Generalized Tenney norms and Tp interval space

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= = 
[[toc]]
=Basics= 

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 ℤ-module, the vector space can be constructed by simply allowing the existence of monzos with real coordinates, hence defining the module over ℝ instead of ℤ 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.

The most important and natural norm which arises in this scenario is the **Tenney norm**, which we will explore below.

=The Tenney Norm (T1 norm)= 

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 log₂(a·b). For a full-limit monzo |a b c d ...>, this norm can be calculated as |log₂(2)·a| + |log₂(3)·b| + |log₂(5)·c| + |log₂(7)·d| + ... . This is a variant of the ordinary L1 norm where each coordinate is weighted in proportion to the log₂ 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 **Tp norms**, with the Tenney norm being designated the **T1 norm**.

Informally speaking, we can obtain the Tenney norm on an interval space for any JI group by applying these three steps:
# if the interval is a smonzo, map it back to its corresponding full-limit monzo
# weight the axis for each prime p by log₂(p)
# take the ordinary L1 norm of the result.

To formalize this idea in its full generality, the Tenney norm of any vector //v// in an interval space with associated JI group G can be expressed as follows:

[[math]]
\left \| \vec{v} \right \|_{\textbf{T1}} = \left \| \mathbf{W} \cdot \mathbf{V} \cdot \vec{v} \right \|_\textbf{1}
[[math]]

where **V** is a [[xenharmonic/Subgroup Mapping Matrices (V-maps)|V-map]] in which the nth column is a monzo expressing the nth basis element of G in a suitable full-limit containing all of G as a subgroup, **W** is a diagonal weighting matrix in which the nth entry in the diagonal is the log₂ of the nth prime in the limit, and the || · ||₁ on the right hand side of the equation is the L1 norm on the resulting full-limit real vector.

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 //v// can be represented by the simpler expression

[[math]]
\left \| \vec{v} \right \|_{\textbf{T1}} = \left \| \mathbf{W} \cdot \vec{v} \right \|_\textbf{1}
[[math]]

where W is a diagonal "weighting matrix" such that the nth entry in the diagonal is the log2 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. Conversely, it is also important to note that that for groups which can't be generated by only primes or prime powers, the unit sphere of the Tenney norm will //not// look like a dilated L1 unit sphere, but rather something altogether different - it may take the shape of a hexagon or some other object.

=Generalized Tenney Norms (Tp norms)= 
A useful generalization of the Tenney norm, called the **Generalized Tenney Norm**, **Tp norm**, or **Tp height**, can be obtained as follows:

[[math]]
\left \| \vec{v} \right \|_{\textbf{Tp}} = \left \| \mathbf{W} \cdot \mathbf{V} \cdot \vec{v} \right \|_\textbf{p}
[[math]]

Note that the || · || norm on the left side of the equation now has a subscript of Tp rather than T1, and that the || · || 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 Lp norm rather than restricting our consideration to the L1 norm. In this scheme the ordinary Tenney norm now becomes the **T1 norm**, and in general we call an interval space that's been given a Tp norm **Tp interval space**.

Tp norms are thus generalizations of the Tenney norm which continue to weight the primes and prime powers as the Tenney norm does, but which use this information to gauge interval complexity such that each interval n/d may no longer have a complexity of log₂(n*d). Furthermore, generalized Tp norms may sometimes differ from the T1 norm in their ranking of intervals by Tp complexity, although the Tp norm of any interval is always bounded by its T1 norm. However, despite these supposed theoretical disadvantages, there can be many indirect implications of using a Tp norm other than T1 which are musically justified, and as such the study of Tp . One such norm we will look at is the T2

Original HTML content:

