Tenney–Euclidean metrics: Difference between revisions

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The '''Tenney-Euclidean metrics''' are {{w|metric (mathematics)|metrics}} defined in Tenney-Euclidean space. These consist of the TE norm, the TE temperamental norm, and the octave-equivalent TE ~~seminorm~~.

The '''Tenney-Euclidean metrics''' are {{w|metric (mathematics)|metrics}} defined in Tenney-Euclidean space. These consist of the TE norm, which measures the [[complexity]] of an [[interval]] in [[just intonation]], the TE temperamental norm, which measures the complexity of an interval ''in a temperament'', and the octave-equivalent TE seminorms of both.

== TE norm ==

~~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 '''v''' = '''a'''''W'', with transpose '''v'''T = ''W'''''a'''T where the T denotes the transpose. Then the dot product of weighted vals is '''vv'''T = '''a'''''W''2'''a'''T, which makes the Euclidean ~~metric on vals, a measure of complexity, to be ‖''~~'v''~~'‖2 = sqrt~~ ('''vv'''~~T~~) = sqrt (''a''22 + ''a''32/(log23)2 + … + ''a''''p''2/(log2''p'')2); dividing this by sqrt (''n''), where ''n'' = π(''p'') is the number of primes to ''p'' gives the '''Tenney-Euclidean complexity'''~~, or~~ '''TE complexity'''.

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

Similarly, if '''b''' is a monzo, then in weighted coordinates the monzo becomes '''m''' = ''W''-1'''b''', and the dot product is '''m'''T'''m''' = '''b'''T''W''-2'''b''', leading to sqrt ('''m'''T'''m''') = sqrt (''b''22 + (log23)2''b''32 + … + (log2''p'')2''b''''p''2); multiplying this by sqrt (''n'') gives the dual RMS norm on monzos, a measure of complexity ~~we may call the '''Tenney-Euclidean norm''', or '''TE norm'''~~.

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 '''v''' = '''a'''''W'', with transpose '''v'''T = ''W'''''a'''T where the T denotes the transpose. Then the dot product of weighted vals is '''vv'''T = '''a'''''W''2'''a'''T, which makes the Euclidean metric on vals, a measure of complexity, to be ‖'''v'''‖2 = sqrt ('''vv'''T) = sqrt (''a''22 + ''a''32/(log23)2 + … + ''a''''p''2/(log2''p'')2); dividing this by sqrt (''n''), where ''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 '''m''' = ''W''-1'''b''', and the dot product is '''m'''T'''m''' = '''b'''T''W''-2'''b''', leading to sqrt ('''m'''T'''m''') = sqrt (''b''22 + (log23)2''b''32 + … + (log2''p'')2''b''''p''2); multiplying this by sqrt (''n'') gives the dual RMS norm on monzos which serves as a measure of complexity.

== Temperamental complexity ==

Suppose now ''A'' is a matrix whose rows are vals defining a ''p''-limit regular temperament. Then the corresponding weighted matrix is ''V'' = ''AW''. The [[Tenney-Euclidean Tuning|TE tuning]] projection matrix is then ''V''+''V'', where ''V''+ is the [[Tenney-Euclidean Tuning|pseudoinverse]]. If the rows of ''V'' (or equivalently, ''A'') are linearly independent, then we have ''V''+ = ''V''T(''VV''T)-1, where ''V''T denotes the transpose. In terms of vals, the tuning [[projection matrix]] is ~~''P'' =~~ ''V''+''V'' = ''V''T(''VV''T)-1''V'' = ''WA''T(''AW''2''A''T)-1''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, '''m'''1T''P'''''m'''2 defines the semidefinite form on weighted monzos, and hence '''b'''1T''W''-1''PW''-1'''b'''2 defines a semidefinite form on unweighted monzos, in terms of the matrix '''P''' = ''W''-1''PW''-1 = ''A''T(''AW''2''A''T)-1''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}} sqrt ('''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 ''V'' = ''AW''. The [[Tenney-Euclidean Tuning|TE tuning]] [[projection matrix]] is then ''P'' = ''V''+''V'', where ''V''+ is the [[Tenney-Euclidean Tuning|pseudoinverse]]. If the rows of ''V'' (or equivalently, ''A'') are linearly independent, then we have ''V''+ = ''V''T(''VV''T)-1, where ''V''T denotes the transpose. In terms of vals, the tuning projection matrix is ''V''+''V'' = ''V''T(''VV''T)-1''V'' = ''WA''T(''AW''2''A''T)-1''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, '''m'''1T''P'''''m'''2 defines the semidefinite form on weighted monzos, and hence '''b'''1T''W''-1''PW''-1'''b'''2 defines a semidefinite form on unweighted monzos, in terms of the matrix '''P''' = ''W''-1''PW''-1 = ''A''T(''AW''2''A''T)-1''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}} sqrt ('''b'''T'''Pb''').

