Tenney–Euclidean metrics: Difference between revisions

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== 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+ = VT(VVT)-1, where VT denotes the transpose. In terms of vals, the tuning projection matrix is P = V+V = VT(VVT)-1V = WAT(AW2AT)-1AW. P is a [[Wikipedia: Positive-definite matrix|positive semidefinite matrix]], so it defines a [[Wikipedia: Definite bilinear form|positive semidefinite bilinear form]]. In terms of weighted monzos m1 and m2, m1TPm2 defines the semidefinite form on weighted monzos, and hence b1TW-1PW-1b2 defines a semidefinite form on unweighted monzos, in terms of the matrix W-1~~WAT(AW2AT)-1AWW~~-1 = AT(AW2AT)-1A ~~= '''P'''~~. From the semidefinite form we obtain an associated [[Wikipedia: Definite quadratic form|semidefinite quadratic form]] bT'''P'''b and from this the [[Wikipedia: Norm (mathematics)|seminorm]] sqrt (bT'''P'''b).

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+ = VT(VVT)-1, where VT denotes the transpose. In terms of vals, the tuning projection matrix is P = V+V = VT(VVT)-1V = WAT(AW2AT)-1AW. P is a [[Wikipedia: Positive-definite matrix|positive semidefinite matrix]], so it defines a [[Wikipedia: Definite bilinear form|positive semidefinite bilinear form]]. In terms of weighted monzos m1 and m2, m1TPm2 defines the semidefinite form on weighted monzos, and hence b1TW-1PW-1b2 defines a semidefinite form on unweighted monzos, in terms of the matrix '''P''' = W-1PW-1 = AT(AW2AT)-1A. From the semidefinite form we obtain an associated [[Wikipedia: Definite quadratic form|semidefinite quadratic form]] bT'''P'''b and from this the [[Wikipedia: Norm (mathematics)|seminorm]] sqrt (bT'''P'''b).

It may be noted that (VVT)-1 = (AW2AT)-1 is the inverse of the [[Wikipedia: 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 [[Wikipedia: Inner product space|inner product]] on the result. Given a monzo b, Ab represents the tempered interval corresponding to b in a basis defined by the mapping A, and ''P'' = (AW2AT)-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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 == 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 [[Wikipedia: Positive-definite matrix|positive semidefinite matrix]], so it defines a [[Wikipedia: 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>Pm<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 W<sup>-1</sup>WA<sup>T</sup>(AW<sup>2</sup>A<sup>T</sup>)<sup>-1</sup>AWW<sup>-1</sup> = A<sup>T</sup>(AW<sup>2</sup>A<sup>T</sup>)<sup>-1</sup>A = '''P'''. From the semidefinite form we obtain an associated [[Wikipedia: Definite quadratic form|semidefinite quadratic form]] b<sup>T</sup>'''P'''b and from this the [[Wikipedia: Norm (mathematics)|seminorm]] sqrt (b<sup>T</sup>'''P'''b).
+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 [[Wikipedia: Positive-definite matrix|positive semidefinite matrix]], so it defines a [[Wikipedia: 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>Pm<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 [[Wikipedia: Definite quadratic form|semidefinite quadratic form]] b<sup>T</sup>'''P'''b and from this the [[Wikipedia: Norm (mathematics)|seminorm]] sqrt (b<sup>T</sup>'''P'''b).
 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 [[Wikipedia: 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 [[Wikipedia: Inner product space|inner product]] on the result. Given a monzo b, Ab represents the tempered interval corresponding to b in a basis defined by the mapping A, and ''P'' = (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.