For each $n \geq 1$, if there are $n$ Woodin cardinals below a measurable cardinal, then every $\Pi^1_{n+1}$ set of reals is determined. Consequently the projective pointclasses at the corresponding finite levels have the regularity consequences obtained from determinacy, including Lebesgue measurability, the Baire property, and the perfect set property for the relevant projective sets.