[squeaky-clean]

> However, digital synthesis does not require this sort of conception at all, and can be constructed without any summing of a harmonic series.

A naive sawtooth algorithm (linear rise from -1.0 to 1.0) will create aliasing and not be a true saw. You cannot filter out this aliasing unless you do this with extreme oversampling. Otherwise you need to synthesize the waveform in an alias-free (or alias minimizing) method.

There's quite a few ways to digitally synthesize a sawtooth, all with some compromise, but they're all based on sine summation theory.

One of the more common ways is to precalculate a table of single-cycle bandlimited waveforms for every 1/3 octave or so, and choose the nearest table index for a given note-frequency being played, and interpolate as needed. (It's essentially mipmapping).