Page  00000001 Perceptual and Analytical Analysis of the effect of the Hammer Impact on the Piano Tones Julien Bensa('), Kristoffer Jensen(2), Richard Kronland-Martinetl' and S0lvi Ystad(l' (1) C.N.R.S. Laboratoire de M6canique et d'Acoustique, 31 chemin J. Aiguier 13402 Marseille FRANCE {bensa, kronland, ystad} @lma.cnrs-mrs.fr (2) DIKU, University of Copenhagen, Universitetsparken 1, 2100 Kobenhavn DENMARK krist@diku.dk ABSTRACT This paper presents a model of the excitation corresponding to the hammer impact in a hybrid piano model. An experimental setup permits the measurement of the signal at the bridge location. The signal is separated into an excitation and resonant contribution. This work focuses on the modeling of the excitation part using both physical and perceptually relevant parameters. The resulting model can be used both in analysis/synthesis and transformation. Important piano performance parameters such as position and velocity of the hammer are inherent in the model. INTRODUCTION The aim of this paper is to model the non-linear behavior of the sound generated by the interaction between a hammer and a piano string. This work is situated in the general analysis-synthesis context, which means that the objective is not only to be able to simulate sounds, but also to perfectly reconstruct a given sound. Many articles on physical modeling of the hammer-string interaction have been published (Boutillon, 1988), (Askenfelt et al., 1993), (Hall, 1992). From a synthesis point of view, the parameters intervening in these models are difficult to fit for a given piano and the sounds generated this way are not identical to the original ones. Signal models offer an alternative approach to the synthesis of piano tones since the main perceptual characteristics of the piano sounds resides in a specific amplitude envelope and in a well established relation between slightly inharmonic partials (Fletcher et al., 1990). These parameters can be estimated from the analysis of real sounds (Kronland-Martinet et al., 1997) but such a synthesis approach hides important relations between the sounds and the physics of the instrument. This makes the control of signal models difficult, although perceptually inspired models can help (Jensen, 1999). Hybrid models combining signal and physical models have shown to be a good compromise between these two approaches, allowing the most important physical features of the sound generator, together with the most important perceptual aspects of the sound to be taken into account. Such hybrid models have been successfully used to resynthezise flute sounds (Ystad, 1998). Synthesis methods for piano tones have been proposed (Van Duyne et al., 1994), (Van Duyne et al., 1995) but the analysissynthesis problem was not considered. In this paper, we propose a hybrid approach to model the effect of the hammer impact on the piano tones based on a source-resonant scheme. The resonant part corresponds to the string and is modeled using a waveguide model. The source or excitation part, which is a consequence of the interaction between the hammer and string, is modeled using a signal synthesis technique closely related to both physical and perceptual parameters. This source resonance separation is valid, even if it is well known that the timbre of the piano tone is mainly due to the interaction of the hammer and the string. Nevertheless, this interaction which generates an "initial condition" for the propagation of waves in the string is not directly perceived. The duration of the interaction is about 3 milliseconds (Askenfelt et al., 1993) and perceptual experiments have shown that removing the first milliseconds of a piano sound doesn't alter its perception (Schaeffer, 1966). The construction of the model together with the estimation of the parameters are based on an experimental setup which boundary conditions are close to the case of a real piano. An accelerometer placed on the bridge measures the vibrations generated by the interaction between the hammer and the string while a laser vibrometer measures the speed at which the hammer hits the string. A waveguide model is constructed to simulate the propagation of the waves in the string. This model makes it possible to remove the "string contribution" from the signal recorded by the accelerometer in order to extract the excitation signal. The non-linear behavior of the excitation signal is further modeled by perceptual and analytical analysis.

