full wave rectification introduces a DC offset as well as distortion. using an analog multiplier without the 90 degree phase shift acts exactly like full wave rectification. (feeding both X and Y inputs acts as a squaring (mathematically, not analog clipping) function. when the wave is positive, the result (X*Y) is positive, and when the input goes negative, the output is also positive, since the square of a negative number is positive. with the 90 degree phase shift, you get frequency doubling without distortion or DC offset because twice per cycle, you have a positive number multiplied by a negative number, giving (with a sine wave, but works for complex waveforms as well) a distortion free frequency doubling. the box i built was so clean i actually had to switch it in and out of circuit while demonstrating it for anybody to know for sure it was working. the output of the multiplier should be followed by a differentiator to restore a flat frequency response (take an integrator circuit and swap the resistor and the cap), or if no lower frequencies are expected to be used, replace the integrator with a differentiator and you still get a 90 degree phase shift between the multiplier inputs, but the frequency response slopes up 3 db per octave instead of down.
the other low distortion method of shifting up or down an octave "back in the day" was the use of serial analog "bucket brigade" delay chips. instead of keeping a constant clock frequency, the clock rate was ramped up or down at a rate of change that either doubled or halved the number of samples exiting the delay device as went in. this is a complicated way of doing it, and actually requires two systems in tandem, with the control voltages to the two systems out of phase to eliminate the clicks and pops when the sawtooth control voltage retraces. the only noticeable noise and distortion in such a system are the delay artifacts that sound like springs going "sproinggg" in the background noise