Abstract Houston's method for summing phonon modes in the Brillouin zone is applied to exclude specular transmission of phonon modes of specific symmetries, thus, modifying the Acoustic Mismatch Model when phonon heat flux is incident from a heavier to a lighter medium. The Houston method is also used to impose conservation of the number of phonons in each direction of high-symmetry, thus modifying the detailed balance theorem and the Diffuse Mismatch Model. Based on the assumption that phonons are in equilibrium at the interface and are transmitted specularly or diffusely by two-phonon elastic processes, interpolation between the modified Acoustic Mismatch Model and the modified Diffuse Mismatch Model has led to a general analytical formalism for low-temperature interface thermal conductance. The Debye temperature, the only parameter in the derived formalism, is expressed as a function of temperature by assimilating numerically obtained specific heat values to the Debye expression for specific heat. Previous measurements of the low-temperature thermal conductance of smooth and rough interfaces between dissimilar materials could be reproduced numerically without adjustment of model parameters, demonstrating the importance of modifications to the Acoustic Mismatch Model and the Diffuse Mismatch Model and supporting the hypothesis that anharmonic processes play a minimal role in heat transport across the interfaces studied below room temperature. The formalism developed is used to study the thermal conductance of the interface between silicon and germanium because of the potential of silicon-germanium nanocomposites for thermoelectric applications.