Generalized Tenney dual norms and Tp tuning space: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2012-08-06 11:15:49 UTC</tt>.

: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2012-08-06 12:10:25 UTC</tt>.

: The original revision id was <tt>~~356530718~~</tt>.

: The original revision id was <tt>356537294</tt>.

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

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

<h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">=Dual Norms=

Given any [[Generalized Tenney Norms and Tp ~~spaces~~|Tp norm]] on an interval space **TpG** associated with a group **G**, we can define a corresponding **dual norm** on the dual space **TpG*** which satisfies the following identity:

Given any [[Generalized Tenney Norms and Tp Interval Space|Tp norm]] on an interval space **TpG** associated with a group **G**, we can define a corresponding **dual Tq* norm** on the dual space **TqG*** which satisfies the following identity:

[[math]]

||f|| = \text{sup}\left \{\frac{|f(\vec{v})|}{||\vec{v}||}: \vec{v} \in \textbf{Lp}\right \}

||f||_\mathbf{Tq*} = \text{sup}\left \{\frac{|f(\vec{v})|}{||\vec{v}||_\mathbf{Tp}}: \vec{v} \in \textbf{Lp}\right \}

[[math]]

for all f in **TpG***.

for all f in **TqG***. This normed space, for which the group of vals on **G** comprise the lattice of covectors with integer coefficients, is called **Tq* Tuning Space.**

In the simplest case where **G** has as its chosen basis only primes and prime powers, ~~and hence~~ || · ||**Tp** is given by

==Prime Power Interval Groups==

In the simplest case where **G** has as its chosen basis only primes and prime powers, || · ||**Tp** is given by

[[math]]

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[[math]]

for weighting matrix **WG**~~, then~~ the dual norm || · ||**Tq*** on **TpG*** is given by

for diagonal weighting matrix **WG**. Then the dual norm || · ||**Tq*** on **TqG*** is given for f in **TqG*** by

[[math]]

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[[math]]

where the coefficients p in Tp and q in Tq* satisfy the relationship 1/p + 1/q = 1.</pre></div>

where the coefficients p in Tp and q in Tq* satisfy the relationship 1/p + 1/q = 1.

The dual of any Tp norm is very similar to the dual of the ordinary Lp norm. The crucial difference to be noted is that the weighting for covectors in tuning space is the inverse of the weighting for vectors in interval space; simple primes are weighted less in interval space but more in tuning space. Unlike the weighting matrix for interval space, the weighting matrix on tuning space is a diagonal matrix in which the nth entry in the diagonal is 1/log2(**G**n), where **G**n is the nth basis element in **G**. We denote such inverse weighted norms with an asterisk, so that the inverse-Tenney weighted Linf norm in tuning space is Tinf*.

For **G** with basis of only primes and prime powers, the dual of the T1 norm is the Tinf* norm, the dual of the Tinf norm is the T1* norm, and the dual of the T2 norm is the T2* norm.

==Arbitrary Interval Groups==

For an arbitrary group **G** with its chosen basis containing intervals other than primes and prime powers, || · ||**Tp** is given by

[[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]]

for a [[Subgroup Mapping Matrices (V-maps)|V-map]] **VG** representing **G** in some full-limit **L** and a diagonal weighting matrix **WL** for **L**. Then if **TpL** represents the full-limit interval space that **G** is embedded in, and **TqL*** is the dual space, the dual norm || · ||**Tq **on **TqG*** is given by

[[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]]

Note that this is the quotient norm induced on the space **TqL***/ker(**VG**), where ker(**VG**) 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 space **TqG*** must be isometrically isomorphic to **TqL***/ker(**VG**).</pre></div>

<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><h1 id="toc0"><a name="Dual Norms"></a>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><h1 id="toc0"><a name="Dual Norms"></a>Dual Norms</h1>

Given any <a class="wiki_link" href="/Generalized%20Tenney%20Norms%20and%20Tp%~~20spaces~~">Tp norm</a> on an interval space TpG associated with a group G, we can define a corresponding dual norm on the dual space TpG* which satisfies the following identity:

Given any <a class="wiki_link" href="/Generalized%20Tenney%20Norms%20and%20Tp%20Interval%20Space">Tp norm</a> on an interval space TpG associated with a group G, we can define a corresponding dual Tq* norm on the dual space TqG* which satisfies the following identity:

<!-- ws:start:WikiTextMathRule:0:

[[math]]&lt;br/&gt;

||f|| = \text{sup}\left \{\frac{|f(\vec{v})|}{||\vec{v}||}: \vec{v} \in \textbf{Lp}\right \}&lt;br/&gt;[[math]]

||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|| = \text{sup}\left \{\frac{|f(\vec{v})|}{||\vec{v}||}: \vec{v} \in \textbf{Lp}\right \}</script>

--><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>

for all f in TpG*.

for all f in TqG*. This normed space, for which the group of vals on G comprise the lattice of covectors with integer coefficients, is called Tq* Tuning Space.

