In this paper, we refine previous results by introducing the former approach in a multi-stage algorithm that also adds heuristics and a stochastic optimization method operating on perceptually motivated objective cost functions. In previous works, a computational sound design approach has been proposed for the parameter estimation problem involving timbre matching by deep learning, which was applied to the synthesis of pipe organ tones. Whenever such models are used for commercial applications, an additional constraint is the time to market, making automation of the sound design process desirable. One of the challenges in computational acoustics is the identification of models that can simulate and predict the physical behavior of a system generating an acoustic signal. The model allowed us to better interpret the earthquake as a stick-slip motion. We have also shown that, the recurrence time of an event, the duration time of an event and the slip size of an earthquake can be controlled by the fractional-order derivative, the fractional-order deflection and the magnitude of the magma strength. It appears that the resonant amplitude and resonant frequency are strongly dependent on the fractional-order damping r, fractional-order friction q, the fractional deflection 𝛼, the nonlinear stiffness coefficient and the fractional viscous coefficient. Moreover, the amplitude-frequency equation for the steady-state solution was established.
The results have shown that the fractional-order derivative can affect the dynamical properties of fault rock, which is characterized by the equivalent damping coefficient and the equivalent stiffness coefficient. The numerical simulation method used in this paper is that of Grünwald-Letnikov based on the generalization of the classical derivative, and the approximately analytical solution obtained by the harmonic balance method. In this work, we examine the dynamical behaviour of the “single mass-springs” model for earthquake subjected to the strength due to the up flow of magma for the period of volcanism, considering the fractional viscous damping force, the fractional weakening friction and fractional power law of elastic force. This article provides an overview of a physics-based piano synthesis beginning with computationally heavy, physically accurate approach followed by a discussion of the approaches that are designed to produce the best possible sound quality for real-time synthesis. On the other hand, efficient real-time models are geared toward composers and musicians who perform at home or onstage. Accurate numerical models can be used by physicists and engineers to understand how the instrument functions or to help piano makers with instrument development. The process of physical modeling starts with first understanding the physical principles, followed by creating accurate numerical models, and finally, finding numerically optimized signal processing models that allow sound synthesis in real time by excluding inaudible phenomena and adding some perceptually important features by using signal processing tricks. While most commercial digital pianos are based on sample playback, it is also possible to reproduce a piano's sound by modeling the physics of the instrument.
The size, weight, and price of grand pianos as well as their relatively simple control surface (i.e., the keyboard), have led to the development of digital counterparts that mimic the sound of acoustic pianos as closely as possible. As a result of their complexity and versatility, pianos are arguably one of the most important instruments in Western music.
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