This paper presents theoretical and experimental results about constrained non-negative matrix factorization (NMF) in a Bayesian framework. A model of superimposed Gaussian components including harmonicity is proposed, while temporal continuity is enforced through an inverse-Gamma Markov chain prior. We then exhibit a space-alternating generalized expectation-maximization (SAGE) algorithm to estimate the parameters. Computational time is reduced by initializing the system with an original variant of multiplicative harmonic NMF, which is described as well. The algorithm is then applied to perform polyphonic piano music transcription. It is compared to other state-of-the-art algorithms, especially NMF-based. Convergence issues are also discussed on a theoretical and experimental point of view. Bayesian NMF with harmonicity and temporal continuity constraints is shown to outperform other standard NMF-based transcription systems, providing a meaningful mid-level representation of the data. However, temporal smoothness has its drawbacks, as far as transients are concerned in particular, and can be detrimental to transcription performance when it is the only constraint used. Possible improvements of the temporal prior are discussed.