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| <h2>IMPORTED REVISION FROM WIKISPACES</h2> | | {{texmap}}{{DISPLAYTITLE:Generalized Tenney norms and T<sub>''p''</sub> interval space}} |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | 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>. |
| : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-06 13:25:55 UTC</tt>.<br>
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| : The original revision id was <tt>356546888</tt>.<br>
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| : The revision comment was: <tt></tt><br>
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| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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| <h4>Original Wikitext content:</h4>
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| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">= =
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| [[toc]]
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| =Basics=
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| 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 or smonzos by embedding the group in a normed vector space, so that the norm of any interval is taken to be its complexity. WE can think of the monzos as forming a ℤ-module, with coordinates given by integers, and the vector space can be constructed by simply allowing real coordinates, hence defining the module over ℝ instead of ℤ and giving it the added structure of being a vector space. In either case, the resulting space is called [[Monzos and Interval Space|interval space]], with the monzos forming the lattice of vectors with integer coordinates.
| | The most important and natural norm which arises in this scenario is the [[Tenney norm]], which we will explore below. |
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| The most important and natural norm which arises in this scenario is the **Tenney norm**, which we will explore below.
| | == Tenney norm (T<sub>1</sub> norm) == |
| | {{Main| Tenney height }} |
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| =The Tenney Norm (T1 norm)=
| | 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 {{nowrap|'''m''' {{=}} {{monzo| ''m''<sub>1</sub> ''m''<sub>2</sub> … ''m''<sub>π (''p'')</sub> … }}}}, with π standing for the {{w|prime-counting function}}, this norm can be calculated as |
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| The **Tenney norm**, also called **Tenney height**, is the norm such that for any monzo representing an interval a/b, the norm of the interval is log<span style="font-size: 10px; vertical-align: sub;">2</span>(a·b). For a full-limit monzo |a b c d ...>, this norm can be calculated as |log<span style="font-size: 80%; vertical-align: sub;">2</span>(2)·a| + |log<span style="font-size: 10px; vertical-align: sub;">2</span>(3)·b| + |log<span style="font-size: 10px; vertical-align: sub;">2</span>(5)·c| + |log<span style="font-size: 10px; vertical-align: sub;">2</span>(7)·d| + ... . This is a variant of the ordinary L1 norm where each coordinate is weighted in proportion to the log<span style="font-size: 10px; vertical-align: sub;">2</span> of the prime that it represents. As we will soon see below, generalizations of the Tenney norm that correspond to other weighted Lp norms also exist; these we will call **Tp norms**, with the Tenney norm being designated the **T1 norm**.
| | <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> |
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| | 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'''. |
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| 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: |
| # if the interval is a smonzo, map it back to its corresponding full-limit monzo
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| # weight the axis for each prime p by log<span style="font-size: 10px; vertical-align: sub;">2</span>(p)
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| # take the ordinary L1 norm of the result.
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| 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:
| | # if the interval is a subgroup monzo, with coordinates in the subgroup basis, map it back to its corresponding full-limit monzo |
| | # weight the axis for each prime ''p'' by log<sub>2</sub>(''p'') |
| | # take the ordinary ''L''<sub>1</sub> norm of the result. |
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| [[math]]
| | 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: |
| \left \| \vec{v} \right \|_{\textbf{T1}}^\textbf{G} = \left \| \mathbf{W_L} \cdot \mathbf{V_\textbf{G}} \cdot \vec{v} \right \|_\textbf{1}
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| [[math]]
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| where **V<span style="font-size: 80%; vertical-align: sub;">G</span>** is a [[xenharmonic/Subgroup Mapping Matrices (V-maps)|V-map]] in which the nth column is a monzo expressing the nth basis element of **G** in a suitable full-limit **L** containing all of **G** as a subgroup, **W<span style="font-size: 80%; vertical-align: sub;">L</span>** is a diagonal weighting matrix in which the nth entry in the diagonal is the log<span style="font-size: 10px; vertical-align: sub;">2</span> of the nth prime in **L**, and the || · ||**<span style="font-size: 80%; vertical-align: sub;">1</span>** on the right hand side of the equation is the L1 norm on the resulting full-limit real vector.
