Tenney–Euclidean tuning: Difference between revisions

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We may also obtain the TE tuning from a projection matrix. P = V+V is the orthogonal projection matrix that maps onto the space spanned by the rows of V. This space corresponds to the temperament, and so does P. However, P is independent of how the temperament is defined; it does not depend on whether the vals are linearly independent, how many of them there are, or whether [[contorsion]] has been removed ~~(or in other words, whether it has been defactored)~~. The tuning map giving the tuning of each prime number is found by multiplying by the JI map: JP where J is the JI map, which is the nearest point in the subspace corresponding to the temperament to J.

We may also obtain the TE tuning from a projection matrix. P = V+V is the orthogonal projection matrix that maps onto the space spanned by the rows of V. This space corresponds to the temperament, and so does P. However, P is independent of how the temperament is defined; it does not depend on whether the vals are linearly independent, how many of them there are, or whether [[contorsion]] has been removed. The tuning map giving the tuning of each prime number is found by multiplying by the JI map: JP where J is the JI map, which is the nearest point in the subspace corresponding to the temperament to J.

We may find the same projection matrix starting from a list of weighted monzos rather than vals. If M is a rank ''n'' matrix whose columns are weighted monzos, and I is the ''n''×''n'' identity matrix, then P = I - MM+ is the same projection matrix as V+V so long as the temperament defined by the vals is the same as the temperament defined by the monzos. Again, it is irrelevant if the monzos are independent or how many of them there are.

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However, the main value of unweighted vals is that the pseudoinverse and projection matrix have rational entries, so that the rows of the matrix are [[fractional monzos]]. The Frobenius projection matrix therefore, like the [[wedgie]], defines a completely canonical object not depending on any arbitrary definition (e.g. how Hermite normal form or LLL reduction is specifically defined) which corresponds one-to-one with temperaments, and which does not depend on whether the monzos or vals from which it is computed are [[Saturation|saturated]] ~~(i.e. [[defactored]])~~. It may be found starting either from a set of vals or a set of commas, since if Q is the projection matrix found by treating monzos in the same way as vals, P = I - Q is the same projection matrix as would be found if starting from a set of vals defining the same temperament.

However, the main value of unweighted vals is that the pseudoinverse and projection matrix have rational entries, so that the rows of the matrix are [[fractional monzos]]. The Frobenius projection matrix therefore, like the [[wedgie]], defines a completely canonical object not depending on any arbitrary definition (e.g. how Hermite normal form or LLL reduction is specifically defined) which corresponds one-to-one with temperaments, and which does not depend on whether the monzos or vals from which it is computed are [[Saturation|saturated]]. It may be found starting either from a set of vals or a set of commas, since if Q is the projection matrix found by treating monzos in the same way as vals, P = I - Q is the same projection matrix as would be found if starting from a set of vals defining the same temperament.

Spelling this out, if A is a matrix whose rows are vals, then P = A+A is a [[wikipedia: Positive-definite matrix|positive-semidefinite]] [[wikipedia: Symmetric matrix|symmetric matrix]] with rational matrix entries, which exactly specifies the regular temperament defined by the vals of A. If B is a matrix with columns of monzos which spans the subspace of interval space containing the commas, then this same matrix P is given by I - BB+.

@@ Line 40: / Line 40: @@
 <math>T = GV = JV^+V</math>
-We may also obtain the TE tuning from a projection matrix. P = V<sup>+</sup>V is the orthogonal projection matrix that maps onto the space spanned by the rows of V. This space corresponds to the temperament, and so does P. However, P is independent of how the temperament is defined; it does not depend on whether the vals are linearly independent, how many of them there are, or whether [[contorsion]] has been removed (or in other words, whether it has been defactored). The tuning map giving the tuning of each prime number is found by multiplying by the JI map: JP where J is the JI map, which is the nearest point in the subspace corresponding to the temperament to J.
+We may also obtain the TE tuning from a projection matrix. P = V<sup>+</sup>V is the orthogonal projection matrix that maps onto the space spanned by the rows of V. This space corresponds to the temperament, and so does P. However, P is independent of how the temperament is defined; it does not depend on whether the vals are linearly independent, how many of them there are, or whether [[contorsion]] has been removed. The tuning map giving the tuning of each prime number is found by multiplying by the JI map: JP where J is the JI map, which is the nearest point in the subspace corresponding to the temperament to J.
 We may find the same projection matrix starting from a list of weighted monzos rather than vals. If M is a rank ''n'' matrix whose columns are weighted monzos, and I is the ''n''×''n'' identity matrix, then P = I - MM<sup>+</sup> is the same projection matrix as V<sup>+</sup>V so long as the temperament defined by the vals is the same as the temperament defined by the monzos. Again, it is irrelevant if the monzos are independent or how many of them there are.
@@ Line 67: / Line 67: @@
 <math>T_\text{F} = G_\text{F} A = J_0 A^+A</math>
-However, the main value of unweighted vals is that the pseudoinverse and projection matrix have rational entries, so that the rows of the matrix are [[fractional monzos]]. The Frobenius projection matrix therefore, like the [[wedgie]], defines a completely canonical object not depending on any arbitrary definition (e.g. how Hermite normal form or LLL reduction is specifically defined) which corresponds one-to-one with temperaments, and which does not depend on whether the monzos or vals from which it is computed are [[Saturation|saturated]] (i.e. [[defactored]]). It may be found starting either from a set of vals or a set of commas, since if Q is the projection matrix found by treating monzos in the same way as vals, P = I - Q is the same projection matrix as would be found if starting from a set of vals defining the same temperament.
+However, the main value of unweighted vals is that the pseudoinverse and projection matrix have rational entries, so that the rows of the matrix are [[fractional monzos]]. The Frobenius projection matrix therefore, like the [[wedgie]], defines a completely canonical object not depending on any arbitrary definition (e.g. how Hermite normal form or LLL reduction is specifically defined) which corresponds one-to-one with temperaments, and which does not depend on whether the monzos or vals from which it is computed are [[Saturation|saturated]]. It may be found starting either from a set of vals or a set of commas, since if Q is the projection matrix found by treating monzos in the same way as vals, P = I - Q is the same projection matrix as would be found if starting from a set of vals defining the same temperament.
 Spelling this out, if A is a matrix whose rows are vals, then P = A<sup>+</sup>A is a [[wikipedia: Positive-definite matrix|positive-semidefinite]] [[wikipedia: Symmetric matrix|symmetric matrix]] with rational matrix entries, which exactly specifies the regular temperament defined by the vals of A. If B is a matrix with columns of monzos which spans the subspace of interval space containing the commas, then this same matrix P is given by I - BB<sup>+</sup>.