Bipolar wave current, with both positive and negative current portions, is used to electrodeposit a nanocrystalline grain size deposit. Polarity Ratio is the ratio of the absolute value of the time integrated amplitude of negative polarity current and positive polarity current. Grain size can be precisely controlled in alloys of two or more chemical components, at least one of which is a metal, and at least one of which is most electro-active. Typically, although not always, the amount of the more electro-active material is preferentially lessened in the deposit during times of negative current. The deposit also exhibits superior macroscopic quality, being relatively crack and void free. Parameters of current density, duration of pulse portions, and composition of the bath are determined with reference to constitutive relations showing grain size as a function of deposit composition, and deposit composition as a function of Polarity Ratio, or, perhaps, a single relation showing grain size as a function of Polarity ratio.