An Alpern tower independent of a given partition

Given a measure-preserving transformation $T$ of a probability space $(X, \mathcal B, \mu)$ and a finite measurable partition $P$ of $X$, we show how to construct an Alpern tower of any height whose base is independent of the partition $P$. That is, given $N \in \N$, there exists a Rohlin tower of height $N$, with base $B$ and error set $E$, so that $B$ is independent of $P$, and $T(E) \subset B$.