Dual Norms

Given any 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:

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

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.

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

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

$\left \| \vec{v} \right \|_{\textbf{Tp}}^\textbf{G} = \left \| \mathbf{W_G} \cdot \vec{v} \right \|_\textbf{p}$

for diagonal weighting matrix WG. Then the dual norm || · ||Tq* on TqG* is given for f in TqG* by

$\left \|f \right \|_{\textbf{Tq*}}^\textbf{G} = \left \| f \cdot \mathbf{W}_\mathbf{G}^{-1} \right \|_\textbf{q}$

where the coefficients p in Tp and q in Tq* satisfy the relationship 1/p + 1/q = 1.

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/log2(Gn), where Gn 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. Likewise, T1 interval space is dual to Tinf* tuning space, and T2 interval space is dual to T2* tuning space.

Arbitrary Interval Groups

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

$\left \| \vec{v} \right \|_{\textbf{T1}}^\textbf{G} = \left \| \mathbf{W_L} \cdot \mathbf{V_\textbf{G}} \cdot \vec{v} \right \|_\textbf{1}$

for a V-map VG representing G in some full-limit L and a diagonal weighting matrix WL for L. Then if TpL represents the full-limit interval space that G is embedded in, and TqL* is the dual space, the dual norm || · ||Tq on TqG* is given by

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

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 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).