EURASIP Journal on Applied Signal Processing 2004:7, 926–933c 2004 Hindawi Publishing Corporation Physical Modeling of the Piano N. Giordano Department of Physics, Purdue University, 525 Northwestern Avenue, West Lafayette, IN 47907-2036, USA Email:
[email protected] M. Jiang Department of Physics, Purdue University, 525 Northwestern Avenue, West Lafayette, IN 47907-2036, USA Department of Computer Science, Montana State University, Bozeman, MT 59715, USA Email:
[email protected] Received 21 June 2003; Revised 27 October 2003 A project aimed at constructing a physical model of the piano is described. Our goal is to calculate the sound produced by the instrument entirely from Newton’s laws. The structure of the model is described along with experiments that augment and test the model calculations. The state of the model and what can be learned from it are discussed. Keywords and phrases: physical modeling, piano. 1. INTRODUCTION This paper describes a long term project by our group aimed at physical modeling of the piano. The theme of this volume, model based sound synthesis of musical instruments, is quite broad, so it is useful to begin by discussing precisely what we mean by the term “physical modeling.” The goal of our project is to use Newton’s laws to describe all aspects of the piano. We aim to use F = ma to calculate the motion of the hammers, strings, and soundboard, and ultimately the sound that reaches the listener. Of course, we are not the first group to take such a New- ton’s law approach to the modeling of a musical instrument. For the piano, there have been such modeling studies of the hammer-string interaction [1, 2, 3, 4, 5, 6, 7, 8, 9], string vi- brations [8, 9, 10], and soundboard motion [11]. (Nice re- views of the physics of the piano are given in [12, 13, 14, 15].) There has been similar modeling of portions of other instru- ments (such as the guitar [16]), and of several other com- plete instruments, including the xylophone and the timpani [17, 18, 19]. Our work is inspired by and builds on this pre- vious work. At this point, we should also mention how our work re- lates to other modeling work, such as the digital waveguide approach, which was recently reviewed in [20]. The digital waveguide method makes extensive use of physics in choos- ing the structure of the algorithm; that is, in choosing the proper filter(s) and delay lines, connectivity, and so forth, to properly match and mimic the Newton’s law equations of motion of the strings, soundboard, and other components of the instrument. However, as far as we can tell, certain fea- tures of the model, such as hammer-string impulse func- tions and the transfer function that ultimately relates the sound pressure to the soundboard motion (and other sim- ilar transfer functions), are taken from experiments on real instruments. This approach is a powerful way to produce re- alistic musical tones efficiently, in real time and in a man- ner that can be played by a human performer. However, this approach cannot address certain questions. For example, it would not be able to predict the sound that would be pro- duced if a radically new type of soundboard was employed, or if the hammers were covered with a completely differ- ent type of material than the conventional felt. The physi- cal modeling method that we describe in this paper can ad- dress such questions. Hence, we view the ideas and method embodied in work of Bank and coworkers [20] (and the ref- erences therein) as complementary to the physical modeling approach that is the focus of our work. In this paper, we describe the route that we have taken to assembling a complete physical model of the piano. This complete model is really composed of interacting sub- models which deal with (1) the motions of the hammers and strings and their interaction, (2) soundboard vibrations, and (3) sound generation by the vibrating soundboard. For each of these submodels we must consider several issues, includ- ing selection and implementation of the computational algo- rithm, determination of the values of the many parameters that are involved, and testing the submodel. After consider- ing each of the submodels, we then describe how they are combined to produce a complete computational piano. The Physical Modeling of the Piano 927 quality of the calculated tones is discussed, along with the lessons we have learned from this work. A preliminary and abbreviated report on this project was given in [21]. 2. OVERALL STRATEGY ANDGOALS One of the first modeling decisions that arises is the question of whether to work in the frequency domain or the time do- main. In many situations, it is simplest and most instructive to work in the frequency domain. For example, an under- standing of the distribution of normal mode frequencies, and the nature of the associated eigenvectors for the body vibra- tions of a violin or a piano soundboard, is very instructive. However, we have chosen to base our modeling in the time domain. We believe that this choice has several advantages. First, the initial excitation—in our case this is the motion of a piano hammer just prior to striking a string—is described most conveniently in the time domain. Second, the interac- tion between various components of the instrument, such as the strings and soundboard, is somewhat simpler when viewed in the time domain, especially when one considers the early “attack” portion of a tone. Third, our ultimate goal is to calculate the room pressure as a function of time, so it is appealing to start in the time domain with the hammer mo- tion and stay in the time domain throughout the calculation, ending with the pressure as would be received by a listener. Our time domain modeling is based on finite difference cal- culations [10] that describe all aspects of the instrument. A second element of strategy involves the determination of the many parameters that are required for describing the piano. Ideally, one would like to determine all of these pa- rameters independently, rather than use them as fitting pa- rameters when comparing the modeling results to real (mea- sured) tones. This is indeed possible for all of the parame- ters. For example, dimensional parameters such as the string diameters and lengths, soundboard dimensions, and bridge positions, can all be measured from a real piano. Likewise, various material properties such as the string stiffness, the elastic moduli of the soundboard, and the acoustical proper- ties of the room in which the numerical piano is located, are well known from very straightforward measurements. For a few quantities, most notably the force-compression charac- teristics of the piano hammers, it is necessary to use separate (and independent) experiments. This brings us to a third element of our modeling strategy—the problem of how to test the calculations. The final output is the sound at the listener, so one could “test” the model by simply evaluating the sounds via listening tests. However, it is very useful to separately test the submod- els. For example, the portion of the model that deals with soundboard vibrations can be tested by comparing its pre- dictions for the acoustic impedance with direct measure- ments [11, 22, 23, 24]. Likewise, the room-soundboard com- putation can be compared with studies of sound production by a harmonically driven soundboard [25]. This approach, involving tests against specially designed experiments, has proven to be extremely valuable. The issue of listening tests brings us to the question of goals, that is, what do we hope to accomplish with such a modeling project? At one level, we would hope that the cal- culated piano tones are realistic and convincing. The model could then be used to explore what various hypothetical pi- anos would sound like. For example, one could imagine con- structing a piano with a carbon fiber soundboard, and it would be very useful to be able to predict its sound ahead of time, or to use the model in the design of the new sound- board. On a different and more philosophical level, one might want to ask questions such as “what are the most im- portant elements involved in making a piano sound like a pi- ano?” We emphasize that it is not our goal to make a real time model, nor do we wish to compete with the tones produced by other modeling methods, such as sampling synthesis and digital waveguide modeling [20]. 