  The fermion mass matrix, in addition to having eigenvalues (masses) which
run, also changes its orientation (rotates) with changing energy scales. This
means that its eigenstates at one scale will no longer be eigenstates at
another scale, leading to effects where fermions of different flavours can
``transmute'' into one another. In this paper, the implications of a rotating
mass matrix are analysed and possible transmuation effects are investigated
both in the Standard Model (SM) and in the so-called Dualized Standard Model
(DSM) that we advocate, arriving at the conclusion that some transmutational
decays such as $\psi \longrightarrow \mu \tau$, $\Upsilon \longrightarrow \mu
\tau$ or $\pi^0 \longrightarrow e \mu$ may be within experimental range, if not
immediately, then in the near future.
