Generalized Tenney dual norms and Tp tuning space: Difference between revisions

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=Dual ~~Norms~~=

== Dual norms ==

Given any [[Generalized_Tenney_Norms_and_Tp_Interval_Space|Tp norm]] on an interval space '''TpG''' associated with a group '''G''', we can define a corresponding '''dual Tq* norm''' on the dual space '''TqG'''* which satisfies the following identity:

<math>||f||_\mathbf{Tq*} = \text{sup}\left \{\frac{|f(\vec{v})|}{||\vec{v}||_\mathbf{Tp}}: \vec{v} \in \textbf{Lp}\right \}</math>

for all f in '''TqG'''*. This normed space, for which the group of vals on '''G''' comprise the lattice of covectors with integer coefficients, is called '''Tq* Tuning Space'''. Other vectors in this space may be interpreted as tuning ~~maps~~ that send intervals in '''G''' to a certain number of cents (or other logarithmic units), although only tuning maps lying near the '''[[JIP]]''' will be of much musical relevance.

for all f in '''TqG'''*. This normed space, for which the group of vals on '''G''' comprise the lattice of covectors with integer coefficients, is called '''Tq* Tuning Space'''. Other vectors in this space may be interpreted as [[tuning map]]s that send intervals in '''G''' to a certain number of cents (or other logarithmic units), although only tuning maps lying near the '''[[JIP]]''' will be of much musical relevance.

Note that this norm enables us to define something like a complexity metric on vals, where vals that are closer to the origin (such as <7 11 16|) are rated less complex than vals which are further from the origin (such as <171 271 397|). Additionally, if this metric is used on tuning maps, we can evaluate the average error for any tuning map '''t''' and the '''JIP''' by looking at the quantity ||'''t''' - '''JIP'''||. As per the definition of dual norm above, the Tq* norm of this vector gives us the maximum Tp-weighted mapping for '''t - JIP''' over all intervals, and hence also gives us the maximum error for '''t''' over all intervals.

==Prime ~~Power Interval Groups~~==

== Prime power interval groups ==

In the simplest case where '''G''' has as its chosen basis only primes and prime powers, || · ||'''Tp''' is given by

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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 ==

For an arbitrary group '''G''' with its chosen basis containing intervals other than primes and prime powers, || · ||'''Tp''' is given by

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<math>\left \|f \right \|_{\textbf{Tq*}}^\textbf{G} = \inf_{n \in \text{ker}(\mathbf{V_G})} \left \{ \left \| (f-n) \cdot \mathbf{W}_\mathbf{L}^{-1} \right \|_\textbf{Tq*}^\mathbf{L} \right \}</math>

Note that this is the quotient norm induced on the space '''TqL'''*/ker('''VG'''), where ker('''VG''') is the set of vals in the V-map ~~that are restricted away~~. This result is due to a corollary of the [http://www.math.unl.edu/~s-bbockel1/928/node25.html Hahn-Banach theorem], which demonstrates that the dual space M* to any subspace M of a Banach space V must be isometrically isomorphic to the quotient space V*/ker(M), where ker(M) is the set of all f in V* such that f(M) = 0. Since our vector space '''TpL''' and our subspace is '''TpG''', this proves that our dual space '''TqG'''* must be isometrically isomorphic to '''TqL'''*/ker('''VG''').

Note that this is the quotient norm induced on the space '''TqL'''*/ker('''VG'''), where ker('''VG''') is the set of vals (or tuning maps) that are restricted away given the V-map. This result is due to a corollary of the [http://www.math.unl.edu/~s-bbockel1/928/node25.html Hahn-Banach theorem], which demonstrates that the dual space M* to any subspace M of a Banach space V must be isometrically isomorphic to the quotient space V*/ker(M), where ker(M) is the set of all f in V* such that f(M) = 0. Since our vector space '''TpL''' and our subspace is '''TpG''', this proves that our dual space '''TqG'''* must be isometrically isomorphic to '''TqL'''*/ker('''VG''').

