Tenney–Euclidean metrics: Difference between revisions

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== TE norm ==

The '''Tenney–Euclidean norm''' ('''TE norm''') or '''Tenney–Euclidean complexity''' ('''TE complexity''') applies to ~~vals~~ as well as to ~~monzos~~.

The '''Tenney–Euclidean norm''' ('''TE norm''') or '''Tenney–Euclidean complexity''' ('''TE complexity''') applies to [[val]]s as well as to [[monzo]]s.

Let us define the val weighting matrix ''W'' to be the {{w|diagonal matrix}} with values 1, 1/log23, 1/log25 … 1/log2''p'' along the diagonal. [[~~Applying~~]] this matrix ~~to a vector~~ scales each entry by the corresponding entry of the diagonal matrix. Given a val '''a''' expressed as a row vector, the corresponding vector in weighted coordinates is {{nowrap|'''v''' {{=}} '''a'''''W''}}, with transpose {{nowrap|'''v'''{{t}} {{=}} ''W'''''a'''{{t}}}} where {{t}} denotes the transpose (writing the vector as a column vector instead of a row vector, as in the mathematical guide). Then the dot product of weighted vals is {{nowrap|'''vv'''{{t}} {{=}} '''a'''''W'' 2'''a'''{{t}}}} (the dot product of a vector with itself, or the sum of the squares of its entries), which makes the Euclidean metric on vals[how?], a measure of complexity, to be {{nowrap|‖'''v'''‖2 {{=}} √('''vv'''{{t}})}} {{nowrap|{{=}} √({{subsup|''a''|2|2}} + {{subsup|''a''|3|2}}/(log23)2 + … + {{subsup|''a''|''p''|2}}/(log2''p'')2)}}; dividing this by √(''n''), where {{nowrap|''n'' {{=}} π(''p'')}} is the number of primes to ''p'', gives the TE norm of a val.

Let us define the val weighting matrix ''W'' to be the {{w|diagonal matrix}} with values 1, 1/log23, 1/log25 … 1/log2''p'' along the diagonal. For the [[harmonic limit|''p''-limit]] prime basis ''Q'' = {{val| 2 3 5 … ''p'' }},

<math>\displaystyle W = \operatorname {diag} (1/\log_2 (Q))</math>

Left-multiplying a vector by this matrix scales each entry by the corresponding entry of the diagonal matrix.

Given a val '''a''' expressed as a row vector, the corresponding vector in weighted coordinates is {{nowrap|'''v''' {{=}} '''a'''''W''}}, with transpose {{nowrap|'''v'''{{t}} {{=}} ''W'''''a'''{{t}}}} where {{t}} denotes the transpose (writing the vector as a column vector instead of a row vector, as in the mathematical guide). Then the dot product of weighted vals is {{nowrap|'''vv'''{{t}} {{=}} '''a'''''W'' 2'''a'''{{t}}}} (the dot product of a vector with itself, or the sum of the squares of its entries), which makes the Euclidean metric on vals[how?], a measure of complexity, to be {{nowrap|‖'''v'''‖2 {{=}} √('''vv'''{{t}})}} {{nowrap|{{=}} √({{subsup|''a''|2|2}} + {{subsup|''a''|3|2}}/(log23)2 + … + {{subsup|''a''|''p''|2}}/(log2''p'')2)}}; dividing this by √(''n''), where {{nowrap|''n'' {{=}} π(''p'')}} is the number of primes to ''p'', gives the TE norm of a val.

Similarly, if '''b''' is a monzo, then in weighted coordinates the monzo becomes {{nowrap|'''m''' {{=}} ''W''{{inv}}'''b'''}}, and the dot product is {{nowrap|'''m'''{{t}}'''m''' {{=}} '''b'''{{t}}''W''-2'''b'''}}, leading to {{nowrap|√('''m'''{{t}}'''m''') {{=}} √({{subsup|''b''|2|2}} + (log23)2{{subsup|''b''|3|2}} + … + (log2''p'')2{{subsup|''b''|''p''|2}})}}; multiplying this by √(''n'') gives the dual RMS norm on monzos which serves as a measure of complexity.