Page  00000002 EXPERIMENTATION Since the sound radiated from the soundboard mainly originates from the bridge vibration, we have designed an experimental setup, which permits the measurement of the vibration of the bridge for various velocities of the hammer. acceleration measurement bridge concrete string s Support hammer laser vibrometer Q nut key voltage control Figure 1. Experimental setup. This setup constitutes a massive concrete support with a very low fundamental frequency compared to the string. On the top of the structure, we have fixed a piece of a bridge taken from an real piano and a block system, which permits to tighten the string. On the other extremity of the structure, we have fixed a hard wood support on which a piece of a nut is screwed. The string is then tightened between the bridge and the nut making our setup close to the conditions of a real piano. To study the influence of the hammer velocity, the string is struck with a hammer linked to a Yamaha Disklavier key. By imposing different voltages to the system, one can control the hammer velocity in a reasonably reproducible way. The precise velocity is measured immediately after escapement using a laser vibrometer pointing at the head of the hammer. The vibration of the string at its mobile extremity is measured with an accelerometer placed on the bridge. We have collected twelve acceleration signals corresponding to twelve hammer velocities linearly spaced between 0.3m-s'1 and 1.3m-s1. A RESYNTHESIS MODEL FOR THE PIANO STRING To model the propagation in the piano string, we use a waveguide model (Smith, 1992), (Ystad et al., 1996). It consists in a loop system with a delay line (D) which corresponds to the time the waves need to go back and forth in the medium, and a filter (F) taking into account dissipation, dispersion phenomena, as well as the boundary conditions. The filters D and F characterize the internal phenomena intervening in the string. The transfer function of the waveguide model is given by S(w) F(w)eD (1 ET() = -oD E(w) 1- F(w)e The input E (the excitation of the model) is a direct consequence of the interaction between the hammer and the string, and its form depends on the velocity of the hammer. The output S of the waveguide model is the integral of the vibration measured by the accelerometer on the bridge of the experimental setup (the velocity of the bridge). From the measured signals we estimate temporal parameters thanks to a parametric method applied to each partial of the real sound. Thanks to analytical relations between those parameters and the parameters of the model, we are able to calculate the filters (Daudet et al., 1999), (Ystad 1998) and the excitation in order to do a perfect resynthesis from a perceptual point of view. GENERAL BEHAVIOR OF THE EXCITATION SIGNAL The behavior of the excitation signal with respect to the velocity of the hammer can be represented by the evolution of its spectral components. Moreover, since the string only selects certain frequencies, we shall study the behavior of the spectral components of the excitation at frequencies corresponding to the partials generated by the string. From the data obtained for twelve values of the hammer velocity, we have extracted a set of twelve excitation signals, using the estimation method described in (Daudet et al., 1999) and (Aramaki et al., 2000). 10in5 Spectral components 3.5 1 I - I - I - I - I - I - I - I - I 2 O cz 3.5 2 3.5 8 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 velocity (m/s) Figure 2. Behavior of the spectral components of the excitation signal as a function of the hammer velocity. Figure 2 shows the evolution of the first 8 components of the excitation as a function of the hammer velocity. We clearly see that the amplitude of the partials regularly increases with the velocity. The mean slope of the curves depends on the rank of the component, showing the non linear effect of the hammer impact. One way of modeling this behavior is by fitting a second order polynomial to each curve. This analysis leads to an additive synthesis model for the excitation signal and it allows a perfect resynthesis of the original sound, for any velocity, from a perceptive point of view but it doesn't allow an intuitive and perceptive control of the sound.

