Generalized Tenney norms and Tp interval space: Difference between revisions
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=Basics= | =Basics= | ||
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 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 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 [[http://en.wikipedia.org/wiki/Integer_lattice|integer lattice]] of vectors with integer coordinates, but where we will allow any vector space norm on ℝⁿ. | 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 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 [[Monzos and Interval Space|interval space]], with the monzos forming the [[http://en.wikipedia.org/wiki/Integer_lattice|integer lattice]] of vectors with integer coordinates, but where we will allow any vector space norm on ℝⁿ. | ||
The most important and natural norm which arises in this scenario is the **Tenney norm**, which we will explore below. | The most important and natural norm which arises in this scenario is the **Tenney norm**, which we will explore below. | ||
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<br /> | <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 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 /> | ||
<br /> | <br /> | ||
The most important and natural norm which arises in this scenario is the <strong>Tenney norm</strong>, which we will explore below.<br /> | The most important and natural norm which arises in this scenario is the <strong>Tenney norm</strong>, which we will explore below.<br /> | ||