This paper presents a method for estimating the tuning and the inharmonicity coefficient of piano tones, from single notes or chord recordings. It is based on the Non-negative Matrix Factorization (NMF) framework, with a parametric model for the dictionary atoms. The key point here is to include as a relaxed constraint the inharmonicity law modelling the frequencies of transverse vibrations for stiff strings. Applications show that this can be used to finely estimate the tuning and the inharmonicity coefficient of several notes, even in the case of high polyphony. The use of NMF makes this method relevant when tasks like music transcription or source/note separation are targeted.