Tenney–Euclidean tuning: Difference between revisions
Rework (see talk). This also introduces the free parameter "s" as I used in my parametric norm (but the idea ultimately comes from Mike's formulation of the Wilson norm) |
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=== Benedetti/Wilson–Euclidean tuning === | === Benedetti/Wilson–Euclidean tuning === | ||
'''Benedetti/Wilson–Euclidean tuning''' ('''BE tuning''') adopts the Benedetti or Wilson weight in place of Tenney weight, based on the dual norm of [[Wilson norm]]<ref group="note">Technically, the Benedetti height is not a norm, and tunings that minimize the maximum Benedetti-weighted damage can be different from those based on the Wilson norm for certain subgroups. However, it is almost always more convenient to simply use the Wilson norm in these cases.</ref>. For {{nowrap|''Q'' {{=}} {{val| 2 3 5 … }}}}, the weighting matrix has the form | '''Benedetti/Wilson–Euclidean tuning''' ('''BE tuning''') adopts the Benedetti or Wilson weight in place of Tenney weight, based on the dual norm of [[Wilson norm]]<ref group="note">Technically, the [[Benedetti height]] is not a norm, and tunings that minimize the maximum Benedetti-weighted damage can be different from those based on the Wilson norm for certain subgroups. However, it is almost always more convenient to simply use the Wilson norm in these cases.</ref>. For {{nowrap|''Q'' {{=}} {{val| 2 3 5 … }}}}, the weighting matrix has the form | ||
$$ W = \operatorname{diag} (1/Q) $$ | $$ W = \operatorname{diag} (1/Q) $$ | ||