Quasi-harmonic circular membranes with a central disc or an isolated ring of different density

We first extend the known class of configurations for an ideal circular membrane with a centrally symmetric disc of different density producing quasi-harmonic vibrations. These configurations minimize the global deviation from harmony of the first four overtones. One of these configurations fits the tabla membrane characteristics. Among the new configurations one corresponds to a membrane having a very high density on a narrow ring next to its outer edge. Improvement in harmony then essentially results from a decrease in the number of overtones. We also determine the most harmonic configurations of a circular vibrating membrane having a different density on a centrally symmetric ring not adjacent to its outer edge. In this set one is remarkable because its first five eigenfrequencies have no degeneracy and are almost exactly harmonic.