This is something I've been toying with in simulation for a bit, and I finally got around to putting it together.
The schematic is a normal state-variable filter but the resistor elements of the filter cutoff have been replaced with diode bridges. Regular silicon rectifier bridges will do. They're controlled by a fourth op-amp which generates a balanced voltage to bias the diodes on and off without causing excessive CV feedthrough.
The only major disadvantage of this circuit is its output amplitude is significantly lower than the input amplitude, but that can be corrected with a fifth op-amp to amplify the signal. This is needed because the diodes saturate if the signal is too loud.
I wanted to keep the op-amp count down to four so a single op-amp package can be used. Most op-amps should work, though a TL074, TL084, etc is preferred.
Typical output voltages are 0.5V peak. Op-amp 2 should have a 10Megaohm resistor across its capacitor for stabilisation.