Generalized Tenney norms and Tp interval space: Difference between revisions
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It can be useful to define a notion of the &quot;complexity&quot; 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 free abelian group of monzos or smonzos by embedding the group in a normed vector space, so that the 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 <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">interval space</a>, with the monzos forming the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Integer_lattice" rel="nofollow">integer lattice</a> of vectors with integer coordinates, but where we will allow any vector space norm on ℝⁿ.<br /> | It can be useful to define a notion of the &quot;complexity&quot; 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 free abelian group of monzos or smonzos by embedding the group in a normed vector space, so that the 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 <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">interval space</a>, with the monzos forming the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Integer_lattice" rel="nofollow">integer lattice</a> of vectors with integer coordinates, but where we will allow any vector space norm on ℝⁿ.<br /> | ||