Generalized Tenney norms and Tp interval space: Difference between revisions
Another pass for style and notation. Obviously we wanna use "n" and "d" for numerator and denominator, and subscripts for monzo elements and the "p" in Lp and Tp |
Another pass for style and notation. Rid of the archaic term V-map "V" in favor of subgroup basis matrix "S" and full-limit JI group is now "J" |
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{{Main| Tenney height }} | {{Main| Tenney height }} | ||
The '''Tenney norm''', also called '''Tenney height''', is the norm such that for any monzo representing an interval ''n''/''d'', the norm of the interval is log<sub>2</sub>(''nd''). For a full-limit monzo {{monzo| ''m''<sub>1</sub> ''m''<sub>2</sub> ''m''<sub> | The '''Tenney norm''', also called '''Tenney height''', is the norm such that for any monzo representing an interval ''n''/''d'', the norm of the interval is log<sub>2</sub>(''nd''). For a full-limit monzo '''m''' = {{monzo| ''m''<sub>1</sub> ''m''<sub>2</sub> … ''m''<sub>π (''p'')</sub> … }}, with π standing for the [[wikipedia: Prime-counting function|prime-counting function]], this norm can be calculated as | ||
<math>\displaystyle \lvert m_1 \log_2 (2) \rvert + \lvert m_2 \log_2 (3) \rvert + \ldots + \lvert m_{\pi (p)} \log_2 (p) \rvert </math> | |||
This is a variant of the ordinary ''L''<sub>1</sub> norm where each coordinate is weighted in proportion to the log<sub>2</sub> of the prime that it represents. As we will soon see below, generalizations of the Tenney norm that correspond to other weighted ''L''<sub>''p''</sub> norms also exist; these we will call '''T<sub>''p''</sub> norms''', with the Tenney norm being designated the '''T<sub>1</sub> norm'''. | |||
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: | ||
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# take the ordinary ''L''<sub>1</sub> norm of the result. | # take the ordinary ''L''<sub>1</sub> norm of the result. | ||
To formalize this idea in its full generality, the Tenney norm of any | To formalize this idea in its full generality, the Tenney norm of any subgroup monzo '''m''' in an interval space with associated JI group G can be expressed as follows: | ||
<math>\ | <math>\displaystyle \lVert \vec m \rVert_{\text{T} 1}^G = \lVert W_J \cdot S_G \cdot \vec m \rVert_1</math> | ||
where | where S<sub>G</sub> is a [[Subgroup basis matrices|subgroup basis matrix]] in which the ''n''-th column is a monzo expressing the ''n''-th basis element of G in a suitable full-limit J containing all of G as a subgroup, W<sub>J</sub> is a diagonal weighting matrix in which the ''n''-th entry in the diagonal is the log<sub>2</sub> of the ''n''-th prime in J, and the ‖ · ‖<sub>1</sub> on the right hand side of the equation is the ''L''<sub>1</sub> norm on the resulting full-limit real vector. | ||
It is notable that, for interval spaces corresponding to a group of monzos where the basis is set to consist of only primes or prime powers, the Tenney norm of any interval | It is notable that, for interval spaces corresponding to a group of monzos where the basis is set to consist of only primes or prime powers, the Tenney norm of any interval m can be represented by the simpler expression | ||
<math>\ | <math>\displaystyle \lVert \vec m \rVert_{\text{T} 1}^G = \lVert W_G \cdot \vec m \rVert_1</math> | ||
where | where W<sub>G</sub> is a diagonal "weighting matrix" such that the ''n''-th entry in the diagonal is the log<sub>2</sub> of the interval represented by the ''n''-th coordinate of the monzo. For Tenney norms on these spaces, the unit sphere will look like the ordinary ''L''<sub>1</sub> unit sphere, but be dilated a different amount along each axis. Conversely, it is also important to note that that for groups for which the basis does ''not'' only consist of primes or prime powers, the unit sphere of the Tenney norm will not look like a dilated ''L''<sub>1</sub> unit sphere at all. | ||
== Generalized Tenney norms (T<sub>''p''</sub> norms) == | == Generalized Tenney norms (T<sub>''p''</sub> norms) == | ||
A useful generalization of the Tenney norm, called the '''generalized Tenney norm''', '''T<sub>''p''</sub> norm''', or '''T<sub>''p''</sub> height''', can be obtained as follows: | A useful generalization of the Tenney norm, called the '''generalized Tenney norm''', '''T<sub>''p''</sub> norm''', or '''T<sub>''p''</sub> height''', can be obtained as follows: | ||
