An important characteristic of the atmospheric dynamics is that it constitutes a relatively small perturbation about dominant balances of hydrostacy, geostrophy and thermal winds established in effect of the Earth gravity, rotation, stably-stratified thermal structure of its atmosphere and the incoming flux of solar energy. Given this specificity, it is compelling to formulate the governing partial differential equations (PDEs) in terms of perturbation variables, defined with respect to an arbitrary "ambient" state of the atmosphere that already satisfies these dominant balances. The role of ambient states is to enhance the efficacy of numerical solution---e.g. by simplifying the design of the initial and boundary conditions and/or improving the conditioning of elliptic boundary value problems---without resorting to linearisation of the system. This talk introduces a new generalised theory, formulated for velocity (vector) perturbations as well as entropy and pressure perturbations about an arbitrary ambient state. The theoretical development is complete, from analytic equations to non-oscillatory forward-in-time finite-volume integrators, including all entries of the linear operator and associated coefficients of the Helmholtz problem arising from the semi-implicit solution procedure.