In the simplest case where G has as its chosen basis only primes and prime powers, ~~and hence~~ || · ||Tp is given by

<h2 id="toc1"><a name="Dual Norms-Prime Power Interval Groups"></a>Prime Power Interval Groups</h2>

In the simplest case where G has as its chosen basis only primes and prime powers, || · ||Tp is given by

<!-- ws:start:WikiTextMathRule:1:

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--><script type="math/tex">\left \| \vec{v} \right \|_{\textbf{Tp}}^\textbf{G} = \left \| \mathbf{W_G} \cdot \vec{v} \right \|_\textbf{p}</script>

for weighting matrix WG~~, then~~ the dual norm || · ||Tq* on TpG* is given by

for diagonal weighting matrix WG. Then the dual norm || · ||Tq* on TqG* is given for f in TqG* by

<!-- ws:start:WikiTextMathRule:2:

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--><script type="math/tex">\left \|f \right \|_{\textbf{Tq*}}^\textbf{G} = \left \| f \cdot \mathbf{W}_\mathbf{G}^{-1} \right \|_\textbf{q}</script>

where the coefficients p in Tp and q in Tq* satisfy the relationship 1/p + 1/q = 1.</body></html></pre></div>

where the coefficients p in Tp and q in Tq* satisfy the relationship 1/p + 1/q = 1.

The dual of any Tp norm is very similar to the dual of the ordinary Lp norm. The crucial difference to be noted is that the weighting for covectors in tuning space is the inverse of the weighting for vectors in interval space; simple primes are weighted less in interval space but more in tuning space. Unlike the weighting matrix for interval space, the weighting matrix on tuning space is a diagonal matrix in which the nth entry in the diagonal is 1/log2(Gn), where Gn is the nth basis element in G. We denote such inverse weighted norms with an asterisk, so that the inverse-Tenney weighted Linf norm in tuning space is Tinf*.

For G with basis of only primes and prime powers, the dual of the T1 norm is the Tinf* norm, the dual of the Tinf norm is the T1* norm, and the dual of the T2 norm is the T2* norm.

<h2 id="toc2"><a name="Dual Norms-Arbitrary Interval Groups"></a>Arbitrary Interval Groups</h2>

For an arbitrary group G with its chosen basis containing intervals other than primes and prime powers, || · ||Tp is given by

<!-- ws:start:WikiTextMathRule:3:

[[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>

for a <a class="wiki_link" href="/Subgroup%20Mapping%20Matrices%20%28V-maps%29">V-map</a> VG representing G in some full-limit L and a diagonal weighting matrix WL for L. Then if TpL represents the full-limit interval space that G is embedded in, and TqL* is the dual space, the dual norm || · ||Tq on TqG* is given by

<!-- ws:start:WikiTextMathRule:4:

[[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>

Note that this is the quotient norm induced on the space TqL*/ker(VG), where ker(VG) 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 space TqG* must be isometrically isomorphic to TqL*/ker(VG).</body></html></pre></div>

