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Spectral evolution of the string force signal under timevarying tension. The model parameters are ˘ f1 = 1000 Hz, B = 2.55×10 −6 , η0 = 0.5 s −1 , η1 = 0 s −1 , η2 = 0.0001 s −1 , η3 = 0 s −1 , ¯ m h = 0.3, ¯ x h = 1/23.
Source publication
In physical modelling synthesis, articulation and tuning are effected via time-variation in one or more parameters. Adopting hammered strings as a test case, this paper develops extended forms of such control, proposing a numerical formulation that affords on-line adjustment of each of its scaled-form parameters, including those featuring in the on...
Context in source publication
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... special interest is the behaviour of the algorithm under timevarying tension, because this involves string modes moving out of and into the normal frequency range of interest. Figure 5 shows the amplitude normalised spectrograms of four simulation output signals. In each simulation, the string was excited with a hard hammer (K = 100000 s −2 , α = 1.2) at t = 0.2 s, and the parameter ˘ f1 was subsequently increased upwards in a linear fashion by a factor of 1.5, and then linearly decreased back to its original value. ...
Citations
... Notable studies have used FDTD methods [52,32,53,34,33,79,54,51], modal methods [211][212][213], and FEM [43,42]. While these works primarily used fully implicit discretization, recent trends have shifted towards linearly implicit or fully explicit methods [74,75,27,81,77,175,[214][215][216], aiming to improve computational efficiency, making real-time applications more feasible. ...
... This approach allows for fully explicit solvers when specific matrix structures [189] are exploited [26,77,28]. The IEQ method has been applied to collisions problems, both lumped and distributed [74,75,27,81,215], as well as to the GE string model [73,78]. Similarly, SAV has been used in the context of collisions [77,214,216,30], and geometric nonlinarities [26,77]. ...
Physics-based sound synthesis of musical instruments has seen growing interest in recent years, as it allows for reproducing realistic and natural sounds while offering great flexibility and minimal storage requirements. This research falls within the scope of the NEMUS project, which is dedicated to the digital reproduction of the sound of ancient stringed instruments using physical modelling techniques. Specifically, this work focuses on the numerical simulation of nonlinear string vibration. Nonlinearities are a critical factor in accurately replicating the sound of real-world instruments. Much of the recent literature has employed energy-based methods to ensure algorithmic stability when nonlinear behaviour is present, often resulting in fully implicit schemes requiring iterative root-finding methods. While effective, these schemes are computationally expensive and introduce additional complexities.
Recent developments in numerical analysis have, in some cases, enabled real-time simulation of strongly nonlinear systems using non-iterative algorithms. However, several challenges remain unresolved. This thesis aims to advance the use of finite-difference time-domain and modal methods to address nonlinearities in string vibration, which capture salient perceptual features. The emphasis is on the efficiency of the algorithms, while also developing a framework for the sound synthesis of nonlinear strings.
The work begins with a comprehensive review of string models and simulation techniques, covering both historical and modern approaches. Linear models are then used as a starting point, allowing for the introduction of impedance-type boundary conditions. The research then investigates typical nonlinear effects in string vibration, such as geometric nonlinearities, collisions, and friction, using newly developed non-iterative approaches, including quadratisation-based methods for conservative forces. These techniques significantly reduce computation times, making real-time simulation feasible for most systems. However, the quality of the simulations is still highly dependent on tailored discretisation choices. The thesis concludes with two case studies that apply these methods to physical models of musical instruments.
... Note that Po involves terms proportional to the vertical and horizontal control input velocities dtwH and dtxo, and P ϕ contains a term due to the changing excitation position (dth ϕ ). Such terms are expected when there is time-variance [11,17]. ...
... Following [8,17], real-time factor (RTF) is defined as the amount of time that passes with the computation of one second of audio output. With the assumption that the control parameters vary relatively slowly over time, in a Matlab simulation these were updated blockwise for every N b samples, employing linear interpolation as in [17]. ...
... Following [8,17], real-time factor (RTF) is defined as the amount of time that passes with the computation of one second of audio output. With the assumption that the control parameters vary relatively slowly over time, in a Matlab simulation these were updated blockwise for every N b samples, employing linear interpolation as in [17]. Average RTFs of a few updates that are part of the simulation loop are listed in Table 2. ...
Collisions are an integral part of the sound production mechanism in a wide variety of musical instruments. In physics-based real-time simulation of such nonlinear phenomena, challenges centred around efficient and accurate root-finding arise. Nonlinearly implicit schemes are normally ill-suited for real-time simulation as they rely on iterative solvers for root-solving. Explicit schemes overcome this issue at the cost of a slightly larger error for a given sample rate. In this paper, for the case of lumped collisions, an alternate approach is proposed by approximating the contact potential curve. The approximation is described, and is shown to lead to a non-iterative update for an energy-stable nonlinearly implicit scheme. The method is first tested on single mass-barrier collision simulations, and then employed in conjunction with a modal string model to simulate hammer-string and slide-string interaction. Results are discussed in comparison with existing approaches, and real-time feasibility is demonstrated.