It may be noted that (''VV''T)-1 = (''AW''2''A''T)-1 is the inverse of the {{w|Gramian matrix|Gram 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 ''P''''T'' = (''AW''2''A''T)-1 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 metrics''' are {{w|metric (mathematics)|metrics}} defined in Tenney-Euclidean space. These consist of the TE norm, the TE temperamental norm, and the octave-equivalent TE seminorm.
+The '''Tenney-Euclidean metrics''' are {{w|metric (mathematics)|metrics}} defined in Tenney-Euclidean space. These consist of the TE norm, which measures the [[complexity]] of an [[interval]] in [[just intonation]], the TE temperamental norm, which measures the complexity of an interval ''in a temperament'', and the octave-equivalent TE seminorms of both.
 == TE norm ==
-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 '''v''' = '''a'''''W'', with transpose '''v'''<sup>T</sup> = ''W'''''a'''<sup>T</sup> where the <sup>T</sup> denotes the transpose. Then the dot product of weighted vals is '''vv'''<sup>T</sup> = '''a'''''W''<sup>2</sup>'''a'''<sup>T</sup>, which makes the Euclidean metric on vals, a measure of complexity, to be ‖'''v'''‖<sub>2</sub> = sqrt ('''vv'''<sup>T</sup>) = sqrt (''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 sqrt (''n''), where ''n'' = π(''p'') is the number of primes to ''p'' gives the '''Tenney-Euclidean complexity''', or '''TE complexity'''.
+The '''Tenney-Euclidean norm''' ('''TE norm''') or '''Tenney-Euclidean complexity''' ('''TE complexity''') applies to vals as well as to monzos.
-Similarly, if '''b''' is a monzo, then in weighted coordinates the monzo becomes '''m''' = ''W''<sup>-1</sup>'''b''', and the dot product is '''m'''<sup>T</sup>'''m''' = '''b'''<sup>T</sup>''W''<sup>-2</sup>'''b''', leading to sqrt ('''m'''<sup>T</sup>'''m''') = sqrt (''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 sqrt (''n'') gives the dual RMS norm on monzos, a measure of complexity we may call the '''Tenney-Euclidean norm''', or '''TE norm'''.
+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 '''v''' = '''a'''''W'', with transpose '''v'''<sup>T</sup> = ''W'''''a'''<sup>T</sup> where the <sup>T</sup> denotes the transpose. Then the dot product of weighted vals is '''vv'''<sup>T</sup> = '''a'''''W''<sup>2</sup>'''a'''<sup>T</sup>, which makes the Euclidean metric on vals, a measure of complexity, to be ‖'''v'''‖<sub>2</sub> = sqrt ('''vv'''<sup>T</sup>) = sqrt (''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 sqrt (''n''), where ''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 '''m''' = ''W''<sup>-1</sup>'''b''', and the dot product is '''m'''<sup>T</sup>'''m''' = '''b'''<sup>T</sup>''W''<sup>-2</sup>'''b''', leading to sqrt ('''m'''<sup>T</sup>'''m''') = sqrt (''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 sqrt (''n'') gives the dual RMS norm on monzos which serves as a measure of complexity.
 == Temperamental complexity ==