3. STRINGS ANDHAMMERS Our model begins with a piano hammer moving freely with a speed v h just prior to making contact with a string (or strings, since most notes involve more than one string). Hence, we ignore the mechanics of the action. This mechanics is, of course, quite important from a player’s perspective, since it determines the touch and feel of the instrument [26]. Nev- ertheless, we will ignore these issues, since (at least to a first approximation) they are not directly relevant to the compo- sition of a piano tone and we simply take v h as an input pa- rameter. Typical values are in the range 1–4 m/s [9]. When a hammer strikes a string, there is an interaction force that is a function of the compression of the hammer felt, y f . This force determines the initial excitation and is thus a crucial factor in the composition of the resulting tone. Considerable effort has been devoted to understanding the hammer-string force [1, 2, 3, 4, 5, 6, 7, 27, 28, 29, 30, 31, 32, 33]. Hammer felt is a very complicated material [34], and there is no “first principles” expression for the hammer- string force relation F h (y f ). Much work has assumed a sim- ple power law function F h ¸ y f = F 0 y p f , (1) where the exponent p is typically in the range 2.5–4 and F 0 is an overall amplitude. This power law form seems to be at least qualitatively consistent with many experiments and we therefore used (1) in our initial modeling calculations. While (1) has been widely used to analyze and inter- pret experiments, and also in previous modeling work, it has been known for some time that the force-compression characteristic of most real piano hammers is not a simple reversible function [7, 27, 28, 29, 30]. Ignoring the hystere- sis has seemed reasonable, since the magnitude of the ir- reversibility is often found to be small. Figure 1 shows the force-compression characteristic for a particular hammer (a Steinway hammer from the note middle C) measured in two different ways. In the type I measurement, the hammer struck a stationary force sensor and the resulting force and felt compression were measured as described in [31]. We see 928 EURASIP Journal on Applied Signal Processing Hammer force characteristics Hammer C4 Type I exp. Type II exp. 0 10 20 F h ( N ) 0 0.2 0.4 0.6 y f (mm) Figure 1: Force-compression characteristics measured for a partic- ular piano hammer measured in two different ways. In the type I ex- periment (dotted curve), the hammer struck a stationary force sen- sor and the resulting force, F h , and felt compression, y f , were mea- sured. The initial hammer velocity was approximately 1 m/s. The solid curve is the measured force-compression relation obtained in a type II measurement, in which the same hammer impacted a pi- ano string. This behavior is described qualitatively by (2), with pa- rameters p = 3.5, F 0 = 1.0 ×10 13 N, 0 = 0.90, and τ 0 = 1.0 ×10 −5 second. The dashed arrows indicate compression/decompression branches. that for a particular value of the felt compression, y f , the force is larger during the compression phase of the hammer- string collision than during decompression. However, this difference is relatively small, generally no more than 10% of the total force. Provided that this hysteresis is ignored, the type I result is described reasonably well by the power law function (1) with p ≈ 3. However, we will see below that (1) is not adequate for our modeling work, and this has led us to consider other forms for F h . In order to shed more light on the hammer-string force, we developed a new experimental approach, which we refer to as a type II experiment, in which the force and felt com- pression are measured as the hammer impacts on a string [32, 35]. Since the string rebounds in response to the ham- mer, the hammer-string contact time in this case is consider- ably longer (by a factor of approximately 3) than in the type I measurement. The force-compression relation found in this type II measurement is also shown in Figure 1. In contrast to the type I measurements, the type II results for F h (y) do not consist of two simple branches (one for compression and an- other for decompression). Instead, the type II result exhibits “loops,” which arise for the following reason. When the ham- mer first contacts the string, it excites pulses that travel to the ends of the string, are reflected at the ends, and then re- turn. These pulses return while the hammer is still in contact with the string, and since they are inverted by the reflection, they cause an extra series of compression/decompression cy- cles for the felt. There is considerable hysteresis during these cycles, much more than might have been expected from the type I result. The overall magnitude of the type II force is also somewhat smaller; the hammer is effectively “softer” under the type II conditions. Since the type II arrangement is the one found in real piano, it is important to use this hammer- force characteristic in modeling. We have chosen to model our hysteretic type II hammer measurements following the proposal of Stulov [30, 33]. He has suggested the form F h ¸ y f (t) = F 0 ¸ g ¸ y f (t) − 0 −∞ t g ¸ y f (t ) exp ¸ −(t −t )/τ 0 dt . (2) Here, τ 0 is a characteristic (memory) time scale associated with the felt, 0 is a measure of the magnitude of the hystere- sis, and y f (t) is the variation of the compression with time. In other words, (2) says that the felt “remembers” its pre- vious compression history over a time of order τ 0 , and that the force is reduced according to how much the felt has been compressed during that period. The inherent nonlinearity of the hammer is specified by the function g(z); Stulov took this to be a power law g(z) = z p . (3) Stulov has compared (2) to measurements with real ham- mers and reported very good agreement using τ 0 , 0 , p, and F 0 as fitting parameters. Our own tests of (2) have not shown such good agreement; we have found that it provides only a qualitative (and in some cases semiquantitative) description of the hysteresis shown in Figure 1 [35]. Nevertheless, it is currently the best mathematical description available for the hysteresis, and we have employed it in our modeling calcula- tions. Our string calculations are based on the equation of mo- tion [8, 10, 36] ∂ 2 y ∂t 2 = c 2 s ¸ ∂ 2 y ∂x 2 − ∂ 4 y ∂x 4 −α 1 ∂y ∂t + α 2 ∂ 3 y ∂t 3 , (4) where y(x, t) is the transverse string displacement at time t and position x along the string. c s ≡ µ/T is the wave speed for an ideal string (with stiffness and damping ignored), with T the tension and µ the mass per unit length of the string. When the parameters , α 1 , and α 2 are zero, this is just the simple wave equation. Equation (4) describes only the po- larization mode for which the string displacement is parallel to the initial velocity of the hammer. The other transverse mode and also the longitudinal mode are both ignored; ex- periments have shown that both of these modes are excited in real piano strings [37, 38, 39], but we will leave them for future modeling work. The term in (4) that is proportional to arises from the stiffness of the string. It turns out that c s = r 2 s E s /ρ s , where r s , E s , and ρ s are the radius, Young’s Physical Modeling of the Piano 929 modulus, and density of the string, respectively, [9, 36]. For typical piano strings, is of order 10 −4 , so the stiffness term in (4) is small, but it cannot be neglected as it produces the well-known effect of stretched octaves [36]. Damping is ac- counted for with the terms involving α 1 and α 2 ; one of these terms is proportional to the string velocity, while the other is proportional to ∂ 3 y/∂t 3 . This combination makes the damp- ing dependent on frequency in a manner close to that ob- served experimentally [8, 10]. Our numerical treatment of the string motion employs a finite difference formulation in which both time t and posi- tion x are discretized in units ∆t s and ∆x s [8, 9, 10, 40]. The string displacement is then y(x, t) ≡ y(i∆x s , n∆t s ) ≡ y(i, n). If the derivatives in (4) are written in finite difference form, this equation can be rearranged to express the string dis- placement at each spatial location i at time step n+1 in terms of the displacement at previous time steps as described by Chaigne and Askenfelt [8, 10]. The equation of motion (4) does not contain the hammer force. This is included by the addition of a term on the right-hand side proportional to F h , which acts at the hammer strike point. Since the ham- mer has a finite width, it is customary to spread this force over a small length of the string [8]. So far as we know, the details of how this force is distributed have never been mea- sured; fortunately our modeling results are not very sensitive to this factor (so long as the effective hammer width is qual- itatively reasonable). With this approach to the string calcu- lation, the need for numerical stability together with the de- sired frequency range require that each string be treated as 50–100 vibrating numerical elements [8, 10]. 