Tenney–Euclidean metrics: Difference between revisions

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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>~~

$$ W = \operatorname {diag} (1/\log_2 (Q)) $$

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

Given a val ''V'' expressed as a row vector, the corresponding row vector in weighted coordinates is {{nowrap| ''V''''W'' {{=}} ''VW'' }}, with transpose {{nowrap| {{subsup|''V''|''W''|T}} {{=}} ''WV''{{t}} }} where {{t}} denotes the transpose. ~~Then the~~ dot product of weighted ~~vals~~ is {{nowrap| ''V''''W''{{subsup|''V''|''W''|T}} {{=}} ''VW''2''V''{{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''''W''‖2 {{=}} sqrt(''V''''W''{{subsup|''V''|''W''|T}}) }} {{nowrap| {{=}} sqrt({{subsup|''v''|1|2}} + {{subsup|''v''|2|2}}/(log23)2 + … + {{subsup|''v''|''n''|2}}/(log2''p'')2) }}, where {{nowrap|''n'' {{=}} π(''p'')}} is the number of primes to ''p''; dividing this by sqrt(''n'') gives the TE norm of a val.

Given a val ''V'' expressed as a row vector, the corresponding row vector in weighted coordinates is {{nowrap| ''V''''W'' {{=}} ''VW'' }}, with transpose {{nowrap| {{subsup|''V''|''W''|T}} {{=}} ''WV''{{t}} }} where {{t}} denotes the transpose. The {{w|dot product}} of a weighted val with itself, or the sum of the squares of its entries, is the squared Euclidean metric of the val, {{nowrap| {{subsup|‖''V''''W''‖|2|2}} {{=}} ''V''''W''{{subsup|''V''|''W''|T}} {{=}} ''VW''2''V''{{t}} }}. Thus the Euclidean metric on the val, a measure of complexity, is {{nowrap| ‖''V''''W''‖2 {{=}} sqrt(''V''''W''{{subsup|''V''|''W''|T}}) }} {{nowrap| {{=}} sqrt({{subsup|''v''|1|2}} + {{subsup|''v''|2|2}}/(log23)2 + … + {{subsup|''v''|''n''|2}}/(log2''p'')2) }}, where {{nowrap|''n'' {{=}} π(''p'')}} is the {{w|prime-counting function}} which records the number of primes to ''p''; dividing this by sqrt(''n'') gives the TE norm of a val.

Similarly, if '''m''' is a monzo, then in weighted coordinates the monzo becomes {{nowrap| '''m'''''W'' {{=}} ''W''{{inv}}'''m''' }}, and the dot product is {{nowrap| {{subsup|'''m'''|''W''|T}}'''m'''''W'' {{=}} '''m'''{{t}}''W''-2'''m''' }}, leading to {{nowrap| sqrt({{subsup|'''m'''|''W''|T}}'''m'''''W'') {{=}} sqrt({{subsup|''m''|1|2}} + (log23)2{{subsup|''m''|2|2}} + … + (log2''p'')2{{subsup|''m''|''n''|2}}) }}; multiplying this by sqrt(''n'') gives the dual RMS norm on monzos which serves as a measure of complexity.

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 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>
+$$ W = \operatorname {diag} (1/\log_2 (Q)) $$
 Right-multiplying a row vector by this matrix scales each entry by the corresponding entry of the diagonal matrix.
-Given a val ''V'' expressed as a row vector, the corresponding row vector in weighted coordinates is {{nowrap| ''V''<sub>''W''</sub> {{=}} ''VW'' }}, with transpose {{nowrap| {{subsup|''V''|''W''|T}} {{=}} ''WV''{{t}} }} where {{t}} denotes the transpose. Then the dot product of weighted vals is {{nowrap| ''V''<sub>''W''</sub>{{subsup|''V''|''W''|T}} {{=}} ''VW''<sup>2</sup>''V''{{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>''W''</sub>‖<sub>2</sub> {{=}} sqrt(''V''<sub>''W''</sub>{{subsup|''V''|''W''|T}}) }} {{nowrap| {{=}} sqrt({{subsup|''v''|1|2}} + {{subsup|''v''|2|2}}/(log<sub>2</sub>3)<sup>2</sup> + … + {{subsup|''v''|''n''|2}}/(log<sub>2</sub>''p'')<sup>2</sup>) }}, where {{nowrap|''n'' {{=}} π(''p'')}} is the number of primes to ''p''; dividing this by sqrt(''n'') gives the TE norm of a val.
+Given a val ''V'' expressed as a row vector, the corresponding row vector in weighted coordinates is {{nowrap| ''V''<sub>''W''</sub> {{=}} ''VW'' }}, with transpose {{nowrap| {{subsup|''V''|''W''|T}} {{=}} ''WV''{{t}} }} where {{t}} denotes the transpose. The {{w|dot product}} of a weighted val with itself, or the sum of the squares of its entries, is the squared Euclidean metric of the val, {{nowrap| {{subsup|‖''V''<sub>''W''</sub>‖|2|2}} {{=}} ''V''<sub>''W''</sub>{{subsup|''V''|''W''|T}} {{=}} ''VW''<sup>2</sup>''V''{{t}} }}. Thus the Euclidean metric on the val, a measure of complexity, is {{nowrap| ‖''V''<sub>''W''</sub>‖<sub>2</sub> {{=}} sqrt(''V''<sub>''W''</sub>{{subsup|''V''|''W''|T}}) }} {{nowrap| {{=}} sqrt({{subsup|''v''|1|2}} + {{subsup|''v''|2|2}}/(log<sub>2</sub>3)<sup>2</sup> + … + {{subsup|''v''|''n''|2}}/(log<sub>2</sub>''p'')<sup>2</sup>) }}, where {{nowrap|''n'' {{=}} π(''p'')}} is the {{w|prime-counting function}} which records the number of primes to ''p''; dividing this by sqrt(''n'') gives the TE norm of a val.
 Similarly, if '''m''' is a monzo, then in weighted coordinates the monzo becomes {{nowrap| '''m'''<sub>''W''</sub> {{=}} ''W''{{inv}}'''m''' }}, and the dot product is {{nowrap| {{subsup|'''m'''|''W''|T}}'''m'''<sub>''W''</sub> {{=}} '''m'''{{t}}''W''<sup>-2</sup>'''m''' }}, leading to {{nowrap| sqrt({{subsup|'''m'''|''W''|T}}'''m'''<sub>''W''</sub>) {{=}} sqrt({{subsup|''m''|1|2}} + (log<sub>2</sub>3)<sup>2</sup>{{subsup|''m''|2|2}} + … + (log<sub>2</sub>''p'')<sup>2</sup>{{subsup|''m''|''n''|2}}) }}; multiplying this by sqrt(''n'') gives the dual RMS norm on monzos which serves as a measure of complexity.