Tenney–Euclidean metrics: Difference between revisions

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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/log23, 1/log25 … 1/log2''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''2'''a'''{{t}}}}, which makes the Euclidean metric on vals, a measure of complexity, to be {{nowrap|‖'''v'''‖2 {{=}} √('''vv'''{{t}})}} {{nowrap|{{=}} √(''a''~~~~2~~~~2~~~~ + ''a''~~~~3~~~~2~~~~/(log23)2 + … + ''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. 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''2'''a'''{{t}}}}, which makes the Euclidean metric on vals, 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''') {{=}} √(''b''~~~~2~~~~2~~~~ + (log23)2''b''~~~~3~~~~2~~~~ + … + (log2''p'')2''b''~~~~''p''~~~~2~~~~)}}; multiplying this by √(''n'') gives the dual RMS norm on monzos which serves as a measure of complexity.

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.

== TE temperamental norm ==

Suppose now ''A'' is a matrix whose rows are vals defining a ''p''-limit regular temperament. Then the corresponding weighted matrix is {{nowrap|''V'' {{=}} ''AW''}}. The [[Tenney–Euclidean Tuning|TE tuning]] [[projection matrix]] is then {{nowrap|''P'' {{=}} ''V''{{mpp}}''V''}}, where ''V''{{mpp}} denotes the {{w|Moore–Penrose pseudoinverse}} of ''V''. If the rows of ''V'' (or equivalently, ''A'') are linearly independent, then we have {{nowrap|''V''{{mpp}} {{=}} ''V''{{t}}(''VV''{{t}}){{inv}}}}. In terms of vals, the tuning projection matrix is {{nowrap|''V''{{mpp}}''V'' {{=}} ''V''{{t}}(''VV''{{t}}){{inv}}''V''}} {{nowrap|{{=}} ''WA''{{t}}(''AW''2''A''{{t}}){{inv}}''AW''}}. ''P'' is a {{w|positive-definite matrix|positive semidefinite matrix}}, so it defines a {{w|definite bilinear form|positive semidefinite bilinear form}}. In terms of weighted monzos '''m'''1 and '''m'''2, {{subsup|'''m'''|1|T}}''P'''''m'''2 defines the semidefinite form on weighted monzos, and hence '''b'''~~~~1~~{{t~~}}''W''{{inv}}''PW''{{inv}}'''b'''2 defines a semidefinite form on unweighted monzos, in terms of the matrix {{nowrap|'''P''' {{=}} ''W''{{inv}}''PW''{{inv}}}} {{nowrap|{{=}} ''A''{{t}}(''AW''2''A''{{t}}){{inv}}''A''}}. From the semidefinite form we obtain an associated {{w|definite quadratic form|semidefinite quadratic form}} '''b'''{{t}}'''Pb''' and from this the {{w|norm (mathematics)|seminorm}} √('''b'''{{t}}'''Pb''').

Suppose now ''A'' is a matrix whose rows are vals defining a ''p''-limit regular temperament. Then the corresponding weighted matrix is {{nowrap|''V'' {{=}} ''AW''}}. The [[Tenney–Euclidean Tuning|TE tuning]] [[projection matrix]] is then {{nowrap|''P'' {{=}} ''V''{{mpp}}''V''}}, where ''V''{{mpp}} denotes the {{w|Moore–Penrose pseudoinverse}} of ''V''. If the rows of ''V'' (or equivalently, ''A'') are linearly independent, then we have {{nowrap|''V''{{mpp}} {{=}} ''V''{{t}}(''VV''{{t}}){{inv}}}}. In terms of vals, the tuning projection matrix is {{nowrap|''V''{{mpp}}''V'' {{=}} ''V''{{t}}(''VV''{{t}}){{inv}}''V''}} {{nowrap|{{=}} ''WA''{{t}}(''AW''2''A''{{t}}){{inv}}''AW''}}. ''P'' is a {{w|positive-definite matrix|positive semidefinite matrix}}, so it defines a {{w|definite bilinear form|positive semidefinite bilinear form}}. In terms of weighted monzos '''m'''1 and '''m'''2, {{subsup|'''m'''|1|T}}''P'''''m'''2 defines the semidefinite form on weighted monzos, and hence {{subsup|'''b'''|1|T}}''W''{{inv}}''PW''{{inv}}'''b'''2 defines a semidefinite form on unweighted monzos, in terms of the matrix {{nowrap|'''P''' {{=}} ''W''{{inv}}''PW''{{inv}}}} {{nowrap|{{=}} ''A''{{t}}(''AW''2''A''{{t}}){{inv}}''A''}}. From the semidefinite form we obtain an associated {{w|definite quadratic form|semidefinite quadratic form}} '''b'''{{t}}'''Pb''' and from this the {{w|norm (mathematics)|seminorm}} √('''b'''{{t}}'''Pb''').

It may be noted that {{nowrap|(''VV''{{t}}){{inv}} {{=}} (''AW''2''A''{{t}}){{inv}}}} is the inverse of the {{w|Gramian matrix}} used to compute [[TE complexity]], and hence is the corresponding Gram matrix for the dual space. Hence '''P''' represents a change of basis defined by the mapping given by the vals combined with an {{w|inner product space|inner product}} on the result. Given a monzo '''b''', ''A'''''b''' represents the tempered interval corresponding to '''b''' in a basis defined by the mapping ''A'', and {{nowrap|''P''''T'' {{=}} (''AW''2''A''{{t}}){{inv}}}} defines a positive-definite quadratic form, and hence a norm, on the tempered interval space with basis defined by ''A''.

