The growl comes from humming the same note that you play, same idea as when you hear someone like Rahsaan Roland Kirk of the guy from Jethro Tull, and a lote of funk players get that rough tone from a flute.
I can just about do it on the flute, but not on the sax. The humming is going to interact with how the reed vibrates, rather than just be added to it. Not sure how you do on a TTS, you'd need a second oscillator to interact with the sample would be my guess to start getting something close