Radical interval: Difference between revisions

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Vectors in interval space, where the coefficients are allowed to be real numbers, do not uniquely correspond to intervals, whereas monzos do. Fractional monzos do also; for each fractional monzo there is one and only one nth root of a positive rational number which corresponds to it.

== Fractional projection ~~maps~~ ==

== Fractional projection matrices ==

A square matrix P is a [[Wikipedia: Projection (linear algebra)|projection]] if P<sup>2</sup> = P. A nontrivial projection, meaning one which is neither the zero matrix nor the identity matrix, has [[Wikipedia: Eigenvalue, eigenvector and eigenspace|eigenvalues]] of both 0 and 1 and no other eigenvalues. If the rows of P represent a tuning of a regular temperament as vectors in either weighted or unweighted [[Monzos and interval space|interval space]], then a comma c of the temperament (in the appropriate coordinates) times P from the left, cP, will be the zero vector. A val of the temperament v, times P on the right, Pv, will satisfy Pv = v.

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== Tunings in terms of fractional monzos ==

If ''n'' is the number of primes π (''p'') less than or equal to ''p'', we may define a unique ''n''×''n'' projection matrix by means of a list of ''n'' - ''r'' commas c and ''r'' '''eigenmonzos''' e. An eigenmonzo is defined as a monzo which is invariant under left multiplication by a fractional monzo projection ~~map~~ P, so that uP = u where u is the eigenmonzo. The name refers to the fact that u is a left [[Wikipedia: Eigenvalue, eigenvector and eigenspace|eigenvector]] for the eigenvalue 1.

If ''n'' is the number of primes π (''p'') less than or equal to ''p'', we may define a unique ''n''×''n'' projection matrix by means of a list of ''n'' - ''r'' commas c and ''r'' '''eigenmonzos''' e. An eigenmonzo is defined as a monzo which is invariant under left multiplication by a fractional monzo projection matrix P, so that uP = u where u is the eigenmonzo. The name refers to the fact that u is a left [[Wikipedia: Eigenvalue, eigenvector and eigenspace|eigenvector]] for the eigenvalue 1.

There are various [[Wikipedia:Matrix norm|matrix norms]], and given a matrix norm, we can define an associated tuning of a regular temperament by taking the matrix with the minimum norm subject to the condition that commas of the temperament are left eigenvectors with eigenvalue 0 (that is, elements of the null space) while vals of the temperament are right eigenvectors with eigenvalue 1. Perhaps the easiest norm to work with is the Frobenius norm, which simply treats a matrix like a vector and takes the square root of the sum of squares of the coefficients of the matrix. The associated tuning is the [[Tenney-Euclidean tuning|Frobenius tuning]], which is the same as the unweighted RMS tuning which can be found using the [[pseudoinverse]]. If ''r'' is the rank of the temperament, the Frobenius norm of the Frobenius tuning is sqrt(''r''), which is minimal; this follows from the [[Wikipedia:Singular value decomposition|singular value decomposition]].

The projection matrix of the Frobenius tuning is symmetrical. Because of this, ''r'' vals spanning the subspace of the temperament can be taken to be ''r'' eigenmonzos instead. The same considerations apply to the projection matrix of TOP-RMS tuning in weighted coordinates. This projection matrix can be transformed to a matrix in unweighted coordinates by monzo weighting the columns and val weighting the rows, but the resulting matrix is no longer symmetrical. However, a val (in unweighted coordinates) can be converted to a left eigenvalue with eigenvalue 1 by doubly val weighing it; that is, val weighting it, and then val weighting it again. By finding fractional monzos which approximate ''r'' such left eigenvalues, a projection ~~map~~ with fractional monzos as rows which approximates TOP-RMS tuning may be obtained. An easy way to do this is to use an equal temperament val to approximate val weighting.

The projection matrix of the Frobenius tuning is symmetrical. Because of this, ''r'' vals spanning the subspace of the temperament can be taken to be ''r'' eigenmonzos instead. The same considerations apply to the projection matrix of TOP-RMS tuning in weighted coordinates. This projection matrix can be transformed to a matrix in unweighted coordinates by monzo weighting the columns and val weighting the rows, but the resulting matrix is no longer symmetrical. However, a val (in unweighted coordinates) can be converted to a left eigenvalue with eigenvalue 1 by doubly val weighing it; that is, val weighting it, and then val weighting it again. By finding fractional monzos which approximate ''r'' such left eigenvalues, a projection matrix with fractional monzos as rows which approximates TOP-RMS tuning may be obtained. An easy way to do this is to use an equal temperament val to approximate val weighting.

For instance, if we take {{val| 19 30 44 53 }} and divide each coordinate twice by the corresponding coordinate of {{val| 31 49 72 87 }} we obtain a fractional eigenmonzo {{val| 19/961 30/2401 11/1296 53/7569 }}, and similarly from the 31et val we have {{val| 1/31 1/49 1/72 1/87 }}. Using these as fractional eigenmonzos gives us a tuning which is already very close (less than 0.01 cents error for the primes) to TOP-RMS. Using 171et instead, the error is less than 0.0002 cents.

