Tenney–Euclidean metrics: Difference between revisions
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== TE norm == | == TE norm == | ||
The ''' | The '''Tenney–Euclidean norm''' ('''TE norm''') or '''Tenney–Euclidean complexity''' ('''TE complexity''') applies to vals as well as to monzos. | ||
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. 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. Then the dot product of weighted vals is {{nowrap|'''vv'''{{t}} {{=}} '''a'''''W''<sup>2</sup>'''a'''{{t}}}}, which makes the Euclidean metric on vals, 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. 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. Then the dot product of weighted vals is {{nowrap|'''vv'''{{t}} {{=}} '''a'''''W''<sup>2</sup>'''a'''{{t}}}}, which makes the Euclidean metric on vals, 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. | ||