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In this paper an experimental method to quantify perceptual differences between acoustic stimuli is presented. The experiments are implemented as a signal-in-noise task, where two sounds are to be discriminated. By adjusting the signal-to-noise ratio (SNR) the difficulty of the sound discrimination is manipulated. If two sounds are very similar already, a low level of added noise (high SNR) makes the discrimination task difficult. For more dissimilar sounds, a higher amount of noise (lower SNR) is needed to affect discriminability. In other words, a strong correlation between SNR and similarity is expected. The experimental noises are generated to have similar spectro-temporal properties to those of the test stimuli. As a study case, the suggested method was used to evaluate recordings of one note played on seven Viennese pianos using (1) non-reverberant sounds (as recorded) and (2) reverberant sounds, where reverberation was added by means of digital convolution. The experimental results of the suggested method were compared with a similarity experiment using the method of triadic comparisons. The results of both methods were significantly correlated with each other. [ABSTRACT FROM AUTHOR]

A method is presented for reconstructing piano hammer forces through appropriate filtering of the measured string velocity. The filter design is based on the analysis of the pulses generated by the hammer blow and propagating along the string. In the five lowest octaves, the hammer force is reconstructed by considering two waves only: the incoming wave from the hammer and its first reflection at the front end. For the higher notes, four- or eight-wave schemes must be considered. The theory is validated on simulated string velocities by comparing imposed and reconstructed forces. The simulations are based on a nonlinear damped stiff string model previously developed by Chabassier, Chaigne, and Joly [J. Acoust. Soc. Am. 134(1), 648-665 (2013)]. The influence of absorption, dispersion, and amplitude of the string waves on the quality of the reconstruction is discussed. Finally, the method is applied to real piano strings. The measured string velocity is compared to the simulated velocity excited by the reconstructed force, showing a high degree of accuracy. A number of simulations are compared to simulated strings excited by a force derived from measurements of mass and acceleration of the hammer head. One application to an historic piano is also presented. [ABSTRACT FROM AUTHOR]

A time-domain global modeling of a grand piano is presented. The string model includes internal losses, stiffness, and geometrical nonlinearity. The hammer-string interaction is governed by a nonlinear dissipative compression force. The soundboard is modeled as a dissipative bidimensional orthotropic Reissner-Mindlin plate where the presence of ribs and bridges is treated as local heterogeneities. The coupling between strings and soundboard at the bridge allows the transmission of both transverse and longitudinal waves to the soundboard. The soundboard is coupled to the acoustic field, whereas all other parts of the structure are supposed to be perfectly rigid. The acoustic field is bounded artificially using perfectly matched layers. The discrete form of the equations is based on original energy preserving schemes. Artificial decoupling is achieved, through the use of Schur complements and Lagrange multipliers, so that each variable of the problem can be updated separately at each time step. The capability of the model is highlighted by series of simulations in the low, medium, and high register, and through comparisons with waveforms recorded on a Steinway D piano. Its ability to account for phantom partials and precursors, consecutive to string nonlinearity and inharmonicity, is particularly emphasized. [ABSTRACT FROM AUTHOR]

In pianos, the transfer of energy from strings to soundboard and the radiation of sound are highly dependent on the dynamical properties of the soundboard. In this paper, a numerical study is conducted for various rib configurations, showing that even slight irregularities in rib spacing can induce a strong localization of the soundboard velocity pattern. The effective vibrating area can be further reduced due to the spatial filtering effect of the bridge. Numerical predictions of modal shapes and operating deflection shapes are confirmed by series of measurements made on upright piano soundboards. Simulations of radiated pressure based on measured and calculated soundboard velocity fields show that localization tends to broaden the cone of directivity and to reduce the number of lobes. [ABSTRACT FROM AUTHOR]

Sounds convey information about the materials composing an object. Stimuli were synthesized using a computer model of impacted plates that varied their material properties: viscoelastic and thermoelastic damping and wave velocity (related to elasticity and mass density). The range of damping properties represented a continuum between materials with predominant viscoelastic and thermoelastic damping (glass and aluminum, respectively). The perceptual structure of the sounds was inferred from multidimensional scaling of dissimilarity judgments and from their categorization as glass or aluminum. Dissimilarity ratings revealed dimensions that were closely related to mechanical properties: a wave-velocity-related dimension associated with pitch and a damping-related dimension associated with timbre and duration. When asked to categorize sounds, however, listeners ignored the cues related to wave velocity and focused on cues related to damping. In both dissimilarity-rating and identification experiments, the results were independent of the material of the mallet striking the plate (rubber or wood). Listeners thus appear to select acoustical information that is reliable for a given perceptual task. Because the frequency changes responsible for detecting changes in wave velocity can also be due to changes in geometry, they are not as reliable for material identification as are damping cues. [ABSTRACT FROM AUTHOR]

