A refining apparatus having magnetically responsive refining elements that can be smaller than the workpiece being refined are disclosed. The refining apparatus can supply a parallel refining motion to the refining element(s) through magnetic coupling forces. The refining apparatus can supply multiple different parallel refining motions to multiple different refining elements solely through magnetic coupling forces to improve refining quality and versatility. New refining methods, refining apparatus, and refining elements disclosed. Methods of refining using frictional refining, chemical refining, tribochemical refining, and electrochemical refining and combinations thereof are disclosed. A refining chamber can be used. New methods of control are refining disclosed. The new magnetic refining methods, apparatus, and magnetically responsive refining elements can help improve yield and lower the cost of manufacture for refining of workpieces having extremely close tolerances such as semiconductors wafers. Refining fluids are preferred. Reactive refining aids are preferred. Electro-refining for adding and removing material is disclosed. A method to use business calculations combined with physical measurements to improve control is discussed. Use of business calculations to change the cost of finishing semiconductor wafers is discussed. The method can help cost of manufacture forecasting for pre-ramp-up, ramp-up, and commercial manufacture. Actively based accounting can be preferred for some applications. New methods and new apparatus for non-steady state refining control are disclosed.