted as tuning maps that send intervals in <strong>G</strong> to a certain number of cents (or other logarithmic units), although only tuning maps lying near the <strong>JIP</strong> will be of much musical relevance.<br /> | |||
<br /> | <br /> | ||
for | Note that this norm enables us to define something like a complexity metric on vals, where vals that are closer to the origin (such as &lt;7 11 16|) are rated less complex than vals which are further from the origin (such as &lt;171 271 397|). Additionally, if this metric is used on tuning maps, we can evaluate the average error for any tuning map <strong>t</strong> and the <strong>JIP</strong> by looking at the quantity ||<strong>t</strong> - <strong>JIP</strong>||. As per the definition of dual norm above, the Tq* norm of this vector gives us the maximum Tp-weighted mapping for <strong>t - JIP</strong> over all intervals, and hence also gives us the maximum error for <strong>t</strong> over all intervals.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:7:&lt;h2&gt; --><h2 id="toc1"><a name="Dual Norms-Prime Power Interval Groups"></a><!-- ws:end:WikiTextHeadingRule:7 -->Prime Power Interval Groups</h2> | <!-- ws:start:WikiTextHeadingRule:7:&lt;h2&gt; --><h2 id="toc1"><a name="Dual Norms-Prime Power Interval Groups"></a><!-- ws:end:WikiTextHeadingRule:7 -->Prime Power Interval Groups</h2> | ||
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[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\left \| \vec{v} \right \|_{\textbf{Tp}}^\textbf{G} = \left \| \mathbf{W_G} \cdot \vec{v} \right \|_\textbf{p}&lt;br/&gt;[[math]] | |||
--><script type="math/tex"> \left \| \vec{v} \right \|_{\textbf{Tp}}^\textbf{G} = \left \| \mathbf{W_G} \cdot \vec{v} \right \|_\textbf{p}</script><!-- ws:end:WikiTextMathRule:1 --><br /> | --><script type="math/tex">\left \| \vec{v} \right \|_{\textbf{Tp}}^\textbf{G} = \left \| \mathbf{W_G} \cdot \vec{v} \right \|_\textbf{p}</script><!-- ws:end:WikiTextMathRule:1 --><br /> | ||
<br /> | <br /> | ||
for diagonal weighting matrix <strong>W<span style="font-size: 80%; vertical-align: sub;">G</span></strong>. Then the dual norm || · ||<strong><span style="font-size: 10px; vertical-align: sub;">Tq*</span></strong> on <strong>Tq<span style="font-size: 10px; vertical-align: super;">G</span></strong>* is given for f in <strong>Tq<span style="font-size: 10px; vertical-align: super;">G</span></strong>* by<br /> | for diagonal weighting matrix <strong>W<span style="font-size: 80%; vertical-align: sub;">G</span></strong>. Then the dual norm || · ||<strong><span style="font-size: 10px; vertical-align: sub;">Tq*</span></strong> on <strong>Tq<span style="font-size: 10px; vertical-align: super;">G</span></strong>* is given for f in <strong>Tq<span style="font-size: 10px; vertical-align: super;">G</span></strong>* by<br /> | ||
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[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\left \|f \right \|_{\textbf{Tq*}}^\textbf{G} = \left \| f \cdot \mathbf{W}_\mathbf{G}^{-1} \right \|_\textbf{q}&lt;br/&gt;[[math]] | |||
--><script type="math/tex"> \left \|f \right \|_{\textbf{Tq*}}^\textbf{G} = \left \| f \cdot \mathbf{W}_\mathbf{G}^{-1} \right \|_\textbf{q}</script><!-- ws:end:WikiTextMathRule:2 --><br /> | --><script type="math/tex">\left \|f \right \|_{\textbf{Tq*}}^\textbf{G} = \left \| f \cdot \mathbf{W}_\mathbf{G}^{-1} \right \|_\textbf{q}</script><!-- ws:end:WikiTextMathRule:2 --><br /> | ||
<br /> | <br /> | ||
where the coefficients p in Tp and q in Tq* satisfy the relationship 1/p + 1/q = 1.<br /> | where the coefficients p in Tp and q in Tq* satisfy the relationship 1/p + 1/q = 1.<br /> | ||
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[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\left \| \vec{v} \right \|_{\textbf{T1}}^\textbf{G} = \left \| \mathbf{W_L} \cdot \mathbf{V_\textbf{G}} \cdot \vec{v} \right \|_\textbf{1}&lt;br/&gt;[[math]] | |||
--><script type="math/tex"> \left \| \vec{v} \right \|_{\textbf{T1}}^\textbf{G} = \left \| \mathbf{W_L} \cdot \mathbf{V_\textbf{G}} \cdot \vec{v} \right \|_\textbf{1}</script><!-- ws:end:WikiTextMathRule:3 --><br /> | --><script type="math/tex">\left \| \vec{v} \right \|_{\textbf{T1}}^\textbf{G} = \left \| \mathbf{W_L} \cdot \mathbf{V_\textbf{G}} \cdot \vec{v} \right \|_\textbf{1}</script><!-- ws:end:WikiTextMathRule:3 --><br /> | ||
<br /> | <br /> | ||
