Comparison between three different Viennese pianos of the nineteenth century

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1 String Instruments: Paper ICA Comparison between three different Viennese pianos of the nineteenth century Antoine Chaigne (a), Matthieu Hennet (b), Juliette Chabassier (c), Marc Duruflé (d) (a) University of Music and Performing Arts Vienna, Austria, (b) ENSTA ParisTech, France, (c) Inria Bordeaux, France, (d) Inria Bordeaux, France, Abstract Measurements are performed on three pianos built in Vienna during the 19th century by three generations of the Streicher family. These selected pianos are representative examples of the evolution of Viennese piano making. The first piano (NS19) was made in 1819 by Nanette Streicher: its structure is close to an harpsichord, with a thin soundboard and a single bridge. The second piano (JBS36) was built by her son, Johann Baptist, in Its soundboard has wider ribs, and its bridge is divided in two parts. The string scaling shows higher tensions compared to NS19. Finally, the third piano () was made by Emil Streicher, Johann Baptist s son, in This piano is larger than the two others. Its soundboard is thicker, again with an increase of tension compared to JBS36. Physical parameters relative to the geometry and material of the constitutive parts of the pianos (strings, hammers, soundboard) are derived from these measurements. These parameters serve here as input data for simulating vibrations of strings and soundboard, thanks to a time-domain model of a piano (Chabassier et al., Jasa 134(1), pp ) which couples together the hammer, the strings, the soundboard and the acoustic field. Fine adjustments of the parameters are made, by comparing measured and simulated waveforms. Further simulations are conducted with systematic variations of selected parameters (hammer mass, string tension, soundboard thickness,...). These variations, which would be hard (or even impossible) to achieve in the reality shed a new light on the links between physical parameters and sound quality, with an historical perspective on the art of piano making. Keywords: Historic pianos, measurements, simulations

2 Comparison between three different Viennese pianos of the nineteenth century 1 Introduction Investigating the acoustics of historic pianos shed useful light on the evolution of piano making. The present work is focused on three pianos successively made by three generations of the Streicher family, a famous dynasty of piano makers in Vienna, and are representative milestones of the transition from harpsichord to modern pianos, during the 19th century [1]. More specifically, the aim of the work presented here is to establish relationships between the development of the physical and geometrical parameters of the pianos, and the consecutive changes in tone quality. The method used is a combination of analysis and synthesis techniques. Measurements on the pianos were conducted in G. Hecher s atelier and in the Technical Museum in Vienna [2]. The significant parameters were extracted from the measurements of sounds and vibrations, using dedicated techniques [3]. These parameters serve as input data of a piano simulation model previously developed by Chabassier [4]. In Section 2, the main features of the piano physics subjected to evolution in the history of piano making are presented and discussed. Measurements and analysis are summarized in Section 3. In Section 4, it is shown how associated simulations can help in describing and predicting the changes in piano tone quality during its evolution. Focus is made here on the vibratory behavior of the piano. The soundboard-air coupling and the radiation properties of the instrument are left for a future study. 2 Physics of the piano and historic evolution T (N) L (mm) f1 (Hz) String A3 NS19 CG28 JBS36 JBS51 STD Piano Phi (mm) Zc (kg/s) String A3 NS19 CG28 JBS36 JBS51 STD Piano Figure 1: Piano evolution. Example of string A3. T : string tension at rest, L: string length, f 1 fundamental frequency, Phi: string diameter, Z c characteristic impedance. NS19: N. Streicher 1819, CG28: C. Graf 1828, JBS36 and JBS51: J.B. Streicher 1836 and 1851, : J.B. Streicher and Son 1873, STD: Steinway D (1977). 2

