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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{texmap}}{{DISPLAYTITLE:Generalized Tenney norms and T<sub>''p''</sub> interval space}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
It can be useful to define a notion of the "[[complexity]]" of an [[interval]], so that small-integer ratios such as 3/2 are less complex and intervals such as 32805/32768 are more complex. This can be accomplished for any {{w|free abelian group}} of ([[subgroup]]) [[Monzos and interval space|monzos]] by {{w|embedding}} the group in a {{w|normed vector space}}, so that the {{w|Norm (mathematics)|norm}} of any interval is taken to be its complexity. The monzos form a ℤ-module, with coordinates given by integers, and the vector space embedding can be constructed by simply allowing real coordinates, hence defining the module over ℝ instead of ℤ and giving it the structure of a vector space. The resulting space is called [[Monzos and interval space|interval space]], with the monzos forming the {{w|integer lattice}} of vectors with integer coordinates, but where we will allow any vector space norm on ℝ<sup>''n''</sup>.
: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2012-08-06 03:59:50 UTC</tt>.<br>
: The original revision id was <tt>356495966</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">=&lt;span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif;"&gt;Basics&lt;/span&gt;=
&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;
&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;It can be useful to define a notion of the "complexity" of an interval, so that small-integer ratios such as 3/2 are less complex and intervals such as 32805/32768 are more complex. This can be accomplished for any free abelian group of monzos or smonzos by embedding the group in a normed vector space, so that the norm of any interval is taken to be its complexity. Alternatively, if one prefers to think of the monzos as forming a &lt;span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 15px;"&gt;ℤ&lt;/span&gt;-module, the vector space can be constructed by simply allowing the existence of monzos with real coordinates, hence defining the module over &lt;span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 15px;"&gt;ℝ&lt;/span&gt; instead of &lt;span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 15px;"&gt;ℤ&lt;/span&gt; and giving it the added structure of being a vector space. In either case, the resulting space is called [[Monzos and Interval Space|interval space]], with the monzos forming the lattice of vectors with integer coordinates.&lt;/span&gt;


&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;The most important and natural norm which arises in this scenario is the Tenney norm, which we will explore below.&lt;/span&gt;
The most important and natural norm which arises in this scenario is the [[Tenney norm]], which we will explore below.


=&lt;span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif;"&gt; &lt;/span&gt;=
== Tenney norm (T<sub>1</sub> norm) ==
=&lt;span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif;"&gt;The Tenney Norm (T1 norm)&lt;/span&gt;=  
{{Main| Tenney height }}


&lt;span style="background-color: #ffffff; color: #222222; display: block;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt;The **Tenney norm**, also called **Tenney height**, is the norm such that for any monzo representing an interval a/b, the norm of the interval is log(a·b). For a full-limit monzo |a b c d ...&gt;, this norm can be calculated as |log&lt;span style="background-color: #ffffff; color: #222222; font-size: 80%; vertical-align: sub;"&gt;2&lt;/span&gt;(2)·a| + |log&lt;span style="font-size: 10px; vertical-align: sub;"&gt;2&lt;/span&gt;(3)·b| + |log&lt;span style="font-size: 10px; vertical-align: sub;"&gt;2&lt;/span&gt;(5)·c| + |log&lt;span style="font-size: 10px; vertical-align: sub;"&gt;2&lt;/span&gt;(7)·d| + ... . This is a variant of the ordinary L1 norm where each coordinate is weighted in proportion to the log&lt;span style="font-size: 10px; vertical-align: sub;"&gt;2&lt;/span&gt; of the prime that it represents. As we will soon see below, generalizations of the Tenney norm that correspond to other weighted Lp norms also exist; these we will call &lt;/span&gt;**Tp norms**, with the Tenney norm being designated the **T1 norm**.&lt;/span&gt;
The '''Tenney norm''', also called '''Tenney height''', is the norm such that for any monzo representing an interval ''n''/''d'', the norm of the interval is log<sub>2</sub>(''nd''). For a full-limit monzo {{nowrap|'''m''' {{=}} {{monzo| ''m''<sub>1</sub> ''m''<sub>2</sub> … ''m''<sub>π (''p'')</sub> … }}}}, with π standing for the {{w|prime-counting function}}, this norm can be calculated as