[[~~category~~:~~math~~]]

[[Category:Math]]

[[Category:Tuning space]]

[[Category:Temperament complexity measures]]

[[Category:Tenney-weighted measures]]

@@ Line 1: / Line 1: @@
-=Dual Norms=
+{{todo|intro|inline=1}}
+== Dual norms ==
 Given any [[Generalized_Tenney_Norms_and_Tp_Interval_Space|Tp norm]] on an interval space '''Tp<sup>G</sup>''' associated with a group '''G''', we can define a corresponding '''dual Tq* norm''' on the dual space '''Tq<sup>G</sup>'''* which satisfies the following identity:
 <math>||f||_\mathbf{Tq*} = \text{sup}\left \{\frac{|f(\vec{v})|}{||\vec{v}||_\mathbf{Tp}}: \vec{v} \in \textbf{Lp}\right \}</math>
-for all f in '''Tq<sup>G</sup>'''*. This normed space, for which the group of vals on '''G''' comprise the lattice of covectors with integer coefficients, is called '''Tq* Tuning Space'''. Other vectors in this space may be interpreted as tuning maps that send intervals in '''G''' to a certain number of cents (or other logarithmic units), although only tuning maps lying near the '''[[JIP]]''' will be of much musical relevance.
+for all f in '''Tq<sup>G</sup>'''*. This normed space, for which the group of vals on '''G''' comprise the lattice of covectors with integer coefficients, is called '''Tq* Tuning Space'''. Other vectors in this space may be interpreted as [[tuning map]]s that send intervals in '''G''' to a certain number of cents (or other logarithmic units), although only tuning maps lying near the '''[[JIP]]''' will be of much musical relevance.
 Note that this norm enables us to define something like a complexity metric on vals, where vals that are closer to the origin (such as &lt;7 11 16|) are rated less complex than vals which are further from the origin (such as &lt;171 271 397|). Additionally, if this metric is used on tuning maps, we can evaluate the average error for any tuning map '''t''' and the '''JIP''' by looking at the quantity ||'''t''' - '''JIP'''||. As per the definition of dual norm above, the Tq* norm of this vector gives us the maximum Tp-weighted mapping for '''t - JIP''' over all intervals, and hence also gives us the maximum error for '''t''' over all intervals.
-==Prime Power Interval Groups==
+== Prime power interval groups ==
 In the simplest case where '''G''' has as its chosen basis only primes and prime powers, || · ||'''<sub>Tp</sub>''' is given by
@@ Line 23: / Line 24: @@
 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 ==
 For an arbitrary group '''G''' with its chosen basis containing intervals other than primes and prime powers, || · ||'''<sub>Tp</sub>''' is given by
@@ Line 32: / Line 33: @@
 <math>\left \|f \right \|_{\textbf{Tq*}}^\textbf{G} = \inf_{n \in \text{ker}(\mathbf{V_G})} \left \{ \left \| (f-n) \cdot \mathbf{W}_\mathbf{L}^{-1} \right \|_\textbf{Tq*}^\mathbf{L} \right \}</math>
-Note that this is the quotient norm induced on the space '''Tq<sup>L</sup>'''*/ker('''V<sub>G</sub>'''), where ker('''V<sub>G</sub>''') is the set of vals in the V-map that are restricted away. This result is due to a corollary of the [http://www.math.unl.edu/~s-bbockel1/928/node25.html Hahn-Banach theorem], which demonstrates that the dual space M* to any subspace M of a Banach space V must be isometrically isomorphic to the quotient space V*/ker(M), where ker(M) is the set of all f in V* such that f(M) = 0. Since our vector space '''Tp<sup>L</sup>''' and our subspace is '''Tp<sup>G</sup>''', this proves that our dual space '''Tq<sup>G</sup>'''* must be isometrically isomorphic to '''Tq<sup>L</sup>'''*/ker('''V<sub>G</sub>''').
+Note that this is the quotient norm induced on the space '''Tq<sup>L</sup>'''*/ker('''V<sub>G</sub>'''), where ker('''V<sub>G</sub>''') is the set of vals (or tuning maps) that are restricted away given the V-map. This result is due to a corollary of the [http://www.math.unl.edu/~s-bbockel1/928/node25.html Hahn-Banach theorem], which demonstrates that the dual space M* to any subspace M of a Banach space V must be isometrically isomorphic to the quotient space V*/ker(M), where ker(M) is the set of all f in V* such that f(M) = 0. Since our vector space '''Tp<sup>L</sup>''' and our subspace is '''Tp<sup>G</sup>''', this proves that our dual space '''Tq<sup>G</sup>'''* must be isometrically isomorphic to '''Tq<sup>L</sup>'''*/ker('''V<sub>G</sub>''').
-[[category:math]]
+[[Category:Math]]
+[[Category:Tuning space]]
+[[Category:Temperament complexity measures]]
+[[Category:Tenney-weighted measures]]
+{{Todo| cleanup }}