| | <math>\displaystyle \lVert \vec m \rVert_{\text{T} 1}^G = \lVert W_J \cdot S_G \cdot \vec m \rVert_1</math> |
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| 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 //v// can be represented by the simpler expression
| | where ''S''<sub>''G''</sub> is a [[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 {{nowrap|‖ · ‖<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. |
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| [[math]]
| | 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 |
| \left \| \vec{v} \right \|_{\textbf{T1}}^\textbf{G} = \left \| \mathbf{W_G} \cdot \vec{v} \right \|_\textbf{1}
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| [[math]]
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| where **W<span style="font-size: 10px; vertical-align: sub;">G</span>** is a diagonal "weighting matrix" such that the nth entry in the diagonal is the log<span style="font-size: 10px; vertical-align: sub;">2</span> of the interval represented by the nth coordinate of the monzo. For Tenney norms on these spaces, the unit sphere will look like the ordinary L1 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 won't look like a dilated L1 unit sphere at all.
| | <math>\displaystyle \lVert \vec m \rVert_{\text{T} 1}^G = \lVert W_G \cdot \vec m \rVert_1</math> |
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| =Generalized Tenney Norms (Tp norms)=
| | 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. |
| A useful generalization of the Tenney norm, called the **Generalized Tenney Norm**, **Tp norm**, or **Tp height**, can be obtained as follows:
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| [[math]]
| | == Generalized Tenney norms (T<sub>''p''</sub> norms) == |
| \left \| \vec{v} \right \|_{\textbf{Tp}}^\textbf{G} = \left \| \mathbf{W_L} \cdot \mathbf{V_\textbf{G}} \cdot \vec{v} \right \|_\textbf{p}
| | 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]]
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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's been given a Tp norm **Tp interval space**. We may sometimes notate this as **Tp<span style="font-size: 80%; vertical-align: super;">G</span>**, where **G** is the associated group the interval space is built around.
| | <math>\displaystyle \lVert \vec m \rVert_{\text{T} p}^G = \lVert W_J \cdot S_G \cdot \vec m \rVert_p</math> |
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| Note that the || · ||**<span style="font-size: 80%; vertical-align: sub;">Tp</span>** norm on the left side of the equation now has a subscript of Tp rather than T1, and that the || · ||**<span style="font-size: 80%; vertical-align: sub;">p</span>** 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 Lp norm rather than restricting our consideration to the L1 norm.
| | 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. |
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| Tp 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<span style="font-size: 10px; vertical-align: sub;">2</span>(n·d). Furthermore, generalized Tp norms may sometimes differ from the T1 norm in their ranking of intervals by Tp complexity, although the Tp 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 Tp norm other than T1 which are theoretically justified; additionally, certain Tp norms are worth using as an approximation to T1 for their strong computational advantages. As such, Tp spaces are musically justified enough to merit further study, despite the surface advantages of sticking exclusively to the T1 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. |
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| =The Tenney-Euclidean Norm (TE norm, T2 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. |
| 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<span style="font-size: 80%; vertical-align: super;">[[Tenney-Euclidean metrics|(1)]][[Tenney-Euclidean temperament measures|(2)]][[Tenney-Euclidean Tuning|(3)]]</span>. It approximates the T1 complexity of many intervals, although notably rates 9/1 as more complex than 15/1.
