Let $m\in \mathbb N$. If there are $m$ Woodin cardinals $\delta_1<\cdots<\delta_m$ and a measurable cardinal $\kappa>\delta_m$, then every $\Pi^1_{m+1}$ subset of $\omega^\omega$ is determined. Since determinacy is preserved under complementation at a fixed level, the same hypothesis gives determinacy for every $\Sigma^1_{m+1}$ subset of $\omega^\omega$. Consequently, if for every $m\in \mathbb N$ there are $m$ Woodin cardinals below a measurable cardinal, then projective determinacy holds.