I may be very stupid for asking this, but is there a simple analogue
circuit in which I can input a frequency (currently 20kHz) and output
half the frequency?
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Not stupid at all, it's been the holy grail for pitch-shifters for
quite some time. I don't believe there's a simple analog solution and
even if you do it digitally in real time by sampling the input with
one clock, doing an A to D conversion on the slices and then doing a D
to A and outputting that analog signal at half the input sample clock
frequency implies sufficient storage to keep the buffer from
overloading.
Think about it like this: You've got a bathtub filling up at one
gallon per second and, at the same time, being drained at two quarts
per second, so eventually it'll overflow.