Vals and tuning space: Difference between revisions
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By [http://en.wikipedia.org/wiki/Embedding embedding] the monzos into a suitable vector space, norms may be placed on the monzos in various ways, turning them into [http://mathworld.wolfram.com/PointLattice.html lattices] in a vector space. Given a vector space norm on a space of ket vectors, the [http://mathworld.wolfram.com/DualNormedSpace.html dual vector space norm] on the space of bra vectors is defined as the least quantity ||V|| making | By [http://en.wikipedia.org/wiki/Embedding embedding] the monzos into a suitable vector space, norms may be placed on the monzos in various ways, turning them into [http://mathworld.wolfram.com/PointLattice.html lattices] in a vector space. Given a vector space norm on a space of ket vectors, the [http://mathworld.wolfram.com/DualNormedSpace.html dual vector space norm] on the space of bra vectors is defined as the least quantity ||V|| making | ||
'''| | '''|⟨V|M⟩| ≤ ||V|| ||M||''' | ||
to be always true. The dual of the [http://mathworld.wolfram.com/L1-Norm.html L1 norm] is the [http://mathworld.wolfram.com/L-Infinity-Norm.html Linfty norm], and the dual space of Tenney interval space is Tenney tuning space. The embedding of monzos into a real normed vector space automatically induces a dual embedding of vals into a corresponding normed vector space, tuning space, in which vals are lattice points. The dual norm to the L2 norm is the L2 norm, and the dual space to Tenney-Euclidean interval space is ''Tenney-Euclidean tuning space''. The Euclidean norm on a val v is given by | to be always true. The dual of the [http://mathworld.wolfram.com/L1-Norm.html L1 norm] is the [http://mathworld.wolfram.com/L-Infinity-Norm.html Linfty norm], and the dual space of Tenney interval space is Tenney tuning space. The embedding of monzos into a real normed vector space automatically induces a dual embedding of vals into a corresponding normed vector space, tuning space, in which vals are lattice points. The dual norm to the L2 norm is the L2 norm, and the dual space to Tenney-Euclidean interval space is ''Tenney-Euclidean tuning space''. The Euclidean norm on a val v is given by | ||