fft - How to add pitch-shifted version of signal to itself?

I want to write an algorithm to add pitch-shifted version of audio signal to itself (like Electro-Harmonix POG does). And I need to do it in realtime. How to implement it?

My guess is: break signal into pieces, get FFT of each piece, copy each of the frequencies in the original signal (for example, for 1 octave up shift copy 20hz to 40hz, 40hz to 80hz and so on), get the iFFT, play it back. But I don't know if I got the right idea. Will it work?