Perceptual similarity between piano notes: Experimental method applicable to reverberant and non-reverberant sounds.

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.

Reconstruction of piano hammer force from string velocity.

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.

Modeling and simulation of a grand piano.

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.

Dynamical properties of piano soundboards.

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.

The psychomechanics of simulated sound sources: material properties of impacted thin plates.

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.

The psychomechanics of simulated sound sources: material properties of impacted bars.

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.

Time-domain simulation of a guitar: model and method.

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.