Generator complexity: Difference between revisions

=Definition=

Suppose A = <0 A₃ A₅ A₇ ... Ap| is the generator mapping val for a rank two temperament with P periods to the octave, and B = <0 B₃ B₅ B₇ ... Bp| is the same val in weighted coordinates. For instance, <0 1 -2 -2| is the generator mapping val for seven limit [[pajara|pajara]], and <0 1/log2(3) -2/log2(5) -2/log2(7)| ≅ <0 0.631 -0.831 -0.712| is the val in weighted coordinates. For any vector V, let max(V) - min(V) = span(V). The ''generator complexity'' of the temperament is P span(B). In the case of pajara, which has two periods to the octave, this would be 2*(0.631 - (-0.861)) = 2.984. This can also be described in terms of the wedgie W of the temperament, as span(2∨W), which is the span of 0 followed by the first n-1 elements of W, where n is the number of primes in the p-limit.  

Generator complexity satisfies the inequality, for any p-limit interval I, G(I) ≤ C KE(I), where C is the generator complexity of the temperament, G(I) is the number of generator steps, times P, required to reach the tempered version of I, and KE(I) is the [[Kees_Height|Kees expressibility]] of I. So for instance, in meantone G(5/4) = 4, since it requires four generator steps to get to 5/4, and KE(5/4) = log2(5). In pajara, G(5/4) = 4 also, since two generator steps are required to get to 5/4 (5/4 = (4/3)^2 * 45/64), and P = 2, so that G(5/4) = 2*2.

This inequality can be used to give an alternative definition of generator complexity: C = sup G(I)/KE(I) over non-octave intervals, where KE(I)>0. A related definition can be extended to higher ranks: since the [[Tenney-Euclidean_metrics#The OETES|OETES]] in the case of a rank two temperament is proportional (albeit with a different proportionality factor for each temperament) to G(I), we can define a complexity measure for any rank of temperament by C = sup OETES(I)/KE(I).

Generator complexity has the nice property that for any MOS of size N, floor(N/(C KE(I))) intervals with pitch class corresponding to I are guaranteed to exist in the MOS. Generator complexity is also useful in making complete searches using [[The_wedgie|the wedgie]] for temperaments below a certain complexity and badness bounds, allowing for a more efficient search.

If m = |m2 m3 m5 ... mp> is a vector with weighted coordinates in interval space, then KE(m), the Kees  expressibility of m, is (|m3 + m5+ ... +mp| + |m3| + |m5| + ... + |mp|)/2. The "2" coordinate, m2, plays  no role in Kees expressibility, so we may replace it with anything we choose. If we replace it with -e3 -e5-...-ep, we may define expressibility in terms of the L1 norm, as || |-e3-e5-...-ep  e3  e5 ... ep>  ||/2.

For any vector space X with a subspace A, we may define a quotient space X/A as the equivalence classes of vectors in X where two vectors are equivalent iff their difference lies in A. Then we have a short exact  sequence 0 → A → X → X/A → 0. Taking the duals of this gives us 0 → (X/A)* → X* → A* → 0. The annihilator  of A is the subspace A⁀ of X* consisting of those functionals f such that <f|A> equals 0; that is, it is the subspace of all the functionals f such that <f|a> equals 0 for every a in A. There is a natural isomorphism between the annihilator A⁀ of A and the dual of the quotient (X/A)*, and also between X*/A⁀ and A*.

Now suppose X is a finite dimensional real normed vector space. A is then also a finite dimensional real normed vector space, inheriting its norm from X, and X/A is a finite dimensional real normed vector space, with a norm given by, for an equivalence class [x], ||[x]|| equals inf {||x + a||, a∈ A}. Algebraically X is (noncanonically) isomorphic to X*, but in general they are no longer isomorphic as normed spaces. Instead, we have the [[dual_norm|dual norm]] on X*, defined by setting, over all nonzero x ∈ X, ||f||* = sup <f|x>/||x||. Under the dual norm X* is also a finite dimensional normed vector space, A⁀ is isometrically isomorphic to (X/A)*, and X*/A⁀ is isometrically isomorphic to A*.

In the situation which concerns us, X is the p-limit interval space of dimension n under a norm of one half times the L1 norm, A is a subspace of dimension n-1, whose coordinates sum to 0; hence A can be described as having the one-dimensional subspace A⁀ = {kJ}, where J is the [[JIP|JIP]], as its annihilator. X has a norm of half the L1 norm, and hence X* has a norm of twice the L∞ norm. The norm on A* is defined by its isomorphism with X*/A⁀; the minimum defining inf {||f + kJ||}  occurs for the value of k where the maximum of f + kJ and minus the minimum of f + kj are the same. In that case, 2||f + kJ||_L∞ = span(f), which is the generator complexity of f. Hence generator complexity is the dual norm for Kees expressibility as a norm on pitch classes.

=STD complexity=

If B = <0 B₃ B₅ B₇ ... Bp| is the generator mapping val in weighted coordinates, and P is the period, then the ''STD complexity'' (a term due to Graham Breed) is P STD(B), where "STD" means the standard deviation. If μ(V) is the mean of the components of the vector V, and J is the [[JIP|JIP]] <1 1 1 ... 1|, then  ₱(V) = V - μ(V)J is the projection of V onto the subspace of vectors with zero mean value. We have STD(V) = √( ₱(V) ∙ ₱(V) / dim(V)), where dim(V) is the dimension of V and the " ∙ " denotes the dot product. If M = [M0, M1] is the [[Temperament_Mapping_Matrices_(M-maps)|mapping matrix]] in weighted coordinates in the standard [[Normal_lists#x-Normal val lists|normal val list]] form, then we may express STD complexity as STDcom(M) = M0[1] STD(M1).

Associated to STD complexity is STD error. If S = ₱(M0) ∧ ₱(M1), then STDerr(M) = √(S ∙ S / dim(M1)*₱(M1) ∙  ₱(M1)).

[[Category:generator]]

[[Category:math]]

[[Category:theory]]