Mechanical polishing of silver offers the greatest amount of control as compared to other cleaning methods, such as chemical dips or electrolytic setups. Lab grade (chemically pure, homogenous particle size) calcium carbonate mixed into a paste with deionized water and applied with soft cotton flannel provides gentle polishing for most* silver. 

*Care must be taken when mechanically polishing silverplate or gilt silver, as these plating layers can be very thin, sometimes only a few microns in thickness, and can be abraded away easily.