A Parametric Model of Piano Tuning

A parametric model of aural tuning of acoustic pianos is presented in this paper. From a few parameters, a whole tessitura model is obtained, that can be applied to any kind of pianos. Because the tuning of piano is strongly linked to the inharmonicity of its strings, a 2-parameter model for the inharmonicity coefficient along the keyboard is introduced. Constrained by piano string design considerations, its estimation requires only a few notes in the bass range. Then, from tuning rules, we propose a 4-parameter model for the fundamental frequency evolution on the whole tessitura, taking into account the model of the inhamonicity coefficient. The global model is applied to 5 different pianos (4 grand pianos and 1 upright piano) to control the quality of the tuning. Besides the generation of tuning reference curves for non-professional tuners, potential applications could include the parametrization of synthesizers, or its use in transcription / source separation algorithm as a physical constraint to increase robustness.