Generalized Tenney norms and Tp interval space: Difference between revisions

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Tp 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&lt;span style="font-size: 10px; vertical-align: sub;"&gt;2&lt;/span&gt;(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 theoretically justified; additionally, certain Tp norms are worth using as an approximation to T1 for their strong computational advantages. As such, Tp spaces are musically justified enough to merit further study, despite the surface advantages of sticking exclusively to the T1 norm.

The T2 norm is often called the **Tenney-Euclidean norm**, **TE norm**, or **TE height**, as it has the same relationship with Euclidean geometry that the T1 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&lt;span style="font-size: 80%; vertical-align: super;"&gt;[[Tenney-Euclidean metrics|(1)]][[Tenney-Euclidean temperament measures|(2)]][[Tenney-Euclidean Tuning|(3)]]&lt;/span&gt;. It approximates the T1 complexity of many intervals, although notably rates 9/1 as more complex than 15/1.

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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 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 &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;br /&gt;

The most important and natural norm which arises in this scenario is the &lt;strong&gt;Tenney norm&lt;/strong&gt;, which we will explore below.&lt;br /&gt;

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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&lt;span style="font-size: 10px; vertical-align: sub;"&gt;2&lt;/span&gt;(a·b). For a full-limit monzo |a b c d ...&amp;gt;, this norm can be calculated as |log&lt;span style="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;strong&gt;Tp norms&lt;/strong&gt;, with the Tenney norm being designated the &lt;strong&gt;T1 norm&lt;/strong&gt;.&lt;br /&gt;

where &lt;strong&gt;W&lt;span style="font-size: 10px; vertical-align: sub;"&gt;G&lt;/span&gt;&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. Conversely, it is also important to note that that for groups for which the basis does &lt;em&gt;not&lt;/em&gt; only consist of primes or prime powers, the unit sphere of the Tenney norm won't look like a dilated L1 unit sphere at all.&lt;br /&gt;

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  A useful generalization of the Tenney norm, called the &lt;strong&gt;Generalized Tenney Norm&lt;/strong&gt;, &lt;strong&gt;Tp norm&lt;/strong&gt;, or &lt;strong&gt;Tp height&lt;/strong&gt;, can be obtained as follows:&lt;br /&gt;

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Tp 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&lt;span style="font-size: 10px; vertical-align: sub;"&gt;2&lt;/span&gt;(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 theoretically justified; additionally, certain Tp norms are worth using as an approximation to T1 for their strong computational advantages. As such, Tp spaces are musically justified enough to merit further study, despite the surface advantages of sticking exclusively to the T1 norm.&lt;br /&gt;

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  The T2 norm is often called the &lt;strong&gt;Tenney-Euclidean norm&lt;/strong&gt;, &lt;strong&gt;TE norm&lt;/strong&gt;, or &lt;strong&gt;TE height&lt;/strong&gt;, as it has the same relationship with Euclidean geometry that the T1 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&lt;span style="font-size: 80%; vertical-align: super;"&gt;&lt;a class="wiki_link" href="/Tenney-Euclidean%20metrics"&gt;(1)&lt;/a&gt;&lt;a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures"&gt;(2)&lt;/a&gt;&lt;a class="wiki_link" href="/Tenney-Euclidean%20Tuning"&gt;(3)&lt;/a&gt;&lt;/span&gt;. It approximates the T1 complexity of many intervals, although notably rates 9/1 as more complex than 15/1.&lt;br /&gt;

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  --&gt;&lt;script type="math/tex"&gt;\left \langle \vec{v},\vec{w} \right \rangle_\textbf{G} = \vec{v}^T \cdot \mathbf{W}^2_\mathbf{L} \cdot \vec{w}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:4 --&gt;&lt;br /&gt;

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