The answer to the original question harks back to the means of
producing "analog" sound on the "digital" speaker of a PC, and it seems
many don't remember how effective this was.

  It is however processor-intensive.  It consists of using PWM to
produce the analogue "levels", switching at a rate too high to be
audible.  You don't need a DAC and you can generally get away with no
filtering (the PC after all, managed to...).

  You don't need to have 256-level (8-bit) accuracy for chime tones
either, the embedded chips in the knick-knacks almost certainly don't.
The residual harmonics of the square waves give a rich "reedy" tone.

  Consider using 16 levels, with say, a 100kHz clocking rate.  That is,
every 10µs, you increment your (4-bit) counter and compare it to the
desired "analog" level, setting the output bit if it is greater.

  No, actually you use my favourite "phase accumulator" algorithm - you
add the "analog" value (0 to 15) to a register, and set the output if
bit 4 is set.  Easy enough?

  Key words include "isosynchronous code".  You do the above at the
regular 10µs interval, interspersed in the code which actually
synthesises the waveshape, which itself is probably a Phase Accumulator
and wave-shape table (lookup or rule-based).
-- 
  Cheers,
        Paul B.