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 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|{{=}} √(''a''<sub>2</sub><sup>2</sup> + ''a''<sub>3</sub><sup>2</sup>/(log<sub>2</sub>3)<sup>2</sup> + … + ''a''<sub>''p''</sub><sup>2</sup>/(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.
-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''') {{=}} √(''b''<sub>2</sub><sup>2</sup> + (log<sub>2</sub>3)<sup>2</sup>''b''<sub>3</sub><sup>2</sup> + … + (log<sub>2</sub>''p'')<sup>2</sup>''b''<sub>''p''</sub><sup>2</sup>)}}; multiplying this by √(''n'') gives the dual RMS norm on monzos which serves as a measure of complexity.
+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.
 == TE temperamental norm ==
-Suppose now ''A'' is a matrix whose rows are vals defining a ''p''-limit regular temperament. Then the corresponding weighted matrix is {{nowrap|''V'' {{=}} ''AW''}}. The [[Tenney–Euclidean Tuning|TE tuning]] [[projection matrix]] is then {{nowrap|''P'' {{=}} ''V''{{mpp}}''V''}}, where ''V''{{mpp}} denotes the {{w|Moore–Penrose pseudoinverse}} of ''V''. If the rows of ''V'' (or equivalently, ''A'') are linearly independent, then we have {{nowrap|''V''{{mpp}} {{=}} ''V''{{t}}(''VV''{{t}}){{inv}}}}. In terms of vals, the tuning projection matrix is {{nowrap|''V''{{mpp}}''V'' {{=}} ''V''{{t}}(''VV''{{t}}){{inv}}''V''}} {{nowrap|{{=}} ''WA''{{t}}(''AW''<sup>2</sup>''A''{{t}}){{inv}}''AW''}}. ''P'' is a {{w|positive-definite matrix|positive semidefinite matrix}}, so it defines a {{w|definite bilinear form|positive semidefinite bilinear form}}. In terms of weighted monzos '''m'''<sub>1</sub> and '''m'''<sub>2</sub>, {{subsup|'''m'''|1|T}}''P'''''m'''<sub>2</sub> defines the semidefinite form on weighted monzos, and hence '''b'''<sub>1</sub>{{t}}''W''{{inv}}''PW''{{inv}}'''b'''<sub>2</sub> defines a semidefinite form on unweighted monzos, in terms of the matrix {{nowrap|'''P''' {{=}} ''W''{{inv}}''PW''{{inv}}}} {{nowrap|{{=}} ''A''{{t}}(''AW''<sup>2</sup>''A''{{t}}){{inv}}''A''}}. From the semidefinite form we obtain an associated {{w|definite quadratic form|semidefinite quadratic form}} '''b'''{{t}}'''Pb''' and from this the {{w|norm (mathematics)|seminorm}} √('''b'''{{t}}'''Pb''').
+Suppose now ''A'' is a matrix whose rows are vals defining a ''p''-limit regular temperament. Then the corresponding weighted matrix is {{nowrap|''V'' {{=}} ''AW''}}. The [[Tenney–Euclidean Tuning|TE tuning]] [[projection matrix]] is then {{nowrap|''P'' {{=}} ''V''{{mpp}}''V''}}, where ''V''{{mpp}} denotes the {{w|Moore–Penrose pseudoinverse}} of ''V''. If the rows of ''V'' (or equivalently, ''A'') are linearly independent, then we have {{nowrap|''V''{{mpp}} {{=}} ''V''{{t}}(''VV''{{t}}){{inv}}}}. In terms of vals, the tuning projection matrix is {{nowrap|''V''{{mpp}}''V'' {{=}} ''V''{{t}}(''VV''{{t}}){{inv}}''V''}} {{nowrap|{{=}} ''WA''{{t}}(''AW''<sup>2</sup>''A''{{t}}){{inv}}''AW''}}. ''P'' is a {{w|positive-definite matrix|positive semidefinite matrix}}, so it defines a {{w|definite bilinear form|positive semidefinite bilinear form}}. In terms of weighted monzos '''m'''<sub>1</sub> and '''m'''<sub>2</sub>, {{subsup|'''m'''|1|T}}''P'''''m'''<sub>2</sub> defines the semidefinite form on weighted monzos, and hence {{subsup|'''b'''|1|T}}''W''{{inv}}''PW''{{inv}}'''b'''<sub>2</sub> defines a semidefinite form on unweighted monzos, in terms of the matrix {{nowrap|'''P''' {{=}} ''W''{{inv}}''PW''{{inv}}}} {{nowrap|{{=}} ''A''{{t}}(''AW''<sup>2</sup>''A''{{t}}){{inv}}''A''}}. From the semidefinite form we obtain an associated {{w|definite quadratic form|semidefinite quadratic form}} '''b'''{{t}}'''Pb''' and from this the {{w|norm (mathematics)|seminorm}} √('''b'''{{t}}'''Pb''').
 It may be noted that {{nowrap|(''VV''{{t}}){{inv}} {{=}} (''AW''<sup>2</sup>''A''{{t}}){{inv}}}} is the inverse of the {{w|Gramian matrix}} used to compute [[TE complexity]], and hence is the corresponding Gram matrix for the dual space. Hence '''P''' represents a change of basis defined by the mapping given by the vals combined with an {{w|inner product space|inner product}} on the result. Given a monzo '''b''', ''A'''''b''' represents the tempered interval corresponding to '''b''' in a basis defined by the mapping ''A'', and {{nowrap|''P''<sub>''T''</sub> {{=}} (''AW''<sup>2</sup>''A''{{t}}){{inv}}}} defines a positive-definite quadratic form, and hence a norm, on the tempered interval space with basis defined by ''A''.