Generalized Tenney norms and Tp interval space: Difference between revisions
ArrowHead294 (talk | contribs) mNo edit summary |
-legacy |
||
| (2 intermediate revisions by one other user not shown) | |||
| Line 1: | Line 1: | ||
{{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>. | ||
| Line 38: | Line 38: | ||
In this scheme the ordinary Tenney norm now becomes the '''T<sub>1</sub> norm''', and in general we call an interval space that has been given a T<sub>''p''</sub> norm '''T<sub>''p''</sub> interval space'''. We may sometimes notate this as T<sub>''p''</sub><sup>G</sup>, where ''G'' is the associated group the interval space is built around. | In this scheme the ordinary Tenney norm now becomes the '''T<sub>1</sub> norm''', and in general we call an interval space that has been given a T<sub>''p''</sub> norm '''T<sub>''p''</sub> interval space'''. We may sometimes notate this as T<sub>''p''</sub><sup>G</sup>, where ''G'' is the associated group the interval space is built around. | ||
Note that the {{nowrap|‖ · ‖<sub>T''p''</sub>}} norm on the left side of the equation now has a subscript of T<sub>''p''</sub> rather than T<sub>1</sub>, and that the {{ | Note that the {{nowrap|‖ · ‖<sub>T''p''</sub>}} norm on the left side of the equation now has a subscript of T<sub>''p''</sub> rather than T<sub>1</sub>, and that the {{nowrap|‖ · ‖<sub>''p''</sub>}} norm on the right side of the equation now has a subscript of ''p'' rather than 1. The generalized Tenney norm can thus be thought of as applying the same three-step process that the Tenney norm does, but where the last step can be an arbitrary ''L''<sub>''p''</sub> norm rather than restricting our consideration to the ''L''<sub>1</sub> norm. | ||
T<sub>''p''</sub> norms are thus generalizations of the Tenney norm which continue to weight the primes and prime powers as the Tenney norm does, but for which an arbitrary interval ''n''/''d'' may no longer have a complexity of log<sub>2</sub>(''nd''). Furthermore, generalized T<sub>''p''</sub> norms may sometimes differ from the T<sub>1</sub> norm in their ranking of intervals by T<sub>''p''</sub> complexity, although the T<sub>''p''</sub> norm of any interval is always bounded by its T<sub>1</sub> norm. However, despite these supposed theoretical disadvantages, there can be many indirect implications of using a T<sub>''p''</sub> norm other than T<sub>1</sub> which are theoretically justified; additionally, certain T<sub>''p''</sub> norms are worth using as an approximation to T<sub>1</sub> for their strong computational advantages. As such, T<sub>''p''</sub> spaces are musically justified enough to merit further study, despite the surface advantages of sticking exclusively to the T<sub>1</sub> norm. | T<sub>''p''</sub> norms are thus generalizations of the Tenney norm which continue to weight the primes and prime powers as the Tenney norm does, but for which an arbitrary interval ''n''/''d'' may no longer have a complexity of log<sub>2</sub>(''nd''). Furthermore, generalized T<sub>''p''</sub> norms may sometimes differ from the T<sub>1</sub> norm in their ranking of intervals by T<sub>''p''</sub> complexity, although the T<sub>''p''</sub> norm of any interval is always bounded by its T<sub>1</sub> norm. However, despite these supposed theoretical disadvantages, there can be many indirect implications of using a T<sub>''p''</sub> norm other than T<sub>1</sub> which are theoretically justified; additionally, certain T<sub>''p''</sub> norms are worth using as an approximation to T<sub>1</sub> for their strong computational advantages. As such, T<sub>''p''</sub> spaces are musically justified enough to merit further study, despite the surface advantages of sticking exclusively to the T<sub>1</sub> norm. | ||
| Line 61: | Line 61: | ||
<math>\displaystyle | <math>\displaystyle | ||
\begin{bmatrix} | |||
1 & 0 & 0 & 0 \\ | |||
0 & 2 & 0 & -1 \\ | |||
0 & -1 & 1 & 0 \\ | |||
\end{ | \end{bmatrix}</math> | ||
Note that the "rows" here are written in kets; this is a convention to signify that each ket, representing a monzo, is actually supposed to represent a column of the matrix as explained in [[Subgroup basis matrices]]. | Note that the "rows" here are written in kets; this is a convention to signify that each ket, representing a monzo, is actually supposed to represent a column of the matrix as explained in [[Subgroup basis matrices]]. | ||
| Line 82: | Line 82: | ||
<math>\displaystyle | <math>\displaystyle | ||
\lVert \vec m \rVert_{\text{T} 1}^{2 \text{.} 9/7 \text{.} 5/3} = \left \lVert | \left\lVert \vec m \right\rVert_{\text{T} 1}^{2 \text{.} 9/7 \text{.} 5/3} = \left \lVert | ||
\begin{bmatrix} | \begin{bmatrix} | ||
\log_2(2) & 0 & 0 & 0\\ | \log_2(2) & 0 & 0 & 0\\ | ||
| Line 88: | Line 88: | ||
0 & 0 & \log_2(5) & 0\\ | 0 & 0 & \log_2(5) & 0\\ | ||
0 & 0 & 0 & \log_2(7) | 0 & 0 & 0 & \log_2(7) | ||
\end{bmatrix} \cdot | \end{bmatrix} \cdot \begin{bmatrix} | ||
1 & 0 & 0 & 0 \\ | |||
0 & 2 & 0 & -1 \\ | |||
0 & -1 & 1 & 0 \\ | |||
\end{ | \end{bmatrix} \cdot \begin{bmatrix} | ||
0 & -2 & 1 \\ | |||
\end{ | \end{bmatrix} | ||
\right \rVert_1 | \right \rVert_1 | ||
</math> | </math> | ||
| Line 100: | Line 100: | ||
which finally resolves to | which finally resolves to | ||
<math>\displaystyle \lVert \vec m \rVert_{\text{T} 1}^{2.9/7.5/3} = \lVert \monzo{ \begin{matrix} 0 & -7.925 & 2.322 & 5.615 \end{matrix} } \rVert_1 = 15.861</math> | <math>\displaystyle \left\lVert \vec m \right\rVert_{\text{T} 1}^{2.9/7.5/3} = \lVert \monzo{ \begin{matrix} 0 & -7.925 & 2.322 & 5.615 \end{matrix} } \rVert_1 = 15.861</math> | ||
Note that {{nowrap|15.861 {{=}} {{!}}0{{!}} + {{!}}−7.925{{!}} + {{!}}2.322{{!}} + {{!}}5.615{{!}}}}, which is the ''L''<sub>1</sub> norm of the vector. | Note that {{nowrap|15.861 {{=}} {{!}}0{{!}} + {{!}}−7.925{{!}} + {{!}}2.322{{!}} + {{!}}5.615{{!}}}}, which is the ''L''<sub>1</sub> norm of the vector. | ||