Tenney–Euclidean tuning: Difference between revisions

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== Frobenius tuning and Frobenius projection map ==

We may also do the same things starting from nonweighted vals. This leads to a different tuning, the '''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]] (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 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 nonweighted vals. This leads to a different tuning, the '''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.

The list of Frobenius generators, GF, is given by:

where J0 is the nonweighted JIP and A is the nonweighted mapping.

The Frobenius tuning map, TF, is given by:

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]] (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 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 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 55: / Line 55: @@
 == Frobenius tuning and Frobenius projection map ==
-We may also do the same things starting from nonweighted vals. This leads to a different tuning, the '''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]] (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 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 nonweighted vals. This leads to a different tuning, the '''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.
+The list of Frobenius generators, G<sub>F</sub>, is given by:
+<math>G_\text{F} = J_0 A^+</math>
+where J<sub>0</sub> is the nonweighted JIP and A is the nonweighted mapping.
+The Frobenius tuning map, T<sub>F</sub>, is given by:
+<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 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]] (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 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 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>.