Generalized Tenney norms and Tp interval space: Difference between revisions
Wikispaces>mbattaglia1 **Imported revision 356539736 - Original comment: ** |
Wikispaces>mbattaglia1 **Imported revision 356539790 - 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 12:32: | : This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2012-08-06 12:32:37 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>356539790</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]] | ||
where **V<span style="font-size: 80%; vertical-align: sub;">G</span>** 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 **L** containing all of G as a subgroup, **W<span style="font-size: 80%; vertical-align: sub;">L</span>** is a diagonal weighting matrix in which the nth entry in the diagonal is the log<span style="font-size: 10px; vertical-align: sub;">2</span> of the nth prime in **L**, and the || · ||**<span style="font-size: 80%; vertical-align: sub;">1</span>** on the right hand side of the equation is the L1 norm on the resulting full-limit real vector. | where **V<span style="font-size: 80%; vertical-align: sub;">G</span>** 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 **L** containing all of **G** as a subgroup, **W<span style="font-size: 80%; vertical-align: sub;">L</span>** is a diagonal weighting matrix in which the nth entry in the diagonal is the log<span style="font-size: 10px; vertical-align: sub;">2</span> of the nth prime in **L**, and the || · ||**<span style="font-size: 80%; vertical-align: sub;">1</span>** 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 | 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 | ||
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--><script type="math/tex">\left \| \vec{v} \right \|_{\textbf{T1}}^\textbf{G} = \left \| \mathbf{W_L} \cdot \mathbf{V_\textbf{G}} \cdot \vec{v} \right \|_\textbf{1}</script><!-- ws:end:WikiTextMathRule:0 --><br /> | --><script type="math/tex">\left \| \vec{v} \right \|_{\textbf{T1}}^\textbf{G} = \left \| \mathbf{W_L} \cdot \mathbf{V_\textbf{G}} \cdot \vec{v} \right \|_\textbf{1}</script><!-- ws:end:WikiTextMathRule:0 --><br /> | ||
<br /> | <br /> | ||
where <strong>V<span style="font-size: 80%; vertical-align: sub;">G</span></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 <strong>G</strong> in a suitable full-limit <strong>L</strong> containing all of G as a subgroup, <strong>W<span style="font-size: 80%; vertical-align: sub;">L</span></strong> is a diagonal weighting matrix in which the nth entry in the diagonal is the log<span style="font-size: 10px; vertical-align: sub;">2</span> of the nth prime in <strong>L</strong>, and the || · ||<strong><span style="font-size: 80%; vertical-align: sub;">1</span></strong> on the right hand side of the equation is the L1 norm on the resulting full-limit real vector.<br /> | where <strong>V<span style="font-size: 80%; vertical-align: sub;">G</span></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 <strong>G</strong> in a suitable full-limit <strong>L</strong> containing all of <strong>G</strong> as a subgroup, <strong>W<span style="font-size: 80%; vertical-align: sub;">L</span></strong> is a diagonal weighting matrix in which the nth entry in the diagonal is the log<span style="font-size: 10px; vertical-align: sub;">2</span> of the nth prime in <strong>L</strong>, and the || · ||<strong><span style="font-size: 80%; vertical-align: sub;">1</span></strong> on the right hand side of the equation is the L1 norm on the resulting full-limit real vector.<br /> | ||
<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 /> | 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 /> | ||