@@ 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:mbattaglia1|mbattaglia1]] and made on <tt>2012-08-06 11:15:49 UTC</tt>.<br>
+: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2012-08-06 12:10:25 UTC</tt>.<br>
-: The original revision id was <tt>356530718</tt>.<br>
+: The original revision id was <tt>356537294</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>
 <h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">=Dual Norms=
-Given any [[Generalized Tenney Norms and Tp spaces|Tp norm]] on an interval space **Tp&lt;span style="font-size: 10px; vertical-align: sub;"&gt;G&lt;/span&gt;** associated with a group **G**, we can define a corresponding **dual norm** on the dual space **Tp&lt;span style="font-size: 10px; vertical-align: sub;"&gt;G&lt;/span&gt;*** which satisfies the following identity:
+Given any [[Generalized Tenney Norms and Tp Interval Space|Tp norm]] on an interval space **Tp&lt;span style="font-size: 10px; vertical-align: super;"&gt;G&lt;/span&gt;** associated with a group **G**, we can define a corresponding **dual Tq* norm** on the dual space **Tq&lt;span style="font-size: 10px; vertical-align: super;"&gt;G&lt;/span&gt;*** which satisfies the following identity:
 [[math]]
-||f|| = \text{sup}\left \{\frac{|f(\vec{v})|}{||\vec{v}||}: \vec{v} \in \textbf{Lp}\right \}
+||f||_\mathbf{Tq*} = \text{sup}\left \{\frac{|f(\vec{v})|}{||\vec{v}||_\mathbf{Tp}}: \vec{v} \in \textbf{Lp}\right \}
 [[math]]
-for all f in **Tp&lt;span style="font-size: 10px; vertical-align: sub;"&gt;G&lt;/span&gt;***.
+for all f in **Tq&lt;span style="font-size: 10px; vertical-align: super;"&gt;G&lt;/span&gt;***. This normed space, for which the group of vals on **G** comprise the lattice of covectors with integer coefficients, is called **Tq* Tuning Space.**
-In the simplest case where **G** has as its chosen basis only primes and prime powers, and hence || · ||**&lt;span style="font-size: 10px; vertical-align: sub;"&gt;Tp&lt;/span&gt;** is given by
+==Prime Power Interval Groups==
+In the simplest case where **G** has as its chosen basis only primes and prime powers, || · ||**&lt;span style="font-size: 10px; vertical-align: sub;"&gt;Tp&lt;/span&gt;** is given by
 [[math]]
@@ Line 21: / Line 22: @@
 [[math]]
-for weighting matrix **W&lt;span style="font-size: 80%; vertical-align: sub;"&gt;G&lt;/span&gt;**, then the dual norm || · ||**&lt;span style="font-size: 10px; vertical-align: sub;"&gt;Tq*&lt;/span&gt;** on **Tp&lt;span style="font-size: 10px; vertical-align: sub;"&gt;G&lt;/span&gt;*** is given by
+for diagonal weighting matrix **W&lt;span style="font-size: 80%; vertical-align: sub;"&gt;G&lt;/span&gt;**. Then the dual norm || · ||**&lt;span style="font-size: 10px; vertical-align: sub;"&gt;Tq*&lt;/span&gt;** on **Tq&lt;span style="font-size: 10px; vertical-align: super;"&gt;G&lt;/span&gt;*** is given for f in **Tq&lt;span style="font-size: 10px; vertical-align: super;"&gt;G&lt;/span&gt;*** by
 [[math]]
@@ Line 27: / Line 28: @@
 [[math]]
-where the coefficients p in Tp and q in Tq* satisfy the relationship 1/p + 1/q = 1.</pre></div>
+where the coefficients p in Tp and q in Tq* satisfy the relationship 1/p + 1/q = 1.
+The dual of any Tp norm is very similar to the dual of the ordinary Lp norm. The crucial difference to be noted is that the weighting for covectors in tuning space is the inverse of the weighting for vectors in interval space; simple primes are weighted less in interval space but more in tuning space. Unlike the weighting matrix for interval space, the weighting matrix on tuning space is a diagonal matrix in which the nth entry in the diagonal is 1/log&lt;span style="font-size: 80%; vertical-align: sub;"&gt;2&lt;/span&gt;(**G**&lt;span style="font-size: 80%; vertical-align: sub;"&gt;n&lt;/span&gt;), where **G**&lt;span style="font-size: 10px; vertical-align: sub;"&gt;n&lt;/span&gt; is the nth basis element in **G**. We denote such inverse weighted norms with an asterisk, so that the inverse-Tenney weighted Linf norm in tuning space is Tinf*.
