Generalized Tenney dual norms and Tp tuning space: Difference between revisions
Wikispaces>mbattaglia1 **Imported revision 356537294 - Original comment: ** |
Wikispaces>mbattaglia1 **Imported revision 356537652 - Original comment: ** |
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The dual of any Tp norm is very similar to the dual of the ordinary Lp norm. The crucial difference to be noted is that the weighting for covectors in tuning space is the inverse of the weighting for vectors in interval space; simple primes are weighted less in interval space but more in tuning space. Unlike the weighting matrix for interval space, the weighting matrix on tuning space is a diagonal matrix in which the nth entry in the diagonal is 1/log<span style="font-size: 80%; vertical-align: sub;">2</span>(**G**<span style="font-size: 80%; vertical-align: sub;">n</span>), where **G**<span style="font-size: 10px; vertical-align: sub;">n</span> is the nth basis element in **G**. We denote such inverse weighted norms with an asterisk, so that the inverse-Tenney weighted Linf norm in tuning space is Tinf*. | The dual of any Tp norm is very similar to the dual of the ordinary Lp norm. The crucial difference to be noted is that the weighting for covectors in tuning space is the inverse of the weighting for vectors in interval space; simple primes are weighted less in interval space but more in tuning space. Unlike the weighting matrix for interval space, the weighting matrix on tuning space is a diagonal matrix in which the nth entry in the diagonal is 1/log<span style="font-size: 80%; vertical-align: sub;">2</span>(**G**<span style="font-size: 80%; vertical-align: sub;">n</span>), where **G**<span style="font-size: 10px; vertical-align: sub;">n</span> is the nth basis element in **G**. We denote such inverse weighted norms with an asterisk, so that the inverse-Tenney weighted Linf norm in tuning space is Tinf*. | ||
For **G** with basis of only primes and prime powers, the dual of the T1 norm is the Tinf* norm, the dual of the Tinf norm is the T1* norm, and the dual of the T2 norm is the T2* norm. | For **G** with basis of only primes and prime powers, the dual of the T1 norm is the Tinf* norm, the dual of the Tinf norm is the T1* norm, and the dual of the T2 norm is the T2* norm. Likewise, T1 interval space is dual to Tinf* tuning space, and T2 interval space is dual to T2* tuning space. | ||
==Arbitrary Interval Groups== | ==Arbitrary Interval Groups== | ||
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The dual of any Tp norm is very similar to the dual of the ordinary Lp norm. The crucial difference to be noted is that the weighting for covectors in tuning space is the inverse of the weighting for vectors in interval space; simple primes are weighted less in interval space but more in tuning space. Unlike the weighting matrix for interval space, the weighting matrix on tuning space is a diagonal matrix in which the nth entry in the diagonal is 1/log<span style="font-size: 80%; vertical-align: sub;">2</span>(<strong>G</strong><span style="font-size: 80%; vertical-align: sub;">n</span>), where <strong>G</strong><span style="font-size: 10px; vertical-align: sub;">n</span> is the nth basis element in <strong>G</strong>. We denote such inverse weighted norms with an asterisk, so that the inverse-Tenney weighted Linf norm in tuning space is Tinf*.<br /> | The dual of any Tp norm is very similar to the dual of the ordinary Lp norm. The crucial difference to be noted is that the weighting for covectors in tuning space is the inverse of the weighting for vectors in interval space; simple primes are weighted less in interval space but more in tuning space. Unlike the weighting matrix for interval space, the weighting matrix on tuning space is a diagonal matrix in which the nth entry in the diagonal is 1/log<span style="font-size: 80%; vertical-align: sub;">2</span>(<strong>G</strong><span style="font-size: 80%; vertical-align: sub;">n</span>), where <strong>G</strong><span style="font-size: 10px; vertical-align: sub;">n</span> is the nth basis element in <strong>G</strong>. We denote such inverse weighted norms with an asterisk, so that the inverse-Tenney weighted Linf norm in tuning space is Tinf*.<br /> | ||
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For <strong>G</strong> with basis of only primes and prime powers, the dual of the T1 norm is the Tinf* norm, the dual of the Tinf norm is the T1* norm, and the dual of the T2 norm is the T2* norm.<br /> | For <strong>G</strong> with basis of only primes and prime powers, the dual of the T1 norm is the Tinf* norm, the dual of the Tinf norm is the T1* norm, and the dual of the T2 norm is the T2* norm. Likewise, T1 interval space is dual to Tinf* tuning space, and T2 interval space is dual to T2* tuning space.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:9:&lt;h2&gt; --><h2 id="toc2"><a name="Dual Norms-Arbitrary Interval Groups"></a><!-- ws:end:WikiTextHeadingRule:9 -->Arbitrary Interval Groups</h2> | <!-- ws:start:WikiTextHeadingRule:9:&lt;h2&gt; --><h2 id="toc2"><a name="Dual Norms-Arbitrary Interval Groups"></a><!-- ws:end:WikiTextHeadingRule:9 -->Arbitrary Interval Groups</h2> | ||