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Figure 2: Schematic view of a grand piano’s mechanism.

This approach, guided by an historic expertise and an empirical approach, can only be judged by the final hearing, and still needs to be adjusted for each piano, in order to adapt to slight variations of materials, boundary conditions, which may profoundly affect the sound of the instrument. In order to reduce the proportion of empiricism, and anticipate the impact of possible changes in the vibrational and acoustical behavior of the instrument, many piano makers have their own research laboratory, oriented towards experimentation but also towards numerical simulation, as shown in figure 3 in the case of Schimmel. Methods and tools that have proven their performance in many areas of engineering (car engineering, meteorology, geophysics, aeronautics...) are now part of the improvement and testing process of various parts of the piano (soundboard modal analysis, spectral analysis of the strings, shape optimization of the cast iron frame...). Some piano makers collaborate with universities or research laboratories, in order to answer specific questions about the instrument (radiation efficiency, characteristics damping time, or boundary conditions at the bridge as, for example, in the collaboration between the pianos Stuart & Sons and the Australian research centre CSIRO: www.cmis.csiro.au/bob.anderssen/stuartpiano/stuartpiano.htm. The approach used by piano makers, however, suffers from one major limitation : although they are able to study in detail the behavior of each part of the instrument, they generally do not consider the coupling between its main elements, which in fact may significantly influence this behavior. In contrary, a comprehensive modeling tool, accounting for all the couplings between the main parts of the instrument, yields a better understanding of the influence of some particular settings on the whole behavior of the piano. It becomes then possible to conduct « virtual experiments », by systematically changing materials, geometries, or some other design parameters, and observe the effect of these changes on the entire vibro-acoustical behavior of the instrument, and ultimately, on the resulting sound. This observation has lead to the present collaboration between the Unité de Mécanique, ENSTA ParisTech, specialized in musical acoustics, and directed by Antoine Chaigne, and the project team POems at ENSTA ParisTech, CNRS and INRIA specialized in the development of numerical methods for wave equations, and directed by Patrick Joly. This collaboration already gave birth in the past to other modeling tools for musical instruments (timpani and guitar), and their success encouraged us to reiterate this effort for the piano.

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(a)

(b)

Figure 3: “Computer assisted piano engineering”, from schimmel-piano.de

In summary, the objective of this project is to contribute to the understanding of the physical mechanisms involved in the generation of the piano sound, using physical modeling and numerical simulation of mechanical and acoustical vibrations, from the hammer’s excitation to sound propagation, including the vibrations of the strings and of the soundboard... This objective is related to the problem of sound synthesis, whose one aim is to generate realistic sounds of given instruments (here, the piano). Many methods reach this goal successfully. A number of them operate in real-time, based on various strategies (pre-recording of some selected representative sounds ; frequency, amplitude or phase modulation ; additive or substractive synthesis...). However, most of them have only little connections with the physics of the instrument. Our wish is not only to reproduce the sound generated by a physical object (the piano) convincingly, but rather to understand how this specific object can generate such a particular sound, by modeling the complete instrument, based on the equations of the physics. Such an approach can be referred to as « physics based sound synthesis ». The principle is to write the equations of the dynamics of the coupled problem in the time domain, and seek a numerical solution. Various methods were used in recent times. With regard to the specific case of the piano, one can mention, for example, the very popular method of digital waveguides, through which a complete model of linear piano was proposed by Nakamura in 1986 and which is reviewed by Bank in 2003. This method is particularly effective and efficient and can be coupled with more traditional methods (as for instance finite differences in Bensa, 2003). The modal method chosen at IRCAM for Modalys (http://forumnet.ircam.fr/701.html, see figure 4) is also an powerful strategy. Another approach is to use the standard tools of numerical analysis for PDEs (finite element, finite differences) to solve the system of equations numerically. The advantage of such an approach is to keep a strong connection to the physical reality, and to make very few a priori assumptions on the behavior of the solution. The intention is to reproduce the attack transients and the extinction of the tones faithfully, thus offering a better understanding of the complex mechanisms that takes place in the vibrating structure. Figure 4: Snapshot of the software Modalys. This approach was adopted in the past to study separate parts of the piano: Boutillon (1988) and Chaigne (1994) 3

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investigated the interaction between the hammer and the strings. Bilbao (2005) was interested in modeling the nonlinear behavior of the strings. Cuenca (2007) proposed a model to explain the coupling between the strings and the soundboard at the bridges. Vyasarayani (2009) and Izadbakhsh (2008) studied the vibration the hammer shank. To our knowledge, there is only one published work (Giordano 2004) which focuses on both the modeling a full piano and its numerical formulation. His model makes use of partial differential equations, and accounts for the involved phenomena : from the initial blow of the hammer to the propagation of sound, including the linear vibration of strings and soundboard. For the discrete formulation of the problem, he use classical numerical analysis tools, namely finite differences in space and time. Our work is aimed at continuing this effort by providing a complete piano model (to our knowledge, it is the most accurate model available today), and a reliable, innovative and accurate numerical method to solve it. These two steps were naturally the two main parts of the work, validated by very realistic numerical simulations.