<math>\ | <math>\displaystyle \lVert \vec m \rVert_{\text{T} p}^G = \lVert W_J \cdot S_G \cdot \vec m \rVert_p</math> | ||
In this scheme the ordinary Tenney norm now becomes the '''T<sub>1</sub> norm''', and in general we call an interval space that | 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 | Note that the ‖ · ‖<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 ‖ · ‖<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. | ||
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The T<sub>2</sub> norm is also the only T<sub>''p''</sub> norm that naturally defines an inner product, given by the matrix multiplication | The T<sub>2</sub> norm is also the only T<sub>''p''</sub> norm that naturally defines an inner product, given by the matrix multiplication | ||
<math>\left \langle \vec | <math>\displaystyle \left \langle \vec v, \vec w \right \rangle_G = \vec v^\mathsf {T} \cdot (S_G^\mathsf {T} \cdot W_J^2 \cdot S_G) \cdot \vec w</math> | ||
The matrix product | The matrix product S<sub>G</sub><sup>T</sup> · W<sub>J</sub><sup>2</sup> · S<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<sub>''p''</sub> 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 | <math>\displaystyle \left \langle \vec v, \vec w \right \rangle_G = \vec v ^\mathsf {T} \cdot W_G^2 \cdot \vec w</math> | ||
In these cases, the relationship between the TE inner product and the ordinary Euclidean inner product becomes apparent; the former is simply a version of the latter which weights the coordinates of each vector. | In these cases, the relationship between the TE inner product and the ordinary Euclidean inner product becomes apparent; the former is simply a version of the latter which weights the coordinates of each vector. | ||
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== Examples == | == Examples == | ||
Say that we are in the 2.9/7.5/3 subgroup, and we want to find the T<sub>1</sub> norm of {{monzo| 0 -2 1 }}. Then we can come up with a | Say that we are in the 2.9/7.5/3 subgroup, and we want to find the T<sub>1</sub> norm of {{monzo| 0 -2 1 }}. Then we can come up with a subgroup basis matrix S<sub>G</sub> for this subgroup in the 7-limit as follows: | ||
<math>\left[ \begin{array}{rrrrrl} | <math>\displaystyle | ||
\left[ \begin{array}{rrrrrl} | |||
| & 1 & 0 & 0 & 0 & \rangle\\ | | & 1 & 0 & 0 & 0 & \rangle\\ | ||
| & 0 & 2 & 0 & -1 & \rangle\\ | | & 0 & 2 & 0 & -1 & \rangle\\ | ||
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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]]. | ||
We can also come up with a weighting matrix for the full-limit | We can also come up with a weighting matrix for the full-limit W<sub>J</sub> as follows: | ||
<math>\begin{bmatrix} | <math>\displaystyle | ||
\begin{bmatrix} | |||
\log_2(2) & 0 & 0 & 0\\ | \log_2(2) & 0 & 0 & 0\\ | ||
0 & \log_2(3) & 0 & 0\\ | 0 & \log_2(3) & 0 & 0\\ | ||
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\end{bmatrix}</math> | \end{bmatrix}</math> | ||
Given these matrices, the T<sub>1</sub> norm of our subgroup basis monzo {{monzo| 0 -2 1 }}, which we will call ''' | Given these matrices, the T<sub>1</sub> norm of our subgroup basis monzo {{monzo| 0 -2 1 }}, which we will call '''m''', can be found by taking the ''L''<sub>1</sub> norm of the resulting real vector W<sub>J</sub> · S<sub>G</sub> · '''m'''. This expression works out to | ||
<math>\ | <math>\displaystyle | ||
\lVert \vec m \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\\ | ||
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| & 0 & -2 & 1 & \rangle | | & 0 & -2 & 1 & \rangle | ||
\end{array} \right] | \end{array} \right] | ||
\right \ | \right \rVert_1 | ||
</math> | |||
which finally resolves to | which finally resolves to | ||
<math>\ | <math>\displaystyle \lVert \vec m \rVert_{\text{T} 1}^{2.9/7.5/3} = \lVert [ \begin{matrix} 0 & -7.925 & 2.322 & 5.615 \end{matrix} \rangle \rVert_1 = 15.861</math> | ||
Note that the value 15.861 is obtained by |0| + |-7.925| + |2.322| + |5.615|, which is the | Note that the value 15.861 is obtained by |0| + |-7.925| + |2.322| + |5.615|, which is the ''L''<sub>1</sub> norm of the vector. | ||
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 ''L''<sub>1</sub> 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. | 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 ''L''<sub>1</sub> 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<sub>2</sub>(245·243) is indeed equal to 15.861. | ||
== See also == | == See also == | ||