Requirements formalization for a mathematical model of the tuning object of the class of polyharmonic acoustic signal generation systems

Abstract

The paper addresses the issue of formalizing requirements for a mathematical model of a tuning object belonging to the class of polyharmonic acoustic signal generation systems, using the piano as an example. It is substantiated that, for evaluating the efficiency of software for automated professional tuning, it is insufficient to focus solely on the maximum similarity between the synthesized sound and that of a real instrument. The model must simultaneously provide adequate reproduction of acoustic signal properties that affect pitch determination and the estimation of spectral component deviations, controllable variation of the tuning state and spatial recording conditions, an unambiguous parametric description, and sufficient computational efficiency. The main approaches to piano sound synthesis are analyzed: sampling, physical modeling, additive synthesis, and neural network models. It is shown that sampling provides high acoustic similarity and low computational complexity but does not offer the required parametric control; physical modeling is flexible but can be excessively complex; and neural network models can provide high synthesis quality but do not guarantee transparent control over individual acoustic characteristics. Based on the comparison, additive synthesis is identified as the most appropriate foundation for the model. A set of 16 critical requirements is formulated, covering the parameterization of the tuning state, multiple strings per note, transverse and longitudinal string vibrations, inharmonicity and its dependence on the physical parameters of a string, non-uniformity and temporal instability of overtones, the dependence of the spectrum on the recording position, decay, attack, secondary spectral components, signal-level normalization, idealization of non-essential properties, reference character of the description, and computational efficiency. The scope of application is defined as follows: the deviation of a note’s fundamental frequency from its nominal value is allowed within ±100 cents, the tone generation duration is limited to three seconds, and the average signal level is normalized to a level corresponding to moderate loudness. The proposed system of requirements defines the domain of adequacy of the future model and provides a basis for its further development, software implementation, and experimental validation.