4. THE SOUNDBOARD Wood is a complicated material [41]. Soundboards are as- sembled from wood that is “quarter sawn,” which means that two of the principal axes of the elastic constant tensor lie in the plane of the board. The equation of motion for such a thin orthotropic plate is [11, 22, 23, 42] ρ b h b ∂ 2 z ∂t 2 = −D x ∂ 4 z ∂x 4 − ¸ D x ν y + D y ν x + 4D xy ∂ 4 z ∂x 2 ∂y 2 −D y ∂ 4 z ∂y 4 + F s (x, y) −β ∂z ∂t , (5) where the rigidity factors are D x = h 3 b E x 12 ¸ 1 −ν x ν y , D y = h 3 b E y 12 ¸ 1 −ν x ν y , D xy = h 3 b G xy 12 . (6) Here, our board lies in the x − y plane and z is its displace- ment. (These x and y directions are, of course, not the same as the x and y coordinates used in describing the string mo- tion.) The soundboard coordinates x and y run perpendic- ular and parallel to the grain of the board. E x and ν x are Young’s modulus and Poisson’s ration for the x direction, and so forth for y, G xy is the shear modulus, h b is the board thickness and ρ b is its density. The values of all elastic con- stants were taken from [41]. In order to model the ribs and bridges, the thickness and rigidity factors are position depen- dent (since these factors are different at the ribs and bridges than on the “bare” board) as described in [11]. There are also some additional terms that enter the equation of mo- tion (5) at the ends of bridges [11, 17, 18, 43]. F s (x, y) is the force from the strings on the bridge. This force acts at the appropriate bridge location; it is proportional to the com- ponent of the string tension perpendicular to the plane of the board, and is calculated from the string portion of the model. Finally, we include a loss term proportional to the parameter β [11]. The physical origin of this term involves elastic losses within the board. We have not attempted to model this physics according to Newton’s laws, but have sim- ply chosen a value of β which yields a quality factor for the soundboard modes which is similar to that observed experi- mentally [11, 24]. 1 Finally, we note that the soundboard “acts back” on the strings, since the bridge moves and the strings are attached to the bridge. Hence, the interaction of strings in a unison group, and also sympathetic string vibrations (with the dampers disengaged from the strings) are included in the model. For the solution of (5), we again employed a finite dif- ference algorithm. The space dimensions x and y were dis- cretized, both in steps of size ∆x b ; this spatial step need not be related to the step size for the string ∆x s . As in our previous work on soundboard modeling [11], we chose ∆x b = 2 cm, since this is just small enough to capture the structure of the board, including the widths of the ribs and bridges. Hence, the board was modeled as ∼ 100 ×100 vibrating elements. The behavior of our numerical soundboard can be judged by calculations of the mechanical impedance, Z, as defined by Z = F v b , (7) where F is an applied force and v b is the resulting sound- board velocity. Here, we assume that F is a harmonic (single frequency) force applied at a point on the bridge and v b is measured at the same point. Figure 2 shows results calculated from our model [11] for the soundboard from an upright pi- ano. Also shown are measurements for a real upright sound- board (with the same dimensions and bridge positions, etc., as in the model). The agreement is quite acceptable, espe- cially considering that parameters such as the dimensions of the soundboard, the position and thickness of the ribs and bridges, and the elastic constants of the board were taken 1 In principle, one might expect the soundboard losses to be frequency dependent, as found for the string. At present there is no good experimental data on this question, so we have chosen the simplest possible model with just a single loss term in (5). 930 EURASIP Journal on Applied Signal Processing Experiment Soundboard impedance Upright piano at middle C Model 100 200 500 1000 2000 5000 10 4 Z ( k g / s ) 100 1000 10 4 f (Hz) Figure 2: Calculated (solid curve) and measured (dotted curve) mechanical impedance for an upright piano soundboard. Here, the force was applied and the board velocity was measured at the point where the string for middle C crosses the bridge. Results from [11, 24]. from either direct measurements or handbook values (e.g., Young’s modulus). 5. THE ROOM Our time domain room modeling follows the work of Bottel- dooren [44, 45]. We begin with the usual coupled equations for the velocity and pressure in the room ρ a ∂v x ∂t = − ∂p ∂x , ρ a ∂v y ∂t = − ∂p ∂y , ρ a ∂v z ∂t = − ∂p ∂z , ∂p ∂t = ρ a c 2 a ¸ − ∂v x ∂x − ∂v y ∂y − ∂v z ∂z , (8) where p is the pressure, the velocity components are v x , v y , and v z , ρ a is the density, and c a is the speed of sound in air. This family of equations is similar in form to an electromag- netic problem, and much is known about how to deal with it numerically. We employ a finite difference approach in which staggered grids in both space and time are used for the pres- sure and velocity. Given a time step ∆t r , the pressure is com- puted at times n∆t r while the velocity is computed at times (n+1/2)∆t r . A similar staggered grid is used for the space co- ordinates, with the pressure calculated on the grid i∆x r , j∆x r , k∆x r , while v x is calculated on the staggered grid (i+1/2)∆x r , j∆x r , and k∆x r . The grids for v y and v z are arranged in a sim- ilar manner, as explained in [44, 45]. Sound is generated in this numerical room by the vibra- tion of the soundboard. We situate the soundboard from the previous section on a plane perpendicular to the z direction in the room, approximately 1 mfromthe nearest parallel wall (i.e., the floor). At each time step the velocity v z of the room air at the surface of the soundboard is set to the calculated soundboard velocity at that instant, as obtained from the soundboard calculation. The room is taken to be a rectangular box with the same acoustical properties for all 6 walls. The walls of the room are modeled in terms of their acoustic impedance, Z, with p = Zv n , (9) where v n is the component of the (air) velocity normal to the wall [46]. Measurements of Z for a number of materials [47] have found that it is typically frequency dependent with the form Z(ω) ≈ Z 0 − iZ ω , (10) where ω is the angular frequency. Incorporating this fre- quency domain expression for the acoustic impedance into our time domain treatment was done in the manner de- scribed in [45]. The time step for the room calculation was ∆t r = 1/22050 ≈ 4.5 × 10 −4 s, as explained in the next section. The choice of spatial step size ∆x r was then influenced by two considerations. First, in order for the finite difference al- gorithm to be numerically stable in three dimensions, one must have ∆x r /( √ 3∆t r ) > c a . Second, it is convenient for the spatial steps for the soundboard and room to be commen- surate. In the calculations described below, the room step size was ∆x r = 4 cm, that is, twice the soundboard step size. When using the calculated soundboard velocity to obtain the room velocity at the soundboard surface, we averaged over 4 soundboard grid points for each room grid point. Typical numerical rooms were 3×4×4 m 3 , and thus contained ∼ 10 6 finite difference elements. Figure 3 shows results for the sound generation by an upright soundboard. Here, the soundboard was driven har- monically at the point where the string for middle C contacts the bridge, and we plot the sound pressure normalized by the board velocity at the driving point [25]. It is seen that the model results compare well with the experiments. This pro- vides a check on both the soundboard and the room models. 