Although coaxial airblast primary atomization has been studied for decades, relatively little attention has been given to three-stream designs; this is especially true for transonic self-pulsating injectors. Herein, the effects of nozzle geometry, grid resolution, modulation, and gas flow rate on the acoustics and spray character within an industrial scale system were investigated computationally using axisymmetric (AS) and three-dimensional (3D) models. Metrics included stream pressure pulsations, spray lift-off, spray angle, and primary droplet length scale, along with the spectral alignment among these parameters. Strong interactions existed between geometry and inner gas (IG) feed rate. Additionally, inner nozzle retraction and outer stream meeting angle were intimately coupled. Particular attention was given to develop correlations for various metrics versus retraction; one such example is that injector flow capacity was found to be linearly proportional to retraction. Higher IG flows were found to widen sprays, bringing the spray in closer to the nozzle face, and reducing droplet length scales. Substantial forced modulation of the IG at its dominant tone did not strongly affect many metrics. Incompressible 3D results were similar to some of the AS results, which affirmed the predictive power by running AS simulations as surrogates. Lastly, normalized droplet size versus normalized distance from the injector followed a strikingly similar trend as that found from prior two-fluid air-slurry calibration work.