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| The T2 norm is also the only Tp norm that naturally defines an inner product, given by the matrix multiplication
| | == Tenney–Euclidean norm (TE norm, T<sub>2</sub> norm) == |
| | {{Main| Tenney–Euclidean metrics }} |
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| [[math]] | | The T<sub>2</sub> norm is often called the '''Tenney–Euclidean norm''', '''TE norm''', or '''TE height''', as it has the same relationship with Euclidean geometry that the T<sub>1</sub> 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 T<sub>1</sub> complexity of many intervals, although notably rates 9/1 as more complex than 15/1. |
| \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}
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| [[math]] | |
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| The matrix product (**V<span style="font-size: 80%; vertical-align: sub;">G</span>**<span style="font-size: 80%; vertical-align: super;">T</span> · **W<span style="font-size: 10px; vertical-align: sub;">L</span>**<span style="font-size: 10px; vertical-align: super;">2</span> · **V<span style="font-size: 10px; vertical-align: sub;">G</span>**) 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//**<span style="font-size: 10px; vertical-align: super;">T</span> denotes a row vector. As was the case with Tp norms in general, this equation simplifies considerably when the group **G** takes as its basis only primes and prime powers, becoming instead | | 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 |
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| [[math]]
| | <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> |
| \left \langle \vec{v},\vec{w} \right \rangle_\textbf{G} = \vec{v}^T \cdot \mathbf{W}^2_\mathbf{L} \cdot \vec{w} | | |
| [[math]]
| | The matrix product {{nowrap|{{subsup|''S''|''G''|T}} · {{subsup|''W''|''J''|2}} · ''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 |
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| | <math>\displaystyle \left \langle \vec v, \vec w \right \rangle_G = \vec v ^\mathsf {T} \cdot W_G^2 \cdot \vec w</math> |
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| 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 subgroup basis matrix ''S''<sub>''G''</sub> for this subgroup in the 7-limit as follows: |
| | |
| | <math>\displaystyle |
| | \begin{bmatrix} |
| | 1 & 0 & 0 & 0 \\ |
| | 0 & 2 & 0 & -1 \\ |
| | 0 & -1 & 1 & 0 \\ |
| | \end{bmatrix}</math> |
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| Say that we're in the 2.9/7.5/3 subgroup, and we want to find the T1 norm of |0 -2 1>. Then we can come up with a V-map **V<span style="font-size: 10px; vertical-align: sub;">G</span>** for this subgroup in the 7-limit as follows:
| | 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]]. |
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| [[math]]
| | We can also come up with a weighting matrix for the full-limit W<sub>J</sub> as follows: |
| \[ \left[ \begin{array}{rrrrrl}
| |
| | & 1 & 0 & 0 & 0 & \rangle\\
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| | & 0 & 2 & 0 & -1 & \rangle\\
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| | & 0 & -1 & 1 & 0 & \rangle
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| \end{array} \right] \]
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| [[math]]
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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 [[Subgroup Mapping Matrices (V-maps)|as explained here]].
| | <math>\displaystyle |
| | \begin{bmatrix} |
| | \log_2(2) & 0 & 0 & 0\\ |
| | 0 & \log_2(3) & 0 & 0\\ |
| | 0 & 0 & \log_2(5) & 0\\ |
| | 0 & 0 & 0 & \log_2(7) |
| | \end{bmatrix}</math> |
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| We can also come up with a weighting matrix for the full-limit **W<span style="font-size: 10px; vertical-align: sub;">L</span>** as follows:
| | 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 {{nowrap|''W''<sub>''J''</sub> · ''S''<sub>''G''</sub> · '''m'''}}. This expression works out to |
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| [[math]]
| | <math>\displaystyle |
| | \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\\ |
| 0 & \log_2(3) & 0 & 0\\ | | 0 & \log_2(3) & 0 & 0\\ |
| 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 \begin{bmatrix} |
| | 1 & 0 & 0 & 0 \\ |
| | 0 & 2 & 0 & -1 \\ |
| | 0 & -1 & 1 & 0 \\ |
| | \end{bmatrix} \cdot \begin{bmatrix} |
| | 0 & -2 & 1 \\ |
| \end{bmatrix} | | \end{bmatrix} |
| [[math]]
| | \right \rVert_1 |
| | </math> |
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| Given these matrices, the T1 norm of our smonzo |0 -2 1>, which we will call **v**, can be found by taking the L1 norm of the resulting real vector **W<span style="font-size: 10px; vertical-align: sub;">L</span>** · **V<span style="font-size: 10px; vertical-align: sub;">G</span>** · **v**. This expression works out to
| | which finally resolves to |
