Page  00000001 Note and Hammer Velocity Dependence of a Piano String Model Based on Coupled Digital Waveguides Julien Bensa(12), Frederic Gibaudan(l), Kristoffer Jensen(3) and Richard Kronland-Martinet(1,2) (1) CNRS-LMA, 31 chemin J. Aiguier 13402 Marseille France (2) CCRMA, Stanford University, CA, USA (3) DIKU, University of Copenhagen, Universitetsparken 1, 2100 Kobenhavn DENMARK email: {bensa, gibaudan, kronland}, Abstract Previous works have shown that a combination of digital waveguide and signal models allows a perfect resynthesis of the sound generated by a string struck by a hammer. In particular the non-linear behavior of the hammer/string interaction is well reproduced. However, in an acoustic piano, two or three strings are struck by the same hammer, and the characteristics of the strings and the hammers are different from the bass to the treble. To take into account these considerations, the model for one piano string is extended so that it can generate notes produced by two and three strings. The behavior of the model parameters (the resonant part and the excitation part) is then studied, with respect to the velocities and the notes played. This research is an essential step in the design of a complete piano model. 1 Introduction This paper aims to extend previous works on piano analysis/synthesis. Bensa et al. (2000) showed that a source/resonator model allows the resynthesis of the vibration generated by a struck string. For that purpose, a digital waveguide simulating the resonator was used together with a signal model simulating the source. Even though the resynthesis is of very good quality for different velocities, this model doesn't allow playing on the full piano range. This is due to the fact that the strings are different (in number, mass, length, etc.). In this paper, the previous model is extended to model the vibrations generated by one, two or three coupled strings at the bridge level of a grand piano. The source is correlated to the energy the hammer inject in the coupled resonator. To calibrate this model a large number of measurements were made for each note of a grand piano. The behavior of the model parameters with respect to both the hammer velocity and the note played has been studied. 2 Design of the resonator model Direct problem. The digital waveguide is used as a resonator model for sound synthesis. This model is able to simulate the wave propagation in the strings at a very low computational cost. This synthesis algorithm was first proposed by Karplus and Strong (1983) and then investigated and improved by Jaffe and Smith (1983). The digital waveguide model represents, in terms of linear filters, the most important features of the propagation of waves between the two extremities of the string, namely the propagation delay, the dissipation and the dispersion phenomena. These filters can be estimated from the analysis of real sounds, constructed from the data of physical parameters, or set manually when used in a musical situation. In the middle and high range of the piano, two or three strings are struck for the same note. The sound produced is the result of a complex coupling between the strings. This coupling leads to phenomena like beats and double decays on the amplitude of the partials, which constitute a very important feature of the piano sound. Therefore, in order to resynthesize the vibration at the bridge level, one elementary waveguide is used for the one-string notes (Bensa et al. 2000) and a coupled digital waveguide model with two or three elementary waveguides is used for the two or three strings notes. The elementary and the two coupled waveguides are described in (Aramaki et al. 2001). The three coupled waveguide is represented in figure 1. Each individual string is modeled using an elementary waveguide (denoted G1, G2, G3). This elementary digital waveguide consists of a loop system that contains a pure delay filter (denoted Di, D2, D3), corresponding to the propagation delay for the wave to go back and forth along the string, and another filter (denoted Fi, F2, F3) which takes into account the dissipation, dispersion phenomena and boundary conditions. To design the coupled model, the output of an elementary waveguide was connected to the

Page  00000002 input of another waveguide using coupling filters. The coupling filters depend on the distance between the strings at the bridge level: in the case of the triplet, the coupling filters Ca of the adjacent strings are equal but different from the coupling filters Ce, between the extreme strings. this method, the modulus and the phase of each filter are obtained in the neighborhood of the resonances. The excitation is then extracted using a deconvolution process with respect to the waveguide transfer function. 3 Design of the source model The source (or excitation part) is modeled in the frequency domain in three parts. These are the static spectrum, which contains the spectral slope and the spectral modulation caused by the hammer impact position. The static spectrum model is linear, with the spectral modulation modeled with the log of a sinusoidal. The spectral tilt, which models the change in brightness when the hammer velocity is changed, and the energy model, which models the increase of energy when the hammer velocity is increased. All source modelization is done (in dB) in the frequency domain on the discrete partial frequencies or indexes. The source model parameters are found using a nonlinear curve fit (More 1977) minimizing perceptual criteria. It has already proven its quality (Bensa et al. 2000), both with respect to sound quality in analysis/synthesis, but also because it has a number of relevant performance parameters, including in particular hammer position, velocity and felt characteristics. Examples of the resulting excitation spectrum for the AO note are shown in Figure 2. It is clear that the brightness tilt model works well for the spectral slope change between the different velocities. Excitation for note AO, pp, mf and ff, org(stipled), model(solid) Figure 