Acoustic vortices have attracted a great deal of attention in recent years due to their numerous applications. We introduce a highly efficient method for the generation of acoustic Bessel vortices in air, using spiral-shaped active diffraction gratings, which can be operated within a broad spectral range of ultrasonic frequencies. Using a single-arm active spiral source, we achieve the simultaneous generation of vortices of different topological charges, well separated among each other along the propagation axis. With a theoretical analysis, numerical simulations, and experiments, we demonstrate some features about spiral diffraction gratings, such as the equivalence between specific diffraction orders of an m-armed spiral and a single-arm spiral, and the annihilation of prescribed diffraction orders by tuning the width to pitch ratio.