For each $n>1$, we construct explicitly a rigid weakly mixing rank-$n$ transformation with homogeneous spectrum of multiplicity $n$. The existence of such transformations was established recently by O. Ageev via Baire category arguments (a new short category proof is also given here). As an application, for any subset $M\subset\mathbb N$ containing 1, a weakly mixing transformation whose essential range for the spectral multiplicity equals $n\cdot M$ is constructed.