-Suppose now ''A'' is a matrix whose rows are vals defining a ''p''-limit regular temperament. Then the corresponding weighted matrix is ''V'' = ''AW''. The [[Tenney-Euclidean Tuning|TE tuning]] projection matrix is then ''V''<sup>+</sup>''V'', where ''V''<sup>+</sup> is the [[Tenney-Euclidean Tuning|pseudoinverse]]. If the rows of ''V'' (or equivalently, ''A'') are linearly independent, then we have ''V''<sup>+</sup> = ''V''<sup>T</sup>(''VV''<sup>T</sup>)<sup>-1</sup>, where ''V''<sup>T</sup> denotes the transpose. In terms of vals, the tuning [[projection matrix]] is ''P'' = ''V''<sup>+</sup>''V'' = ''V''<sup>T</sup>(''VV''<sup>T</sup>)<sup>-1</sup>''V'' = ''WA''<sup>T</sup>(''AW''<sup>2</sup>''A''<sup>T</sup>)<sup>-1</sup>''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>, '''m'''<sub>1</sub><sup>T</sup>''P'''''m'''<sub>2</sub> defines the semidefinite form on weighted monzos, and hence '''b'''<sub>1</sub><sup>T</sup>''W''<sup>-1</sup>''PW''<sup>-1</sup>'''b'''<sub>2</sub> defines a semidefinite form on unweighted monzos, in terms of the matrix '''P''' = ''W''<sup>-1</sup>''PW''<sup>-1</sup> = ''A''<sup>T</sup>(''AW''<sup>2</sup>''A''<sup>T</sup>)<sup>-1</sup>''A''. From the semidefinite form we obtain an associated {{w|definite quadratic form|semidefinite quadratic form}} '''b'''<sup>T</sup>'''Pb''' and from this the {{w|norm (mathematics)|seminorm}} sqrt ('''b'''<sup>T</sup>'''Pb''').
+Suppose now ''A'' is a matrix whose rows are vals defining a ''p''-limit regular temperament. Then the corresponding weighted matrix is ''V'' = ''AW''. The [[Tenney-Euclidean Tuning|TE tuning]] [[projection matrix]] is then ''P'' = ''V''<sup>+</sup>''V'', where ''V''<sup>+</sup> is the [[Tenney-Euclidean Tuning|pseudoinverse]]. If the rows of ''V'' (or equivalently, ''A'') are linearly independent, then we have ''V''<sup>+</sup> = ''V''<sup>T</sup>(''VV''<sup>T</sup>)<sup>-1</sup>, where ''V''<sup>T</sup> denotes the transpose. In terms of vals, the tuning projection matrix is ''V''<sup>+</sup>''V'' = ''V''<sup>T</sup>(''VV''<sup>T</sup>)<sup>-1</sup>''V'' = ''WA''<sup>T</sup>(''AW''<sup>2</sup>''A''<sup>T</sup>)<sup>-1</sup>''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>, '''m'''<sub>1</sub><sup>T</sup>''P'''''m'''<sub>2</sub> defines the semidefinite form on weighted monzos, and hence '''b'''<sub>1</sub><sup>T</sup>''W''<sup>-1</sup>''PW''<sup>-1</sup>'''b'''<sub>2</sub> defines a semidefinite form on unweighted monzos, in terms of the matrix '''P''' = ''W''<sup>-1</sup>''PW''<sup>-1</sup> = ''A''<sup>T</sup>(''AW''<sup>2</sup>''A''<sup>T</sup>)<sup>-1</sup>''A''. From the semidefinite form we obtain an associated {{w|definite quadratic form|semidefinite quadratic form}} '''b'''<sup>T</sup>'''Pb''' and from this the {{w|norm (mathematics)|seminorm}} sqrt ('''b'''<sup>T</sup>'''Pb''').
 It may be noted that (''VV''<sup>T</sup>)<sup>-1</sup> = (''AW''<sup>2</sup>''A''<sup>T</sup>)<sup>-1</sup> is the inverse of the {{w|Gramian matrix|Gram 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 ''P''<sub>''T''</sub> = (''AW''<sup>2</sup>''A''<sup>T</sup>)<sup>-1</sup> defines a positive-definite quadratic form, and hence a norm, on the tempered interval space with basis defined by ''A''.