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| [[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> |
| \left \| \vec{v} \right \|_\mathbf{T1}^\mathbf{2.\frac{9}{7}.\frac{5}{3}} = \left \| | |
| \begin{bmatrix} | |
| \log_2(2) & 0 & 0 & 0\\
| |
| 0 & \log_2(3) & 0 & 0\\
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| 0 & 0 & \log_2(5) & 0\\
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| 0 & 0 & 0 & \log_2(7)
| |
| \end{bmatrix} \cdot \left[ \begin{array}{rrrrrl}
| |
| | & 1 & 0 & 0 & 0 & \rangle\\
| |
| | & 0 & 2 & 0 & -1 & \rangle\\
| |
| | & 0 & -1 & 1 & 0 & \rangle
| |
| \end{array} \right] \cdot \left[ \begin{array}{rrrrl} | |
| | & 0 & -2 & 1 & \rangle
| |
| \end{array} \right]
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| \right \|_\mathbf{1}
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| [[math]]
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| which finally resolves to | | Note that {{nowrap|15.861 {{=}} {{!}}0{{!}} + {{!}}−7.925{{!}} + {{!}}2.322{{!}} + {{!}}5.615{{!}}}}, which is the ''L''<sub>1</sub> norm of the vector. |
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| [[math]]
| | 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 {{nowrap|log<sub>2</sub>(245 · 243)}} is indeed equal to 15.861. |
| \left \| \vec{v} \right \|_\mathbf{T1}^\mathbf{2.\frac{9}{7}.\frac{5}{3}} = \left \| |0 \;-7.925 \;\; 2.322 \;\; 5.615 \rangle \right \|_\mathbf{1} = 15.861
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| [[math]]
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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.
| | == See also == |
| | * [[Generalized Tenney dual norms and Tp tuning space|Generalized Tenney dual norms and T<sub>''p''</sub> tuning space]] |
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| To confirm this, we can put smonzo |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(n·d) 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.</pre></div>
| | [[Category:Math]] |
| <h4>Original HTML content:</h4>
| | [[Category:Interval space]] |
| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Generalized Tenney Norms and Tp Interval Space</title></head><body><!-- ws:start:WikiTextHeadingRule:9:&lt;h1&gt; --><h1 id="toc0"><!-- ws:end:WikiTextHeadingRule:9 --> </h1>
| | [[Category:Interval complexity measures]] |
| <!-- ws:start:WikiTextTocRule:21:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --><div style="margin-left: 1em;"><a href="#toc0"> </a></div>
| | [[Category:Tenney-weighted measures]] |
| <!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --><div style="margin-left: 1em;"><a href="#Basics">Basics</a></div>
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| <!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --><div style="margin-left: 1em;"><a href="#The Tenney Norm (T1 norm)">The Tenney Norm (T1 norm)</a></div>
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| <!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --><div style="margin-left: 1em;"><a href="#Generalized Tenney Norms (Tp norms)">Generalized Tenney Norms (Tp norms)</a></div>
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| <!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --><div style="margin-left: 1em;"><a href="#The Tenney-Euclidean Norm (TE norm, T2 norm)">The Tenney-Euclidean Norm (TE norm, T2 norm)</a></div>
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| <!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --><div style="margin-left: 1em;"><a href="#Examples">Examples</a></div>
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| <!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --></div>
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| <!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextHeadingRule:11:&lt;h1&gt; --><h1 id="toc1"><a name="Basics"></a><!-- ws:end:WikiTextHeadingRule:11 -->Basics</h1>
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| <br />
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| It can be useful to define a notion of the &quot;complexity&quot; 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 or smonzos by embedding the group in a normed vector space, so that the norm of any interval is taken to be its complexity. WE can think of the monzos as forming a ℤ-module, with coordinates given by integers, and the vector space can be constructed by simply allowing real coordinates, hence defining the module over ℝ instead of ℤ and giving it the added structure of being a vector space. In either case, the resulting space is called <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">interval space</a>, with the monzos forming the lattice of vectors with integer coordinates.<br />
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| <br />
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| The most important and natural norm which arises in this scenario is the <strong>Tenney norm</strong>, which we will explore below.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:13:&lt;h1&gt; --><h1 id="toc2"><a name="The Tenney Norm (T1 norm)"></a><!-- ws:end:WikiTextHeadingRule:13 -->The Tenney Norm (T1 norm)</h1>