1. Three coupled elementary waveguide. The input E (the excitation of the model) is a direct consequence of the interaction between the hammer and the strings and corresponds to the amount of energy transferred to each partial of the resonant model. Its shape mainly depends on the velocity and the position of the hammer impact. The excitation is the same for each elementary waveguide, since it is supposed that the hammer strikes the strings in the same way. The output S corresponds to the vibration at the bridge level. Inverse problem. The calibration of our model has to be made carefully since the aim of this work is to make analysis-synthesis. The parameters are estimated using data collected on a real piano with an experimental setup described in part 4. The estimation method is partly similar to the one described in (Aramaki et al. 2001). With the analysis of only one signal (measured at the bridge level), all the parameters of the model are estimated. The analysis method can be summarized as follow. First, each partial of the measured signal is isolated using band-pass filtering. Then, a parametric method, the Steiglitz-McBride method (Steiglitz and McBride 1965), is used in the time domain to estimate the temporal parameters. As the sound is assumed to be a sum of exponentially decaying sinusoids, in the case of three strings, three amplitudes, damping coefficients and frequencies are extracted for each partial. In a second step, the analytical systems between those temporal parameters and the filters' parameters are solved. For three strings, unfortunately, only numerical solution can be found. With M a, 0) -Q. C-O 1500 2000 2500 frequency (Hz) 4000 Figure 2. mf, ff and pp excitations for the AO note. Estimated (dashed) and modeled (solid). 4 Experiment To collect real signals an accelerometer is positioned on the bridge of a Disklavier piano, close to the struck strings. The vibrations of the bridge are measured for each note of the piano with a medium velocity of the hammer. For some

Page  00000003 particular notes, several signals with different hammer velocities were recorded. MIDI velocity was measured and the relation between the velocity measured in MIDI code and the velocity of the hammer (in m/s) was determined to be linear by measuring velocities of four hammers with a photonic sensor for different MIDI velocities. 5 Behavior of the resonator With respect to the hammer velocity. Like most musical instruments, the spectral content of the piano sound depends on the playing style. This behavior, which is often nonlinear, is very important and it changes the sound in a way so that when the pianist plays forte, the sounds high partials contain more energy. In the piano case, this non-linearity is directly linked to the hammer-string interaction. Thus, it is the excitation and not the resonator that is responsible for this non-linearity. The source/resonance separation assumes that the resonant part characterizing the strings is independent of the hammer velocity (neglecting the nonlinearity of the strings). Figure 3 shows the amplitude of a filter of the resonant part. This filter is represented for different hammer velocities of the note AO. the amplitude weakly decrease. This non-linear behavior is not directly linked to the hammer string contact, but instead it is due to the non-linearity in the string itself: the larger the amplitude is, the more the internal losses are important. With respect to the note played. For a string of length L, the transformations of the wave during a back and forth propagation (2L long) are simulated by the loop filters representing the resonator. Since the strings have different lengths, the filters are normalized and compared between different notes. Therefore the elementary filter is considered, dF=Fl/(2L) However, this normalization is not entirely (physically) correct, since the filter F also takes into account losses due to energy transfer at the bridge level, which is independent of the string length. Figure 4 (top) shows filter 6F for notes A2, A4, F5, A6 (average filters for the triplets of strings). //:-\: ' ': ~\.~~AA2 S0.98.-o 0 0.96 F5 A4 A6 C,) 0 E 0.99 i, 0.97 ' 0.96 0.95 0.94 0.93 0.92 0.91 0.9 0 500 1000 1500 frequency (Hz) low elocity I 0.941 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 frequency (Hz) high velocity 2000 2500 Fl1# 0 E D1#\ AO 0.9L 0 1000 2000 3000 4000 frequency (Hz) 5000 6000 Figure 3. Amplitude of the filter F corresponding to the note AO as a function of the frequency and for several hammer velocities. The damping, which is due to the intrinsic losses within the string (viscoelastic, thermoelastic, etc.) and to the energy transfer to the bridge and to the soundboard (which also depends on frequency), is weak in the piano string. The bridge interaction probably is the cause of the irregularities of the filters. The amplitude of the filter (modulus) decreases while the frequency increases. Finally, by considering the behavior of this filter with respect to the hammer velocities, it is verified that their amplitudes are similar, which validates the use of this approach. Nevertheless, if the evolution of the higher partials with respect to hammer velocities is observed, it is noticed that Figure 4. Top: Amplitude of the filter 6F for the notes A2, A4, F5, A6. Bottom: Amplitude of the filter 6F for the notes AO, D1#, F1#. The amplitude of the filter seems to decrease as a function of the notes. In fact, the normalization gives more importance to energy transfer losses (caused by the bridge) for shorter strings, which entails an artificial decrease of 6F modulus. Internal string losses and energy transfer losses at the bridge should ideally be separated in order to correctly compare the filters. Figure 4 (bottom) shows 6F for three notes (AO, D1#, Fl#) corresponding to wrapped strings. The behavior of the amplitude has an opposite behavior than the high notes. This is explained by the difference in wrapping of the low note strings, which entails strong differences in the linear masses.

