Tenney–Euclidean metrics: Difference between revisions

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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 {{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.

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 PT = (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 {{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.

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 (tTPTt) where t is the image of a monzo b by t = Ab.

== Octave equivalent TE seminorm ==

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Similarly, starting from the monzo {{monzo| -1 1 0 }} for 3/2, we may multiply this by '''P''', obtaining {{val| -0.8793 0.9957 1.9526 }}, and taking the dot product of this with {{monzo| -1 1 0 }} gives 1.875 with square root 1.3693, which is T (3/2).

We can, however, map the monzos to elements of a rank ''r'' abelian group (where ''r'' is the rank of the temperament) which abstractly represents the elements of the temperament without regard to tuning, the [[abstract regular temperament]]. If b is a monzo, this mapping is given by Ab. Hence we have A{{monzo| 1 -5 3 }} maps to {{monzo| 0 0 }} for the interval associated to 250/243, and A{{monzo| -1 1 0 }} maps to {{monzo| 9 13 }} for the interval assciated to 3/2. This is the number of steps needed to get to 3/2 in 15et and 22et respectively. We now may obtain a matrix defining the temperamental norm on this abstract temperament by ''P'' = (AW2AT)-1, which is approximately

We can, however, map the monzos to elements of a rank ''r'' abelian group (where ''r'' is the rank of the temperament) which abstractly represents the elements of the temperament without regard to tuning, the [[abstract regular temperament]]. If b is a monzo, this mapping is given by Ab. Hence we have A{{monzo| 1 -5 3 }} maps to {{monzo| 0 0 }} for the interval associated to 250/243, and A{{monzo| -1 1 0 }} maps to {{monzo| 9 13 }} for the interval assciated to 3/2. This is the number of steps needed to get to 3/2 in 15et and 22et respectively. We now may obtain a matrix defining the temperamental norm on this abstract temperament by PT = (AW2AT)-1, which is approximately

[175.3265 -120.0291]

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[-120.0291 82.1730]

Using this, we find the temperamental norm of {{monzo| 9 13 }} to be sqrt ([9 13]''P''[9 13]T) ~ sqrt (1.875) ~ 1.3693, identical to the temperamental seminorm of 3/2. Note however that while '''P''' does not depend on the choice of basis vals for the temperament, ''P'' does; if we choose [{{val| 1 2 3 }}, {{val| 0 -3 -5 }}] for our basis instead, then 3/2 is represented by {{monzo| 1 -3 }} and ''P'' changes coordinates to produce the same final result of temperamental complexity.

Using this, we find the temperamental norm of {{monzo| 9 13 }} to be sqrt ([9 13]PT[9 13]T) ~ sqrt (1.875) ~ 1.3693, identical to the temperamental seminorm of 3/2. Note however that while '''P''' does not depend on the choice of basis vals for the temperament, PT does; if we choose [{{val| 1 2 3 }}, {{val| 0 -3 -5 }}] for our basis instead, then 3/2 is represented by {{monzo| 1 -3 }} and PT changes coordinates to produce the same final result of temperamental complexity.

If instead we want the OETES, we may remove the first row of [{{val| 1 2 3 }}, {{val| 0 -3 -5 }}], leaving just [{{val| 0 -3 -5 }}]. If we now call this 1×3 matrix A, then ''P'' = (AW2AT)-1 is a 1×1 matrix; in effect a scalar, with value [{{val| 0.1215588 }}]. Multiplying a monzo b by A on the left gives a 1×1 matrix Ab whose value is the number of generator steps of porcupine (of size a tempered 10/9) it takes to get to the octave class to which b belongs. Performing the multiplication and taking the square root, we conclude the OE complexity is simply proportional to this number of generator steps.

If instead we want the OETES, we may remove the first row of [{{val| 1 2 3 }}, {{val| 0 -3 -5 }}], leaving just [{{val| 0 -3 -5 }}]. If we now call this 1×3 matrix A, then PT = (AW2AT)-1 is a 1×1 matrix; in effect a scalar, with value [{{val| 0.1215588 }}]. Multiplying a monzo b by A on the left gives a 1×1 matrix Ab whose value is the number of generator steps of porcupine (of size a tempered 10/9) it takes to get to the octave class to which b belongs. Performing the multiplication and taking the square root, we conclude the OE complexity is simply proportional to this number of generator steps.

