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The general relationship between the macroscopic theory of crystal elasticity and a recently introduced microscopic formalism is established for primitive crystals. The relationship is more simple and direct than the corresponding relationship between the macroscopic theory and the Born-Huang formalism, which has to be established via sound-wave propagation. Additional conditions are derived for the Born-Huang formalism which remove an inconsistency in the microscopic theory and confirm that purely nearest-neighbor interactions must be central. The new macroscopic-microscopic relationship is applied to an illustrative study of the simple cubic structure which also shows that three of the six third-order elastic constants of the alkali halides depend on long-range interactions.