Knowledge of the diffuse-field coherence between array sensors is a basic assumption for a wide range of array processing applications. Explicit relations previously existed only for omnidirectional and first-order directional sensors, or a restricted arrangement of differential patterns. We present a closed-form formulation of the theoretical coherence function between arbitrary directionally band-limited sensors for the general cases that a) the responses of the individual sensors are known or estimated, and the coherence needs to be known for an arbitrary arrangement, and b) that no information on the sensor directionality or on array geometry exists, but calibration measurements around the array are available.