6. PUTTINGIT ALL TOGETHER Our model involves several distinct but coupled sub- systems—the hammers/strings, the soundboard, and the room—and it is useful to review how they fit together com- putationally. The calculation begins by giving some initial velocity to a particular hammer. This hammer then strikes a string (or strings), and they interact through either (1) or (2). This sets the string(s) for that note into motion, and these Physical Modeling of the Piano 931 Experiment Sound generation Soundboard driven at C4 Model 0.5 1 2 5 10 20 p / v b ( a r b . u n i t s ) 20 100 10 3 10 4 Frequency (Hz) Figure 3: Results for the sound pressure normalized by the sound- board velocity for an upright piano soundboard: calculated (solid curve) and measured (dotted curve). The board was driven at the point where the string for middle C crosses the bridge. Results from [25]. in turn act on the bridge and soundboard. As we have al- ready mentioned, the vibrations of each component of our model are calculated with a finite difference algorithm, each with an associated time step. Since the systems are coupled— that is, the strings drive the soundboard, the soundboard acts back on the strings, and the soundboard drives the room— it would be computationally simpler to use the same value of the time step for all three subsystems. However, the equa- tion of motion for the soundboard is highly dispersive, and stability requirements demand a much smaller time step for the soundboard than is needed for string and room simula- tions. Given the large number of room elements, this would greatly (and unnecessarily) slow down the calculation. We have therefore chosen to instead make the various time steps commensurate, with ∆t r = 1 22050 ¸ s, ∆t s = ∆t r 4 , ∆t b = ∆t s 6 , (11) where the subscripts correspond to the room (r), string (s), and soundboard (b). To explain this hierarchy, we first note that the roomtime step is chosen to be compatible with com- mon audio hardware and software; 1/∆t r is commensurate with the data rates commonly used in CD sound formats. We then see that each room time step contains 4 string time steps; that is, the string algorithm makes 4 iterations for each iteration of the room model. Likewise, each string time step contains 6 soundboard steps. The overall computational speed is currently somewhat less than “real time.” With a typical personal computer (clock speed 1 GHz), a 1 minute simulation requires approximately 30 minutes of computer time. Of course, this gap will nar- row in the future in accord with Moore’s law. In addition, the model should transfer easily to a cluster (i.e., multi-CPU) machine. We have also explored an alternative approach to the room modeling involving a ray tracing approach [48]. Ray tracing allows one to express the relationship between soundboard velocity and sound pressure as a multiparame- ter map, involving approximately 10 4 parameters. The values of these parameters can be precalculated and stored, resulting in about an order of magnitude speed-up in the calculation as compared to the room algorithm described above. 7. ANALYSIS OF THE RESULTS: WHAT HAVE WE LEARNEDANDWHERE DOWE GONEXT? In the previous section, we saw that a real-time Newton’s law simulation of the piano is well within reach. While such a simulation would certainly be interesting, it is not a primary goal of our work. We instead wish to use the modeling to learn about the instrument. With that in mind, we now con- sider the quality of the tones calculated with the current ver- sion of the model. In our initial modeling, we employed power law ham- mers described by (1) with parameters based on type I hammer experiments by our group [31]. The results were disappointing—it is hard to accurately describe the tones in words, but they sounded distinctly plucked and somewhat metallic. While we cannot include our calculated sounds as part of this paper, they are available on our website http://www.physics.purdue.edu/piano. After many modeling calculations, we came to the conclusion that the hammer model—for example, the power lawdescription (1)—was the problem. Note that we do not claim that power law ham- mers must always give unsatisfactory results. Our point is that when the power law parameters are chosen to fit the type I behavior of real hammers, the calculated tones are poor. It is certainly possible (and indeed, likely) that power law param- eters that will yield good piano tones can be found. How- ever, based on our experience, it seems that these parameters should be viewed as fitting parameters, as they may not ac- curately describe any real hammers. This led us to the type II hammer experiments described above, and to a description of the hammer-string force in terms of the Stulov function (2), with parameters (τ 0 , 0 , etc.) taken from these type II experiments [35]. The results were much improved. While they are not yet “Steinway quality,” it is our opinion that the calculated tones could be mistaken for a real piano. In that sense, they pass a sort of acoustical Turing test. Our conclusion is that the hammers are an es- sential part of the instrument. This is hardly a revolutionary result. However, based on our modeling, we can also make a somewhat stronger statement: in order to obtain a real- istic piano tone, the modeling should be based on hammer parameters observed in type II measurements, with the hys- teresis included in the model. There are a number of issues that we plan to address in the future. (1) The hammer portion of the model still 932 EURASIP Journal on Applied Signal Processing needs attention. Our experiments [35] indicate that while the Stulov function does provide a qualitative description of the hammer force hysteresis, there are significant quan- titative differences. It may be necessary to develop a bet- ter functional description to replace the Stulov form. (2) As it currently stands, our string model includes only one polarization mode, corresponding to vibrations parallel to the initial hammer velocity. It is well known that the other transverse polarization mode can be important [37]. This can be readily included, but will require a more general soundboard model since the two transverse modes couple through the motion of the bridge. (3) The soundboard of a real piano is supported by a case. Measurements in our laboratory indicate that the case acceleration can be as large as 5% or so of the soundboard acceleration, so the sound emitted by the case is considerable. (4) We plan to refine the room model. Our current room model is certainly a very crude approximation to a realistic room. Real rooms have wall coverings of various types (with differing values of the acoustic impedances), and contain chairs and other objects. At our current level of sophistication, it appears that the hammers are more of a limitation than the room model, but this may well change as the hammer modeling is improved. In conclusion, we have made good progress in developing a physical model of the piano. It is now possible to produce realistic tones using Newton’s laws with realistic and inde- pendently determined instrument parameters. Further im- provements of the model seem quite feasible. We believe that physical modeling can provide new insights into the piano, and that similar approaches can be applied to other instru- ments. ACKNOWLEDGMENTS We thank P. Muzikar, T. Rossing, A. Tubis, and G. Weinre- ich for many helpful and critical discussions. We also are in- debted to A. Korty, J. Winans II, J. Millis, S. Dietz, J. Jourdan, J. Roberts, and L. Reuff for their contributions to our piano studies. This work was supported by National Science Foun- dation (NSF) through Grant PHY-9988562. REFERENCES [1] D. E. Hall, “Piano string excitation in the case of small ham- mer mass,” Journal of the Acoustical Society of America, vol. 79, no. 1, pp. 141–147, 1986. [2] D. E. Hall, “Piano string excitation II: General solution for a hard narrow hammer,” Journal of the Acoustical Society of America, vol. 81, no. 2, pp. 535–546, 1987. [3] D. E. Hall, “Piano string excitation III: General solution for a soft narrow hammer,” Journal of the Acoustical Society of America, vol. 81, no. 2, pp. 547–555, 1987. [4] D. E. Hall and A. Askenfelt, “Piano string excitation V: Spectra for real hammers and strings,” Journal of the Acoustical Society of America, vol. 83, no. 4, pp. 1627–1638, 1988. [5] D. E. Hall, “Piano string excitation. 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Brooke, “Synthesis of guitar tones from fundamental parameters relating to con- struction,” Proceedings of the Institute of Acoustics, vol. 12, no. 1, pp. 757–764, 1990. [17] A. Chaigne and V. Doutaut, “Numerical simulations of xy- lophones. I. Time-domain modeling of the vibrating bars,” Journal of the Acoustical Society of America, vol. 101, no. 1, pp. 539–557, 1997. [18] V. Doutaut, D. Matignon, and A. Chaigne, “Numerical simu- lations of xylophones. II. Time-domain modeling of the res- onator and of the radiated sound pressure,” Journal of the Acoustical Society of America, vol. 104, no. 3, pp. 1633–1647, 1998. [19] L. Rhaouti, A. Chaigne, and P. Joly, “Time-domain model- ing and numerical simulation of a kettledrum,” Journal of the Acoustical Society of America, vol. 105, no. 6, pp. 3545–3562, 1999. [20] B. Bank, F. Avanzini, G. Borin, G. De Poli, F. Fontana, and D. Rocchesso, “Physically informed signal processing meth- ods for piano sound synthesis: a research overview,” EURASIP Journal on Applied Signal Processing, vol. 2003, no. 10, pp. 941–952, 2003. [21] N. Giordano, M. Jiang, and S. Dietz, “Experimental and com- putational studies of the piano,” in Proc. 17th International Congress on Acoustics, vol. 4, Rome, Italy, September 2001. [22] J. Kindel and I.