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Spherical microphone arrays provide a flexible solution to obtaining higher-order directivity patterns, which are useful in audio recording and reproduction. A general systematic approach to the design of directivity patterns for spherical microphone arrays is introduced in this paper. The directivity patterns are obtained by optimizing a cost function which is a convex combination of a front-back energy ratio and a smoothness term. Most of the standard directivity patterns i.e. omnidirectional, cardioid, subcardioid, hypercardioid and supercardioid are particular solutions of this optimization problem with specific values of two free parameters: the angle of the frontal sector, and the convex combination factor. By varying these two parameters, more general solutions of practical use are obtained.