&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;In its full generality, the Tenney norm of any interval space can be expressed as follows:&lt;/span&gt;
<math>\displaystyle \lvert m_1 \log_2 (2) \rvert + \lvert m_2 \log_2 (3) \rvert + \ldots + \lvert m_{\pi (p)} \log_2 (p) \rvert </math>


&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;
This is a variant of the ordinary ''L''<sub>1</sub> norm where each coordinate is weighted in proportion to the log<sub>2</sub> of the prime that it represents. As we will soon see below, generalizations of the Tenney norm that correspond to other weighted ''L''<sub>''p''</sub> norms also exist; these we will call '''T<sub>''p''</sub> norms''', with the Tenney norm being designated the '''T<sub>1</sub> norm'''.
&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;[[math]]&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block;"&gt;
&lt;span style="font-family: arial,sans-serif;"&gt;\left \| \vec{v} \right \|_{\textbf{T1}} = \left \| \mathbf{W} \cdot \mathbf{V} \cdot \vec{v} \right \|_1&lt;/span&gt;
&lt;span style="font-family: arial,sans-serif;"&gt;[[math]]&lt;/span&gt;


&lt;span style="font-family: arial,sans-serif;"&gt;where &lt;/span&gt;**V** is a [[xenharmonic/Subgroup Mapping Matrices (V-maps)|V-map]] in which the columns are monzos express
Informally speaking, we can obtain the Tenney norm on an interval space for any JI group by applying these three steps:
&lt;/span&gt;&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;
&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;For interval spaces corresponding to a group of monzos where the basis is set to consist of only primes or prime powers, the Tenney norm of any interval v can be represented by the expression&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;
&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;[[math]]&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;
\left \| \vec{v} \right \|_{\textbf{T1}} = \left \| \mathbf{W} \cdot \vec{v} \right \|_1
&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;[[math]]&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;
&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;where **W** is a diagonal "weighting matrix" such that the nth entry in the diagonal is the log&lt;span style="font-size: 10px; vertical-align: sub;"&gt;2&lt;/span&gt; of the interval represented by the nth coordinate of the monzo. For Tenney norms on these spaces, the unit sphere will look like the ordinary L1 unit sphere, but be dilated a different amount along each axis.&lt;/span&gt;


&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;
# if the interval is a subgroup monzo, with coordinates in the subgroup basis, map it back to its corresponding full-limit monzo
&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block;"&gt;
# weight the axis for each prime ''p'' by log<sub>2</sub>(''p'')
&lt;/span&gt;</pre></div>
# take the ordinary ''L''<sub>1</sub> norm of the result.
<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 Norms and Tp Interval Space&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Basics"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;&lt;span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif;"&gt;Basics&lt;/span&gt;&lt;/h1&gt;
To formalize this idea in its full generality, the Tenney norm of any subgroup monzo '''m''' in an interval space with associated JI group ''G'' can be expressed as follows:
&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;&lt;br /&gt;
 
&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;It can be useful to define a notion of the &amp;quot;complexity&amp;quot; of an interval, so that small-integer ratios such as 3/2 are less complex and intervals such as 32805/32768 are more complex. This can be accomplished for any free abelian group of monzos or smonzos by embedding the group in a normed vector space, so that the norm of any interval is taken to be its complexity. Alternatively, if one prefers to think of the monzos as forming a &lt;span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 15px;"&gt;ℤ&lt;/span&gt;-module, the vector space can be constructed by simply allowing the existence of monzos with real coordinates, hence defining the module over &lt;span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 15px;"&gt;ℝ&lt;/span&gt; instead of &lt;span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 15px;"&gt;ℤ&lt;/span&gt; and giving it the added structure of being a vector space. In either case, the resulting space is called &lt;a class="wiki_link" href="/Monzos%20and%20Interval%20Space"&gt;interval space&lt;/a&gt;, with the monzos forming the lattice of vectors with integer coordinates.&lt;/span&gt;&lt;br /&gt;
<math>\displaystyle \lVert \vec m \rVert_{\text{T} 1}^G = \lVert W_J \cdot S_G \cdot \vec m \rVert_1</math>
&lt;br /&gt;
 