For a more substantial example we need to consider at least a rank three temperament, so let us turn to 7-limit marvel, the 7-limit temperament tempering out 225/224. The 2×4 matrix of monzos whose first row represents 2 and whose second row 225/224 is [{{monzo| 1 0 0 0 }}, {{monzo| -5 2 2 -1 }}]. If we denote log2 of the odd primes by p3, p5, p7 etc, then the monzo weighting of this matrix is M = [{{monzo| 1 0 0 0 }}, {{monzo| -5 2p3 2p5 -p7 }}], and P = I - MM+ = [{{monzo| 1 0 0 0 }}, {{monzo| 0 4(p5)2+(p7)2 -4(p3)(p5) 2(p3)(p7) }}/H, {{monzo| 0 -4(p3)(p5) 4(p3)2+(p7)2 2(p5)(p7) }}/H, {{monzo| 0 2(p3)(p7) 2(p5)(p7) 4((p3)2+(p5)2) }}/H], where H = 4(p3)2+4(p5)2+(p7)2. On the other hand, we may start from the normal val list for the temperament, which is [{{val| 1 0 0 -5 }}, {{val| 0 1 0 2 }}, {{val| 0 0 1 2 }}]. Removing the first row gives [{{val| 0 1 0 2 }}, {{val| 0 0 1 2 }}], and val weighting this gives C = [{{val| 0 1/p3 0 2/p7 }}, {{val| 0 0 1/p5 2/p7 }}]. Then P = C+C is precisely the same matrix we obtained before.