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 == TE norm ==
-The '''Tenney–Euclidean norm''' ('''TE norm''') or '''Tenney–Euclidean complexity''' ('''TE complexity''') applies to vals as well as to monzos.
+The '''Tenney–Euclidean norm''' ('''TE norm''') or '''Tenney–Euclidean complexity''' ('''TE complexity''') applies to [[val]]s as well as to [[monzo]]s.
-Let us define the val weighting matrix ''W'' to be the {{w|diagonal matrix}} with values 1, 1/log<sub>2</sub>3, 1/log<sub>2</sub>5 … 1/log<sub>2</sub>''p'' along the diagonal. [[Applying]] this matrix to a vector scales each entry by the corresponding entry of the diagonal matrix. Given a val '''a''' expressed as a row vector, the corresponding vector in weighted coordinates is {{nowrap|'''v''' {{=}} '''a'''''W''}}, with transpose {{nowrap|'''v'''{{t}} {{=}} ''W'''''a'''{{t}}}} where {{t}} denotes the transpose (writing the vector as a column vector instead of a row vector, as in the mathematical guide). Then the dot product of weighted vals is {{nowrap|'''vv'''{{t}} {{=}} '''a'''''W''&#x200A;<sup>2</sup>'''a'''{{t}}}} (the dot product of a vector with itself, or the sum of the squares of its entries), which makes the Euclidean metric on vals<sup>[how?]</sup>, a measure of complexity, to be {{nowrap|‖'''v'''‖<sub>2</sub> {{=}} √('''vv'''{{t}})}} {{nowrap|{{=}} √({{subsup|''a''|2|2}} + {{subsup|''a''|3|2}}/(log<sub>2</sub>3)<sup>2</sup> + … + {{subsup|''a''|''p''|2}}/(log<sub>2</sub>''p'')<sup>2</sup>)}}; dividing this by √(''n''), where {{nowrap|''n'' {{=}} π(''p'')}} is the number of primes to ''p'', gives the TE norm of a val.
+Let us define the val weighting matrix ''W'' to be the {{w|diagonal matrix}} with values 1, 1/log<sub>2</sub>3, 1/log<sub>2</sub>5 … 1/log<sub>2</sub>''p'' along the diagonal. For the [[harmonic limit|''p''-limit]] prime basis ''Q'' = {{val| 2 3 5 … ''p'' }},
+<math>\displaystyle W = \operatorname {diag} (1/\log_2 (Q))</math>
+Left-multiplying a vector by this matrix scales each entry by the corresponding entry of the diagonal matrix.
+Given a val '''a''' expressed as a row vector, the corresponding vector in weighted coordinates is {{nowrap|'''v''' {{=}} '''a'''''W''}}, with transpose {{nowrap|'''v'''{{t}} {{=}} ''W'''''a'''{{t}}}} where {{t}} denotes the transpose (writing the vector as a column vector instead of a row vector, as in the mathematical guide). Then the dot product of weighted vals is {{nowrap|'''vv'''{{t}} {{=}} '''a'''''W''&#x200A;<sup>2</sup>'''a'''{{t}}}} (the dot product of a vector with itself, or the sum of the squares of its entries), which makes the Euclidean metric on vals<sup>[how?]</sup>, a measure of complexity, to be {{nowrap|‖'''v'''‖<sub>2</sub> {{=}} √('''vv'''{{t}})}} {{nowrap|{{=}} √({{subsup|''a''|2|2}} + {{subsup|''a''|3|2}}/(log<sub>2</sub>3)<sup>2</sup> + … + {{subsup|''a''|''p''|2}}/(log<sub>2</sub>''p'')<sup>2</sup>)}}; dividing this by √(''n''), where {{nowrap|''n'' {{=}} π(''p'')}} is the number of primes to ''p'', gives the TE norm of a val.
 Similarly, if '''b''' is a monzo, then in weighted coordinates the monzo becomes {{nowrap|'''m''' {{=}} ''W''{{inv}}'''b'''}}, and the dot product is {{nowrap|'''m'''{{t}}'''m''' {{=}} '''b'''{{t}}''W''<sup>-2</sup>'''b'''}}, leading to {{nowrap|√('''m'''{{t}}'''m''') {{=}} √({{subsup|''b''|2|2}} + (log<sub>2</sub>3)<sup>2</sup>{{subsup|''b''|3|2}} + … + (log<sub>2</sub>''p'')<sup>2</sup>{{subsup|''b''|''p''|2}})}}; multiplying this by √(''n'') gives the dual RMS norm on monzos which serves as a measure of complexity.