+For **G** with basis of only primes and prime powers, the dual of the T1 norm is the Tinf* norm, the dual of the Tinf norm is the T1* norm, and the dual of the T2 norm is the T2* norm.
+==Arbitrary Interval Groups==
+For an arbitrary group **G** with its chosen basis containing intervals other than primes and prime powers, || · ||**&lt;span style="font-size: 10px; vertical-align: sub;"&gt;Tp&lt;/span&gt;** is given by
+[[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]]
+for a [[Subgroup Mapping Matrices (V-maps)|V-map]] **V&lt;span style="font-size: 80%; vertical-align: sub;"&gt;G&lt;/span&gt;** representing **G** in some full-limit **L** and a diagonal weighting matrix **W&lt;span style="font-size: 10px; vertical-align: sub;"&gt;L&lt;/span&gt;** for **L**. Then if **Tp&lt;span style="font-size: 10px; vertical-align: super;"&gt;L&lt;/span&gt;** represents the full-limit interval space that **G** is embedded in, and **Tq&lt;span style="font-size: 10px; vertical-align: super;"&gt;L&lt;/span&gt;*** is the dual space, the dual norm || · ||**&lt;span style="font-size: 10px; vertical-align: sub;"&gt;Tq &lt;/span&gt;**on **Tq&lt;span style="font-size: 80%; vertical-align: super;"&gt;G&lt;/span&gt;*** is given by
+[[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]]
+Note that this is the quotient norm induced on the space **Tq&lt;span style="font-size: 10px; vertical-align: super;"&gt;L&lt;/span&gt;***/ker(**V&lt;span style="font-size: 10px; vertical-align: sub;"&gt;G&lt;/span&gt;**), where ker(**V&lt;span style="font-size: 10px; vertical-align: sub;"&gt;G&lt;/span&gt;**) 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 space **Tq&lt;span style="font-size: 10px; vertical-align: super;"&gt;G&lt;/span&gt;*** must be isometrically isomorphic to **Tq&lt;span style="font-size: 10px; vertical-align: super;"&gt;L&lt;/span&gt;***/ker(**V&lt;span style="font-size: 10px; vertical-align: sub;"&gt;G&lt;/span&gt;**).</pre></div>
 <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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Generalized Tenney Dual Norms and Tp Tuning Space&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:3:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Dual Norms"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:3 --&gt;Dual Norms&lt;/h1&gt;
+<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Generalized Tenney Dual Norms and Tp Tuning Space&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:5:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Dual Norms"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:5 --&gt;Dual Norms&lt;/h1&gt;
-  Given any &lt;a class="wiki_link" href="/Generalized%20Tenney%20Norms%20and%20Tp%20spaces"&gt;Tp norm&lt;/a&gt; on an interval space &lt;strong&gt;Tp&lt;span style="font-size: 10px; vertical-align: sub;"&gt;G&lt;/span&gt;&lt;/strong&gt; associated with a group &lt;strong&gt;G&lt;/strong&gt;, we can define a corresponding &lt;strong&gt;dual norm&lt;/strong&gt; on the dual space &lt;strong&gt;Tp&lt;span style="font-size: 10px; vertical-align: sub;"&gt;G&lt;/span&gt;&lt;/strong&gt;* which satisfies the following identity:&lt;br /&gt;
+  Given any &lt;a class="wiki_link" href="/Generalized%20Tenney%20Norms%20and%20Tp%20Interval%20Space"&gt;Tp norm&lt;/a&gt; on an interval space &lt;strong&gt;Tp&lt;span style="font-size: 10px; vertical-align: super;"&gt;G&lt;/span&gt;&lt;/strong&gt; associated with a group &lt;strong&gt;G&lt;/strong&gt;, we can define a corresponding &lt;strong&gt;dual Tq* norm&lt;/strong&gt; on the dual space &lt;strong&gt;Tq&lt;span style="font-size: 10px; vertical-align: super;"&gt;G&lt;/span&gt;&lt;/strong&gt;* which satisfies the following identity:&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:0:
 [[math]]&amp;lt;br/&amp;gt;
-||f|| = \text{sup}\left \{\frac{|f(\vec{v})|}{||\vec{v}||}: \vec{v} \in \textbf{Lp}\right \}&amp;lt;br/&amp;gt;[[math]]