Page  00000004 6 Behavior of the excitation model The goal is now to investigate the evolution of the source model parameters (the static spectrum, spectral tilt and energy parameters) as a function of the pitch and velocity. The parameter estimation seems stable and reliable, and certainly good enough to be used in piano synthesis. The energy and brightness (Beauchamp 1982) for all notes and velocities are shown in Figure 5. 0Energy: AO (o), F1 (x), B1 (+), G2 (*), C3 (sq), G3 (di), D4 (v), E5 (A), F6 (<) The static spectrums for five notes are shown in Figure 6 (left). The spectral slopes are rising with the notes, if calculated in Hz. The spectral modulation strength is falling with pitch, whereas the spectral modulation rate (corresponding to the position of the hammer) is constant (up to G3) around the 8th partial, as expected. The spectral tilts for five notes are shown for pp, mf and ff, corresponding to the extreme and middle velocities, in Figure 6 (right). It is clear that the slope is falling for the pp and rising for theff in all cases. The mf slope is close to zero for all notes. In general, the spectral tilt is well modeled by a straight line, pivoting around a fixed point (which is constant at about 300 Hz, except for the high notes). 7 Conclusion S-1 -3C E 4 -4C -5C -6C 30 25 U) a) 20 0)15 S10 a n a ~" o 0 1 2 3 4 velocity (m/s) Excitation brightness: 5 6 7 This paper has shown that the model consisting of a resonant part and excitation part allows the resynthesis of all notes of the piano. The sounds obtained are perceptually indistinguishable from the originals. The resonant model simulates the coupling effect, which is perceptually very important. The filters obtained are stable, even if the variations in the physical parameters of the string do not yet vpermit the use of one universal filter. The source is well - I extracted using a deconvolution process. It corresponds very well to the source model consisting of three parts (static ie excitation for spectrum, spectral tilt and energy). The validity of the (symbol), model source model was verified for different notes and velocities. In order to reduce the number of parameters, a model of the note evolution is necessary, which this work has shown to pp and ff (solid), mf (+) be possible. 0 1 2 3 4 velocity (m/s) Figure 5. Energy and brightness for th different velocities and notes. Estimated (solid). Static Spectrum,: AO, B1, C3, D4, F6 o 0 00 a; _ -50 Ca 100 S,, 0 I -50 c -100 m- -100 S0 S-50 Ca -100 0 1000 2000 3000 4000 5000 6000 frequency (Hz) frequency (Hz) Spectral tilt A OL, S40 a 20 a -20 S-40 S40 420 140 20 0 1000 2000 3000 4000 5000 6000 frequency (Hz) E -20 0--20 -0 1000 2000 3000 4000 5000 6000 References Aramaki M., J. Bensa, L. Daudet, Ph. Guillemain, R. KronlandMartinet (2001), Resynthesis of coupled piano strings vibrations based on physical modeling, to be published in the Journal of New Music Research, Ed. Swets & Zeitlinger. Beauchamp, J., (1982), Synthesis by spectral amplitude and "Brightness" matching of analyzed musical instrument tones. J. Acoust. Eng. Soc., Vol. 30, No. 6, 396-406. Bensa J., K. Jensen, R. Kronland-Martinet, S. Ystad (2000), Perceptual and Analytical Analysis of the effect of the Hammer Impact on the Piano Tones, Proceedings of the International Computer Music Conference. International Computer Music Association, pp. 58-61. Jaffe D. and J.O. Smith (1983), Extensions of the Karplus-Strong Plucked String Algorithm, Computer Music Journal 7(2):56 -69. Karplus K.and A. Strong, (1983), Digital Synthesis of Plucked String and Drum Timbres, Computer Music Journal 7(2):43 -55. More, J.J., (1977) The Levenberg-Marquardt Algorithm: Implementation and Theory, Numerical Analysis, ed. G.A. Watson, Springer-Verlag, pp. 105-116. Steiglitz K. and L. E. McBride (1965), A technique for the Identification of Linear Systems, IEEE Trans. Automatic Control AC-10: pp. 461-464. Figure 6. Static Spectrum (left) and Spectral Tilt (forpp, mf (+) andff) for five notes. The estimated values, denoted with a symbol, and the model values (solid) are close in all cases, showing that the source model works well (for these perceptually important parameters) in the full playing range of the piano for all velocities. The asymptotic behavior (the change in the excitation is weaker for the high velocities) is clear for all notes.