&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;The most important and natural norm which arises in this scenario is the Tenney norm, which we will explore below.&lt;/span&gt;&lt;br /&gt;
where ''S''<sub>''G''</sub> is a [[subgroup basis matrix]] in which the ''n''-th column is a monzo expressing the ''n''-th basis element of ''G'' in a suitable full-limit ''J'' containing all of ''G'' as a subgroup, ''W''<sub>''J''</sub> is a diagonal weighting matrix in which the ''n''-th entry in the diagonal is the log<sub>2</sub> of the ''n''-th prime in ''J'', and the {{nowrap|‖ · ‖<sub>1</sub>}} on the right hand side of the equation is the ''L''<sub>1</sub> norm on the resulting full-limit real vector.
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;&lt;span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif;"&gt; &lt;/span&gt;&lt;/h1&gt;
It is notable that, for interval spaces corresponding to a group of monzos where the basis is set to consist of only primes or prime powers, the Tenney norm of any interval ''m'' can be represented by the simpler expression
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="The Tenney Norm (T1 norm)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;&lt;span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif;"&gt;The Tenney Norm (T1 norm)&lt;/span&gt;&lt;/h1&gt;
 
&lt;br /&gt;
<math>\displaystyle \lVert \vec m \rVert_{\text{T} 1}^G = \lVert W_G \cdot \vec m \rVert_1</math>
&lt;span style="background-color: #ffffff; color: #222222; display: block;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt;The &lt;strong&gt;Tenney norm&lt;/strong&gt;, also called &lt;strong&gt;Tenney height&lt;/strong&gt;, is the norm such that for any monzo representing an interval a/b, the norm of the interval is log(a·b). For a full-limit monzo |a b c d ...&amp;gt;, this norm can be calculated as |log&lt;span style="background-color: #ffffff; color: #222222; font-size: 80%; vertical-align: sub;"&gt;2&lt;/span&gt;(2)·a| + |log&lt;span style="font-size: 10px; vertical-align: sub;"&gt;2&lt;/span&gt;(3)·b| + |log&lt;span style="font-size: 10px; vertical-align: sub;"&gt;2&lt;/span&gt;(5)·c| + |log&lt;span style="font-size: 10px; vertical-align: sub;"&gt;2&lt;/span&gt;(7)·d| + ... . This is a variant of the ordinary L1 norm where each coordinate is weighted in proportion to the log&lt;span style="font-size: 10px; vertical-align: sub;"&gt;2&lt;/span&gt; of the prime that it represents. As we will soon see below, generalizations of the Tenney norm that correspond to other weighted Lp norms also exist; these we will call &lt;/span&gt;&lt;strong&gt;Tp norms&lt;/strong&gt;, with the Tenney norm being designated the &lt;strong&gt;T1 norm&lt;/strong&gt;.&lt;/span&gt;&lt;br /&gt;
 
&lt;br /&gt;
where ''W''<sub>''G''</sub> is a diagonal "weighting matrix" such that the ''n''-th entry in the diagonal is the log<sub>2</sub> of the interval represented by the ''n''-th coordinate of the monzo. For Tenney norms on these spaces, the unit sphere will look like the ordinary ''L''<sub>1</sub> unit sphere, but be dilated a different amount along each axis. Conversely, it is also important to note that that for groups for which the basis does ''not'' only consist of primes or prime powers, the unit sphere of the Tenney norm will not look like a dilated ''L''<sub>1</sub> unit sphere at all.
&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;In its full generality, the Tenney norm of any interval space can be expressed as follows:&lt;/span&gt;&lt;br /&gt;
 
&lt;br /&gt;
== Generalized Tenney norms (T<sub>''p''</sub> norms) ==
&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;&lt;br /&gt;
A useful generalization of the Tenney norm, called the '''generalized Tenney norm''', '''T<sub>''p''</sub> norm''', or '''T<sub>''p''</sub> height''', can be obtained as follows:
&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;&lt;a class="wiki_link" href="/math"&gt;math&lt;/a&gt;&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block;"&gt;&lt;br /&gt;
 
&lt;span style="font-family: arial,sans-serif;"&gt;\left \| \vec{v} \right \|_{\textbf{T1}} = \left \| \mathbf{W} \cdot \mathbf{V} \cdot \vec{v} \right \|_1&lt;/span&gt;&lt;br /&gt;
<math>\displaystyle \lVert \vec m \rVert_{\text{T} p}^G = \lVert W_J \cdot S_G \cdot \vec m \rVert_p</math>
&lt;span style="font-family: arial,sans-serif;"&gt;&lt;a class="wiki_link" href="/math"&gt;math&lt;/a&gt;&lt;/span&gt;&lt;br /&gt;
 
