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

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

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.

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

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.

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

Note that this is the quotient norm induced on the space '''Tq^L'''*/ker('''V_G'''), where ker('''V_G''') 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^L''' and our subspace is '''Tp^G''', this proves that our dual space '''Tq^G'''* must be isometrically isomorphic to '''Tq^L'''*/ker('''V_G''').