+||f||_\mathbf{Tq*} = \text{sup}\left \{\frac{|f(\vec{v})|}{||\vec{v}||_\mathbf{Tp}}: \vec{v} \in \textbf{Lp}\right \}&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;||f|| = \text{sup}\left \{\frac{|f(\vec{v})|}{||\vec{v}||}: \vec{v} \in \textbf{Lp}\right \}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:0 --&gt;&lt;br /&gt;
+  --&gt;&lt;script type="math/tex"&gt;||f||_\mathbf{Tq*} = \text{sup}\left \{\frac{|f(\vec{v})|}{||\vec{v}||_\mathbf{Tp}}: \vec{v} \in \textbf{Lp}\right \}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:0 --&gt;&lt;br /&gt;
 &lt;br /&gt;
-for all f in &lt;strong&gt;Tp&lt;span style="font-size: 10px; vertical-align: sub;"&gt;G&lt;/span&gt;&lt;/strong&gt;*.&lt;br /&gt;
+for all f in &lt;strong&gt;Tq&lt;span style="font-size: 10px; vertical-align: super;"&gt;G&lt;/span&gt;&lt;/strong&gt;*. This normed space, for which the group of vals on &lt;strong&gt;G&lt;/strong&gt; comprise the lattice of covectors with integer coefficients, is called &lt;strong&gt;Tq* Tuning Space.&lt;/strong&gt;&lt;br /&gt;
 &lt;br /&gt;
-In the simplest case where &lt;strong&gt;G&lt;/strong&gt; has as its chosen basis only primes and prime powers, and hence || · ||&lt;strong&gt;&lt;span style="font-size: 10px; vertical-align: sub;"&gt;Tp&lt;/span&gt;&lt;/strong&gt; is given by&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:7:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Dual Norms-Prime Power Interval Groups"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:7 --&gt;Prime Power Interval Groups&lt;/h2&gt;
+ In the simplest case where &lt;strong&gt;G&lt;/strong&gt; has as its chosen basis only primes and prime powers, || · ||&lt;strong&gt;&lt;span style="font-size: 10px; vertical-align: sub;"&gt;Tp&lt;/span&gt;&lt;/strong&gt; is given by&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:1:
@@ Line 46: / Line 67: @@
   --&gt;&lt;script type="math/tex"&gt;\left \| \vec{v} \right \|_{\textbf{Tp}}^\textbf{G} = \left \| \mathbf{W_G} \cdot \vec{v} \right \|_\textbf{p}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:1 --&gt;&lt;br /&gt;
 &lt;br /&gt;
-for weighting matrix &lt;strong&gt;W&lt;span style="font-size: 80%; vertical-align: sub;"&gt;G&lt;/span&gt;&lt;/strong&gt;, then the dual norm || · ||&lt;strong&gt;&lt;span style="font-size: 10px; vertical-align: sub;"&gt;Tq*&lt;/span&gt;&lt;/strong&gt; on &lt;strong&gt;Tp&lt;span style="font-size: 10px; vertical-align: sub;"&gt;G&lt;/span&gt;&lt;/strong&gt;* is given by&lt;br /&gt;
+for diagonal weighting matrix &lt;strong&gt;W&lt;span style="font-size: 80%; vertical-align: sub;"&gt;G&lt;/span&gt;&lt;/strong&gt;. Then the dual norm || · ||&lt;strong&gt;&lt;span style="font-size: 10px; vertical-align: sub;"&gt;Tq*&lt;/span&gt;&lt;/strong&gt; on &lt;strong&gt;Tq&lt;span style="font-size: 10px; vertical-align: super;"&gt;G&lt;/span&gt;&lt;/strong&gt;* is given for f in &lt;strong&gt;Tq&lt;span style="font-size: 10px; vertical-align: super;"&gt;G&lt;/span&gt;&lt;/strong&gt;* by&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:2:
@@ Line 53: / Line 74: @@
   --&gt;&lt;script type="math/tex"&gt;\left \|f \right \|_{\textbf{Tq*}}^\textbf{G} = \left \| f \cdot \mathbf{W}_\mathbf{G}^{-1} \right \|_\textbf{q}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:2 --&gt;&lt;br /&gt;
 &lt;br /&gt;
-where the coefficients p in Tp and q in Tq* satisfy the relationship 1/p + 1/q = 1.&lt;/body&gt;&lt;/html&gt;</pre></div>
+where the coefficients p in Tp and q in Tq* satisfy the relationship 1/p + 1/q = 1.&lt;br /&gt;
+&lt;br /&gt;
+The dual of any Tp norm is very similar to the dual of the ordinary Lp norm. The crucial difference to be noted is that the weighting for covectors in tuning space is the inverse of the weighting for vectors in interval space; simple primes are weighted less in interval space but more in tuning space. Unlike the weighting matrix for interval space, the weighting matrix on tuning space is a diagonal matrix in which the nth entry in the diagonal is 1/log&lt;span style="font-size: 80%; vertical-align: sub;"&gt;2&lt;/span&gt;(&lt;strong&gt;G&lt;/strong&gt;&lt;span style="font-size: 80%; vertical-align: sub;"&gt;n&lt;/span&gt;), where &lt;strong&gt;G&lt;/strong&gt;&lt;span style="font-size: 10px; vertical-align: sub;"&gt;n&lt;/span&gt; is the nth basis element in &lt;strong&gt;G&lt;/strong&gt;. We denote such inverse weighted norms with an asterisk, so that the inverse-Tenney weighted Linf norm in tuning space is Tinf*.&lt;br /&gt;