&lt;br /&gt;
In this scheme the ordinary Tenney norm now becomes the '''T<sub>1</sub> norm''', and in general we call an interval space that has been given a T<sub>''p''</sub> norm '''T<sub>''p''</sub> interval space'''. We may sometimes notate this as T<sub>''p''</sub><sup>G</sup>, where ''G'' is the associated group the interval space is built around.
&lt;span style="font-family: arial,sans-serif;"&gt;where &lt;/span&gt;&lt;strong&gt;V&lt;/strong&gt; is a &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Subgroup%20Mapping%20Matrices%20%28V-maps%29"&gt;V-map&lt;/a&gt; in which the columns are monzos express&lt;br /&gt;
 
&lt;/span&gt;&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;&lt;br /&gt;
Note that the {{nowrap|‖ · ‖<sub>T''p''</sub>}} norm on the left side of the equation now has a subscript of T<sub>''p''</sub> rather than T<sub>1</sub>, and that the {{nowrap|‖ · ‖<sub>''p''</sub>}} norm on the right side of the equation now has a subscript of ''p'' rather than 1. The generalized Tenney norm can thus be thought of as applying the same three-step process that the Tenney norm does, but where the last step can be an arbitrary ''L''<sub>''p''</sub> norm rather than restricting our consideration to the ''L''<sub>1</sub> norm.
&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;For interval spaces corresponding to a group of monzos where the basis is set to consist of only primes or prime powers, the Tenney norm of any interval v can be represented by the expression&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;&lt;br /&gt;
 
&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;&lt;a class="wiki_link" href="/math"&gt;math&lt;/a&gt;&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;&lt;br /&gt;
T<sub>''p''</sub> norms are thus generalizations of the Tenney norm which continue to weight the primes and prime powers as the Tenney norm does, but for which an arbitrary interval ''n''/''d'' may no longer have a complexity of log<sub>2</sub>(''nd''). Furthermore, generalized T<sub>''p''</sub> norms may sometimes differ from the T<sub>1</sub> norm in their ranking of intervals by T<sub>''p''</sub> complexity, although the T<sub>''p''</sub> norm of any interval is always bounded by its T<sub>1</sub> norm. However, despite these supposed theoretical disadvantages, there can be many indirect implications of using a T<sub>''p''</sub> norm other than T<sub>1</sub> which are theoretically justified; additionally, certain T<sub>''p''</sub> norms are worth using as an approximation to T<sub>1</sub> for their strong computational advantages. As such, T<sub>''p''</sub> spaces are musically justified enough to merit further study, despite the surface advantages of sticking exclusively to the T<sub>1</sub> norm.
\left \| \vec{v} \right \|_{\textbf{T1}} = \left \| \mathbf{W} \cdot \vec{v} \right \|_1&lt;br /&gt;
 
&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;&lt;a class="wiki_link" href="/math"&gt;math&lt;/a&gt;&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;&lt;br /&gt;
== Tenney–Euclidean norm (TE norm, T<sub>2</sub> norm) ==
&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;where &lt;strong&gt;W&lt;/strong&gt; is a diagonal &amp;quot;weighting matrix&amp;quot; such that the nth entry in the diagonal is the log&lt;span style="font-size: 10px; vertical-align: sub;"&gt;2&lt;/span&gt; of the interval represented by the nth coordinate of the monzo. For Tenney norms on these spaces, the unit sphere will look like the ordinary L1 unit sphere, but be dilated a different amount along each axis.&lt;/span&gt;&lt;br /&gt;
{{Main| Tenney–Euclidean metrics }}
&lt;br /&gt;
 
&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;&lt;br /&gt;
The T<sub>2</sub> norm is often called the '''Tenney–Euclidean norm''', '''TE norm''', or '''TE height''', as it has the same relationship with Euclidean geometry that the T<sub>1</sub> norm has with taxicab geometry. Applications involving the TE norm tend to be easy to compute, such as the host of TE-related metrics defined for rating temperaments<sup>[[Tenney–Euclidean metrics|(1)]][[Tenney–Euclidean temperament measures|(2)]][[Tenney–Euclidean tuning|(3)]]</sup>. It approximates the T<sub>1</sub> complexity of many intervals, although notably rates 9/1 as more complex than 15/1.
&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block;"&gt;&lt;br /&gt;
 
&lt;/span&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
The T<sub>2</sub> norm is also the only T<sub>''p''</sub> norm that naturally defines an inner product, given by the matrix multiplication
 
