Tenney–Euclidean tuning: Difference between revisions
→Motivation: add the most important motivations to a new paragraph at the top |
Note on "Benedetti" tuning |
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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 height]]. 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 | ||
$$ W = \operatorname{diag} (1/Q) $$ | $$ W = \operatorname{diag} (1/Q) $$ | ||
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This sends monzos for 50/49, 64/63 etc. to the unison monzo, and vals for 10et, 12et, and 22et to themselves. | This sends monzos for 50/49, 64/63 etc. to the unison monzo, and vals for 10et, 12et, and 22et to themselves. | ||
== Notes == | |||
<references group="note"/> | |||
[[Category:Terms]] | [[Category:Terms]] | ||
[[Category:Math]] | [[Category:Math]] | ||
[[Category:Regular temperament tuning]] | [[Category:Regular temperament tuning]] | ||