3 During the 19th century, the most important trend in the evolution of piano making is the continuous increase in string tension at rest T (see Figure 1). The intended goal was to increase the sound power of the instrument, in order to keep an appropriate dynamical balance with orchestras of increasing size. As a consequence, the length L and diameter Φ of the strings were regularly increasing, while the fundamental frequency f 1 also was tuned slightly higher with time. Another consequence is the increase of the characteristic impedance Z c of the strings which, in turn, induces a stronger coupling with the soundboard. In connection with the increase in tension, the thickness of the soundboard increases, in order to withstand the action of the strings. However, acting on the soundboard s thickness only was not sufficient in this respect. Therefore, the piano makers were forced to use first metal bars (around 185) and metallic frames (after 187) in order to alleviate the traction of the strings. Last, but not least, the excitation mechanism of the strings (mainly the hammer and the shank) were made progressively heavier, in order to increase the amplitude of the hammer striking force. All these structural modifications have important consequences on the timbral properties of the piano tones. In terms of analysis parameters, the effects can be seen on the following quantities: energies, energy ratio, temporal and spectral envelopes, damping factors, soundboard modes and transient spectrum, amplitude of string displacement and consecutive nonlinear effects, hammer force pulse (shape, duration, amplitude), fine structure of the spectra: phantom partials, inharmonicity. Below is a short survey of the main physical phenomena and analysis quantities modified by the evolution of construction parameters in the history of piano making (see also [5]): Hammer-string contact. The main physical parameters involved are: hammer mass and velocity, string mass and tension. These parameters have consequences on hammer force duration, shape and magnitude. For example: the shorter the force pulse, the broader is the excited spectrum. Wave propagation on the string. For constant length L and fundamental frequency f 1 (constant wave speed c = 2L f 1 ), and for a constant material (metal density ρ), the changes in diameter influence both the inharmonicity and the nonlinear coefficients. The nonlinear coefficients govern the density and amplitudes of the phantom partials [6]. String-soundboard coupling. In case of low tension, the vibratory energy transferred from strings to soundboard in ancient instruments is weak, compared to modern pianos. As a consequence, the energy dissipation in the strings is low, and the decay times are longer. Soundboard-air coupling. Due to the regularly increase of soundboard thickness during the nineteenth century, the critical frequency decreases. This implies that the upper frequency limit from which the radiation efficiency becomes significant decreases. In the spectral domain, this is reflected by a relative higher damping of the higher frequencies for the thicker soundboards. This property can be seen on both the soundboard acceleration and sound spectra (see Section 3). In the time-domain, it was shown in a previous work that the Soundboard Energy Ratio (SER) is high for thin soundboards during the initial transients, because of a high kinetic energy, but that this energy decreases faster than for 3

4 thick soundboards [7]. In terms of acoustic energy, the magnitude of the Acoustic Energy Ratio (AER) increases substantially by increasing the global stiffness of the soundboard, without increasing its mass. This is practically achieved by stiffening the soundboard with ribs, and giving it some curvature (or crown). Spectral content of the transients. The soundboard is a key element that gives to each piano its unique character. This is due, in particular, to its eigenmodes. Each played note excites a subset of eigenmodes, depending on the location of the coupling point at the bridge. The excited modes are present in the transients all the piano tones. They are damped more rapidly than the string s partials, because of the dissipation in the wood. 3 Measurements and analysis The analysis performed on the selected pianos (NS19, JBS36 and ) is made according to the physical principles summarized in the previous section. An overview of the most representative results are presented in this Section. For clarity, the differences between the two extreme pianos (NS19 and ) are emphasized, but the properties of the intermediate piano (JBS36) are in between. 3.1 String excitation and hammer force Hammer force (N) NS Time (ms) Figure 2: Measured hammer forces. Comparison between NS19 and pianos. Note D 3. As a first result, one can notice in Figure 2 that the amplitudes of the hammer force pulses are significantly different, for the same note and for comparable impact velocity. Here, the hammer forces and striking velocities are derived from measurements of the string velocity, following the method presented in [3]. For the NS19 and pianos, the duration are the same, but the magnitude of the more recent piano () is nearly eight times higher than the older one (NS19). This can be partly explained by the hammer mass ratio ( 2), and partly by the tension ratio ( 4). In case of low tension, the whole string is pulled by the hammer and the reaction of the string balancing the hammer action is low. This figure illustrates the fact that, in the evolution of its construction, the piano allows more and more mechanical power to be fed 4

5 into the strings. This is one condition for increasing the sound power. 3.2 Time and spectral analysis Magnitude (a. u.) Time (s) Magnitude (db) NS Frequency (khz) Figure 3: Measured temporal (left) and spectral (right) envelopes of soundboard accelerations. Note A3. Comparison between NS19 (red) and piano (black). Figure 3 shows the measured temporal and spectral envelopes of the soundboard acceleration for the note A3 played at comparable levels on the NS19 and JBS73 pianos, respectively. In the time-domain, whatever the recording points on the soundboard, the piano shows higher amplitudes and shorter decay times than the NS19. In the spectral domain, when the spectrum is normalized with respect to the highest partial s magnitude, one can see that the spectrum of the NS19 piano (calculated over one second duration) shows relatively more energy in the high frequencies. In other words, the spectral envelope is broader. One can roughly say that the NS19 spectrum is still close to the one of a harpsichord, whereas the is closer to a modern piano, which is confirmed by auditory evaluation. 3.3 Soundboard properties and transient analysis.8 Soundboard impulse - JBS36-4 Magnitude (a. u.) Amplitude (db) Time (s) Frequency (khz) Figure 4: Experimental modal analysis of the soundboards. Example JBS36. As mentioned in Section 2, the main functions of the soundboard are to ensure appropriate 5

