Generalized Tenney norms and Tp interval space: Difference between revisions
m →Examples: <math> fix |
smonzo |
||
| Line 2: | Line 2: | ||
=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 | 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 (subgroup) monzos 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. | ||
| Line 12: | Line 12: | ||
Informally speaking, we can obtain the Tenney norm on an interval space for any JI group by applying these three steps: | Informally speaking, we can obtain the Tenney norm on an interval space for any JI group by applying these three steps: | ||
<ol><li>if the interval is a | <ol><li>if the interval is a subgroup monzo, with coordinates in the subgroup basis, map it back to its corresponding full-limit monzo</li><li>weight the axis for each prime p by log<span style="font-size: 10px; vertical-align: sub;">2</span>(p)</li><li>take the ordinary L1 norm of the result.</li></ol> | ||
To formalize this idea in its full generality, the Tenney norm of any vector ''v'' in an interval space with associated JI group '''G''' can be expressed as follows: | To formalize this idea in its full generality, the Tenney norm of any vector ''v'' in an interval space with associated JI group '''G''' can be expressed as follows: | ||