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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In this scheme the ordinary Tenney norm now becomes the '''T1 norm''', and in general we call an interval space that has been given a T''p'' norm '''T''p'' interval space'''. We may sometimes notate this as T''p''G, where ''G'' is the associated group the interval space is built around.

Note that the {{nowrap|‖ · ‖T''p''}} norm on the left side of the equation now has a subscript of T''p'' rather than T1, and that the {{~~nowrap‖~~ · ‖''p''}} 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''''p'' norm rather than restricting our consideration to the ''L''1 norm.

Note that the {{nowrap|‖ · ‖T''p''}} norm on the left side of the equation now has a subscript of T''p'' rather than T1, and that the {{nowrap|‖ · ‖''p''}} 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''''p'' norm rather than restricting our consideration to the ''L''1 norm.

T''p'' 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 log2(''nd''). Furthermore, generalized T''p'' norms may sometimes differ from the T1 norm in their ranking of intervals by T''p'' complexity, although the T''p'' norm of any interval is always bounded by its T1 norm. However, despite these supposed theoretical disadvantages, there can be many indirect implications of using a T''p'' norm other than T1 which are theoretically justified; additionally, certain T''p'' norms are worth using as an approximation to T1 for their strong computational advantages. As such, T''p'' spaces are musically justified enough to merit further study, despite the surface advantages of sticking exclusively to the T1 norm.

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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>.
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 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 {{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.
+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.