Writing a continuous model The choice of a relevant model is based on both the existing literature and experimental observations. Its degree of refinement largely depends on the anticipated future exploration one intends to conduct. It should also be determined in view of the anticipated computational burden. In order to address some of the remaining open questions, and also for collecting experimental signals for analysis and comparisons with future simulations, we performed series of measurements on two different pianos. The first, shown in figure 6(a) (left), is a strung soundboard of an upright Pleyel piano, made available to us by ITEMM (http://www.itemm.fr) (notice that the soundboard is extracted from the rim and that the hammer mechanism and keyboard are missing). The second, shown in figure 6(b), is a Steinway D grand piano in normal playing situation (we simply removed the lid), put at our disposal by IRCAM (http://www.ircam.fr). This latter piano was chosen in this work as the « model » piano for geometrical and material data, in order to compare measurements performed on it with numerical simulations. Six reference notes were chosen and measured (marked in blue in the figure of the Steinway D on the right) and were particularly examined throughout this work. On the figure, the shape of the soundboard is materialized in green, and the reference axes are in red. A first insight about the specificities of the piano sound which we want to account for in our modeling can already be given through the comparison of two simple measurements: the acceleration of the soundboard at the point indicated AcTa (see figure 5) when the note C2 is played, the first time for a mezzo piano dynamical level (mp), the second for a fortissimo sound (ff). Two spectrograms (time - frequency diagrams) of the obtained data are displayed In figures 5, for the first seconds after the hammer’s strike, with a frequency rang between 0 and 9 kHz.

(0, 2)

AcTa

(0, 1)

(0, 0)

D�1 C2

F3

C�5

G6 C7

Figure 5: Geometry chosen on the Steinway D

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(a) ITEMM strung soundboard of an upright Pleyel piano

(b) IRCAM Steinway D grand piano

Figure 6: Upright and grand pianos used for the experiment

(a) C2 note, mp dynamic level

(b) C2 note, ff dynamic level

Figure 7: Upright and grand pianos used for the experiment

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• The measurement stops well before the end of vibration, which goes on for more than 20 seconds, • The signal has a frequency content close to a « harmonic » distribution, reflecting the nearly periodic nature of the sound : a consequence of the specific vibration of the string. In fact, a closer analysis shows a slight inharmonicity which comes mainly from the stiffness of the string, • The frequency content of a bass note as C2 can extend up to 9kHz, • The high-frequency components are attenuated faster than low-frequency components, which is due to damping phenomena present in strings, hammer felt, soundboard material and acoustic radiation. • There is a great difference between the two spectrograms obtained at two different dynamic levels. Clearly, treble components are more present in the ff spectrogram, which also shows some blurred zones suggesting the presence of non harmonic partials, with a different order of magnitude for the damping. This large dynamic spectral enrichment is the mark of a nonlinear behavior in the instrument, and we will see in fact that both the hammer and strings are subjected to nonlinear phenomena. In view of these observations, it seems essential to write a model that captures the nonlinear character of hammers and strings, the frequency dependence of the damping, and of course the radiation of the soundboard in the surrounding air. We wish to provide a priori estimates for the solutions of this coupled and, sometimes, nonlinear problem, by means of energy identities for each part of the piano, then in the presence of coupling. Besides the physical interest brought by these identities, we will use them to ensure the numerical stability of the discrete problem, a non trivial task. • A special effort has been given to the string model, which is represented as a stiff string subjected to large deformations, and viscoelastic dissipation. In addition to the prominent transverse vibration, we model the propagation of a shear wave, a consequence of the stiffness of the string, using a prestressed Timoshenko beam model. Finally, we take the longitudinal vibration into account through a nonlinear model based on the exact description of the geometric deformations. A model of nonplanar vibration is proposed. • The hammer model reflects the nonlinear behavior with hysteresis of the felt, via a power law. • The soundboard is modeled as a thick orthotropic Reissner Mindlin plate, whose heterogeneity allows us to consider the effect of ribs and bridges. Using a modal decomposition we assume a « diagonal » damping, that is intrinsic to each mode. The radiation of the soundboard is governed by the standard structural acoustics equations, where the equation of the structure, written in the modal basis, is coupled to the equations of sound propagation in air, expressed in terms of velocity and pressure. The rim of the piano is then considered as a rigid obstacle. • The strings are coupled to the soundboard at the bridge. The bridge is considered as a rigid body only moving in the direction orthogonal to the table. The velocities of strings and soundboard are equal at the bridge. We take advantage of the slight angle formed by the two sides of the strings with the horizontal plane, in order to model the transmission of both transverse and longitudinal components of the string tension to the soundboard. This allows the model to account for the « precursor » and « phantom partials », specific to the piano sound. Choirs of strings are considered in this model, as well as the duplex scales, the una corda playing and the aliquot system. • The complete model of piano combines all the above mentioned features. It is written as a variational formulation, before its discretization. Lagrange multipliers are introduced to materialize the forces applied by the strings at the bridge, and incorporated in the variational formulation in order to address the coupling conditions in a mathematical « weak » sense.

