hm.. very broad question..
few hints nonetheless!
Take a look at what analogue or virtual analogue synths do, they too have squares and sawtooths as basic-waveform, but they generally do more than that. Common synths are substractive synths, meaning they use filters to shape the spectrum of the raw waveform. Filters are however somewhat overrated imho, ppl should focus more on the raw waveform to get interesting shapes.
Stuff you can do with the raw waveform:
sync: at each x samples, reset the phase.
if this is your wave: abcdefghijklmnopqrstuvwxyz012345
and you sync it with 8, the result is:
abcdefghabcdefghabcdefghabcdefgh
Mix: create a number of different waveforms, and mix them together, each with its own intensity-level, note that you could optionally invert each waveform and phaseshift each waveform. With the right waveforms, this mixing can act as filtering, but way more detailed compared with 'real' filters!
Octave-shift: Double the wavelength = octave down, halve the wavelength = octave up.
for octave up, this:
abcdefgh
becomes:
acegaceg
There are plenty more things to try, even FM! If you don't care for formulas and algos, then simply draw the waveforms with some tool orso.. or pick any soundeditor, load some sound, optionally edit it, take 32 samples out, optionally loop it, optionally xfade it, and use that..
Esp. with soundgeneration, the sky is the limit. Don't bother with sound-laws, just find all kinda ways to deform the wave, work with it in a visual way. Don't be affraid for a curse given out by the IRCAM-elite for doing unofficial things with a graph that could be used as sound.