-C. Wang, “Modal analysis and finite ele- ment analysis of a piano soundboard,” in Proc. 5th Interna- tional Modal Analysis Conference, pp. 1545–1549, Union Col- lege, Schenectady, NY, USA, 1987. [23] J. Kindel, “Modal analysis and finite element analysis of a piano soundboard,” M.S. thesis, University of Cincinnati, Cincinnati, Ohio, USA, 1989. Physical Modeling of the Piano 933 [24] N. Giordano, “Mechanical impedance of a piano sound- board,” Journal of the Acoustical Society of America, vol. 103, no. 4, pp. 2128–2133, 1998. [25] N. Giordano, “Sound production by a vibrating piano sound- board: Experiment,” Journal of the Acoustical Society of Amer- ica, vol. 104, no. 3, pp. 1648–1653, 1998. [26] A. Askenfelt and E. V. Jansson, “From touch to string vibra- tions. II. The motion of the key and hammer,” Journal of the Acoustical Society of America, vol. 90, no. 5, pp. 2383–2393, 1991. [27] T. Yanagisawa, K. Nakamura, and H. Aiko, “Experimental study on force-time curve during the contact between ham- mer and piano string,” Journal of the Acoustical Society of Japan, vol. 37, pp. 627–633, 1981. [28] T. Yanagisawa and K. Nakamura, “Dynamic compression characteristics of piano hammer,” Transactions of Musical Acoustics Technical Group Meeting of the Acoustic Society of Japan, vol. 1, pp. 14–17, 1982. [29] T. Yanagisawa and K. Nakamura, “Dynamic compression characteristics of piano hammer felt,” Journal of the Acous- tical Society of Japan, vol. 40, pp. 725–729, 1984. [30] A. Stulov, “Hysteretic model of the grand piano hammer felt,” Journal of the Acoustical Society of America, vol. 97, no. 4, pp. 2577–2585, 1995. [31] N. Giordano and J. P. Winans II, “Piano hammers and their force compression characteristics: does a power law make sense?,” Journal of the Acoustical Society of America, vol. 107, no. 4, pp. 2248–2255, 2000. [32] N. Giordano and J. P. Millis, “Hysteretic behavior of pi- ano hammers,” in Proc. International Symposium on Musi- cal Acoustics, D. Bonsi, D. Gonzalez, and D. Stanzial, Eds., pp. 237–240, Perugia, Umbria, Italy, September 2001. [33] A. Stulov and A. M¨ agi, “Piano hammer: Theory and experi- ment,” in Proc. International Symposium on Musical Acoustics, D. Bonsi, D. Gonzalez, and D. Stanzial, Eds., pp. 215–220, Pe- rugia, Umbria, Italy, September 2001. [34] J. I. Dunlop, “Nonlinear vibration properties of felt pads,” Journal of the Acoustical Society of America, vol. 88, no. 2, pp. 911–917, 1990. [35] N. Giordano and J. P. Millis, “Using physical modeling to learn about the piano: New insights into the hammer-string force,” in Proc. International Congress on Acoustics, S. Furui, H. Kanai, and Y. Iwaya, Eds., pp. III–2113, Kyoto, Japan, April 2004. [36] N. H. Fletcher and T. D. Rossing, The Physics of Musical In- struments, Springer-Verlag, New York, NY, USA, 1991. [37] G. Weinreich, “Coupled piano strings,” Journal of the Acous- tical Society of America, vol. 62, no. 6, pp. 1474–1484, 1977. [38] M. Podlesak and A. R. Lee, “Dispersion of waves in piano strings,” Journal of the Acoustical Society of America, vol. 83, no. 1, pp. 305–317, 1988. [39] N. Giordano and A. J. Korty, “Motion of a piano string: lon- gitudinal vibrations and the role of the bridge,” Journal of the Acoustical Society of America, vol. 100, no. 6, pp. 3899–3908, 1996. [40] N. Giordano, Computational Physics, Prentice-Hall, Upper Saddle River, NJ, USA, 1997. [41] V. Bucur, Acoustics of Wood, CRC Press, Boca Raton, Fla, USA, 1995. [42] S. G. Lekhnitskii, Anisotropic Plates, Gordon and Breach Sci- ence Publishers, New York, NY, USA, 1968. [43] J. W. S. Rayleigh, Theory of Sound, Dover, NewYork, NY, USA, 1945. [44] D. Botteldooren, “Acoustical finite-difference time-domain simulation in a quasi-Cartesian grid,” Journal of the Acoustical Society of America, vol. 95, no. 5, pp. 2313–2319, 1994. [45] D. Botteldooren, “Finite-difference time-domain simulation of low-frequency room acoustic problems,” Journal of the Acoustical Society of America, vol. 98, no. 6, pp. 3302–3308, 1995. [46] P. M. Morse and K. U. Ingard, Theoretical Acoustics, Princeton University Press, Princeton, NJ, USA, 1986. [47] L. L. Beranek, “Acoustic impedance of commercial materials and the performance of rectangular rooms with one treated surface,” Journal of the Acoustical Society of America, vol. 12, pp. 14–23, 1940. [48] M. Jiang, “Room acoustics and physical modeling of the piano,” M.S. thesis, Purdue University, West Lafayette, Ind, USA, 1999. N. Giordano obtained his Ph.D. from Yale University in 1977, and has been at the De- partment of Physics at Purdue University since 1979. His research interests include mesoscopic and nanoscale physics, compu- tational physics, and musical acoustics. He is the author of the textbook Computational Physics (Prentice-Hall, 1997). He also col- lects and restores antique pianos. M. Jiang has a B.S. degree in physics (1997) from Peking University, China, and M.S. degrees in both physics and computer sci- ence (1999) from Purdue University. Some of the work described in this paper was part of his physics M.S. thesis. After graduation, he worked as a software engineer for two years, developing Unix kernel software and device drivers. In 2002, he moved to Boze- man, Montana, where he is now pursuing a Ph.D. in computer science in Montana State University. Minghui’s current research interests include the design of algorithms, compu- tational geometry, and biological modeling and bioinformatics. various material properties such as the string stiffness. soundboard dimensions. we ignore the mechanics of the action. especially when one considers the early “attack” portion of a tone. 29. it has been known for some time that the force-compression characteristic of most real piano hammers is not a simple reversible function [7. 32. 30]. When a hammer strikes a string. STRINGS AND HAMMERS One of the first modeling decisions that arises is the question of whether to work in the frequency domain or the time domain. This brings us to a third element of our modeling strategy—the problem of how to test the calculations. one might want to ask questions such as “what are the most important elements involved in making a piano sound like a piano?” We emphasize that it is not our goal to make a real time model. since (at least to a first approximation) they are not directly relevant to the composition of a piano tone and we simply take vh as an input parameter. and also in previous modeling work. it is simplest and most instructive to work in the frequency domain. 31. the interaction between various components of the instrument. can all be measured from a real piano. Second. Our time domain modeling is based on finite difference calculations [10] that describe all aspects of the instrument. so one could “test” the model by simply evaluating the sounds via listening tests. In the type I measurement. along with the lessons we have learned from this work. dimensional parameters such as the string diameters and lengths. 28. 28. Our model begins with a piano hammer moving freely with a speed vh just prior to making contact with a string (or strings. 22. For example. since most notes involve more than one string). For example. A second element of strategy involves the determination of the many parameters that are required for describing the piano. First. This is indeed possible for all of the parameters. involving tests against specially designed experiments. 2. and the nature of the associated eigenvectors for the body vibrations of a violin or a piano soundboard. an understanding of the distribution of normal mode frequencies. For a few quantities. 27. one could imagine constructing a piano with a carbon fiber soundboard. 30. 23. the room-soundboard computation can be compared with studies of sound production by a harmonically driven soundboard [25]. has proven to be extremely valuable. 5. Likewise. nor do we wish to compete with the tones produced by other modeling methods. one would like to determine all of these parameters independently. rather than use them as fitting parameters when comparing the modeling results to real (measured) tones. there is an interaction force that is a function of the compression of the hammer felt. p (1) where the exponent p is typically in the range 2. Nevertheless. 