I've been messing around with software synth via the PCM trick and I feel like there's sort of a fundamental problem that's detracting a lot from the quality of what comes out the back end, needlessly so, not to mention making it uncomfortable to the listener's ear.
As far as I can tell, you're feeding the 5512.5Hz signal into the audio version of a "nearest neighbor" sampler to upsample it to the host OS' audio hardware sample rate, which is typically gonna be either 44100Hz or 48000Hz. That means repeating the sample values as many times as necessary to fill the gaps. In this case it'd repeat the same value around 8 times. This of course produces stairsteps on the 44100Hz+ "curve".
The issue with that is that modern audio hardware has really good fidelity and it's going to reproduce those sharp stairsteps very faithfully, effectively overlaying a constant 5512.5Hz triangle wave on the original wave that pulsates its amplitude based on how big the deltas between samples are in the original wave.
Like, if the original wave was a 262Hz middle-C sine wave tone, you'd get a 5512.5Hz buzz overlaid that gets strongest each time the wave crosses 0, where the rate of change is maximal. It crosses 0 twice per cycle, so you're getting an audible crackling buzz at 524Hz.
That's not what would have happened on old analog audio output circuitry. There would be ramping time between levels, some sort of acceleration, deceleration, and maybe overshoot too, as the line settled at its new level. Basically the audio version of an LCD pixel's "response time" or "gray-to-gray time". The overlaid triangle wave would be closer to a sine wave, more of a tone than a buzz, still present and still distorting the intended wave a bit, but much less harsh to the ear.
I dunno what options you have to change the upsampling. If you're doing it by hand, your options are open, but if you're relying on hardware, you might need to experiment. You probably don't want to change it over to using full interpolation, because that'll make it hard for people to do deliberate buzzing.
If it were me, I'd try doing a manual nearest-neighbor upsample to 22100Hz (edit: or maybe just 11050Hz) to retain the low-resolution stairsteps you'd expect the low clock rates to produce on retro hardware, but I'd let the host OS/hardware do interpolation to upsample it the rest of the way to the true output rate of 44100Hz or 48000Hz, which will simulate the ramping up/down time. This'd take the edge off, for the sake of the listener's ear.