Raising the indices of the Hodge operator

Question

Let $(M,g)$ be a Riemannian $n$-manifold; $\mu$ is the volume form associated with $g$, and $\star$ is the Hodge operator induced by $\mu$. Given a $k$-form $\beta$ on $M$, the $(n-k)$-multivector $B$ (alternating contravariant tensor on $M$) is obtained by raising the indices of $\star \beta$ with $g$. Using components one has $$\beta_{i_1...i_k}= \frac{1}{(n-k)!}\mu_{i_1...i_k \,i_{k+1} ... i_n}B^{i_{k+1} ...i_n} .$$ I was wondering if the operation $\beta\mapsto B$ has a specific name and what the most common notation is. I would guess $(\star\beta)^{\sharp}$, but maybe there is something more compact...