Tenney–Euclidean tuning: Difference between revisions

Line 35:

Applying the weighted ~~vals~~ to the generators ~~gives~~

Applying the weighted val list to the generators, The TE tuning map is given by

~~where T is the TE tuning map.~~

We may also obtain the TE tuning from a projection map. P = V+V is the orthogonal projection map 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 map starting from a list of weighted monzos rather than vals. If M is a rank ''n'' matrix whose ~~rows~~ are weighted monzos, and I is the ''n''×''n'' identity matrix, then P = I - M+M is the same projection map 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.

We may find the same projection map 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 map 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.

== Pure octaves TE tuning ==

Line 51:

Line 49:

== Frobenius projection map ==

We may also do the same things starting from unweighted vals. This leads to a different tuning, the [[Fractional monzos|Frobenius tuning]], which is perfectly functional but has less theoretical justification than TE tuning. However, if greater weight needs to be attached to the larger primes than TE tuning attaches, Frobenius tuning may be preferred; people who feel that larger primes require more tuning care than smaller ones may well prefer it. However the main value of unweighted vals is that the pseudoinverse and projection map have rational entries, so that the rows of the matrix are [[fractional monzos]]. The Frobenius projection map therefore, like the [[wedgie]], defines a completely canonical object not depending on any arbitrary definition (eg how Hermite normal form or LLL reduction is specifically defined) which corresponds 1-1 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 map found by treating monzos in the same way as vals, P = I - Q is the same projection map as would be found if starting from a set of vals defining the same temperament.

We may also do the same things starting from unweighted vals. This leads to a different tuning, the [[Fractional monzos|Frobenius tuning]], which is perfectly functional but has less theoretical justification than TE tuning. However, if greater weight needs to be attached to the larger primes than TE tuning attaches, Frobenius tuning may be preferred; people who feel that larger primes require more tuning care than smaller ones may well prefer it. However the main value of unweighted vals is that the pseudoinverse and projection map have rational entries, so that the rows of the matrix are [[fractional monzos]]. The Frobenius projection map 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 map found by treating monzos in the same way as vals, P = I - Q is the same projection map as would be found if starting from a set of vals defining the same temperament.

Spelling this out, if V is a matrix whose rows are vals, then P = V+V 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 V. If M is a matrix with ~~rows~~ of monzos which spans the subspace of interval space containing the commas, then this same matrix P is given by I - M+M.

Spelling this out, if V is a matrix whose rows are vals, then P = V+V 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 V. If M 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 - MM+.

If the vals defining V are linearly independent, then P = VT(VVT)-1V. If the ~~rows~~ of M are independent, then we likewise have P = I - MT~~(MM~~T)-1M.

If the vals defining V are linearly independent, then P = VT(VVT)-1V. If the columns of M are independent, then we likewise have P = I - M(MTM)-1MT.

== Examples ==

@@ Line 35: / Line 35: @@
 <math>\vec{g} = JV^+</math>
-Applying the weighted vals to the generators gives
+Applying the weighted val list to the generators, The TE tuning map is given by
 <math>T = \vec{g}V = JV^+V</math>
-where T is the TE tuning map.
 We may also obtain the TE tuning from a projection map. P = V<sup>+</sup>V is the orthogonal projection map 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 map starting from a list of weighted monzos rather than vals. If M is a rank ''n'' matrix whose rows are weighted monzos, and I is the ''n''×''n'' identity matrix, then P = I - M<sup>+</sup>M is the same projection map 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.
+We may find the same projection map 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 map 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.
 == Pure octaves TE tuning ==
@@ Line 51: / Line 49: @@
 == Frobenius projection map ==
-We may also do the same things starting from unweighted vals. This leads to a different tuning, the [[Fractional monzos|Frobenius tuning]], which is perfectly functional but has less theoretical justification than TE tuning. However, if greater weight needs to be attached to the larger primes than TE tuning attaches, Frobenius tuning may be preferred; people who feel that larger primes require more tuning care than smaller ones may well prefer it. However the main value of unweighted vals is that the pseudoinverse and projection map have rational entries, so that the rows of the matrix are [[fractional monzos]]. The Frobenius projection map therefore, like the [[wedgie]], defines a completely canonical object not depending on any arbitrary definition (eg how Hermite normal form or LLL reduction is specifically defined) which corresponds 1-1 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 map found by treating monzos in the same way as vals, P = I - Q is the same projection map as would be found if starting from a set of vals defining the same temperament.
+We may also do the same things starting from unweighted vals. This leads to a different tuning, the [[Fractional monzos|Frobenius tuning]], which is perfectly functional but has less theoretical justification than TE tuning. However, if greater weight needs to be attached to the larger primes than TE tuning attaches, Frobenius tuning may be preferred; people who feel that larger primes require more tuning care than smaller ones may well prefer it. However the main value of unweighted vals is that the pseudoinverse and projection map have rational entries, so that the rows of the matrix are [[fractional monzos]]. The Frobenius projection map 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 map found by treating monzos in the same way as vals, P = I - Q is the same projection map as would be found if starting from a set of vals defining the same temperament.
-Spelling this out, if V is a matrix whose rows are vals, then P = V<sup>+</sup>V 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 V. If M is a matrix with rows of monzos which spans the subspace of interval space containing the commas, then this same matrix P is given by I - M<sup>+</sup>M.
+Spelling this out, if V is a matrix whose rows are vals, then P = V<sup>+</sup>V 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 V. If M 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 - MM<sup>+</sup>.
-If the vals defining V are linearly independent, then P = V<sup>T</sup>(VV<sup>T</sup>)<sup>-1</sup>V. If the rows of M are independent, then we likewise have P = I - M<sup>T</sup>(MM<sup>T</sup>)<sup>-1</sup>M.
+If the vals defining V are linearly independent, then P = V<sup>T</sup>(VV<sup>T</sup>)<sup>-1</sup>V. If the columns of M are independent, then we likewise have P = I - M(M<sup>T</sup>M)<sup>-1</sup>M<sup>T</sup>.
 == Examples ==