+&lt;br /&gt;
+For &lt;strong&gt;G&lt;/strong&gt; with basis of only primes and prime powers, the dual of the T1 norm is the Tinf* norm, the dual of the Tinf norm is the T1* norm, and the dual of the T2 norm is the T2* norm.&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:9:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Dual Norms-Arbitrary Interval Groups"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:9 --&gt;Arbitrary Interval Groups&lt;/h2&gt;
+ For an arbitrary group &lt;strong&gt;G&lt;/strong&gt; with its chosen basis containing intervals other than primes and prime powers, || · ||&lt;strong&gt;&lt;span style="font-size: 10px; vertical-align: sub;"&gt;Tp&lt;/span&gt;&lt;/strong&gt; is given by&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextMathRule:3:
+[[math]]&amp;lt;br/&amp;gt;
+\left \| \vec{v} \right \|_{\textbf{T1}}^\textbf{G} = \left \| \mathbf{W_L} \cdot \mathbf{V_\textbf{G}} \cdot \vec{v} \right \|_\textbf{1}&amp;lt;br/&amp;gt;[[math]]
+ --&gt;&lt;script type="math/tex"&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;/script&gt;&lt;!-- ws:end:WikiTextMathRule:3 --&gt;&lt;br /&gt;
+&lt;br /&gt;
+for a &lt;a class="wiki_link" href="/Subgroup%20Mapping%20Matrices%20%28V-maps%29"&gt;V-map&lt;/a&gt; &lt;strong&gt;V&lt;span style="font-size: 80%; vertical-align: sub;"&gt;G&lt;/span&gt;&lt;/strong&gt; representing &lt;strong&gt;G&lt;/strong&gt; in some full-limit &lt;strong&gt;L&lt;/strong&gt; and a diagonal weighting matrix &lt;strong&gt;W&lt;span style="font-size: 10px; vertical-align: sub;"&gt;L&lt;/span&gt;&lt;/strong&gt; for &lt;strong&gt;L&lt;/strong&gt;. Then if &lt;strong&gt;Tp&lt;span style="font-size: 10px; vertical-align: super;"&gt;L&lt;/span&gt;&lt;/strong&gt; represents the full-limit interval space that &lt;strong&gt;G&lt;/strong&gt; is embedded in, and &lt;strong&gt;Tq&lt;span style="font-size: 10px; vertical-align: super;"&gt;L&lt;/span&gt;&lt;/strong&gt;* is the dual space, the dual norm || · ||&lt;strong&gt;&lt;span style="font-size: 10px; vertical-align: sub;"&gt;Tq &lt;/span&gt;&lt;/strong&gt;on &lt;strong&gt;Tq&lt;span style="font-size: 80%; vertical-align: super;"&gt;G&lt;/span&gt;&lt;/strong&gt;* is given by&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextMathRule:4:
+[[math]]&amp;lt;br/&amp;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 \}&amp;lt;br/&amp;gt;[[math]]
+ --&gt;&lt;script type="math/tex"&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;/script&gt;&lt;!-- ws:end:WikiTextMathRule:4 --&gt;&lt;br /&gt;
+&lt;br /&gt;
+Note that this is the quotient norm induced on the space &lt;strong&gt;Tq&lt;span style="font-size: 10px; vertical-align: super;"&gt;L&lt;/span&gt;&lt;/strong&gt;*/ker(&lt;strong&gt;V&lt;span style="font-size: 10px; vertical-align: sub;"&gt;G&lt;/span&gt;&lt;/strong&gt;), where ker(&lt;strong&gt;V&lt;span style="font-size: 10px; vertical-align: sub;"&gt;G&lt;/span&gt;&lt;/strong&gt;) is the set of vals in the V-map that are restricted away. This result is due to a corollary of the &lt;a class="wiki_link_ext" href="http://www.math.unl.edu/~s-bbockel1/928/node25.html" rel="nofollow" target="_blank"&gt;Hahn-Banach theorem&lt;/a&gt;, which demonstrates that the space &lt;strong&gt;Tq&lt;span style="font-size: 10px; vertical-align: super;"&gt;G&lt;/span&gt;&lt;/strong&gt;* must be isometrically isomorphic to &lt;strong&gt;Tq&lt;span style="font-size: 10px; vertical-align: super;"&gt;L&lt;/span&gt;&lt;/strong&gt;*/ker(&lt;strong&gt;V&lt;span style="font-size: 10px; vertical-align: sub;"&gt;G&lt;/span&gt;&lt;/strong&gt;).&lt;/body&gt;&lt;/html&gt;</pre></div>