Generalized Tenney norms and Tp interval space: Difference between revisions

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~~{{Legacy}}~~{{texmap}}{{DISPLAYTITLE:Generalized Tenney norms and T''p'' interval space}}

{{texmap}}{{DISPLAYTITLE:Generalized Tenney norms and T''p'' interval space}}

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 {{w|free abelian group}} of ([[subgroup]]) [[Monzos and interval space|monzos]] by {{w|embedding}} the group in a {{w|normed vector space}}, so that the {{w|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 {{w|integer lattice}} of vectors with integer coordinates, but where we will allow any vector space norm on ℝ''n''.

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	~~{{Legacy}}~~{{texmap}}{{DISPLAYTITLE:Generalized Tenney norms and T<sub>''p''</sub> interval space}}		{{texmap}}{{DISPLAYTITLE:Generalized Tenney norms and T<sub>''p''</sub> interval space}}
	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 {{w\|free abelian group}} of ([[subgroup]]) [[Monzos and interval space\|monzos]] by {{w\|embedding}} the group in a {{w\|normed vector space}}, so that the {{w\|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 {{w\|integer lattice}} of vectors with integer coordinates, but where we will allow any vector space norm on ℝ<sup>''n''</sup>.		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 {{w\|free abelian group}} of ([[subgroup]]) [[Monzos and interval space\|monzos]] by {{w\|embedding}} the group in a {{w\|normed vector space}}, so that the {{w\|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 {{w\|integer lattice}} of vectors with integer coordinates, but where we will allow any vector space norm on ℝ<sup>''n''</sup>.