for a <a class="wiki_link" href="/Subgroup%20Mapping%20Matrices%20%28V-maps%29">V-map</a> <strong>V<span style="font-size: 80%; vertical-align: sub;">G</span></strong> representing <strong>G</strong> in some full-limit <strong>L</strong> and a diagonal weighting matrix <strong>W<span style="font-size: 10px; vertical-align: sub;">L</span></strong> for <strong>L</strong>. Then if <strong>Tp<span style="font-size: 10px; vertical-align: super;">L</span></strong> represents the full-limit interval space that <strong>G</strong> is embedded in, and <strong>Tq<span style="font-size: 10px; vertical-align: super;">L</span></strong>* is the dual space, the dual norm || · ||<strong><span style="font-size: 10px; vertical-align: sub;">Tq </span></strong>on <strong>Tq<span style="font-size: 80%; vertical-align: super;">G</span></strong>* is given by<br /> | for a <a class="wiki_link" href="/Subgroup%20Mapping%20Matrices%20%28V-maps%29">V-map</a> <strong>V<span style="font-size: 80%; vertical-align: sub;">G</span></strong> representing <strong>G</strong> in some full-limit <strong>L</strong> and a diagonal weighting matrix <strong>W<span style="font-size: 10px; vertical-align: sub;">L</span></strong> for <strong>L</strong>. Then if <strong>Tp<span style="font-size: 10px; vertical-align: super;">L</span></strong> represents the full-limit interval space that <strong>G</strong> is embedded in, and <strong>Tq<span style="font-size: 10px; vertical-align: super;">L</span></strong>* is the dual space, the dual norm || · ||<strong><span style="font-size: 10px; vertical-align: sub;">Tq </span></strong>on <strong>Tq<span style="font-size: 80%; vertical-align: super;">G</span></strong>* is given by<br /> | ||
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[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\left \|f \right \|_{\textbf{Tq*}}^\textbf{G} = \inf_{n \in \text{ker}(\mathbf{V_G})} \left \{ \left \| (f-n) \cdot \mathbf{W}_\mathbf{L}^{-1} \right \|_\textbf{Tq*}^\mathbf{L} \right \}&lt;br/&gt;[[math]] | |||
--><script type="math/tex"> \left \|f \right \|_{\textbf{Tq*}}^\textbf{G} = \inf_{n \in \text{ker}(\mathbf{V_G})} \left \{ \left \| (f-n) \cdot \mathbf{W}_\mathbf{L}^{-1} \right \|_\textbf{Tq*}^\mathbf{L} \right \}</script><!-- ws:end:WikiTextMathRule:4 --><br /> | --><script type="math/tex">\left \|f \right \|_{\textbf{Tq*}}^\textbf{G} = \inf_{n \in \text{ker}(\mathbf{V_G})} \left \{ \left \| (f-n) \cdot \mathbf{W}_\mathbf{L}^{-1} \right \|_\textbf{Tq*}^\mathbf{L} \right \}</script><!-- ws:end:WikiTextMathRule:4 --><br /> | ||
<br /> | <br /> | ||
Note that this is the quotient norm induced on the space <strong>Tq<span style="font-size: 10px; vertical-align: super;">L</span></strong>*/ker(<strong>V<span style="font-size: 10px; vertical-align: sub;">G</span></strong>), where ker(<strong>V<span style="font-size: 10px; vertical-align: sub;">G</span></strong>) is the set of vals in the V-map that are restricted away. This result is due to a corollary of the <a class="wiki_link_ext" href="http://www.math.unl.edu/~s-bbockel1/928/node25.html" rel="nofollow" target="_blank">Hahn-Banach theorem</a>, which demonstrates that the dual space M* to any subspace M of a Banach space V must be isometrically isomorphic to the quotient space V*/ker(M), where ker(M) is the set of all f in V* such that f(M) = 0. Since our vector space <strong>Tp<span style="font-size: 10px; vertical-align: super;">L</span></strong> and our subspace is <strong>Tp<span style="font-size: 10px; vertical-align: super;">G</span></strong>, this proves that our dual space <strong>Tq<span style="font-size: 10px; vertical-align: super;">G</span></strong>* must be isometrically isomorphic to <strong>Tq<span style="font-size: 10px; vertical-align: super;">L</span></strong>*/ker(<strong>V<span style="font-size: 10px; vertical-align: sub;">G</span></strong>).</body></html></pre></div> | Note that this is the quotient norm induced on the space <strong>Tq<span style="font-size: 10px; vertical-align: super;">L</span></strong>*/ker(<strong>V<span style="font-size: 10px; vertical-align: sub;">G</span></strong>), where ker(<strong>V<span style="font-size: 10px; vertical-align: sub;">G</span></strong>) is the set of vals in the V-map that are restricted away. This result is due to a corollary of the <a class="wiki_link_ext" href="http://www.math.unl.edu/~s-bbockel1/928/node25.html" rel="nofollow" target="_blank">Hahn-Banach theorem</a>, which demonstrates that the dual space M* to any subspace M of a Banach space V must be isometrically isomorphic to the quotient space V*/ker(M), where ker(M) is the set of all f in V* such that f(M) = 0. Since our vector space <strong>Tp<span style="font-size: 10px; vertical-align: super;">L</span></strong> and our subspace is <strong>Tp<span style="font-size: 10px; vertical-align: super;">G</span></strong>, this proves that our dual space <strong>Tq<span style="font-size: 10px; vertical-align: super;">G</span></strong>* must be isometrically isomorphic to <strong>Tq<span style="font-size: 10px; vertical-align: super;">L</span></strong>*/ker(<strong>V<span style="font-size: 10px; vertical-align: sub;">G</span></strong>).</body></html></pre></div> | ||