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| <br />
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| The <strong>Tenney norm</strong>, also called <strong>Tenney height</strong>, is the norm such that for any monzo representing an interval a/b, the norm of the interval is log<span style="font-size: 10px; vertical-align: sub;">2</span>(a·b). For a full-limit monzo |a b c d ...&gt;, this norm can be calculated as |log<span style="font-size: 80%; vertical-align: sub;">2</span>(2)·a| + |log<span style="font-size: 10px; vertical-align: sub;">2</span>(3)·b| + |log<span style="font-size: 10px; vertical-align: sub;">2</span>(5)·c| + |log<span style="font-size: 10px; vertical-align: sub;">2</span>(7)·d| + ... . This is a variant of the ordinary L1 norm where each coordinate is weighted in proportion to the log<span style="font-size: 10px; vertical-align: sub;">2</span> of the prime that it represents. As we will soon see below, generalizations of the Tenney norm that correspond to other weighted Lp norms also exist; these we will call <strong>Tp norms</strong>, with the Tenney norm being designated the <strong>T1 norm</strong>.<br />
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| <br />
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| Informally speaking, we can obtain the Tenney norm on an interval space for any JI group by applying these three steps:<br />
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| <ol><li>if the interval is a smonzo, 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><br />
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| To formalize this idea in its full generality, the Tenney norm of any vector <em>v</em> in an interval space with associated JI group <strong>G</strong> can be expressed as follows:<br />
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| <br />
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| <!-- ws:start:WikiTextMathRule:0:
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| [[math]]&lt;br/&gt;
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| \left \| \vec{v} \right \|_{\textbf{T1}}^\textbf{G} = \left \| \mathbf{W_L} \cdot \mathbf{V_\textbf{G}} \cdot \vec{v} \right \|_\textbf{1}&lt;br/&gt;[[math]]
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| --><script type="math/tex">\left \| \vec{v} \right \|_{\textbf{T1}}^\textbf{G} = \left \| \mathbf{W_L} \cdot \mathbf{V_\textbf{G}} \cdot \vec{v} \right \|_\textbf{1}</script><!-- ws:end:WikiTextMathRule:0 --><br />
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| <br />
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| where <strong>V<span style="font-size: 80%; vertical-align: sub;">G</span></strong> is a <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Subgroup%20Mapping%20Matrices%20%28V-maps%29">V-map</a> in which the nth column is a monzo expressing the nth basis element of <strong>G</strong> in a suitable full-limit <strong>L</strong> containing all of <strong>G</strong> as a subgroup, <strong>W<span style="font-size: 80%; vertical-align: sub;">L</span></strong> is a diagonal weighting matrix in which the nth entry in the diagonal is the log<span style="font-size: 10px; vertical-align: sub;">2</span> of the nth prime in <strong>L</strong>, and the || · ||<strong><span style="font-size: 80%; vertical-align: sub;">1</span></strong> on the right hand side of the equation is the L1 norm on the resulting full-limit real vector.<br />
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| <br />
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| 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 <em>v</em> can be represented by the simpler expression<br />
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| <br />
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| <!-- ws:start:WikiTextMathRule:1:
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| [[math]]&lt;br/&gt;
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| \left \| \vec{v} \right \|_{\textbf{T1}}^\textbf{G} = \left \| \mathbf{W_G} \cdot \vec{v} \right \|_\textbf{1}&lt;br/&gt;[[math]]
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| --><script type="math/tex">\left \| \vec{v} \right \|_{\textbf{T1}}^\textbf{G} = \left \| \mathbf{W_G} \cdot \vec{v} \right \|_\textbf{1}</script><!-- ws:end:WikiTextMathRule:1 --><br />
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| <br />
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| where <strong>W<span style="font-size: 10px; vertical-align: sub;">G</span></strong> is a diagonal &quot;weighting matrix&quot; such that the nth entry in the diagonal is the log<span style="font-size: 10px; vertical-align: sub;">2</span> of the interval represented by the nth coordinate of the monzo. For Tenney norms on these spaces, the unit sphere will look like the ordinary L1 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 <em>not</em> only consist of primes or prime powers, the unit sphere of the Tenney norm won't look like a dilated L1 unit sphere at all.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:15:&lt;h1&gt; --><h1 id="toc3"><a name="Generalized Tenney Norms (Tp norms)"></a><!-- ws:end:WikiTextHeadingRule:15 -->Generalized Tenney Norms (Tp norms)</h1>
