Generalized Tenney norms and Tp interval space: Difference between revisions
Wikispaces>mbattaglia1 **Imported revision 356539790 - Original comment: ** |
Wikispaces>mbattaglia1 **Imported revision 356542386 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | 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 12: | : This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2012-08-06 12:51:59 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>356542386</tt>.<br> | ||
: The revision comment was: <tt></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> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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[[math]] | [[math]] | ||
Given these matrices, the T1 norm of our smonzo |0 -2 1>, which we will call **v**, can be found by taking the L1 norm of the resulting real vector **W<span style="font-size: 10px; vertical-align: sub;">L</span>** · **V<span style="font-size: 10px; vertical-align: sub;">G</span>** · **v**. This real vector works out to |0 -7.925 2.322 5.615>, and its L1 norm is |0| + |-7.925| + |2.322| + |5.615| = 15.861. This is the T1 norm of |0 -2 1> on the 2.5/3.9/7 group. | Given these matrices, the T1 norm of our smonzo |0 -2 1>, which we will call **v**, can be found by taking the L1 norm of the resulting real vector **W<span style="font-size: 10px; vertical-align: sub;">L</span>** · **V<span style="font-size: 10px; vertical-align: sub;">G</span>** · **v**. This expression works out to | ||
[[math]] | |||
\left \| \vec{v} \right \|_\mathbf{T1}^\mathbf{2.\nicefrac{9}{7}.\nicefrac{5}{3}} | |||
\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]] | |||
real vector works out to |0 -7.925 2.322 5.615>, and its L1 norm is |0| + |-7.925| + |2.322| + |5.615| = 15.861. This is the T1 norm of |0 -2 1> on the 2.5/3.9/7 group. | |||
To confirm this, we can put smonzo |0 -2 1> back into rational form to see that it represents the interval 245/243. As the L1 norm is supposed to give log(n·d) for any interval n/d, we can confirm that we have the right answer above by noting that log(245·243) is indeed equal to 15.861.</pre></div> | To confirm this, we can put smonzo |0 -2 1> back into rational form to see that it represents the interval 245/243. As the L1 norm is supposed to give log(n·d) for any interval n/d, we can confirm that we have the right answer above by noting that log(245·243) is indeed equal to 15.861.</pre></div> | ||
<h4>Original HTML content:</h4> | <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 Norms and Tp Interval Space</title></head><body><!-- ws:start:WikiTextHeadingRule: | <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 Norms and Tp Interval Space</title></head><body><!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc0"><!-- ws:end:WikiTextHeadingRule:8 --> </h1> | ||
<!-- ws:start:WikiTextTocRule: | <!-- ws:start:WikiTextTocRule:20:&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:20 --><!-- ws:start:WikiTextTocRule:21: --><div style="margin-left: 1em;"><a href="#toc0"> </a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --><div style="margin-left: 1em;"><a href="#Basics">Basics</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --><div style="margin-left: 1em;"><a href="#The Tenney Norm (T1 norm)">The Tenney Norm (T1 norm)</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --><div style="margin-left: 1em;"><a href="#Generalized Tenney Norms (Tp norms)">Generalized Tenney Norms (Tp norms)</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --><div style="margin-left: 1em;"><a href="#The Tenney-Euclidean Norm (TE norm, T2 norm)">The Tenney-Euclidean Norm (TE norm, T2 norm)</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --><div style="margin-left: 1em;"><a href="#Examples">Examples</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc1"><a name="Basics"></a><!-- ws:end:WikiTextHeadingRule:10 -->Basics</h1> | ||
<br /> | <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 /> | 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 /> | ||
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The most important and natural norm which arises in this scenario is the <strong>Tenney norm</strong>, which we will explore below.<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 /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc2"><a name="The Tenney Norm (T1 norm)"></a><!-- ws:end:WikiTextHeadingRule:12 -->The Tenney Norm (T1 norm)</h1> | ||
<br /> | <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 log<span style="font-size: 10px; vertical-align: sub;">2</span>(a·b). For a full-limit monzo |a b c d ...&gt;, this norm can be calculated as |log<span style="font-size: 80%; vertical-align: sub;">2</span>(2)·a| + |log<span style="font-size: 10px; vertical-align: sub;">2</span>(3)·b| + |log<span style="font-size: 10px; vertical-align: sub;">2</span>(5)·c| + |log<span style="font-size: 10px; vertical-align: sub;">2</span>(7)·d| + ... . This is a variant of the ordinary L1 norm where each coordinate is weighted in proportion to the log<span style="font-size: 10px; vertical-align: sub;">2</span> 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 /> | 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<span style="font-size: 10px; vertical-align: sub;">2</span>(a·b). For a full-limit monzo |a b c d ...&gt;, this norm can be calculated as |log<span style="font-size: 80%; vertical-align: sub;">2</span>(2)·a| + |log<span style="font-size: 10px; vertical-align: sub;">2</span>(3)·b| + |log<span style="font-size: 10px; vertical-align: sub;">2</span>(5)·c| + |log<span style="font-size: 10px; vertical-align: sub;">2</span>(7)·d| + ... . This is a variant of the ordinary L1 norm where each coordinate is weighted in proportion to the log<span style="font-size: 10px; vertical-align: sub;">2</span> 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 /> | ||