Page  00000003 PERCEPTUAL MODEL OF THE EXCITATION SIGNAL The second model of the excitation signal proposed here aims at defining a few perceptive and physically related parameters in order to control the synthesized sound. The model is created from a prior idea of what the excitation consists of. It is divided into three parts, The energy The static spectrum The brightness tilt The energy is a function of the velocity of the hammer, whereas the static spectrum (mean spectral envelope) depends on, for instance, the position, but not on the velocity of the hammer. The spectral tilt models the increase of brightness as a function of the hammer velocity. This separation not only limits the number of parameters but it also suggests a way to control the excitation with the physical parameters at hand, such as the speed and the position of the hammer. The excitation model is shown in figure 3. 12)Ei(w) (E (W)) = 1 log( Eio) 12 i A (3) The static excitation model has two parameters; the spectral slope and the position of the hammer, which defines the silent modes of the sound. The silent modes could also have been taken into account in the waveguide model (Guillemain et al. 1997). The spectral envelope of the static excitation is modeled as a second order polynomial to account for the flattening in the high frequencies. The logarithm of a sinusoid, the frequency of which depends on the position of the hammer, is superposed on this polynomial. The resulting mean spectrum of the excitation is shown in figure 4. Static Excitation 01 1 1 ----1----1-I -10 -20 -30 -40 --50 Static Spectrum Brightness tilt Gain \\ k~ /\ Il iI\\ i \ J I\11 -9 0 1 1 1 1 1 1 1 1 5 10 15 20 25 30 35 partial index Figure 4. Static excitation: original (dashed), modeled (solid). 40 Brightness tilt The increase of energy in the high partials due to the compression of the felt is represented by the brightness tilt, Figure 3. Excitation model. Separation The separation is made by first normalizing the 12 excitations with the L2 norm (the square root of the energy). Then the log of the normalized excitation is taken, and finally the mean excitation is subtracted from the normalized excitations. The excitation can be recreated by the following formula, B, (w) =log(E(W)) - (E,(o)) -A (4) This function, which represents the change in brightness as a function of the velocity, can be seen in figure 5. Briahtness tilt E(O)=- AeB (a)+(El()) (2) where A is the energy, B(co) is the spectral tilt due to the different velocities of the hammer and <Ei(c)> is the mean logarithmic spectrum of the excitations, which is independent of the hammer speed. Energy The square root of the energy is linearly proportional to the hammer velocity, and is easily modeled as a linear function. Static excitation The static excitation, which is the mean of the log of the normalized excitations is calculated as, -g 0 S-5 Figure 5. Brightness tilt due to the different hammer velocities (shown to the right in m/s).

Page  00000004 The spectral tilt can be successfully modeled by a linear model, the tilt of which is changed with the hammer velocity. The difference in slope is greater for the lower velocities than for the higher ones. The brightness tilt is modeled as, Bt(Wk)=(BM - Bm e-Pv)" k (5 where k is the partial index, v is the velocity and Bm is a constant defining the minimum brightness. Since the constant 1j is positive, this function tends asymptotically with increasing v towards BM, setting the maximum brightness. This model is consistent with the compression characteristics of the felt (Boutillon 1988). Perceptual fit of the model The coefficients of the excitation model are found by minimizing the brightness error e= B-B (6 where ||.|12is the L2 norm and, B = E (kE(wk) kT(ok)) / (E(ok)" T(kw,)) (7: k k is the centroid, which is closely correlated with the perception of brightness (Beauchamp 1982), (McAdams et al. 1995). Care must be taken to include the effect of the transfer function of the waveguide model in the calculation of the coefficients of the excitation model, as otherwise, the resulting brightness could have been biased by the transfer function. CONCLUSION This paper has shown that the non-linear behavior of the sound generated by the interaction between a hammer and a piano string is efficiently modeled using hybrid techniques. It has been shown that waveguide models can be excited by a signal model, the parameters of which are closely related to perceptual criteria. This makes the control of the hybrid model easy since the control parameters are correlated to both physical characteristics of the system and perceptual effects. The parameters of the model are estimated from the analysis of real sounds, making the resynthesis of a given sound possible. For that purpose, techniques have been developed separating the signal measured on an experimental setup into an excitation contribution and a resonant contribution. Each contribution is then modeled independently; the resonance by a waveguide model, the excitation by a physically related signal model, the parameters of which are calculated by minimizing perceptual criteria. The final model allows analysis-synthesis of real sounds and transformations by acting on the parameters. Both physical (length, tension, material of the string; position and velocity of the hammer) and perceptual (brightness, temporal and spectral behavior) characteristics of the source can be modified. This work is part of a more general project of piano modeling and it is currently extended to models of the coupling of strings and radiation of the soundboard. REFERENCES Boutillon, X. 1988, "Model for piano hammers: ) Experimental determination and digital simulation", J. Acoust. Soc. Am., 83(2): 746-754. 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