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| A useful generalization of the Tenney norm, called the <strong>Generalized Tenney Norm</strong>, <strong>Tp norm</strong>, or <strong>Tp height</strong>, can be obtained as follows:<br />
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| <br />
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| <!-- ws:start:WikiTextMathRule:2:
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| [[math]]&lt;br/&gt;
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| \left \| \vec{v} \right \|_{\textbf{Tp}}^\textbf{G} = \left \| \mathbf{W_L} \cdot \mathbf{V_\textbf{G}} \cdot \vec{v} \right \|_\textbf{p}&lt;br/&gt;[[math]]
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| --><script type="math/tex">\left \| \vec{v} \right \|_{\textbf{Tp}}^\textbf{G} = \left \| \mathbf{W_L} \cdot \mathbf{V_\textbf{G}} \cdot \vec{v} \right \|_\textbf{p}</script><!-- ws:end:WikiTextMathRule:2 --><br />
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| <br />
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| In this scheme the ordinary Tenney norm now becomes the <strong>T1 norm</strong>, and in general we call an interval space that's been given a Tp norm <strong>Tp interval space</strong>. We may sometimes notate this as <strong>Tp<span style="font-size: 80%; vertical-align: super;">G</span></strong>, where <strong>G</strong> is the associated group the interval space is built around.<br />
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| <br />
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| Note that the || · ||<strong><span style="font-size: 80%; vertical-align: sub;">Tp</span></strong> norm on the left side of the equation now has a subscript of Tp rather than T1, and that the || · ||<strong><span style="font-size: 80%; vertical-align: sub;">p</span></strong> 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 Lp norm rather than restricting our consideration to the L1 norm.<br />
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| <br />
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| Tp 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<span style="font-size: 10px; vertical-align: sub;">2</span>(n·d). Furthermore, generalized Tp norms may sometimes differ from the T1 norm in their ranking of intervals by Tp complexity, although the Tp 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 Tp norm other than T1 which are theoretically justified; additionally, certain Tp norms are worth using as an approximation to T1 for their strong computational advantages. As such, Tp spaces are musically justified enough to merit further study, despite the surface advantages of sticking exclusively to the T1 norm.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:17:&lt;h1&gt; --><h1 id="toc4"><a name="The Tenney-Euclidean Norm (TE norm, T2 norm)"></a><!-- ws:end:WikiTextHeadingRule:17 -->The Tenney-Euclidean Norm (TE norm, T2 norm)</h1>
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| The T2 norm is often called the <strong>Tenney-Euclidean norm</strong>, <strong>TE norm</strong>, or <strong>TE height</strong>, 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<span style="font-size: 80%; vertical-align: super;"><a class="wiki_link" href="/Tenney-Euclidean%20metrics">(1)</a><a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures">(2)</a><a class="wiki_link" href="/Tenney-Euclidean%20Tuning">(3)</a></span>. It approximates the T1 complexity of many intervals, although notably rates 9/1 as more complex than 15/1.<br />
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| <br />
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| The T2 norm is also the only Tp norm that naturally defines an inner product, given by the matrix multiplication<br />
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| <br />
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| <!-- ws:start:WikiTextMathRule:3:
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| [[math]]&lt;br/&gt;
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| \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}&lt;br/&gt;[[math]]
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| --><script type="math/tex">\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}</script><!-- ws:end:WikiTextMathRule:3 --><br />
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| <br />