6 transmission of energy from the strings at the input, and radiation of sound in the acoustic field at the output (see also [5]). Besides this, the soundboard itself is a resonator with eigenmodes depending on its geometry, material and boundary conditions. The initial transient of piano tones is usually sharp, so that it excites a large number of soundboard modes. The amplitude of the excited eigenmodes depend on the location of the coupling point (more precisely, coupling area) between the string and the soundboard. Because of both the finite duration of the string pulse and finite coupling area, the number of excited modes is always smaller that the number of modes predicted by a numerical model of the soundboard. In order to detect the soundboard modes involved in the coupling of one given note, the soundboard was excited at the bridge coupling area with an impact hammer, all the strings being muted with felt strips. The resulting acceleration of the soundboard in this area is analyzed with the help of high-resolution spectral methods (see Figure 4). An important part of the soundboard modes 6 Amplitude (db) Frequency (khz) Figure 5: Spectral analysis of the measured transients. Soundboard acceleration excited with a D 3 note. Piano. excited by impulses are present in the normal excitation by string forces, as seen in Figure 5. These modes are damped more rapidly that the strings partials. Therefore the mean value of their amplitude are significantly smaller than those of the strings (-2 to -3 db in the presented D 3 example). However, removing them shows that they are clearly audible, especially in the higher range (above the note C4) where the strings partials are progressively more spaced apart from each other. Since the internal damping in the wooden soundboard increases with frequency, it becomes hard in practice to detect the soundboard modes in transients above 1kHz. 4 Simulations The analysis of piano signals, for the same note played on different pianos, yields global differences due to simultaneous variations between several construction parameters. One interest of simulations is to examine the effect of each variation separately, which allows to quantify its relative weigth in the observed global differences. In addition, fine and systematic variations can be done for each parameter. Finally, the previously developed method of energy analysis yields useful informations on the whole piano, for one given played note, irrespective of the observation point (microphone or accelerometer) [7]. 6

7 4.1 Simulation model The model used in the simulations is extensively described in several papers [4], [8]. In this model, damped stiff nonlinear strings excited by a hammer are coupled to the soundboard, via the bridge. The soundboard is coupled to the external air. The set of equations describing the system is solved in the time domain String excitation 1.5 String velocity (m/s) Time (ms) Figure 6: Comparison between measured (solid) and simulated (dashed) string velocity. String A3. Piano NS19. In the simulations presented in this Section, the virtual strings are excited by hammer force pulses derived from measurements [3]. The relevance of the force pulses is validated by comparing the resulting simulated string motion with the measurements (see Figure 6) Soundboard modeling and eigenmodes Figure 7: Three soundboards. (Left) NS19. (Center) JBS36. (Right). The evolution of the piano is reflected in the design of the soundboard. In addition to increasing thickness, the soundboard is reinforced by ribs in order to increase its rigidity, without increasing 7

too much its mass. This can be viewed in Figure 7, as well as the presence of an apron on the keyboard side. Also the width of the soundboard increase, due to the increasing number of notes.

8 too much its mass. This can be viewed in Figure 7, as well as the presence of an apron on the keyboard side. Also the width of the soundboard increase, due to the increasing number of notes. As shown in [7], increasing the rigidity contributes to enhance the radiated sound power. In the present work, each soundboard was meshed using the gmsh free software [9]. A Figure 8: Examples of soundboard modes. JBS36. modal analysis is performed up to 6-7 khz, which represents more than 2 modes. Some examples of modes are shown in Figure 8. The coupling with the strings is made in the modal space, after addition of a damping term in each modal equation. The law of damping with frequency is derived from experiments. 4.2 Energy analysis Magnitude (db) NS19 Magnitude (db) NS Frequency (khz) Frequency (khz) Figure 9: (Left) Spectral analysis of the simulated soundboard energy. (Right) Phantom partials. Note D 3. Comparison between NS19 and pianos. As shown in [7], one attractive feature of the energy analysis of the piano tones is to provide information irrespective of the location of the observations points (acceleration, pressure). Calculating the energy terms in simulations are straightforward since, for each time step, the values of the quantities of interest (velocity, pressure,..) are available on each point of the discrete spatial grid. In experiments, gathering so many data is technically feasible, using 8

