Harmonic sinusoidal models are an essential tool for music audio signal analysis. Bayesian harmonic models are particularly interesting, since they allow the joint exploitation of various priors on the model parameters. However existing inference methods often rely on specific prior distributions and remain computationally demanding for realistic data. In this article, we investigate a generic inference method based on approximate factorization of the joint posterior into a product of independent distributions on small subsets of parameters. We discuss the conditions under which this factorization holds true and propose two criteria to choose these subsets adaptively. We evaluate the resulting performance experimentally for the task of multiple pitch estimation using different levels of factorization.