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| The matrix product (<strong>V<span style="font-size: 80%; vertical-align: sub;">G</span></strong><span style="font-size: 80%; vertical-align: super;">T</span> · <strong>W<span style="font-size: 10px; vertical-align: sub;">L</span></strong><span style="font-size: 10px; vertical-align: super;">2</span> · <strong>V<span style="font-size: 10px; vertical-align: sub;">G</span></strong>) itself is a positive definite matrix, and as such defines the inner product for the TE norm. Note that this setup represents the vectors <strong><em>v</em></strong> and <strong><em>w</em></strong> by column vectors, so that <strong><em>v</em></strong><span style="font-size: 10px; vertical-align: super;">T</span> denotes a row vector. As was the case with Tp norms in general, this equation simplifies considerably when the group <strong>G</strong> takes as its basis only primes and prime powers, becoming instead<br />
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| <br />
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| <!-- ws:start:WikiTextMathRule:4:
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| [[math]]&lt;br/&gt;
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| \left \langle \vec{v},\vec{w} \right \rangle_\textbf{G} = \vec{v}^T \cdot \mathbf{W}^2_\mathbf{L} \cdot \vec{w}&lt;br/&gt;[[math]]
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| --><script type="math/tex">\left \langle \vec{v},\vec{w} \right \rangle_\textbf{G} = \vec{v}^T \cdot \mathbf{W}^2_\mathbf{L} \cdot \vec{w}</script><!-- ws:end:WikiTextMathRule:4 --><br />
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| <br />
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| 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.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:19:&lt;h1&gt; --><h1 id="toc5"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:19 -->Examples</h1>
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| <br />
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| Say that we're in the 2.9/7.5/3 subgroup, and we want to find the T1 norm of |0 -2 1&gt;. Then we can come up with a V-map <strong>V<span style="font-size: 10px; vertical-align: sub;">G</span></strong> for this subgroup in the 7-limit as follows:<br />
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| <br />
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| <!-- ws:start:WikiTextMathRule:5:
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| [[math]]&lt;br/&gt;
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| \[ \left[ \begin{array}{rrrrrl}&lt;br /&gt;
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| | &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; \rangle\\&lt;br /&gt;
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| | &amp; 0 &amp; 2 &amp; 0 &amp; -1 &amp; \rangle\\&lt;br /&gt;
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| | &amp; 0 &amp; -1 &amp; 1 &amp; 0 &amp; \rangle&lt;br /&gt;
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| \end{array} \right] \]&lt;br/&gt;[[math]]
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| --><script type="math/tex">\[ \left[ \begin{array}{rrrrrl}
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| | & 1 & 0 & 0 & 0 & \rangle\\
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| | & 0 & 2 & 0 & -1 & \rangle\\
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| | & 0 & -1 & 1 & 0 & \rangle
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| \end{array} \right] \]</script><!-- ws:end:WikiTextMathRule:5 --><br />
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| <br />
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| Note that the &quot;rows&quot; 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 <a class="wiki_link" href="/Subgroup%20Mapping%20Matrices%20%28V-maps%29">as explained here</a>.<br />
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| <br />
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| We can also come up with a weighting matrix for the full-limit <strong>W<span style="font-size: 10px; vertical-align: sub;">L</span></strong> as follows:<br />
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| <br />
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| <!-- ws:start:WikiTextMathRule:6:
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| [[math]]&lt;br/&gt;
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| \begin{bmatrix}&lt;br /&gt;
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| \log_2(2) &amp; 0 &amp; 0 &amp; 0\\&lt;br /&gt;
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| 0 &amp; \log_2(3) &amp; 0 &amp; 0\\&lt;br /&gt;
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| 0 &amp; 0 &amp; \log_2(5) &amp; 0\\&lt;br /&gt;
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| 0 &amp; 0 &amp; 0 &amp; \log_2(7)&lt;br /&gt;
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| \end{bmatrix}&lt;br/&gt;[[math]]
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| --><script type="math/tex">\begin{bmatrix}
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| \log_2(2) & 0 & 0 & 0\\
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| 0 & \log_2(3) & 0 & 0\\
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| 0 & 0 & \log_2(5) & 0\\
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| 0 & 0 & 0 & \log_2(7)
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| \end{bmatrix}</script><!-- ws:end:WikiTextMathRule:6 --><br />