Sound can convey information about the materials composing an object that are often not directly available to the visual system. Material and geometric properties of synthesized impacted bars with a tube resonator were varied, their perceptual structure was inferred from multidimensional scaling of dissimilarity judgments, and the psychophysical relations between the two were quantified. Constant cross-section bars varying in mass density and viscoelastic damping coefficient were synthesized with a physical model in experiment 1. A two-dimensional perceptual space resulted, and the dimensions were correlated with the mechanical parameters after applying a power-law transformation. Variable cross-section bars varying in length and viscoelastic damping coefficient were synthesized in experiment 2 with two sets of lengths creating high- and low-pitched bars. In the low-pitched bars, there was a coupling between the bar and the resonator that modified the decay characteristics. Perceptual dimensions again corresponded to the mechanical parameters. A set of potential temporal, spectral, and spectrotemporal correlates of the auditory representation were derived from the signal. The dimensions related to mass density and bar length were correlated with the frequency of the lowest partial and are related to pitch perception. The correlate most likely to represent the viscoelastic damping coefficient across all three stimulus sets is a linear combination of a decay constant derived from the temporal envelope and the spectral center of gravity derived from a cochlear representation of the signal. These results attest to the perceptual salience of energy-loss phenomena in sound source behavior. [ABSTRACT FROM AUTHOR]

This paper presents a three-dimensional time-domain numerical model of the vibration and acoustic radiation from a guitar. The model involves the transverse displacement of the string excited by a force pulse, the flexural motion of the soundboard, and the sound radiation. A specific spectral method is used for solving the Kirchhoff-Love's dynamic top plate model for a damped, heterogeneous orthotropic material. The air-plate interaction is solved with a fictitious domain method, and a conservative scheme is used for the time discretization. Frequency analysis is performed on the simulated sound pressure and plate velocity waveforms in order to evaluate quantitatively the transfer of energy through the various components of the coupled system: from the string to the soundboard and from the soundboard to the air. The effects of some structural changes in soundboard thickness and cavity volume on the produced sounds are presented and discussed. Simulations of the same guitar in three different cases are also performed: "in vacuo," in air with a perfectly rigid top plate, and in air with an elastic top plate. This allows comparisons between structural, acoustic, and structural-acoustic modes of the instrument. Finally, attention is paid to the evolution with time of the spatial pressure field. This shows, in particular, the complex evolution of the directivity pattern in the near field of the instrument, especially during the attack. [ABSTRACT FROM AUTHOR]

A time-domain model for the flexural vibrations of damped plates was presented in a companion paper [Part I, J. Acoust. Soc. Am. 109, 1422-1432 (2001)]. In this paper (Part II), the damped-plate model is extended to impact excitation, using Hertz's law of contact, and is solved numerically in order to synthesize sounds. The numerical method is based on the use of a finite-difference scheme of second order in time and fourth order in space. As a consequence of the damping terms, the stability and dispersion properties of this scheme are modified, compared to the undamped case. The numerical model is used for the time-domain simulation of vibrations and sounds produced by impact on isotropic and orthotropic plates made of various materials (aluminum, glass, carbon fiber and wood). The efficiency of the method is validated by comparisons with analytical and experimental data. The sounds produced show a high degree of similarity with real sounds and allow a clear recognition of each constitutive material of the plate without ambiguity. © 2001 Acoustical Society of America. [ABSTRACT FROM AUTHOR]

A time-domain formulation for the flexural vibrations in damped rectangular isotropic and orthotropic plates is developed, in order to investigate transient excitation of plates by means of sound synthesis. The model includes three basic mechanisms of damping (thermoelasticity, viscoelasticity and radiation) using a general differential operator. The four rigidity factors of the plate are modified by perturbation terms, each term corresponding to one specific damping mechanism. The first damping term is derived from the coupling between the thermoelastic stress-strain relations and the heat diffusion equation. The second term is obtained from the general differential formulation of viscoelasticity. The third term is obtained through a Padé approximation of the damping factor which governs the coupling of the plate with the surrounding air. The decay factors predicted by the model reproduce adequately the dependence on both dimensions and frequency of the decay factors measured on rectangular plates of various sizes and thicknesses made of four different materials (aluminum, glass, carbon fiber, and wood). The numerical resolution of the complete problem, including initial and boundary conditions, and the comparison between real and simulated sounds are presented in a companion paper [J. Acoust. Soc. Am. 109, 1433-1447 (2000)]. © 2001 Acoustical Society of America. [ABSTRACT FROM AUTHOR]