6. we would hope that the calculated piano tones are realistic and convincing. This force determines the initial excitation and is thus a crucial factor in the composition of the resulting tone. is somewhat simpler when viewed in the time domain. most notably the force-compression characteristics of the piano hammers. The model could then be used to explore what various hypothetical pianos would sound like. We see . the initial excitation—in our case this is the motion of a piano hammer just prior to striking a string—is described most conveniently in the time domain. since it determines the touch and feel of the instrument [26]. 3. is very instructive. However. This approach. For example. the hammer struck a stationary force sensor and the resulting force and felt compression were measured as described in [31]. ending with the pressure as would be received by a listener. we have chosen to base our modeling in the time domain. 33]. We believe that this choice has several advantages. such as the strings and soundboard. Considerable effort has been devoted to understanding the hammer-string force [1. our ultimate goal is to calculate the room pressure as a function of time. since the magnitude of the irreversibility is often found to be small. Ideally. On a different and more philosophical level. it is necessary to use separate (and independent) experiments. For example. Hammer felt is a very complicated material [34]. 27. quite important from a player’s perspective. A preliminary and abbreviated report on this project was given in [21]. y f . and the acoustical properties of the room in which the numerical piano is located. 2.5–4 and F0 is an overall amplitude. Likewise. so it is appealing to start in the time domain with the hammer motion and stay in the time domain throughout the calculation. of course. the portion of the model that deals with soundboard vibrations can be tested by comparing its predictions for the acoustic impedance with direct measurements [11.Physical Modeling of the Piano quality of the calculated tones is discussed. Typical values are in the range 1–4 m/s [9]. are well known from very straightforward measurements. it is very useful to separately test the submodels. the elastic moduli of the soundboard. Ignoring the hysteresis has seemed reasonable. and bridge positions. 4. 29. that is. However. While (1) has been widely used to analyze and interpret experiments. or to use the model in the design of the new soundboard. and it would be very useful to be able to predict its sound ahead of time. OVERALL STRATEGY AND GOALS 927 The issue of listening tests brings us to the question of goals. Third. such as sampling synthesis and digital waveguide modeling [20]. what do we hope to accomplish with such a modeling project? At one level. 3. Hence. Figure 1 shows the force-compression characteristic for a particular hammer (a Steinway hammer from the note middle C) measured in two different ways. 7. This power law form seems to be at least qualitatively consistent with many experiments and we therefore used (1) in our initial modeling calculations. we will ignore these issues. This mechanics is. The final output is the sound at the listener. and there is no “first principles” expression for the hammerstring force relation Fh (y f ). In many situations. Much work has assumed a simple power law function Fh y f = F0 y f . 24]. Instead. 35]. The term in (4) that is proportional to arises from the stiffness of the string. 0 = 0. and τ0 = 1. 38.” which arise for the following reason. The inherent nonlinearity of the hammer is specified by the function g(z). they cause an extra series of compression/decompression cy- Stulov has compared (2) to measurements with real hammers and reported very good agreement using τ0 . 0 0. t) is the transverse string displacement at time t and position x along the string. This behavior is described qualitatively by (2). (3) that for a particular value of the felt compression.928 EURASIP Journal on Applied Signal Processing cles for the felt. Our string calculations are based on the equation of motion [8. Young’s . The force-compression relation found in this type II measurement is also shown in Figure 1. When the hammer first contacts the string. 36] 2 ∂2 y ∂4 y ∂y ∂3 y 2 ∂ y − α1 = cs − + α2 3 . this difference is relatively small. Equation (4) describes only the polarization mode for which the string displacement is parallel to the initial velocity of the hammer. the force is larger during the compression phase of the hammerstring collision than during decompression. p. We have chosen to model our hysteretic type II hammer measurements following the proposal of Stulov [30. The overall magnitude of the type II force is also somewhat smaller. When the parameters . we developed a new experimental approach. and then return. cs ≡ µ/T is the wave speed for an ideal string (with stiffness and damping ignored). The initial hammer velocity was approximately 1 m/s. and since they are inverted by the reflection. where rs . There is considerable hysteresis during these cycles. were measured. Es . the type II results for Fh (y) do not consist of two simple branches (one for compression and another for decompression). The other transverse mode and also the longitudinal mode are both ignored. and that the force is reduced according to how much the felt has been compressed during that period.90. generally no more than 10% of the total force. ∂t 2 ∂x2 ∂x4 ∂t ∂t (4) where y(x. However. y f . These pulses return while the hammer is still in contact with the string. He has suggested the form Fh y f (t) = F0 g y f (t) − 0 Type II exp. In the type I experiment (dotted curve).4 0. much more than might have been expected from the type I result.6 −∞ 20 Hammer force characteristics Hammer C4 Fh (N) 10 Type I exp. (2) says that the felt “remembers” its previous compression history over a time of order τ0 . The solid curve is the measured force-compression relation obtained in a type II measurement. It turns out that cs = rs2 Es /ρs . and ρs are the radius. and α2 are zero. τ0 is a characteristic (memory) time scale associated with the felt. the hammer struck a stationary force sensor and the resulting force. and y f (t) is the variation of the compression with time. y f .0 × 10−5 second. Provided that this hysteresis is ignored. 39].0 × 1013 N. 33]. in which the same hammer impacted a piano string. with T the tension and µ the mass per unit length of the string. experiments have shown that both of these modes are excited in real piano strings [37. it excites pulses that travel to the ends of the string. Since the string rebounds in response to the hammer.2 y f (mm) 0. in which the force and felt compression are measured as the hammer impacts on a string [32. (2) Figure 1: Force-compression characteristics measured for a particular piano hammer measured in two different ways. and F0 as fitting parameters. In order to shed more light on the hammer-string force. Since the type II arrangement is the one found in real piano. 0 is a measure of the magnitude of the hysteresis. 10. The dashed arrows indicate compression/decompression branches. and this has led us to consider other forms for Fh . the hammer is effectively “softer” under the type II conditions. we will see below that (1) is not adequate for our modeling work. which we refer to as a type II experiment. are reflected at the ends. the hammer-string contact time in this case is considerably longer (by a factor of approximately 3) than in the type I measurement. this is just the simple wave equation. F0 = 1. Here. 0 . α1 . 0 t g y f (t ) exp − (t − t )/τ0 dt . Our own tests of (2) have not shown such good agreement. we have found that it provides only a qualitative (and in some cases semiquantitative) description of the hysteresis shown in Figure 1 [35]. the type I result is described reasonably well by the power law function (1) with p ≈ 3. it is currently the best mathematical description available for the hysteresis. Nevertheless. and we have employed it in our modeling calculations. and felt compression. it is important to use this hammerforce characteristic in modeling. the type II result exhibits “loops. with parameters p = 3. However. In other words. In contrast to the type I measurements. Stulov took this to be a power law g(z) = z p . Fh . but we will leave them for future modeling work.5. as defined by Z= (5) F . We have not attempted to model this physics according to Newton’s laws. but have simply chosen a value of β which yields a quality factor for the soundboard modes which is similar to that observed experimentally [11. of course. The agreement is quite acceptable. Since the hammer has a finite width. n∆ts ) ≡ y(i. This is included by the addition of a term on the right-hand side proportional to Fh . etc. 9. we include a loss term proportional to the parameter β [11]. 