@@ Line 9: / Line 9: @@
 Vectors in interval space, where the coefficients are allowed to be real numbers, do not uniquely correspond to intervals, whereas monzos do. Fractional monzos do also; for each fractional monzo there is one and only one nth root of a positive rational number which corresponds to it.
-== Fractional projection maps ==
+== Fractional projection matrices ==
 A square matrix P is a [[Wikipedia: Projection (linear algebra)|projection]] if P<sup>2</sup> = P. A nontrivial projection, meaning one which is neither the zero matrix nor the identity matrix, has [[Wikipedia: Eigenvalue, eigenvector and eigenspace|eigenvalues]] of both 0 and 1 and no other eigenvalues. If the rows of P represent a tuning of a regular temperament as vectors in either weighted or unweighted [[Monzos and interval space|interval space]], then a comma c of the temperament (in the appropriate coordinates) times P from the left, cP, will be the zero vector. A val of the temperament v, times P on the right, Pv, will satisfy Pv = v.
@@ Line 17: / Line 17: @@
 == Tunings in terms of fractional monzos ==
-If ''n'' is the number of primes π (''p'') less than or equal to ''p'', we may define a unique ''n''×''n'' projection matrix by means of a list of ''n'' - ''r'' commas c and ''r'' '''eigenmonzos''' e. An eigenmonzo is defined as a monzo which is invariant under left multiplication by a fractional monzo projection map P, so that uP = u where u is the eigenmonzo. The name refers to the fact that u is a left [[Wikipedia: Eigenvalue, eigenvector and eigenspace|eigenvector]] for the eigenvalue 1.
+If ''n'' is the number of primes π (''p'') less than or equal to ''p'', we may define a unique ''n''×''n'' projection matrix by means of a list of ''n'' - ''r'' commas c and ''r'' '''eigenmonzos''' e. An eigenmonzo is defined as a monzo which is invariant under left multiplication by a fractional monzo projection matrix P, so that uP = u where u is the eigenmonzo. The name refers to the fact that u is a left [[Wikipedia: Eigenvalue, eigenvector and eigenspace|eigenvector]] for the eigenvalue 1.
 There are various [[Wikipedia:Matrix norm|matrix norms]], and given a matrix norm, we can define an associated tuning of a regular temperament by taking the matrix with the minimum norm subject to the condition that commas of the temperament are left eigenvectors with eigenvalue 0 (that is, elements of the null space) while vals of the temperament are right eigenvectors with eigenvalue 1. Perhaps the easiest norm to work with is the Frobenius norm, which simply treats a matrix like a vector and takes the square root of the sum of squares of the coefficients of the matrix. The associated tuning is the [[Tenney-Euclidean tuning|Frobenius tuning]], which is the same as the unweighted RMS tuning which can be found using the [[pseudoinverse]]. If ''r'' is the rank of the temperament, the Frobenius norm of the Frobenius tuning is sqrt(''r''), which is minimal; this follows from the [[Wikipedia:Singular value decomposition|singular value decomposition]].
-The projection matrix of the Frobenius tuning is symmetrical. Because of this, ''r'' vals spanning the subspace of the temperament can be taken to be ''r'' eigenmonzos instead. The same considerations apply to the projection matrix of TOP-RMS tuning in weighted coordinates. This projection matrix can be transformed to a matrix in unweighted coordinates by monzo weighting the columns and val weighting the rows, but the resulting matrix is no longer symmetrical. However, a val (in unweighted coordinates) can be converted to a left eigenvalue with eigenvalue 1 by doubly val weighing it; that is, val weighting it, and then val weighting it again. By finding fractional monzos which approximate ''r'' such left eigenvalues, a projection map with fractional monzos as rows which approximates TOP-RMS tuning may be obtained. An easy way to do this is to use an equal temperament val to approximate val weighting.
+The projection matrix of the Frobenius tuning is symmetrical. Because of this, ''r'' vals spanning the subspace of the temperament can be taken to be ''r'' eigenmonzos instead. The same considerations apply to the projection matrix of TOP-RMS tuning in weighted coordinates. This projection matrix can be transformed to a matrix in unweighted coordinates by monzo weighting the columns and val weighting the rows, but the resulting matrix is no longer symmetrical. However, a val (in unweighted coordinates) can be converted to a left eigenvalue with eigenvalue 1 by doubly val weighing it; that is, val weighting it, and then val weighting it again. By finding fractional monzos which approximate ''r'' such left eigenvalues, a projection matrix with fractional monzos as rows which approximates TOP-RMS tuning may be obtained. An easy way to do this is to use an equal temperament val to approximate val weighting.
 For instance, if we take {{val| 19 30 44 53 }} and divide each coordinate twice by the corresponding coordinate of {{val| 31 49 72 87 }} we obtain a fractional eigenmonzo {{val| 19/961 30/2401 11/1296 53/7569 }}, and similarly from the 31et val we have {{val| 1/31 1/49 1/72 1/87 }}. Using these as fractional eigenmonzos gives us a tuning which is already very close (less than 0.01 cents error for the primes) to TOP-RMS. Using 171et instead, the error is less than 0.0002 cents.