Tenney–Euclidean metrics: Difference between revisions

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The '''Tenney-Euclidean metrics''' are ~~[[Wikipedia: Metric~~ (mathematics)|metrics]] defined in Tenney-Euclidean space. These consist of the TE norm, the 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, the TE temperamental norm, and the octave-equivalent TE seminorm.

== TE norm ==

Let us define the val weighting matrix W to be the ~~[[wikipedia:diagonal matrix~~|diagonal matrix]] with values 1, 1/log23, 1/log25 … 1/log2p along the diagonal. Given a val "a" expressed as a row vector, the corresponding vector in weighted coordinates is v = aW, with transpose vT = WaT where the T denotes the transpose. Then the dot product of weighted vals is vvT = aW2aT, which makes the Euclidean metric on vals, a measure of complexity, to be ||v||2 = sqrt (vvT) = sqrt (a22 + a32/(log23)2 + … + ap2/(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'''. Similarly, if b is a monzo, then in weighted coordinates the monzo becomes m = W-1b, and the dot product is mTm = bTW-2b, leading to sqrt (mTm) = sqrt (b22 + (log23)2b32 + … + (log2''p'')2b''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 = aW, with transpose vT = WaT where the T denotes the transpose. Then the dot product of weighted vals is vvT = aW2aT, which makes the Euclidean metric on vals, a measure of complexity, to be ||v||2 = sqrt (vvT) = sqrt (a22 + a32/(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'''. Similarly, if "b" is a monzo, then in weighted coordinates the monzo becomes m = W-1b, and the dot product is mTm = bTW-2b, leading to sqrt (mTm) = sqrt (b22 + (log23)2b32 + … + (log2''p'')2b''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'''.

== 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 '''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).

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 {{w|positive-definite matrix|positive semidefinite matrix}}, so it defines a {{w|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 {{w|definite quadratic form|semidefinite quadratic form}} bT'''P'''b and from this the {{w|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.

It may be noted that (VVT)-1 = (AW2AT)-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, 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.

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 ~~[[Wikipedia: Quotient~~ space (linear algebra)|quotient space]] of the full vector space by the commatic subspace such that T(x) = 0 is now a ~~[[Wikipedia: Normed vector space~~|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 (tT''P''t) where t is the image of a monzo b by t = Ab.

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 (tT''P''t) where t is the image of a monzo b by t = Ab.

== 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-1B 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(MTM)-1MT = I - W-1B(BTW-2B)-1BTW-1, and mTPm = bTW-1PW-1b, or bT(W-2 - W-2B(BTW-2B)-1BTW-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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[[Category:Tenney-weighted measures]]

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-The '''Tenney-Euclidean metrics''' are [[Wikipedia: Metric (mathematics)|metrics]] defined in Tenney-Euclidean space. These consist of the TE norm, the 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, the TE temperamental norm, and the octave-equivalent TE seminorm.
 == TE norm ==
-Let us define the val weighting matrix W to be the [[wikipedia:diagonal matrix|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 = aW, with transpose v<sup>T</sup> = Wa<sup>T</sup> where the <sup>T</sup> denotes the transpose. Then the dot product of weighted vals is vv<sup>T</sup> = aW<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'''. 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 = aW, with transpose v<sup>T</sup> = Wa<sup>T</sup> where the <sup>T</sup> denotes the transpose. Then the dot product of weighted vals is vv<sup>T</sup> = aW<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'''. 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'''.
 == 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 '''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).
+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>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 {{w|definite quadratic form|semidefinite quadratic form}} b<sup>T</sup>'''P'''b and from this the {{w|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.
+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, 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.
-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 [[Wikipedia: Quotient space (linear algebra)|quotient space]] of the full vector space by the commatic subspace such that T(x) = 0 is now a [[Wikipedia: Normed vector space|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''t) where t is the image of a monzo b by t = Ab.
+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''t) where t is the image of a monzo b by t = Ab.
 == 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>Pm = 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 ==
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 [[Category:Tenney-weighted measures]]
-{{Todo| reduce mathslang }}
+{{Todo| reduce mathslang | improve readability }}