I bridge the current pricing kernel framework with the early partial-moment pricing models of the beta framework, thereby reconciling and clarifying these bodies of literature. I argue for the inclusion of powers of min and max functions within a generalized kernel, and form a generalized beta model. Polynomial kernels and the kernel underpinning the partial-moment analogue of the Sharpe-Lintner CAPM are nested. I derive the partial-moment analogue to the Black CAPM, thus completing a theoretical parallelism, and compare the kernel-implied and canonical risk-neutral probabilities. A new model involving both lower and upper partial-moments, accommodating various kernel shapes present in the literature, is developed in the context of preference regularity conditions.