Tenney–Euclidean tuning: Difference between revisions
Refine the definition (spell out the weighting matrix), and improve section titles |
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Another way to enforce the pure octave is by adding the constraint before the optimization process. This is the '''CTE tuning'''. The result, under the constraint of pure octaves, remains TE optimal. | Another way to enforce the pure octave is by adding the constraint before the optimization process. This is the '''CTE tuning'''. The result, under the constraint of pure octaves, remains TE optimal. | ||
== Frobenius tuning and Frobenius projection matrix == | == Otherwise normed tunings == | ||
=== Frobenius tuning and Frobenius projection matrix === | |||
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. | 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. | ||
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If the vals defining A are linearly independent, then P = A<sup>T</sup>(AA<sup>T</sup>)<sup>-1</sup>A. If the columns of B are independent, then we likewise have P = I - B(B<sup>T</sup>B)<sup>-1</sup>B<sup>T</sup>. | If the vals defining A are linearly independent, then P = A<sup>T</sup>(AA<sup>T</sup>)<sup>-1</sup>A. If the columns of B are independent, then we likewise have P = I - B(B<sup>T</sup>B)<sup>-1</sup>B<sup>T</sup>. | ||
=== Benedetti-Euclidean tuning === | |||
'''Benedetti-Euclidean tuning''' ('''BE tuning''') adopts the Benedetti weight in place of Tenney weight, based on the dual norm of [[Benedetti height]]. For Q = {{val| 2 3 5 … }}, the weighting matrix has the form | |||
<math>W = \operatorname {diag} (1/Q)</math> | |||
== Examples == | == Examples == | ||