36]. our board lies in the x − y plane and z is its displacement. At present there is no good experimental data on this question.. y) − β . In order to model the ribs and bridges. hb is the board thickness and ρb is its density. There are also some additional terms that enter the equation of motion (5) at the ends of bridges [11. Ex and νx are Young’s modulus and Poisson’s ration for the x direction. while the other is proportional to ∂3 y/∂t 3 . and the elastic constants of the board were taken 1 In principle. 12 1 − νx ν y h3 E y b . For the solution of (5). Fs (x. not the same as the x and y coordinates used in describing the string mo- . The behavior of our numerical soundboard can be judged by calculations of the mechanical impedance. one might expect the soundboard losses to be frequency dependent. Soundboards are assembled from wood that is “quarter sawn. is of order 10−4 . Z. 24]. the position and thickness of the ribs and bridges. (These x and y directions are. The values of all elastic constants were taken from [41]. The physical origin of this term involves elastic losses within the board.1 Finally. 17. as found for the string. it is proportional to the component of the string tension perpendicular to the plane of the board. this spatial step need not be related to the step size for the string ∆xs . THE SOUNDBOARD 929 tion. y) is the force from the strings on the bridge. vb (7) Wood is a complicated material [41]. the details of how this force is distributed have never been measured. the need for numerical stability together with the desired frequency range require that each string be treated as 50–100 vibrating numerical elements [8. The equation of motion (4) does not contain the hammer force. As in our previous work on soundboard modeling [11]. respectively. 18. n). 12 1 − νx ν y h3 Gxy b .) The soundboard coordinates x and y run perpendicular and parallel to the grain of the board.Physical Modeling of the Piano modulus. this equation can be rearranged to express the string displacement at each spatial location i at time step n+1 in terms of the displacement at previous time steps as described by Chaigne and Askenfelt [8. 10]. Finally. This force acts at the appropriate bridge location. we assume that F is a harmonic (single frequency) force applied at a point on the bridge and vb is measured at the same point. the board was modeled as ∼ 100 × 100 vibrating elements. it is customary to spread this force over a small length of the string [8]. including the widths of the ribs and bridges. especially considering that parameters such as the dimensions of the soundboard. and density of the string. 4. Hence. as in the model). Gxy is the shear modulus. one of these terms is proportional to the string velocity. but it cannot be neglected as it produces the well-known effect of stretched octaves [36].” which means that two of the principal axes of the elastic constant tensor lie in the plane of the board. ∂y ∂t ∂4 z ∂4 z where the rigidity factors are Dx = Dy = Dxy = h 3 Ex b . Here. and also sympathetic string vibrations (with the dampers disengaged from the strings) are included in the model. since the bridge moves and the strings are attached to the bridge. since this is just small enough to capture the structure of the board. 40]. The string displacement is then y(x. The space dimensions x and y were discretized. and so forth for y. With this approach to the string calculation. we chose ∆xb = 2 cm. the interaction of strings in a unison group. Hence. The equation of motion for such a thin orthotropic plate is [11. we again employed a finite difference algorithm. so the stiffness term in (4) is small. 23. Our numerical treatment of the string motion employs a finite difference formulation in which both time t and position x are discretized in units ∆ts and ∆xs [8. 10]. Also shown are measurements for a real upright soundboard (with the same dimensions and bridge positions. Damping is accounted for with the terms involving α1 and α2 . Here. both in steps of size ∆xb . 42] ρb hb ∂2 z ∂t 2 = −Dx − Dx ν y + D y νx + 4Dxy ∂x4 ∂x2 ∂y 2 ∂z ∂4 z − D y 4 + Fs (x. Figure 2 shows results calculated from our model [11] for the soundboard from an upright piano. For typical piano strings. fortunately our modeling results are not very sensitive to this factor (so long as the effective hammer width is qualitatively reasonable). 12 (6) where F is an applied force and vb is the resulting soundboard velocity. This combination makes the damping dependent on frequency in a manner close to that observed experimentally [8. the thickness and rigidity factors are position dependent (since these factors are different at the ribs and bridges than on the “bare” board) as described in [11]. t) ≡ y(i∆xs . [9. 10]. If the derivatives in (4) are written in finite difference form. 22. so we have chosen the simplest possible model with just a single loss term in (5). 43]. 10. So far as we know. which acts at the hammer strike point. and is calculated from the string portion of the model. we note that the soundboard “acts back” on the strings. The choice of spatial step size ∆xr was then influenced by two considerations. We situate the soundboard from the previous section on a plane perpendicular to the z direction in the room. PUTTING IT ALL TOGETHER Our model involves several distinct but coupled subsystems—the hammers/strings. Sound is generated in this numerical room by the vibration of the soundboard. and they interact through either (1) or (2). v y . This provides a check on both the soundboard and the room models.e. ∂t ∂y ∂p ∂vz ρa =− . This family of equations is similar in form to an electromagnetic problem. ∂t ∂x ∂v y ∂p ρa =− . and these . Here. the pressure is computed at times n∆tr while the velocity is computed at times (n + 1/2)∆tr . THE ROOM Our time domain room modeling follows the work of Botteldooren [44. 45]. We employ a finite difference approach in which staggered grids in both space and time are used for the pressure and velocity. Figure 3 shows results for the sound generation by an upright soundboard. When using the calculated soundboard velocity to obtain the room velocity at the soundboard surface. and k∆xr . Z. in order for the finite difference algorithm to be numerically stable in three dimensions. 24]. The room is taken to be a rectangular box with the same acoustical properties for all 6 walls. approximately 1 m from the nearest parallel wall (i. The grids for v y and vz are arranged in a similar manner. ∂t ∂z ∂p ∂vx ∂v y ∂vz 2 = ρa ca − − − . as explained in the next section. that is. with p = Zvn . (9) Experiment 5000 2000 Z (kg/s) 1000 500 Model Soundboard impedance 200 Upright piano at middle C 100 100 1000 f (Hz) 104 where vn is the component of the (air) velocity normal to the wall [46]. The walls of the room are modeled in terms of their acoustic impedance. the force was applied and the board velocity was measured at the point where the string for middle C crosses the bridge. This hammer then strikes a string (or strings). At each time step the velocity vz of the room air at the surface of the soundboard is set to the calculated soundboard velocity at that instant. Measurements of Z for a number of materials [47] have found that it is typically frequency dependent with the form Z(ω) ≈ Z0 − iZ . where ω is the angular frequency.. and thus contained ∼ 106 finite difference elements. the velocity components are vx . It is seen that the model results compare well with the experiments. and the room—and it is useful to review how they fit together computationally. Results from [11. Young’s modulus). and we plot the sound pressure normalized by the board velocity at the driving point [25]. 45]. In the calculations described below. ∂t ∂x ∂y ∂z ρa (8) where p is the pressure. Typical numerical rooms were 3 × 4 × 4 m3 . one √ must have ∆xr /( 3∆tr ) > ca .. Given a time step ∆tr . from either direct measurements or handbook values (e. The time step for the room calculation was ∆tr = 1/22050 ≈ 4. we averaged over 4 soundboard grid points for each room grid point. ρa is the density. while vx is calculated on the staggered grid (i+1/2)∆xr .5 × 10−4 s.g. the floor). it is convenient for the spatial steps for the soundboard and room to be commensurate. the soundboard was driven harmonically at the point where the string for middle C contacts the bridge. 6. the soundboard. and vz . twice the soundboard step size. ω (10) Figure 2: Calculated (solid curve) and measured (dotted curve) mechanical impedance for an upright piano soundboard. j∆xr . 