Okay, since you do seem interested, I will try to explain with more detail:
When the frequency of an oscillator stays constant, its wavelength stays constant.
With phase modulation synthesis, both carrier and modulator stay at the same frequency always. Their wavelengths always match, they always stay synchronized.
With frequency modulation, the frequency of the modulator is constant, but the frequency of the carrier is always changing. The wavelength of the carrier is much more complicated in this case, it is an integral related to the changing frequency over time.
So, with a perfect sine wave modulator, the "ideal" mathematical version, the wavelength of the carrier adds back up to where it started; there's equal time spent in lower frequency and higher frequency and it balances out.
With an integer approximation of a sine wave modulator, they only approximately balance. It's similar to how
Euler integration step by step of a physical trajectory will have an error compared to the ideal continuous integral.
What happens when the wavelength of the modulator does not match the wavelength of the carrier? Its phase shifts a little bit each wavelength. How much it shifts depends on how much error. The amount of error is a bit chaotic with these kinds of systems, which is what I mean when I say every different note is going to have a different speed of phase shifting.
Anyhow, if the phase shifts, the timbre of the sound changes. If the phase is shifting slowly, you get a slowly changing sound. If it shifts quickly, you get a quickly changing sound.
There do exist ways to make integer approximations where the errors will balance (and even systems that don't tend to have a few "lucky" notes that are stable), but it requires careful design with this in mind. You're not likely going to get something that does that in the SNES or FDS.
Does this explain what the problem is and how it occurs?
Thank you very much for the detailed explaination, Rainwarrior. This is very much appreciated.
So the problem does not come from FM itself, but from the rounding necessary to store a digital sine wave. Isn't is possible to make the error perfectly symetric, for example by having a sine table which is perfectly symetric ?
About the FDS; I undetsand there is less bits for the modulator, however it is still possible to have a perfecly symetric signal for the modulator despite the lower precision, isn't it? The issue is that I do not think it is possible to amplitude of the modulator dynamically as a note sustain, so this cannot approach what FM synthesis does, I guess, even without accounting the shifting from accumulating errors.