I think I can explain the trick even easier and hopefully that clears up your confusion. Simply store a complex number s which represents the current output of

I think I can explain the trick even easier and hopefully that clears up your confusion. Simply store a complex number s which represents the current output of the synth (cos on the real part, sin on the imag). At each time update (the sample rate), multiply s by a pre-computed constant c (which is also complex valued). You just need for c^(i/sample_rate) to equal e^(i*freq/2pi). It does not matter that freq/2pi doesn't line up with the sample rate. The trick is wholly dependent on just keeping track of the state s and reliably multiplying it at a known update frequency.