Tenney–Euclidean tuning: Difference between revisions

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<math>\displaystyle T = GV = JV^+V</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 [[Mathematical theory of 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, {{nowrap|''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 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 [[Mathematical theory of 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, {{nowrap|''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 ''V'' is a matrix whose rows are vals, then {{nowrap|''P'' {{=}} ''V''{{+}}''V''}} is a {{w|Positive-definite matrix|positive-semidefinite}} {{w|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 {{nowrap|''I'' − ''MM''{{+}}}}.

@@ Line 76: / Line 76: @@
 <math>\displaystyle T = GV = JV^+V</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 [[Mathematical theory of 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, {{nowrap|''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 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 [[Mathematical theory of 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, {{nowrap|''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 ''V'' is a matrix whose rows are vals, then {{nowrap|''P'' {{=}} ''V''{{+}}''V''}} is a {{w|Positive-definite matrix|positive-semidefinite}} {{w|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 {{nowrap|''I'' − ''MM''{{+}}}}.