Julien Bensa

The simulation of piano string vibration: From physical models to finite difference schemes and digital waveguides

A model of transverse piano string vibration, second order in time, which models frequency-dependent loss and dispersion effects is presented here. This model has many desirable properties, in particular that it can be written as a well-posed initial-boundary value problem (permitting stable finite difference schemes) and that it may be directly related to a digital waveguide model, a digital filter-based algorithm which can be used for musical sound synthesis. Techniques for the extraction of model parameters from experimental data over the full range of the grand piano are discussed, as is the link between the model parameters and the filter responses in a digital waveguide. Simulations are performed. Finally, the waveguide model is extended to the case of several coupled strings.

A hybrid resynthesis model for hammer-string interaction of piano tones

This paper presents a source/resonator model of hammer-string interaction that produces realistic piano sound. The source is generated using a subtractive signal model. Digital waveguides are used to simulate the propagation of waves in the resonator. This hybrid model allows resynthesis of the vibration measured on an experimental setup. In particular, the nonlinear behavior of the hammer-string interaction is taken into account in the source model and is well reproduced. The behavior of the model parameters (the resonant part and the excitation part) is studied with respect to the velocities and the notes played. This model exhibits physically and perceptually related parameters, allowing easy control of the sound produced. This research is an essential step in the design of a complete piano model.

Parameter fitting for piano sound synthesis by physical modeling

A difficult issue in the synthesis of piano tones by physical models is to choose the values of the parameters governing the hammer-string model. In fact, these parameters are hard to estimate from static measurements, causing the synthesis sounds to be unrealistic. An original approach that estimates the parameters of a piano model, from the measurement of the string vibration, by minimizing a perceptual criterion is proposed. The minimization process that was used is a combination of a gradient method and a simulated annealing algorithm, in order to avoid convergence problems in case of multiple local minima. The criterion, based on the tristimulus concept, takes into account the spectral energy density in three bands, each allowing particular parameters to be estimated. The optimization process has been run on signals measured on an experimental setup. The parameters thus estimated provided a better sound quality than the one obtained using a global energetic criterion. Both the sounds attack and its brightness were better preserved. This quality gain was obtained for parameter values very close to the initial ones, showing that only slight deviations are necessary to make synthetic sounds closer to the real ones.

The wave digital piano hammer

For sound synthesis purposes, the vibration of a piano string may be simply modeled using bidirectional delay lines or digital waveguides which transport traveling wavelike signals in both directions. Such a digital wave‐type formulation, in addition to yielding a particularly computationally efficient simulation routine, also possesses other important advantages. In particular, it is possible to couple the delay lines to a nonlinear exciting mechanism (the hammer) without compromising stability; in fact, if the hammer and string are lossless, their digital counterparts will be exactly lossless as well. The key to this good property (which can be carried over to other nonlinear elements in musical systems) is that all operations are framed in terms of the passive scattering of discrete signals in the network, the sum of the squares of which serves as a discrete‐time Lyapunov function for the system as a whole. Simulations are presented.

An allpass filter design method with application to piano string synthesis

A nonparametric allpass filter design method for matching a desired group delay as a function of frequency is presented. The technique is useful in physical modeling synthesis of musical instruments exhibiting dispersive wave propagation in which different frequency bands travel at different speeds. While current group delay filter design methods suffer from numerical difficulties except at low filter orders, the technique presented here is numerically robust, producing an allpass filter in cascaded biquad form, and with the filter poles following a smooth loop within the unit circle. The technique was inspired by the observation that a pole‐zero pair arranged in allpass form has 2π total group delay when integrated around the unit circle, regardless of the pole location. To match a given group delay characteristic, the method divides the frequency axis into sections containing 2π total group delay, and assigns a pole‐zero allpass pair to each. In this way, the method incorporates an order selection techn...

Stiff piano string modeling: Computational comparison between finite differences and digital waveguide

As is well known, digital waveguides offer a computationally efficient, and physically motivated, means of simulating wave propagation in strings. The method is based on sampling the traveling wave solution to the ideal wave equation and linearly filtering this solution to simulate dispersive effects due to stiffness and frequency‐dependent loss; such digital filters may terminate the waveguide or be embedded along its length. For strings of high stiffness, however, dispersion filters can be difficult to design and expensive to implement. It is shown how high‐quality time‐domain terminating filters may be derived from given frequency‐domain specifications which depend on the model parameters. Particular attention is paid to the problem of phase approximation, which, in the case of high stiffness, is strongly nonlinear. Finally, in the interest of determining the limits of applicability of digital waveguide techniques, we make a comparison with more conventional finite difference schemes, in terms of compu...

Piano string modeling: From partial differential equations to digital wave‐guide model

A new class of partial differential equations (PDE) is proposed for transverse vibration in stiff, lossy strings, such as piano strings. While only second‐order in time, it models both frequency‐dependent losses and dispersion effects. By restricting the time‐order to 2, valuable advantages are achieved: First, the frequency‐domain analysis is simplified, making it easy to obtain explicit formulas for dispersion and loss versus frequency; for the same reason, exact bounds on sampling in associated finite‐difference‐schemes (FDS) can be derived. Second, it can be shown that the associated FDS is ‘‘well posed’’ in the sense that it is stable, in the limit, as the sampling period goes to zero. Finally, the new PDE class can be used as a starting point for digital wave‐guide modeling [a digital wave‐guide factors one‐dimensional wave propagation as purely lossless throughout the length of the string, with losses and dispersion lumped in a low‐order digital filter at the string endpoint(s)]. We perform numeric...

Piano sound synthesis: Relating physical phenomena to sound quality

The piano is a complex instrument with a large number of mechanical elements, the majority of which contribute to the sound production. The physical characteristics of these elements together with their interaction influence the timbre of the piano sound. Thus, in order to give a precise description of the behavior of this instrument and effectuate a satisfactory sound synthesis, the totality of the physical phenomena that are part of the sound production ideally should be taken into account. However, due to the complexity of the sound production system, this is not possible. Still, works on piano sound synthesis have made it possible to model (usually non‐real‐time) high quality piano sounds. These models represent a great tool when studying the perception of piano timbre. We therefore propose, thanks to sound modeling and subjective studies, to determine the perceptual effect of phenomena involved in a sound production system. This approach hopefully will give a better understanding of the relation betw...