Generalized Tenney norms and Tp interval space

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=<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif;">Basics</span>= 
<span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;">
</span><span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;">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 <span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 15px;">ℤ</span>-module, the vector space can be constructed by simply allowing the existence of monzos with real coordinates, hence defining the module over <span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 15px;">ℝ</span> instead of <span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 15px;">ℤ</span> 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.</span>
<span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;">The most important and natural norm which arises in this scenario is the Tenney norm, which we will explore below.</span>
=<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif;">The Tenney Norm (T1 norm)</span>= 

<span style="background-color: #ffffff; color: #222222; display: block;"><span style="font-family: arial,sans-serif;">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 ...>, this norm can be calculated as |log</span><span style="font-family: arial,sans-serif; font-size: 10px; vertical-align: sub;">2</span><span style="font-family: arial,sans-serif;">(2)·a| + |log</span><span style="font-family: arial,sans-serif; font-size: 10px; vertical-align: sub;">2</span><span style="font-family: arial,sans-serif;">(3)·b| + |log</span><span style="font-family: arial,sans-serif; font-size: 10px; vertical-align: sub;">2</span><span style="font-family: arial,sans-serif;">(5)·c| + |log</span><span style="font-family: arial,sans-serif; font-size: 10px; vertical-align: sub;">2</span><span style="font-family: arial,sans-serif;">(7)·d| + ... . This is a variant of the ordinary L1 norm where each coordinate is weighted in proportion to the log</span><span style="font-family: arial,sans-serif; font-size: 10px; vertical-align: sub;">2</span><span style="font-family: arial,sans-serif;"> 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 </span>**Tp norms**, with the Tenney norm being designated the **T1 norm**.</span>

<span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;">In its full generality, the Tenney norm of any interval space can be expressed as follows:</span><span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;">
\left \| \vec{v} \right \|_{\textbf{T1}} = \left \| \mathbf{W} \cdot \mathbf{V} \cdot \vec{v} \right \|_1

where **V** is a [[xenharmonic/Subgroup Mapping Matrices (V-maps)|V-map]] in which the columns are monzos express</span><span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;">
</span><span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;">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</span><span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;">
</span><span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;">\left \| \vec{v} \right \|_{\textbf{T1}} = \left \| \mathbf{W} \cdot \vec{v} \right \|_1</span><span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;">
</span><span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;">
</span><span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;">where **W** is a diagonal "weighting matrix" such that the nth entry in the diagonal is the log<span style="font-size: 10px; vertical-align: sub;">2</span> 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.</span>

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<html><head><title>Generalized Tenney Norms and Tp Interval Space</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Basics"></a><!-- ws:end:WikiTextHeadingRule:0 --><span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif;">Basics</span></h1>
 <span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"><br />
</span><span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;">It can be useful to define a notion of the &quot;complexity&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 <span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 15px;">ℤ</span>-module, the vector space can be constructed by simply allowing the existence of monzos with real coordinates, hence defining the module over <span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 15px;">ℝ</span> instead of <span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 15px;">ℤ</span> and giving it the added structure of being a vector space. In either case, the resulting space is called <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">interval space</a>, with the monzos forming the lattice of vectors with integer coordinates.</span><br />
<span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;">The most important and natural norm which arises in this scenario is the Tenney norm, which we will explore below.</span><br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="The Tenney Norm (T1 norm)"></a><!-- ws:end:WikiTextHeadingRule:2 --><span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif;">The Tenney Norm (T1 norm)</span></h1>
 <br />
<span style="background-color: #ffffff; color: #222222; display: block;"><span style="font-family: arial,sans-serif;">The <strong>Tenney norm</strong>, also called <strong>Tenney height</strong>, 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</span><span style="font-family: arial,sans-serif; font-size: 10px; vertical-align: sub;">2</span><span style="font-family: arial,sans-serif;">(2)·a| + |log</span><span style="font-family: arial,sans-serif; font-size: 10px; vertical-align: sub;">2</span><span style="font-family: arial,sans-serif;">(3)·b| + |log</span><span style="font-family: arial,sans-serif; font-size: 10px; vertical-align: sub;">2</span><span style="font-family: arial,sans-serif;">(5)·c| + |log</span><span style="font-family: arial,sans-serif; font-size: 10px; vertical-align: sub;">2</span><span style="font-family: arial,sans-serif;">(7)·d| + ... . This is a variant of the ordinary L1 norm where each coordinate is weighted in proportion to the log</span><span style="font-family: arial,sans-serif; font-size: 10px; vertical-align: sub;">2</span><span style="font-family: arial,sans-serif;"> 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 </span><strong>Tp norms</strong>, with the Tenney norm being designated the <strong>T1 norm</strong>.</span><br />
<br />
<span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;">In its full generality, the Tenney norm of any interval space can be expressed as follows:</span><span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"><br />
\left \| \vec{v} \right \|_{\textbf{T1}} = \left \| \mathbf{W} \cdot \mathbf{V} \cdot \vec{v} \right \|_1<br />
<br />
where <strong>V</strong> is a <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Subgroup%20Mapping%20Matrices%20%28V-maps%29">V-map</a> in which the columns are monzos express</span><span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"><br />
</span><span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;">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</span><span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"><br />
</span><span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;">\left \| \vec{v} \right \|_{\textbf{T1}} = \left \| \mathbf{W} \cdot \vec{v} \right \|_1</span><span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"><br />
</span><span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"><br />
</span><span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;">where <strong>W</strong> is a diagonal &quot;weighting matrix&quot; such that the nth entry in the diagonal is the log<span style="font-size: 10px; vertical-align: sub;">2</span> 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.</span></body></html>