A kettledrum is made of a circular elastic membrane stretched over an enclosed air cavity. It is set into vibration by the impact of the mallet. The motion of the membrane is coupled with both the external and internal sound field. A time-domain modeling of this instrument is presented which describes the motion of the mallet and its nonlinear interaction with the membrane, the transverse displacement of the membrane, and the sound pressure inside and outside the cavity. Based on a variational formulation of the problem, which uses the pressure jump over the boundaries of the instrument as a new variable, a numerical scheme is derived by means of three-dimensional finite elements. Higher-order absorbing conditions are used to simulate the free space. The validity of the model is illustrated by successive snapshots showing the pressure fields and the displacement of the membrane. In addition, time histories of energetic quantities help in explaining how the energy is balanced between mallet, membrane, and acoustic field in real instruments. Simulated external pressures show particularly good agreement with the sound field radiated by real instruments in both time and frequency domains. © 1999 Acoustical Society of America. [ABSTRACT FROM AUTHOR]

This paper presents a time-domain modeling for the sound pressure radiated by a xylophone and, more generally, by mallet percussion instruments such as the marimba and vibraphone, using finite difference methods. The time-domain model used for the one-dimensional (1-D) flexural vibrations of a nonuniform bar has been described in a previous paper by Chaigne and Doutaut [J. Acoust. Soc. Am. 101, 539-557 (1997)] and is now extended to the modeling of the sound-pressure field radiated by the bar coupled with a 1-D tubular resonator. The bar is viewed as a linear array of equivalent oscillating spheres. A fraction of the bar field excites the tubular resonator which, in turn, radiates sound with a certain delay. In the present model, the open end of the resonator is represented by an equivalent pulsating sphere. The total sound field is obtained by summing the respective contributions of the bar and tube. Particular care is given for defining a valid approximation of the radiation impedance, both in continuous and discrete time domain, on the basis of Kreiss's theory. The model is successful in reproducing the main features of real instruments: sharp attack, tuning of the bar, directivity, tone color, and aftersound due to the bar-resonator coupling. © 1998 Acoustical Society of America. [ABSTRACT FROM AUTHOR]

A time-domain modeling of xylophone bars excited by the blow of a mallet is presented. The flexural vibrations of the bar, with nonuniform cross section, are modeled by a one-dimensional Euler-Bernoulli equation, modified by the addition of two damping terms for the modeling of losses and a restoring force for the modeling of the stiffness of the suspending cord. The action of the mallet against the bar is described by Hertz's law of contact for linear elastic bodies. This action appears as a force density term on the right-hand side of the bending wave equation. The model is completed by the equation of motion for the mallet, and by free-free boundary conditions for the bar. The bending wave equation of the bar is put into a numerical form by means of an implicit finite-difference scheme, which ensures a sufficient spatial resolution for an accurate tuning of the bar. The geometrical, elastic, and damping parameters of the model are derived from experiments carried out on actual xylophones and mallets. The validity of the numerical model is confirmed by three different procedures: First, a comparison is made between numerical results and analytical solutions. The second series of tests consist of examining the effects of the number of spatial steps on the convergence of the solution. Finally, various comparisons are made between measured and simulated impact forces and bar accelerations. The present model reproduces adequately the main features of a real instrument. The most significant physical parameters of bars, mallets and players' actions can be controlled independently for producing a remarkable variety of tones. © 1997 Acoustical Society of America. [ABSTRACT FROM AUTHOR]

A physical model of the piano string, using finite difference methods, has recently been developed. [Chaigne and Askenfelt, J. Acoust. Soc. Am. 95, 1112-1118 (1994)]. The model is based on the fundamental equations of a damped, stiff string interacting with a nonlinear hammer, from which a numerical finite difference scheme is derived. In the present study, the performance of the model is evaluated by systematic comparisons between measured and simulated piano tones. After a verification of the accuracy of the method, the model is used as a tool for systematically exploring the influence of string stiffness, relative striking position, and hammer-string mass ratio on string waveforms and spectra. [ABSTRACT FROM AUTHOR]

