An improved alignable pulley or sheave assembly includes: (a) a shaft having a rotational axis and an exterior surface which has a portion that is spherically shaped, (b) a pulley having a perimeter and a central opening with an interior surface that is configured so as to allow the pulley to be mounted on shaft's spherical shaped portion, and (c) a means for locking the shaft and pulley together such that they rotate together while also enabling the pulley to tilt from side to side with respect to a plane that lies perpendicular to the shaft's rotational axis.