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You can make two narrow band filters at two frequencies. You pulse one and then the other. This simulates the response of striking a bell or other resonant physical object. To get the right overtone quality you may have to follow the filter with some nonlinear circuit. Or, have several pulsed filters each on an overtone of the lowest frequency one.
I think the easiest way is to buy a IC for that. At Conrad you can buy this IC and oly need to plugin a speaker to it. This IC costs only 2€. I think this is the easiest way to create a ding-dong for doors.