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 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 {{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.
+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<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''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<sub>T</sub>t) where t is the image of a monzo b by t = Ab.
 == Octave equivalent TE seminorm ==
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 Similarly, starting from the monzo {{monzo| -1 1 0 }} for 3/2, we may multiply this by '''P''', obtaining {{val| -0.8793 0.9957 1.9526 }}, and taking the dot product of this with {{monzo| -1 1 0 }} gives 1.875 with square root 1.3693, which is T (3/2).
-We can, however, map the monzos to elements of a rank ''r'' abelian group (where ''r'' is the rank of the temperament) which abstractly represents the elements of the temperament without regard to tuning, the [[abstract regular temperament]]. If b is a monzo, this mapping is given by Ab. Hence we have A{{monzo| 1 -5 3 }} maps to {{monzo| 0 0 }} for the interval associated to 250/243, and A{{monzo| -1 1 0 }} maps to {{monzo| 9 13 }} for the interval assciated to 3/2. This is the number of steps needed to get to 3/2 in 15et and 22et respectively. We now may obtain a matrix defining the temperamental norm on this abstract temperament by ''P'' = (AW<sup>2</sup>A<sup>T</sup>)<sup>-1</sup>, which is approximately
+We can, however, map the monzos to elements of a rank ''r'' abelian group (where ''r'' is the rank of the temperament) which abstractly represents the elements of the temperament without regard to tuning, the [[abstract regular temperament]]. If b is a monzo, this mapping is given by Ab. Hence we have A{{monzo| 1 -5 3 }} maps to {{monzo| 0 0 }} for the interval associated to 250/243, and A{{monzo| -1 1 0 }} maps to {{monzo| 9 13 }} for the interval assciated to 3/2. This is the number of steps needed to get to 3/2 in 15et and 22et respectively. We now may obtain a matrix defining the temperamental norm on this abstract temperament by P<sub>T</sub> = (AW<sup>2</sup>A<sup>T</sup>)<sup>-1</sup>, which is approximately
 [175.3265 -120.0291]
@@ Line 35: / Line 35: @@
 [-120.0291 82.1730]
-Using this, we find the temperamental norm of {{monzo| 9 13 }} to be sqrt ([9 13]''P''[9 13]<sup>T</sup>) ~ sqrt (1.875) ~ 1.3693, identical to the temperamental seminorm of 3/2. Note however that while '''P''' does not depend on the choice of basis vals for the temperament, ''P'' does; if we choose [{{val| 1 2 3 }}, {{val| 0 -3 -5 }}] for our basis instead, then 3/2 is represented by {{monzo| 1 -3 }} and ''P'' changes coordinates to produce the same final result of temperamental complexity.
+Using this, we find the temperamental norm of {{monzo| 9 13 }} to be sqrt ([9 13]P<sub>T</sub>[9 13]<sup>T</sup>) ~ sqrt (1.875) ~ 1.3693, identical to the temperamental seminorm of 3/2. Note however that while '''P''' does not depend on the choice of basis vals for the temperament, P<sub>T</sub> does; if we choose [{{val| 1 2 3 }}, {{val| 0 -3 -5 }}] for our basis instead, then 3/2 is represented by {{monzo| 1 -3 }} and P<sub>T</sub> changes coordinates to produce the same final result of temperamental complexity.
-If instead we want the OETES, we may remove the first row of [{{val| 1 2 3 }}, {{val| 0 -3 -5 }}], leaving just [{{val| 0 -3 -5 }}]. If we now call this 1×3 matrix A, then ''P'' = (AW<sup>2</sup>A<sup>T</sup>)<sup>-1</sup> is a 1×1 matrix; in effect a scalar, with value [{{val| 0.1215588 }}]. Multiplying a monzo b by A on the left gives a 1×1 matrix Ab whose value is the number of generator steps of porcupine (of size a tempered 10/9) it takes to get to the octave class to which b belongs. Performing the multiplication and taking the square root, we conclude the OE complexity is simply proportional to this number of generator steps.
+If instead we want the OETES, we may remove the first row of [{{val| 1 2 3 }}, {{val| 0 -3 -5 }}], leaving just [{{val| 0 -3 -5 }}]. If we now call this 1×3 matrix A, then P<sub>T</sub> = (AW<sup>2</sup>A<sup>T</sup>)<sup>-1</sup> is a 1×1 matrix; in effect a scalar, with value [{{val| 0.1215588 }}]. Multiplying a monzo b by A on the left gives a 1×1 matrix Ab whose value is the number of generator steps of porcupine (of size a tempered 10/9) it takes to get to the octave class to which b belongs. Performing the multiplication and taking the square root, we conclude the OE complexity is simply proportional to this number of generator steps.
 For a more substantial example we need to consider at least a rank three temperament, so let us turn to 7-limit marvel, the 7-limit temperament tempering out 225/224. The 2×4 matrix of monzos whose first row represents 2 and whose second row 225/224 is [{{monzo| 1 0 0 0 }}, {{monzo| -5 2 2 -1 }}]. If we denote log<sub>2</sub> of the odd primes by p3, p5, p7 etc, then the monzo weighting of this matrix is M = [{{monzo| 1 0 0 0 }}, {{monzo| -5 2p3 2p5 -p7 }}], and P = I - MM<sup>+</sup> = [{{monzo| 1 0 0 0 }}, {{monzo| 0 4(p5)<sup>2</sup>+(p7)<sup>2</sup> -4(p3)(p5) 2(p3)(p7) }}/H, {{monzo| 0 -4(p3)(p5) 4(p3)<sup>2</sup>+(p7)<sup>2</sup> 2(p5)(p7) }}/H, {{monzo| 0 2(p3)(p7) 2(p5)(p7) 4((p3)<sup>2</sup>+(p5)<sup>2</sup>) }}/H], where H = 4(p3)<sup>2</sup>+4(p5)<sup>2</sup>+(p7)<sup>2</sup>. On the other hand, we may start from the normal val list for the temperament, which is [{{val| 1 0 0 -5 }}, {{val| 0 1 0 2 }}, {{val| 0 0 1 2 }}]. Removing the first row gives [{{val| 0 1 0 2 }}, {{val| 0 0 1 2 }}], and val weighting this gives C = [{{val| 0 1/p3 0 2/p7 }}, {{val| 0 0 1/p5 2/p7 }}]. Then P = C<sup>+</sup>C is precisely the same matrix we obtained before.