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| <br />
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| Given these matrices, the T1 norm of our smonzo |0 -2 1&gt;, which we will call <strong>v</strong>, can be found by taking the L1 norm of the resulting real vector <strong>W<span style="font-size: 10px; vertical-align: sub;">L</span></strong> · <strong>V<span style="font-size: 10px; vertical-align: sub;">G</span></strong> · <strong>v</strong>. This expression works out to<br />
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| <br />
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| <!-- ws:start:WikiTextMathRule:7:
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| [[math]]&lt;br/&gt;
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| \left \| \vec{v} \right \|_\mathbf{T1}^\mathbf{2.\frac{9}{7}.\frac{5}{3}} = \left \|&lt;br /&gt;
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| \begin{bmatrix}&lt;br /&gt;
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| \log_2(2) &amp; 0 &amp; 0 &amp; 0\\&lt;br /&gt;
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| 0 &amp; \log_2(3) &amp; 0 &amp; 0\\&lt;br /&gt;
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| 0 &amp; 0 &amp; \log_2(5) &amp; 0\\&lt;br /&gt;
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| 0 &amp; 0 &amp; 0 &amp; \log_2(7)&lt;br /&gt;
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| \end{bmatrix} \cdot \left[ \begin{array}{rrrrrl}&lt;br /&gt;
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| | &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; \rangle\\&lt;br /&gt;
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| | &amp; 0 &amp; 2 &amp; 0 &amp; -1 &amp; \rangle\\&lt;br /&gt;
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| | &amp; 0 &amp; -1 &amp; 1 &amp; 0 &amp; \rangle&lt;br /&gt;
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| \end{array} \right] \cdot \left[ \begin{array}{rrrrl}&lt;br /&gt;
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| | &amp; 0 &amp; -2 &amp; 1 &amp; \rangle&lt;br /&gt;
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| \end{array} \right]&lt;br /&gt;
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| \right \|_\mathbf{1}&lt;br/&gt;[[math]]
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| --><script type="math/tex">\left \| \vec{v} \right \|_\mathbf{T1}^\mathbf{2.\frac{9}{7}.\frac{5}{3}} = \left \|
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| \begin{bmatrix}
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| \log_2(2) & 0 & 0 & 0\\
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| 0 & \log_2(3) & 0 & 0\\
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| 0 & 0 & \log_2(5) & 0\\
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| 0 & 0 & 0 & \log_2(7)
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| \end{bmatrix} \cdot \left[ \begin{array}{rrrrrl}
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| | & 1 & 0 & 0 & 0 & \rangle\\
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| | & 0 & 2 & 0 & -1 & \rangle\\
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| | & 0 & -1 & 1 & 0 & \rangle
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| \end{array} \right] \cdot \left[ \begin{array}{rrrrl}
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| | & 0 & -2 & 1 & \rangle
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| \end{array} \right]
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| \right \|_\mathbf{1}</script><!-- ws:end:WikiTextMathRule:7 --><br />
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| <br />
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| which finally resolves to<br />
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| <br />
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| <!-- ws:start:WikiTextMathRule:8:
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| [[math]]&lt;br/&gt;
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| \left \| \vec{v} \right \|_\mathbf{T1}^\mathbf{2.\frac{9}{7}.\frac{5}{3}} = \left \| |0 \;-7.925 \;\; 2.322 \;\; 5.615 \rangle \right \|_\mathbf{1} = 15.861&lt;br/&gt;[[math]]
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| --><script type="math/tex">\left \| \vec{v} \right \|_\mathbf{T1}^\mathbf{2.\frac{9}{7}.\frac{5}{3}} = \left \| |0 \;-7.925 \;\; 2.322 \;\; 5.615 \rangle \right \|_\mathbf{1} = 15.861</script><!-- ws:end:WikiTextMathRule:8 --><br />
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| <br />
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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.<br />
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| <br />
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| To confirm this, we can put smonzo |0 -2 1&gt; back into rational form to see that it represents the interval 245/243. As the L1 norm is supposed to give log(n·d) 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.</body></html></pre></div>
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