The wavelengths and directions of dimensional variations in metal ribbons quenched by Newtonian cooling from the melt are estimated for the melt-spinning and the melt-extraction processes. The main sources of dimensional variation are the vibration and distortion of the melt puddle from which the ribbon is solidified and the variation of the Newtonian coefficient of cooling along the ribbon-roller interface. Puddle vibration and distortion can be controlled by using the melt-extraction process or a melt-constrained melt-spinning process. Variation of the Newtonian coefficient of cooling along the ribbon-roller interface can be diminished by operating under vacuum or in a low-atomic-number rare-gas atmosphere and by using a smooth roller that is cleaned during each roller revolution.