[BertoldVdb]

I worked on this long ago, the main problem with the poor sound quality is that the self-mixing process is non-linear, a good approximation is squaring the output signal.

To improve the quality you can pre-distort the output signal. Taking the square root works quite well, but expands the bandwidth significantly (infinitely, in theory). There is a lot of literature on pre-distortion with bandwidth constraints for telecom power amplifier linearisation. You will also need a linear amplifier to power the array.

The ultrasonic transducers used in this post are very narrowband, having a resonance peak of merely a few 100Hz. You can reduce the Q factor with resistive loading but the output power significantly drops. It seemed these transducers quickly start making an audible whining noise when used for continuous transmission at higher powers. I don't know what caused that, apart from this effect they seemed to hold up for essentially infinite duration.

Using a larger wideband ultrasonic transducer instead of an array of small narrowband transducers again increases the sound quality a lot. We did not find a commercial supplier of such transducer for a reasonable cost, but made some improvised custom electrostatic ones with conductive foil. There is a lot of literature on how to construct ultrasonic transducers but this is not my field.

You will not be able to play bass notes due to physics, the power required would be insane.