The first attempt to generate musical sounds by solving the equations of vibrating strings by means of finite difference methods (FDM) was made by Hiller and Ruiz [J. Audio Eng. Soc. 19, 462-472 (1971)]. It is shown here how this numerical approach and the underlying physical model can be improved in order to simulate the motion of the piano string with a high degree of realism. Starting from the fundamental equations of a damped, stiff string interacting with a nonlinear hammer, a numerical finite difference scheme is derived, from which the time histories of string displacement and velocity for each point of the string are computed in the time domain. The interacting force between hammer and string, as well as the force acting on the bridge, are given by the same scheme. The performance of the model is illustrated by a few examples of simulated string waveforms. A brief discussion of the aspects of numerical stability and dispersion with reference to the proper choice of sampling parameters is also included. [ABSTRACT FROM AUTHOR]

A general method has been previously developed by O. Beslin et al. [J. Acoust. Soc. Am. 89, 1868 (A) (1991)], using a variational method to describe the motion of a perforated plate, and a integral formulation for the modeling of the consecutive radiated sound pressure. This model is now adapted to the particular case of a guitar, whose top plate is represented by a rectangular orthotropic plate with hole and back cavity. The sound pressure computation is carried out in the frequency domain, for a point force acting at the bridge. This yields the pressure versus force transfer function H(ω), from which the corresponding impulse response h(t) can be derived by means of inverse Fourier transforms. Finally, the guitar tones are obtained by convolving h(t) with a time-domain formulation of the force F(t) imparted to the bridge by a guitar string, using a finite difference method previously developed by A. Chaigne [J. Acoust. 5(2), 181-211 (1992)]. This model is used for investigating the influence of various plate parameters on timbre and sound quality of the instrument. [ABSTRACT FROM AUTHOR]

The tones of existing instruments are simulated in the time domain by means of standard finite difference methods. The string model previously applied to the guitar [A. Chaigne, J. Acoust. Soc. Am. Suppl. 1 88, S188 (1990)] and more recently to the piano [A. Chaigne, A. Askenfelt, and E. Jansson, J. Acoust. Soc. Am. 89, 1878(A) (1991)] has been extended to the violin, the xylophone bars, and tympani membranes. The general procedure, which is similar for each type of instrument, will be presented. It consists first in deriving a numerical scheme from the continuous differential equations governing the motion of the vibrating systems including the interaction with an exciter (hammer, bow, mallets,...). In a second step, the mathematical properties of these discrete approximations are evaluated with special emphasis laid on stability, accuracy, and dispersion problems. Simulated waveforms are computed with realistic values of the parameters derived from measurements on existing instruments. These waveforms as well as their spectral contents are then systematically compared with those of real tones. This time-domain approach is used in order to test the influence of various physical parameters on the transients of strings and percussive sounds in which the nonlinear effects are essential. The paper will be illustrated by sound synthesis examples of various instruments obtained with this technique. [ABSTRACT FROM AUTHOR]

Numerical experiments have been made on the piano string with a discrete model using a finite difference calculation procedure. The string model was previously applied to the guitar [A. Chaigne, J. Acoust. Soc. Am. Suppl. 1 88, S188 (1990)] and is now tested with experimentally obtained data for the piano. The numerical experiments include systematic variations of parameters such as hammer velocity and nonlinearity of the felt hammer and their influence on generated waveforms and spectra. Earlier physical experiments on pianos have provided data on the velocity of the hammer and the string. In addition, the hammerstring interaction force has been estimated by measuring the compression of the hammer felt during string contact. Recorded string tones and measured waveforms will be presented and compared with typical examples of those calculated. [ABSTRACT FROM AUTHOR]

In an attempt to simulate guitar sounds, a discrete model was developed that consisted of real strings with stiffness and internal damping, the player's finger, the resonance box, and the modelization of the sound pressure. The vibrating string equation was solved in the time domain using the finite-difference method. The player's figure was modeled as a force density term in the string equation, which accounted for the initial plucking conditions. The resonance box was modeled as impedancelike boundary conditions at one end of the string. These end conditions were obtained under the assumption that the driving force exerted by the string at its ends excites a set of second-order resonances in parallel, the first resonance of the plate being coupled with the air resonance of the box. Finally, the sound pressure was modeled as the contribution of a set of monopoles, where each monopole corresponds to one given resonance. The radiation efficiency of each monopole was derived from a curve fitting on a guitar sound-pressure spectrum measured in an anechoic chamber. The waveforms obtained are in good agreement with experimental waveforms measured on real guitars. Sound examples will be presented high-lighting the realism of the simulation. [ABSTRACT FROM AUTHOR]