5.930 104 EURASIP Journal on Applied Signal Processing j∆xr . k∆xr . A similar staggered grid is used for the space coordinates. the room step size was ∆xr = 4 cm. This sets the string(s) for that note into motion. as explained in [44. Second. The calculation begins by giving some initial velocity to a particular hammer. as obtained from the soundboard calculation. First. Here. with the pressure calculated on the grid i∆xr . We begin with the usual coupled equations for the velocity and pressure in the room ∂p ∂vx =− . and ca is the speed of sound in air. Incorporating this frequency domain expression for the acoustic impedance into our time domain treatment was done in the manner described in [45]. and much is known about how to deal with it numerically. the equation of motion for the soundboard is highly dispersive. it seems that these parameters should be viewed as fitting parameters. the power law description (1)—was the problem. involving approximately 104 parameters. Results from [25]. The board was driven at the point where the string for middle C crosses the bridge. we now consider the quality of the tones calculated with the current version of the model. To explain this hierarchy. string (s). We have also explored an alternative approach to the room modeling involving a ray tracing approach [48]. Given the large number of room elements. based on our modeling. and soundboard (b). 0 . the strings drive the soundboard. we can also make a somewhat stronger statement: in order to obtain a realistic piano tone. they pass a sort of acoustical Turing test. they are available on our website http://www. 6 ∆tr = (11) where the subscripts correspond to the room (r).physics. This led us to the type II hammer experiments described above. We instead wish to use the modeling to learn about the instrument.” With a typical personal computer (clock speed 1 GHz). 1/∆tr is commensurate with the data rates commonly used in CD sound formats. it is not a primary goal of our work. We then see that each room time step contains 4 string time steps. the model should transfer easily to a cluster (i.edu/piano. Ray tracing allows one to express the relationship between soundboard velocity and sound pressure as a multiparameter map. likely) that power law parameters that will yield good piano tones can be found. in turn act on the bridge and soundboard. As we have already mentioned.5 20 100 103 Frequency (Hz) 104 Figure 3: Results for the sound pressure normalized by the soundboard velocity for an upright piano soundboard: calculated (solid curve) and measured (dotted curve). In addition. After many modeling calculations.e. but they sounded distinctly plucked and somewhat metallic.. Likewise. each with an associated time step. Our conclusion is that the hammers are an essential part of the instrument. (1) The hammer portion of the model still . While they are not yet “Steinway quality. we employed power law hammers described by (1) with parameters based on type I hammer experiments by our group [31]. the vibrations of each component of our model are calculated with a finite difference algorithm. with parameters (τ0 . We have therefore chosen to instead make the various time steps commensurate. the calculated tones are poor. we came to the conclusion that the hammer model—for example. 7. multi-CPU) machine. as they may not accurately describe any real hammers. The results were much improved.Physical Modeling of the Piano 20 Sound generation 10 p/vb (arb. Note that we do not claim that power law hammers must always give unsatisfactory results. a 1 minute simulation requires approximately In the previous section. However. this would greatly (and unnecessarily) slow down the calculation. With that in mind. The values of these parameters can be precalculated and stored. each string time step contains 6 soundboard steps. and to a description of the hammer-string force in terms of the Stulov function (2). There are a number of issues that we plan to address in the future. However. In that sense. Our point is that when the power law parameters are chosen to fit the type I behavior of real hammers. However. and the soundboard drives the room— it would be computationally simpler to use the same value of the time step for all three subsystems.” it is our opinion that the calculated tones could be mistaken for a real piano. and stability requirements demand a much smaller time step for the soundboard than is needed for string and room simulations. 22050 ∆t ∆ts = r . based on our experience. the modeling should be based on hammer parameters observed in type II measurements. While we cannot include our calculated sounds as part of this paper. While such a simulation would certainly be interesting. The results were disappointing—it is hard to accurately describe the tones in words. It is certainly possible (and indeed. etc. This is hardly a revolutionary result. In our initial modeling. resulting in about an order of magnitude speed-up in the calculation as compared to the room algorithm described above. the string algorithm makes 4 iterations for each iteration of the room model. units) Soundboard driven at C4 931 30 minutes of computer time. with the hysteresis included in the model. Of course. The overall computational speed is currently somewhat less than “real time.purdue. that is. we first note that the room time step is chosen to be compatible with common audio hardware and software. this gap will narrow in the future in accord with Moore’s law. 4 ∆ts ∆tb = .) taken from these type II experiments [35]. the soundboard acts back on the strings. with 1 s. we saw that a real-time Newton’s law simulation of the piano is well within reach. ANALYSIS OF THE RESULTS: WHAT HAVE WE LEARNED AND WHERE DO WE GO NEXT? 5 2 Experiment 1 Model 0. Since the systems are coupled— that is. 1987. pp. Askenfelt. VI: Nonlinear modeling. S. “Experimental and computational studies of the piano. 1997. there are significant quantitative differences. pp. vol. vol. Bank. “Piano string excitation V: Spectra for real hammers and strings. 535–546. Application to plucked stringed instruments.” Journal of the Acoustical Society of America. “Modal analysis and finite element analysis of a piano soundboard. G.” Journal of the Acoustical Society of America. In conclusion. A. “Design and tone in the mechanoacoustic piano. 17th International Congress on Acoustics. Further improvements of the model seem quite feasible.” Appl. 105. and P. 100. vol. Hall. Conklin Jr. and S. no. vol. Jiang. 1987. vol. [13] H. pp. [14] H. NY. pp. Korty. 6. pp. “On the use of finite differences for musical synthesis. . Chaigne. E. pp. Time-domain modeling of the resonator and of the radiated sound pressure. G. J.” EURASIP Journal on Applied Signal Processing. EURASIP Journal on Applied Signal Processing [6] H. “Numerical simulations of xylophones. 1987. “Acoustics of pianos. 12. 102. Weinreich for many helpful and critical discussions. Our current room model is certainly a very crude approximation to a realistic room. 1631–1640. we have made good progress in developing a physical model of the piano. Matignon. A. 2. Giordano. 30. pp. 539–557. [23] J. I. [5] D. Piano structure. Chaigne and V. I. 4. It may be necessary to develop a better functional description to replace the Stulov form. “Simple model of a piano soundboard. vol. Rome. [9] A. Walker. but will require a more general soundboard model since the two transverse modes couple through the motion of the bridge. “Physically informed signal processing methods for piano sound synthesis: a research overview.” Journal of the Acoustical Society of America. vol. 99. [15] H. pp. [3] D. and D. 1286–1298. 5. Brooke. Rocchesso. [8] A. Tubis. Richardson. 3. Rossing. Winans II. no. 695–708. 2003. 1. “Numerical simulations of piano strings. J. 95. Cincinnati. 3. Union College. Chaigne. 1. P. This work was supported by National Science Foundation (NSF) through Grant PHY-9988562. no. Dietz. no. E. 1989. “Piano string excitation. Kindel and I. 3286–3296. 2. E. D. E. pp.. F. Borin. ACKNOWLEDGMENTS We thank P.” Journal of the Acoustical Society of America.” Proceedings of the Institute of Acoustics.” Journal of the Acoustical Society of America. 2. our string model includes only one polarization mode. 1988.” Journal d’Acoustique. 95–105. no. 101. 1996. II. no. pp. no. thesis. 79. pp. Part II. Schenectady. 1. and contain chairs and other objects. vol. 547–555. 81. Acoustics. vol. 1627–1638. vol. pp.” Journal of the Acoustical Society of America. We believe that physical modeling can provide new insights into the piano.” Journal of the Acoustical Society of America. 81. (3) The soundboard of a real piano is supported by a case. This can be readily included. A physical model for a struck string using finite difference method. Piano strings and scale design.S. 1999. Joly. [21] N. and that similar approaches can be applied to other instruments. 757–764. Hall. no. [12] H. 1987. 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