Generalized Tenney norms and Tp interval space: Difference between revisions
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<math>\left \| \vec{v} \right \|_{\textbf{Tp}}^\textbf{G} = \left \| \mathbf{W_L} \cdot \mathbf{V_\textbf{G}} \cdot \vec{v} \right \|_\textbf{p}</math> | <math>\left \| \vec{v} \right \|_{\textbf{Tp}}^\textbf{G} = \left \| \mathbf{W_L} \cdot \mathbf{V_\textbf{G}} \cdot \vec{v} \right \|_\textbf{p}</math> | ||
In this scheme the ordinary Tenney norm now becomes the '''T1 norm''', and in general we call an interval space that' | 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''<sup>G</sup>''', where '''G''' is the associated group the interval space is built around. | ||
Note that the || · ||'''< | Note that the || · ||'''<sub>T''p''</sub>''' norm on the left side of the equation now has a subscript of T''p'' rather than T1, and that the || · ||'''<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''p'' norm rather than restricting our consideration to the L1 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 log<sub>2</sub>(''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. | |||
== Tenney-Euclidean norm (TE norm, T2 norm) == | == Tenney-Euclidean norm (TE norm, T2 norm) == | ||
The T2 norm is often called the '''Tenney-Euclidean norm''', '''TE norm''', or '''TE height''', as it has the same relationship with Euclidean geometry that the T1 norm has with taxicab geometry. Applications involving the TE norm tend to be easy to compute, such as the host of TE-related metrics defined for rating temperaments< | The T2 norm is often called the '''Tenney-Euclidean norm''', '''TE norm''', or '''TE height''', as it has the same relationship with Euclidean geometry that the T1 norm has with taxicab geometry. Applications involving the TE norm tend to be easy to compute, such as the host of TE-related metrics defined for rating temperaments<sup>[[Tenney-Euclidean metrics|(1)]][[Tenney-Euclidean temperament measures|(2)]][[Tenney-Euclidean tuning|(3)]]</sup>. It approximates the T1 complexity of many intervals, although notably rates 9/1 as more complex than 15/1. | ||
The T2 norm is also the only | The T2 norm is also the only T''p'' norm that naturally defines an inner product, given by the matrix multiplication | ||
<math>\left \langle \vec{v},\vec{w} \right \rangle_\textbf{G} = \vec{v}^T \cdot \left(\mathbf{V}^T_\mathbf{G} \cdot \mathbf{W}^2_\mathbf{L} \cdot \mathbf{V}_\mathbf{G} \right) \cdot \vec{w}</math> | <math>\left \langle \vec{v},\vec{w} \right \rangle_\textbf{G} = \vec{v}^T \cdot \left(\mathbf{V}^T_\mathbf{G} \cdot \mathbf{W}^2_\mathbf{L} \cdot \mathbf{V}_\mathbf{G} \right) \cdot \vec{w}</math> | ||
The matrix product | The matrix product '''V'''<sub>'''G'''</sub><sup>T</sup> · '''W'''<sub>'''L'''</sub><sup>2</sup> · '''V'''<sub>'''G'''</sub> itself is a positive definite matrix, and as such defines the inner product for the TE norm. Note that this setup represents the vectors '''''v''''' and '''''w''''' by column vectors, so that '''''v'''''<sup>T</sup> denotes a row vector. As was the case with T''p'' norms in general, this equation simplifies considerably when the group '''G''' takes as its basis only primes and prime powers, becoming instead | ||
<math>\left \langle \vec{v},\vec{w} \right \rangle_\textbf{G} = \vec{v}^T \cdot \mathbf{W}^2_\mathbf{L} \cdot \vec{w}</math> | <math>\left \langle \vec{v},\vec{w} \right \rangle_\textbf{G} = \vec{v}^T \cdot \mathbf{W}^2_\mathbf{L} \cdot \vec{w}</math> | ||
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== Examples == | == Examples == | ||
Say that we | Say that we are in the 2.9/7.5/3 subgroup, and we want to find the T1 norm of {{monzo| 0 -2 1 }}. Then we can come up with a V-map '''V<sub>G</sub>''' for this subgroup in the 7-limit as follows: | ||
<math>\left[ \begin{array}{rrrrrl} | <math>\left[ \begin{array}{rrrrrl} | ||
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\end{array} \right]</math> | \end{array} \right]</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 [[ | 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]]. | ||
We can also come up with a weighting matrix for the full-limit '''W< | We can also come up with a weighting matrix for the full-limit '''W<sub>L</sub>''' as follows: | ||
<math>\begin{bmatrix} | <math>\begin{bmatrix} | ||
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\end{bmatrix}</math> | \end{bmatrix}</math> | ||
Given these matrices, the T1 norm of our subgroup basis monzo |0 -2 1 | Given these matrices, the T1 norm of our subgroup basis monzo {{monzo| 0 -2 1 }}, which we will call '''v''', can be found by taking the L1 norm of the resulting real vector '''W<sub>L</sub>''' · '''V<sub>G</sub>''' · '''v'''. This expression works out to | ||
<math>\left \| \vec{v} \right \|_\mathbf{T1}^\mathbf{2.9/7.5/3} = \left \| | <math>\left \| \vec{v} \right \|_\mathbf{T1}^\mathbf{2.9/7.5/3} = \left \| | ||
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Note that the value 15.861 is obtained by |0| + |-7.925| + |2.322| + |5.615|, which is the L1 norm of the vector. | Note that the value 15.861 is obtained by |0| + |-7.925| + |2.322| + |5.615|, which is the L1 norm of the vector. | ||
To confirm this, we can put the subgroup basis monzo |0 -2 1 | To confirm this, we can put the subgroup basis monzo {{monzo| 0 -2 1 }} back into rational form to see that it represents the interval 245/243. As the L1 norm is supposed to give log(''nd'') for any interval ''n''/''d'', we can confirm that we have the right answer above by noting that log(245·243) is indeed equal to 15.861. | ||
[[Category:Regular temperament theory]] | [[Category:Regular temperament theory]] | ||