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.. and none of the three schematics would work anyway because piezo discs don't conduct DC so the op-amp inputs would 'float'. You would have to add some voltage to the non-inverting inputs.

If you are sure the background would be quiet and can tolerate false alarms if it isn't, even momentarily, use a piezo (or electret) followed by a high gain op-amp stage, it doesn't matter if the output clips. Follow that with your rectifier or a triggerable timer.

If you know the frequency you are detecting or are willing to 'tune' the detector with a trimpot, you could consider a PLL like the NE567 which could be driven directly from the pick-up and give you a simple high or low output logic signal. It would do the amplifying and filtering for you in just a few components.

Brian.