9 scanning vibrometers and microphones, or dedicated sensors and probes. One can mention, for example, the derivation of radiated sound power from intensity measurements. However, such measurements usually are long and cumbersome, and the results are often limited in frequency due to the dimensions of the experimental spatial grid. In this respect, simulations can be seen as a valuable alternative and/or complementary approach. Figure 9 shows a comparative spectral analysis of the soundboard energy for the simulated note D 3 played on the pianos NS19 and. The left figure shows that the NS19 spectrum has more energy above 3 khz than the, a result which is coherent with the experimental spectral envelope shown in Figure 3 for the note A3 played on the same pianos. Zooming on this spectrum around 1 khz shows a number of peaks interleaved between the string s partials. Accurate measurements of the additional peaks shows that the frequencies of these peaks are all equal to the sum of two of the strings partials. These peaks are the so-called phantom partials resulting from quadratic non-linearity consecutive to the finite displacement of the string. In the present example, the magnitude of the string s displacement is larger for the NS19 piano than for the piano, which causes a relative higher amplitude for the phantom partials. Finally, the time evolution of the soundboard energy is shown in Soundboard Energy Ratio 1-5 NS Time (s) SER NS Time (s) Figure 1: Soundboard energy ratio (SER) vs time. Simulated notes A3 and D 3. Comparison between NS19 and pianos. Figure 1 for the two simulated notes A3 and D 3 played on the NS19 and pianos. In this Figure, the soundboard energy is normalized by the total energy imparted to the piano for the played note, thus giving the Soundboard Energy Ratio (SER). Here again, one can see that the temporal envelopes are coherent with the experimental results shown in Figure 3. The SER is higher for the piano, but the decay times are shorter than for the NS19. As detailed in Section 2, this result is a global consequence of differences in hammer excitation, string tension and soundboard properties of both pianos. 5 Conclusion In this paper, some significant differences between three Viennese pianos built during the 19th century were exhibited and quantified. With the help of dedicated simulations, the specific role of hammer, strings and soundboard parameters in these differences were underlined and 9

10 discussed. The simulations also enhance the fact that time and spectral differences are not only due to the properties of the isolated elements, but also on their coupling. In addition, the simulations provide us with energetic quantities which yield a global view on the properties of the instruments. Future studies will be devoted to the soundboard-air coupling in connection with the properties of the acoustic field surrounding the instruments. Acknowledgements This work was supported by the Lise-Meitner-Fellowship M1653-N3 of the Austrian Science Fund (FWF). The author wishes to thank Alex Mayer (MDW), Caroline Haas and Michael Kirchweger (Technical Museum Vienna), and Gert Hecher (Das Klavier-Atelier, Vienna) for their help in the measurements. The simulations presented in this paper were carried out using the PLAFRIM experimental platform, being developed under the Inria PlaFRIM development action with support from Bordeaux INP, LABRI and IMB and other entities: Conseil Régional d Aquitaine, Université de Bordeaux and CNRS (and ANR in accordance to the programme d investissements d avenir (see References [1] Donhauser, P.; Langer, A. Streicher: Drei Generationen Klavierbau in Wien (in German). Verlag Christoph Dohr, Köln (Germany), 214. [2] Chaigne, A.; Hecher, G. Measurements on historic pianos. Proceedings of the Third Vienna Talk on Music Acoustics, Vienna, Austria, September 215, p 115. [3] Chaigne, A. Reconstructing the piano hammer force from measurements and filtering of the string velocity, J. Acoust. Soc. Am., Vol 139 (4), Pt. 2, 216, p 211. [4] Chabassier, J.; Chaigne, A.; Joly, P. Modeling and simulation of a grand piano. J. Acoust. Soc. Am., Vol 134 (1), 213, pp [5] Chaigne, A.; Kergomard, J. Acoustics of Musical Instruments, Springer, New York (USA), 216. [6] Conklin, H. A. Generation of partials due to nonlinear mixing in a stringed instrument. J. Acoust. Soc. Am., Vol 15, 1999, pp [7] Chaigne, A.; Chabassier, J.; Duruflé, M. Energy Analysis of Structural Changes in Pianos. Proceedings of the Third Vienna Talk on Music Acoustics, Vienna, Austria, September 215, pp [8] Chabassier, J.; Chaigne, A.; Joly, P. Time domain simulation of a piano. Part 1: model description. European Series in Appl. and Ind. Math:M2AN, Vol 48 (5), pp [9] Geuzaine, C.; Remacle, J.F. Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. International Journal for Numerical Methods in Engineering, Vol 79 (11), 29, pp

Acoustics of pianos: An update of recent results

Acoustics of pianos: An update of recent results Antoine Chaigne Department of Music Acoustics (IWK) University of Music and Performing Arts Vienna (MDW) chaigne@mdw.ac.at Projekt Nr P29386-N30 1 Summary