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where <strong>W<span style="font-size: 10px; vertical-align: sub;">G</span></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. Conversely, it is also important to note that that for groups for which the basis does <em>not</em> 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.<br /> | where <strong>W<span style="font-size: 10px; vertical-align: sub;">G</span></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. Conversely, it is also important to note that that for groups for which the basis does <em>not</em> 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.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc3"><a name="Generalized Tenney Norms (Tp norms)"></a><!-- ws:end:WikiTextHeadingRule:14 -->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 /> | 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 /> | <br /> | ||
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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<span style="font-size: 10px; vertical-align: sub;">2</span>(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.<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 for which an arbitrary interval n/d may no longer have a complexity of log<span style="font-size: 10px; vertical-align: sub;">2</span>(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.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc4"><a name="The Tenney-Euclidean Norm (TE norm, T2 norm)"></a><!-- ws:end:WikiTextHeadingRule:16 -->The Tenney-Euclidean Norm (TE norm, T2 norm)</h1> | ||
The T2 norm is often called the <strong>Tenney-Euclidean norm</strong>, <strong>TE norm</strong>, or <strong>TE height</strong>, 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<span style="font-size: 80%; vertical-align: super;"><a class="wiki_link" href="/Tenney-Euclidean%20metrics">(1)</a><a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures">(2)</a><a class="wiki_link" href="/Tenney-Euclidean%20Tuning">(3)</a></span>. It approximates the T1 complexity of many intervals, although notably rates 9/1 as more complex than 15/1.<br /> | The T2 norm is often called the <strong>Tenney-Euclidean norm</strong>, <strong>TE norm</strong>, or <strong>TE height</strong>, 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<span style="font-size: 80%; vertical-align: super;"><a class="wiki_link" href="/Tenney-Euclidean%20metrics">(1)</a><a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures">(2)</a><a class="wiki_link" href="/Tenney-Euclidean%20Tuning">(3)</a></span>. It approximates the T1 complexity of many intervals, although notably rates 9/1 as more complex than 15/1.<br /> | ||
<br /> | <br /> | ||
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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.<br /> | 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.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:18:&lt;h1&gt; --><h1 id="toc5"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:18 -->Examples</h1> | ||
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Say that we're in the 2.9/7.5/3 subgroup, and we want to find the T1 norm of |0 -2 1&gt;. Then we can come up with a V-map <strong>V<span style="font-size: 10px; vertical-align: sub;">G</span></strong> for this subgroup in the 7-limit as follows:<br /> | Say that we're in the 2.9/7.5/3 subgroup, and we want to find the T1 norm of |0 -2 1&gt;. Then we can come up with a V-map <strong>V<span style="font-size: 10px; vertical-align: sub;">G</span></strong> for this subgroup in the 7-limit as follows:<br /> | ||
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\end{bmatrix}</script><!-- ws:end:WikiTextMathRule:6 --><br /> | \end{bmatrix}</script><!-- ws:end:WikiTextMathRule:6 --><br /> | ||
<br /> | <br /> | ||
Given these matrices, the T1 norm of our smonzo |0 -2 1&gt;, which we will call <strong>v</strong>, can be found by taking the L1 norm of the resulting real vector <strong>W<span style="font-size: 10px; vertical-align: sub;">L</span></strong> · <strong>V<span style="font-size: 10px; vertical-align: sub;">G</span></strong> · <strong>v</strong>. This real vector works out to |0 -7.925 2.322 5.615&gt;, and its L1 norm is |0| + |-7.925| + |2.322| + |5.615| = 15.861. This is the T1 norm of |0 -2 1&gt; on the 2.5/3.9/7 group.<br /> | Given these matrices, the T1 norm of our smonzo |0 -2 1&gt;, which we will call <strong>v</strong>, can be found by taking the L1 norm of the resulting real vector <strong>W<span style="font-size: 10px; vertical-align: sub;">L</span></strong> · <strong>V<span style="font-size: 10px; vertical-align: sub;">G</span></strong> · <strong>v</strong>. This expression works out to<br /> | ||
<br /> | |||
<!-- ws:start:WikiTextMathRule:7: | |||
[[math]]&lt;br/&gt; | |||
\left \| \vec{v} \right \|_\mathbf{T1}^\mathbf{2.\nicefrac{9}{7}.\nicefrac{5}{3}}&lt;br /&gt; | |||
\begin{bmatrix}&lt;br /&gt; | |||
\log_2(2) &amp; 0 &amp; 0 &amp; 0\\&lt;br /&gt; | |||
0 &amp; \log_2(3) &amp; 0 &amp; 0\\&lt;br /&gt; | |||
0 &amp; 0 &amp; \log_2(5) &amp; 0\\&lt;br /&gt; | |||
0 &amp; 0 &amp; 0 &amp; \log_2(7)&lt;br /&gt; | |||
\end{bmatrix}&lt;br/&gt;[[math]] | |||
--><script type="math/tex">\left \| \vec{v} \right \|_\mathbf{T1}^\mathbf{2.\nicefrac{9}{7}.\nicefrac{5}{3}} | |||
\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}</script><!-- ws:end:WikiTextMathRule:7 --><br /> | |||
<br /> | |||
real vector works out to |0 -7.925 2.322 5.615&gt;, and its L1 norm is |0| + |-7.925| + |2.322| + |5.615| = 15.861. This is the T1 norm of |0 -2 1&gt; on the 2.5/3.9/7 group.<br /> | |||
<br /> | <br /> | ||
To confirm this, we can put smonzo |0 -2 1&gt; back into rational form to see that it represents the interval 245/243. As the L1 norm is supposed to give log(n·d) for any interval n/d, we can confirm that we have the right answer above by noting that log(245·243) is indeed equal to 15.861.</body></html></pre></div> | To confirm this, we can put smonzo |0 -2 1&gt; back into rational form to see that it represents the interval 245/243. As the L1 norm is supposed to give log(n·d) for any interval n/d, we can confirm that we have the right answer above by noting that log(245·243) is indeed equal to 15.861.</body></html></pre></div> | ||