Writing a continuous model The difficulties of transforming a given continuous model into a discrete form arise first from the specificity of each problem, requiring the development of appropriate numerical methods. Further difficulties are due to the coupling between different parts of the system under consideration. An essential prerequisite is to guarantee the numerical stability of the problem, which can be lost if nonlinearities and couplings are not treated properly. In the context of the piano, the notes can sound up to 30 seconds. In addition, the spectral content often reaches to 10 kHz and more. It is thus usual to seek for numerical solutions containing up to 10 millions of time steps. Long-term stability is therefore mandatory.

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For these reasons, we have tried to obtain a priori estimates for the numerical solution, from a discrete equivalent of the energy identities obtained in the continuous case. Such a method has proved its efficiency in the past, for the modeling of other musical instruments exhibiting similar numerical difficulties (timpani : Rhaouti et al 1999, guitar : Derveaux et al 2003). Our goal is also to keep, in the numerical formulation, the reciprocal nature of the various coupling situations encountered in the continuous problem, while providing a good resolution efficiency. Finally, special care was taken to limit the numerical dispersion, which is an intrinsic and almost anavoidable drawback of any numerical method. Numerical dispersion can alter the simulation results in terms of frequency estimates. The requirements in this respect are very high in the context of musical acoustics, due to the sensitivity of the human ear. The space discretization is based on variational formulations of each subproblem (strings, soundboard and acoustics. After this procedure, a conformal Galerkin approximation is performed, which guarantees the stability of the semi-discrete problem. We use a high-order finite elements technique for the strings and acoustics problems. The acoustic degrees of freedom are taken in concordance with the points of numerical integration, in order to carry out mass lumping which reduces the cost of three-dimensional calculations. For the soundboard problem, because of the particular form of the damping (mode by mode), we choose as approximation space the space spanned by the first eigenmodes of the spatial operator. Since these modes are not known analytically, they are obtained by solving an eigenvalue problem with the high order finite element method. Time discretization must be performed with the aim to preserve, or attenuate, a discrete energy, from one time step to the next. Since the difficulties can be very different for each subproblem, we use specific and adapted numerical methods in time: • The discretization of the acoustic problem is done with an explicit leap-frog scheme, because of the large number of unknowns. For a realistic piano mesh, the inherent stability condition to this scheme imposes, a time step of 1 millisecond. Such a value is reasonable since the time step corresponding to the usual audio sampling rate is nearly equal to 20 milliseconds. The PML (perfectly matched layers) technique artificially truncates the computational domain. Despite the use of a massively parallel code, the resolution of the acoustic part of the problem remains the most expensive step in terms of computation time. • The formulation of the soundboard problem in the eigenmodes basis reduces it to a set of decoupled ordinary differential equations. Assuming that the interaction terms (originating from the strings and the sound pressure jump) are constant during a time step, it is possible to solve them analytically, ie without introducing any additional approximation. • The time discretization of the system of equations governing the strings vibration is probably the most novel and innovative part of our method. The most popular conservative schemes for wave equations are the family of thetaschemes, which have two drawbacks in the context of the piano string: first, they were originally designed for linear equations, and secondly, the less dispersive schemes among this family are subject to a CFL (stability) condition, which might be too restrictive in our case. Therefore we decided on the one hand to adopt a different discretization strotegies for the linear and the nonlinear part of the system respectively. On the other hand, we decided to develop, for the linear part, a dedicated variety of theta-schemes that enables us to ensure both stability and precision in the cases where two waves with very different speeds coexist within the same system of equations, as it is the case for piano strings. For the nonlinear part, we needed to develop new schemes based on the expression of a gradient approach, which ensures the conservation of a discrete energy, within the framework of a class of equations called « Hamiltonian systems of wave equations ». A priori estimates were proposed for these hybrid schemes. • Finally, the discretization of the coupling terms was also guided by the necessity of preserving a discrete energy. We choose to treat the couplings in a centered manner in order to preserve the order of accuracy of the methods chosen on independent problems. Moreover, the scheme involves implicit terms, in order to avoid a new stability condition (related to the coupling) and artificial numerical dissipation. This concerns the coupling of the hammer with the strings, but also the coupling between the strings and the table and finally the vibroacoustic coupling. • Last but not least, work was necessary to optimize the efficiency of the computer code, using adapted Lagrange multipliers and Schur complements. In this way, the update of the unknowns of each subsystem is made separately at each time step.

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