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, 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.

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 ''as mapped by a temperament'', and the octave-equivalent TE seminorms of both.

== TE norm ==

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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~~ ==

== 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 ''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''.

Denoting the temperament-defined, or temperamental, seminorm by ''T''(''x''), the subspace of interval space such that ''T''(''x'') = 0 contains a lattice consisting of the commas of the temperament, which is a sublattice of the lattice of monzos. The {{w|quotient space (linear algebra)|quotient space}} of the full vector space by the commatic subspace such that ''T''(''x'') = 0 is now a {{w|normed vector space}} with norm given by ''T'', in which the intervals of the regular temperament define a lattice. The norm ''T'' on these lattice points is the '''temperamental norm''' or '''temperamental complexity''' of the intervals of the regular temperament; in terms of the basis defined by ''A'', it is sqrt ('''t'''T''P''''T'''''t''') where '''t''' is the image of a monzo '''b''' by '''t''' = ''A'''''b'''.

Denoting the temperament-defined, or temperamental, seminorm by ''T''(''x''), the subspace of interval space such that ''T''(''x'') = 0 contains a lattice consisting of the commas of the temperament, which is a sublattice of the lattice of monzos. The {{w|quotient space (linear algebra)|quotient space}} of the full vector space by the commatic subspace such that ''T''(''x'') = 0 is now a {{w|normed vector space}} with norm given by ''T'', in which the intervals of the regular temperament define a lattice. The norm ''T'' on these lattice points is the '''TE temperamental norm''' or '''TE temperamental complexity''' of the intervals of the regular temperament; in terms of the basis defined by ''A'', it is sqrt ('''t'''T''P''''T'''''t''') where '''t''' is the image of a monzo '''b''' by '''t''' = ''A'''''b'''.

== Octave equivalent TE seminorm ==

== Octave-equivalent TE seminorm ==

Instead of starting from a matrix of vals, we may start from a matrix of monzos. If ''B'' is a matrix with columns of monzos spanning the commas of a regular temperament, then ''M'' = ''W''-1''B'' is the corresponding weighted matrix. ''Q'' = ''MM''+ is a projection matrix dual to ''P'' = ''I'' - ''Q'', where ''I'' is the identity matrix, and ''P'' is the same symmetric matrix as in the previous section. If the rows define a basis for the commas of the temperament, and are therefore linearly independent, then ''P'' = ''I'' - ''M''(''M''T''M'')-1''M''T = ''I'' - ''W''-1''B''(''B''T''W''-2''B'')-1''B''TW-1, and '''m'''T''P'''''m''' = '''b'''T''W''-1''PW''-1'''b''', or '''b'''T(''W''-2 - ''W''-2''B''(''B''T''W''-2''B'')-1''B''T''W''-2)'''b''', so that the terms inside the parenthesis define a formula for '''P''' in terms of the matrix of monzos ''B''.

To define the '''octave equivalent Tenney-Euclidean seminorm''', or '''OETES''', we simply add a column {{monzo| 1 0 0 … 0 }} representing 2 to the matrix ''B''. An alternative procedure is to find the [[Normal lists #Normal val list|normal val list]], and remove the first val from the list, corresponding to the octave or some fraction thereof, and proceed as in the previous section on temperamental complexity. This seminorm is a measure of the octave-equivalent complexity of a given ''p''-limit rational interval in terms of the ''p''-limit regular temperament given by ''A''.

To define the '''octave-equivalent Tenney-Euclidean seminorm''', or '''OETES''', we simply add a column {{monzo| 1 0 0 … 0 }} representing 2 to the matrix ''B''. An alternative procedure is to find the [[Normal lists #Normal val list|normal val list]], and remove the first val from the list, corresponding to the octave or some fraction thereof, and proceed as in the previous section on temperamental complexity. This seminorm is a measure of the octave-equivalent complexity of a given ''p''-limit rational interval in terms of the ''p''-limit regular temperament given by ''A''.

