Tenney–Euclidean metrics: Difference between revisions

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== 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 {{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''').

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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 == 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 {{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''').
+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''.