<math>\displaystyle \left \langle \vec v, \vec w  \right \rangle_G = \vec v^\mathsf {T} \cdot (S_G^\mathsf {T} \cdot W_J^2 \cdot S_G) \cdot \vec w</math>
 
The matrix product {{nowrap|{{subsup|''S''|''G''|T}} · {{subsup|''W''|''J''|2}} · ''S''<sub>''G''</sub>}} itself is a positive definite matrix, and as such defines the inner product for the TE norm. Note that this setup represents the vectors '''v''' and '''w''' by column vectors, so that '''v'''<sup>T</sup> denotes a row vector. As was the case with T<sub>''p''</sub> norms in general, this equation simplifies considerably when the group G takes as its basis only primes and prime powers, becoming instead
 
<math>\displaystyle \left \langle \vec v, \vec w \right \rangle_G = \vec v ^\mathsf {T} \cdot W_G^2 \cdot \vec w</math>
 
In these cases, the relationship between the TE inner product and the ordinary Euclidean inner product becomes apparent; the former is simply a version of the latter which weights the coordinates of each vector.
 
== Examples ==
Say that we are in the 2.9/7.5/3 subgroup, and we want to find the T<sub>1</sub> norm of {{monzo| 0 -2 1 }}. Then we can come up with a subgroup basis matrix ''S''<sub>''G''</sub> for this subgroup in the 7-limit as follows:
 
<math>\displaystyle
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 2 & 0 & -1 \\
0 & -1 & 1 & 0 \\
\end{bmatrix}</math>
 
Note that the "rows" here are written in kets; this is a convention to signify that each ket, representing a monzo, is actually supposed to represent a column of the matrix as explained in [[Subgroup basis matrices]].
 
We can also come up with a weighting matrix for the full-limit W<sub>J</sub> as follows:
 
<math>\displaystyle
\begin{bmatrix}
\log_2(2) & 0 & 0 & 0\\
0 & \log_2(3) & 0 & 0\\
0 & 0 & \log_2(5) & 0\\
0 & 0 & 0 & \log_2(7)
\end{bmatrix}</math>
 
Given these matrices, the T<sub>1</sub> norm of our subgroup basis monzo {{monzo| 0 -2 1 }}, which we will call '''m''', can be found by taking the ''L''<sub>1</sub> norm of the resulting real vector {{nowrap|''W''<sub>''J''</sub> · ''S''<sub>''G''</sub> · '''m'''}}. This expression works out to
 
<math>\displaystyle
\left\lVert \vec m \right\rVert_{\text{T} 1}^{2 \text{.} 9/7 \text{.} 5/3} = \left \lVert
\begin{bmatrix}
\log_2(2) & 0 & 0 & 0\\
0 & \log_2(3) & 0 & 0\\
0 & 0 & \log_2(5) & 0\\
0 & 0 & 0 & \log_2(7)
\end{bmatrix} \cdot \begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 2 & 0 & -1 \\
0 & -1 & 1 & 0 \\
\end{bmatrix} \cdot \begin{bmatrix}
0 & -2 & 1 \\
\end{bmatrix}
\right \rVert_1
</math>
 
which finally resolves to
 
<math>\displaystyle \left\lVert \vec m \right\rVert_{\text{T} 1}^{2.9/7.5/3} = \lVert \monzo{ \begin{matrix} 0 & -7.925 & 2.322 & 5.615 \end{matrix} } \rVert_1 = 15.861</math>
 
Note that {{nowrap|15.861 {{=}} {{!}}0{{!}} + {{!}}−7.925{{!}} + {{!}}2.322{{!}} + {{!}}5.615{{!}}}}, which is the ''L''<sub>1</sub> norm of the vector.
 
To confirm this, we can put the subgroup basis monzo {{monzo| 0 -2 1 }} back into rational form to see that it represents the interval 245/243. As the ''L''<sub>1</sub> norm is supposed to give log(''nd'') for any interval ''n''/''d'', we can confirm that we have the right answer above by noting that {{nowrap|log<sub>2</sub>(245 · 243)}} is indeed equal to 15.861.
 
== See also ==
* [[Generalized Tenney dual norms and Tp tuning space|Generalized Tenney dual norms and T<sub>''p''</sub> tuning space]]
 
[[Category:Math]]
[[Category:Interval space]]
[[Category:Interval complexity measures]]
[[Category:Tenney-weighted measures]]
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