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Calibration precision is currently a limiting systematic in 21 cm cosmology experiments. While there are innumerable calibration approaches, most can be categorized as either `sky-based,' relying on an extremely accurate model of astronomical foreground emission, or `redundant,' requiring a precisely regular array with near-identical antenna response patterns. Both of these classes of calibration are inflexible to the realities of interferometric measurement. In practice, errors in the foreground model, antenna position offsets, and beam response inhomogeneities degrade calibration performance and contaminate the cosmological signal. Here we show that sky-based and redundant calibration can be unified into a highly general and physically motivated calibration framework based on a Bayesian statistical formalism. Our new framework includes sky and redundant calibration as special cases but can additionally support relaxing the rigid assumptions implicit in those approaches. Furthermore, we present novel calibration techniques such as redundant calibration for arrays with no redundant baselines, representing an alternative calibration method for imaging arrays such as the MWA Phase I. These new calibration approaches could mitigate systematics and reduce calibration error, thereby improving the precision of cosmological measurements.