<html><head><title>Generalized Tenney Norms and Tp Interval Space</title></head><body><!-- ws:start:WikiTextHeadingRule:3:&lt;h1&gt; --><h1 id="toc0"><!-- ws:end:WikiTextHeadingRule:3 --> </h1>
 <!-- ws:start:WikiTextTocRule:11:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --><div style="margin-left: 1em;"><a href="#toc0"> </a></div>
<!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --><div style="margin-left: 1em;"><a href="#Basics">Basics</a></div>
<!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --><div style="margin-left: 1em;"><a href="#The Tenney Norm (T1 norm)">The Tenney Norm (T1 norm)</a></div>
<!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --><div style="margin-left: 1em;"><a href="#Generalized Tenney Norms (Tp norms)">Generalized Tenney Norms (Tp norms)</a></div>
<!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --></div>
<!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextHeadingRule:5:&lt;h1&gt; --><h1 id="toc1"><a name="Basics"></a><!-- ws:end:WikiTextHeadingRule:5 -->Basics</h1>
 <br />
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 ℤ-module, the vector space can be constructed by simply allowing the existence of monzos with real coordinates, hence defining the module over ℝ instead of ℤ 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.<br />
<br />
The most important and natural norm which arises in this scenario is the <strong>Tenney norm</strong>, which we will explore below.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:7:&lt;h1&gt; --><h1 id="toc2"><a name="The Tenney Norm (T1 norm)"></a><!-- ws:end:WikiTextHeadingRule:7 -->The Tenney Norm (T1 norm)</h1>
 <br />
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 log₂(a·b). For a full-limit monzo |a b c d ...&gt;, this norm can be calculated as |log₂(2)·a| + |log₂(3)·b| + |log₂(5)·c| + |log₂(7)·d| + ... . This is a variant of the ordinary L1 norm where each coordinate is weighted in proportion to the log₂ 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 <strong>Tp norms</strong>, with the Tenney norm being designated the <strong>T1 norm</strong>.<br />
<br />
Informally speaking, we can obtain the Tenney norm on an interval space for any JI group by applying these three steps:<br />
<ol><li>if the interval is a smonzo, map it back to its corresponding full-limit monzo</li><li>weight the axis for each prime p by log₂(p)</li><li>take the ordinary L1 norm of the result.</li></ol><br />
To formalize this idea in its full generality, the Tenney norm of any vector <em>v</em> in an interval space with associated JI group G can be expressed as follows:<br />
<br />
<!-- ws:start:WikiTextMathRule:0:
[[math]]&lt;br/&gt;
\left \| \vec{v} \right \|_{\textbf{T1}} = \left \| \mathbf{W} \cdot \mathbf{V} \cdot \vec{v} \right \|_\textbf{1}&lt;br/&gt;[[math]]
 --><script type="math/tex">\left \| \vec{v} \right \|_{\textbf{T1}} = \left \| \mathbf{W} \cdot \mathbf{V} \cdot \vec{v} \right \|_\textbf{1}</script><!-- ws:end:WikiTextMathRule:0 --><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 nth column is a monzo expressing the nth basis element of G in a suitable full-limit containing all of G as a subgroup, <strong>W</strong> is a diagonal weighting matrix in which the nth entry in the diagonal is the log₂ of the nth prime in the limit, and the || · ||₁ on the right hand side of the equation is the L1 norm on the resulting full-limit real vector.<br />
<br />
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 <em>v</em> can be represented by the simpler expression<br />
<br />
<!-- ws:start:WikiTextMathRule:1:
[[math]]&lt;br/&gt;
\left \| \vec{v} \right \|_{\textbf{T1}} = \left \| \mathbf{W} \cdot \vec{v} \right \|_\textbf{1}&lt;br/&gt;[[math]]
 --><script type="math/tex">\left \| \vec{v} \right \|_{\textbf{T1}} = \left \| \mathbf{W} \cdot \vec{v} \right \|_\textbf{1}</script><!-- ws:end:WikiTextMathRule:1 --><br />
<br />
where W is a diagonal &quot;weighting matrix&quot; such that the nth entry in the diagonal is the log2 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. Conversely, it is also important to note that that for groups which can't be generated by only primes or prime powers, the unit sphere of the Tenney norm will <em>not</em> look like a dilated L1 unit sphere, but rather something altogether different - it may take the shape of a hexagon or some other object.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:9:&lt;h1&gt; --><h1 id="toc3"><a name="Generalized Tenney Norms (Tp norms)"></a><!-- ws:end:WikiTextHeadingRule:9 -->Generalized Tenney Norms (Tp norms)</h1>
 A useful generalization of the Tenney norm, called the <strong>Generalized Tenney Norm</strong>, <strong>Tp norm</strong>, or <strong>Tp height</strong>, can be obtained as follows:<br />
<br />
<!-- ws:start:WikiTextMathRule:2:
[[math]]&lt;br/&gt;
\left \| \vec{v} \right \|_{\textbf{Tp}} = \left \| \mathbf{W} \cdot \mathbf{V} \cdot \vec{v} \right \|_\textbf{p}&lt;br/&gt;[[math]]
 --><script type="math/tex">\left \| \vec{v} \right \|_{\textbf{Tp}} = \left \| \mathbf{W} \cdot \mathbf{V} \cdot \vec{v} \right \|_\textbf{p}</script><!-- ws:end:WikiTextMathRule:2 --><br />
<br />
Note that the || · || norm on the left side of the equation now has a subscript of Tp rather than T1, and that the || · || 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 Lp norm rather than restricting our consideration to the L1 norm. In this scheme the ordinary Tenney norm now becomes the <strong>T1 norm</strong>, and in general we call an interval space that's been given a Tp norm <strong>Tp interval space</strong>.<br />
<br />
Tp norms are thus generalizations of the Tenney norm which continue to weight the primes and prime powers as the Tenney norm does, but which use this information to gauge interval complexity such that each interval n/d may no longer have a complexity of log₂(n*d). Furthermore, generalized Tp norms may sometimes differ from the T1 norm in their ranking of intervals by Tp complexity, although the Tp norm of any interval is always bounded by its T1 norm. However, despite these supposed theoretical disadvantages, there can be many indirect implications of using a Tp norm other than T1 which are musically justified, and as such the study of Tp . One such norm we will look at is the T2</body></html>