Generalized Tenney norms and Tp interval space: Difference between revisions

It can be useful to define a notion of the "[[complexity]]" of an [[interval]], so that small-integer ratios such as 3/2 are less complex and intervals such as 32805/32768 are more complex. This can be accomplished for any [[wikipedia: Free abelian group|free abelian group]] of ([[subgroup]]) [[Monzos and interval space|monzos]] by [[wikipedia: Embedding|embedding]] the group in a [[wikipedia: Normed vector space|normed vector space]], so that the [[wikipedia: Norm (mathematics)|norm]] of any interval is taken to be its complexity. The monzos form a ℤ-module, with coordinates given by integers, and the vector space embedding can be constructed by simply allowing real coordinates, hence defining the module over ℝ instead of ℤ and giving it the structure of a vector space. The resulting space is called [[Monzos and interval space|interval space]], with the monzos forming the [[wikipedia: Integer lattice|integer lattice]] of vectors with integer coordinates, but where we will allow any vector space norm on ℝ^''n''.