== Examples ==

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-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.
+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 ''as mapped by a temperament'', and the octave-equivalent TE seminorms of both.
 == TE norm ==
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 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 ==
+== 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 ''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''.
-Denoting the temperament-defined, or temperamental, seminorm by ''T''(''x''), the subspace of interval space such that ''T''(''x'') = 0 contains a lattice consisting of the commas of the temperament, which is a sublattice of the lattice of monzos. The {{w|quotient space (linear algebra)|quotient space}} of the full vector space by the commatic subspace such that ''T''(''x'') = 0 is now a {{w|normed vector space}} with norm given by ''T'', in which the intervals of the regular temperament define a lattice. The norm ''T'' on these lattice points is the '''temperamental norm''' or '''temperamental complexity''' of the intervals of the regular temperament; in terms of the basis defined by ''A'', it is sqrt ('''t'''<sup>T</sup>''P''<sub>''T''</sub>'''t''') where '''t''' is the image of a monzo '''b''' by '''t''' = ''A'''''b'''.
+Denoting the temperament-defined, or temperamental, seminorm by ''T''(''x''), the subspace of interval space such that ''T''(''x'') = 0 contains a lattice consisting of the commas of the temperament, which is a sublattice of the lattice of monzos. The {{w|quotient space (linear algebra)|quotient space}} of the full vector space by the commatic subspace such that ''T''(''x'') = 0 is now a {{w|normed vector space}} with norm given by ''T'', in which the intervals of the regular temperament define a lattice. The norm ''T'' on these lattice points is the '''TE temperamental norm''' or '''TE temperamental complexity''' of the intervals of the regular temperament; in terms of the basis defined by ''A'', it is sqrt ('''t'''<sup>T</sup>''P''<sub>''T''</sub>'''t''') where '''t''' is the image of a monzo '''b''' by '''t''' = ''A'''''b'''.
-== Octave equivalent TE seminorm ==
+== Octave-equivalent TE seminorm ==
 Instead of starting from a matrix of vals, we may start from a matrix of monzos. If ''B'' is a matrix with columns of monzos spanning the commas of a regular temperament, then ''M'' = ''W''<sup>-1</sup>''B'' is the corresponding weighted matrix. ''Q'' = ''MM''<sup>+</sup> is a projection matrix dual to ''P'' = ''I'' - ''Q'', where ''I'' is the identity matrix, and ''P'' is the same symmetric matrix as in the previous section. If the rows define a basis for the commas of the temperament, and are therefore linearly independent, then ''P'' = ''I'' - ''M''(''M''<sup>T</sup>''M'')<sup>-1</sup>''M''<sup>T</sup> = ''I'' - ''W''<sup>-1</sup>''B''(''B''<sup>T</sup>''W''<sup>-2</sup>''B'')<sup>-1</sup>''B''<sup>T</sup>W<sup>-1</sup>, and '''m'''<sup>T</sup>''P'''''m''' = '''b'''<sup>T</sup>''W''<sup>-1</sup>''PW''<sup>-1</sup>'''b''', or '''b'''<sup>T</sup>(''W''<sup>-2</sup> - ''W''<sup>-2</sup>''B''(''B''<sup>T</sup>''W''<sup>-2</sup>''B'')<sup>-1</sup>''B''<sup>T</sup>''W''<sup>-2</sup>)'''b''', so that the terms inside the parenthesis define a formula for '''P''' in terms of the matrix of monzos ''B''.
-To define the '''octave equivalent Tenney-Euclidean seminorm''', or '''OETES''', we simply add a column {{monzo| 1 0 0 … 0 }} representing 2 to the matrix ''B''. An alternative procedure is to find the [[Normal lists #Normal val list|normal val list]], and remove the first val from the list, corresponding to the octave or some fraction thereof, and proceed as in the previous section on temperamental complexity. This seminorm is a measure of the octave-equivalent complexity of a given ''p''-limit rational interval in terms of the ''p''-limit regular temperament given by ''A''.
+To define the '''octave-equivalent Tenney-Euclidean seminorm''', or '''OETES''', we simply add a column {{monzo| 1 0 0 … 0 }} representing 2 to the matrix ''B''. An alternative procedure is to find the [[Normal lists #Normal val list|normal val list]], and remove the first val from the list, corresponding to the octave or some fraction thereof, and proceed as in the previous section on temperamental complexity. This seminorm is a measure of the octave-equivalent complexity of a given ''p''-limit rational interval in terms of the ''p''-limit regular temperament given by ''A''.
 == Examples ==