EURASIP Journal on Applied Signal Processing 2004:7, 926–933 c 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 the instrument. However, as far as we can tell, certain fea- tures of the model, such as hammer-string impulse func- This paper describes a long term project by our group aimed tions and the transfer function that ultimately relates the at physical modeling of the piano. The theme of this volume, sound pressure to the soundboard motion (and other sim- model based sound synthesis of musical instruments, is quite ilar transfer functions), are taken from experiments on real broad, so it is useful to begin by discussing precisely what instruments. This approach is a powerful way to produce re- we mean by the term “physical modeling.” The goal of our alistic musical tones efficiently, in real time and in a man- project is to use Newton’s laws to describe all aspects of the ner that can be played by a human performer. However, this piano. We aim to use F = ma to calculate the motion of the approach cannot address certain questions. For example, it hammers, strings, and soundboard, and ultimately the sound would not be able to predict the sound that would be pro- that reaches the listener. duced if a radically new type of soundboard was employed, Of course, we are not the first group to take such a New- or if the hammers were covered with a completely differ- ton’s law approach to the modeling of a musical instrument. ent type of material than the conventional felt. The physi- For the piano, there have been such modeling studies of the cal modeling method that we describe in this paper can ad- hammer-string interaction [1, 2, 3, 4, 5, 6, 7, 8, 9], string vi- dress such questions. Hence, we view the ideas and method brations [8, 9, 10], and soundboard motion [11]. (Nice re- embodied in work of Bank and coworkers [20] (and the ref- views of the physics of the piano are given in [12, 13, 14, 15].) erences therein) as complementary to the physical modeling There has been similar modeling of portions of other instru- approach that is the focus of our work. ments (such as the guitar [16]), and of several other com- In this paper, we describe the route that we have taken plete instruments, including the xylophone and the timpani to assembling a complete physical model of the piano. [17, 18, 19]. Our work is inspired by and builds on this pre- This complete model is really composed of interacting sub- vious work. models which deal with (1) the motions of the hammers and At this point, we should also mention how our work re- strings and their interaction, (2) soundboard vibrations, and lates to other modeling work, such as the digital waveguide (3) sound generation by the vibrating soundboard. For each approach, which was recently reviewed in [20]. The digital of these submodels we must consider several issues, includ- waveguide method makes extensive use of physics in choos- ing selection and implementation of the computational algo- ing the structure of the algorithm; that is, in choosing the rithm, determination of the values of the many parameters proper filter(s) and delay lines, connectivity, and so forth, that are involved, and testing the submodel. After consider- to properly match and mimic the Newton’s law equations of ing each of the submodels, we then describe how they are motion of the strings, soundboard, and other components of combined to produce a complete computational piano. The Physical Modeling of the Piano 927 quality of the calculated tones is discussed, along with the The issue of listening tests brings us to the question of lessons we have learned from this work. A preliminary and goals, that is, what do we hope to accomplish with such a abbreviated report on this project was given in [21]. 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- 2. OVERALL STRATEGY AND GOALS anos would sound like. For example, one could imagine con- structing a piano with a carbon fiber soundboard, and it One of the first modeling decisions that arises is the question would be very useful to be able to predict its sound ahead of of whether to work in the frequency domain or the time do- time, or to use the model in the design of the new sound- main. In many situations, it is simplest and most instructive board. On a different and more philosophical level, one to work in the frequency domain. For example, an under- might want to ask questions such as “what are the most im- standing of the distribution of normal mode frequencies, and portant elements involved in making a piano sound like a pi- the nature of the associated eigenvectors for the body vibra- ano?” We emphasize that it is not our goal to make a real time tions of a violin or a piano soundboard, is very instructive. model, nor do we wish to compete with the tones produced However, we have chosen to base our modeling in the time by other modeling methods, such as sampling synthesis and domain. We believe that this choice has several advantages. digital waveguide modeling [20]. 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- 3. STRINGS AND HAMMERS tion between various components of the instrument, such Our model begins with a piano hammer moving freely with a as the strings and soundboard, is somewhat simpler when speed v just prior to making contact with a string (or strings, viewed in the time domain, especially when one considers h since most notes involve more than one string). Hence, we the early “attack” portion of a tone. Third, our ultimate goal ignore the mechanics of the action. This mechanics is, of is to calculate the room pressure as a function of time, so it is course, quite important from a player’s perspective, since it appealing to start in the time domain with the hammer mo- determines the touch and feel of the instrument [26]. Nev- tion and stay in the time domain throughout the calculation, ertheless, we will ignore these issues, since (at least to a first ending with the pressure as would be received by a listener. approximation) they are not directly relevant to the compo- Our time domain modeling is based on finite difference cal- sition of a piano tone and we simply take v as an input pa- culations [10] that describe all aspects of the instrument. h rameter. Typical values are in the range 1–4 m/s [9]. A second element of strategy involves the determination When a hammer strikes a string, there is an interaction of the many parameters that are required for describing the force that is a function of the compression of the hammer piano. Ideally, one would like to determine all of these pa- felt, y f . This force determines the initial excitation and is rameters independently, rather than use them as fitting pa- thus a crucial factor in the composition of the resulting tone. rameters when comparing the modeling results to real (mea- Considerable effort has been devoted to understanding the sured) tones. This is indeed possible for all of the parame- hammer-string force [1, 2, 3, 4, 5, 6, 7, 27, 28, 29, 30, 31, ters. For example, dimensional parameters such as the string 32, 33]. Hammer felt is a very complicated material [34], diameters and lengths, soundboard dimensions, and bridge and there is no “first principles” expression for the hammer- positions, can all be measured from a real piano. Likewise, ff string force relation Fh(y f ). Much work has assumed a sim- various material properties such as the string sti ness, the ple power law function elastic moduli of the soundboard, and the acoustical proper- ties of the room in which the numerical piano is located, are p Fh y f = F0 y ,(1) well known from very straightforward measurements. For a f few quantities, most notably the force-compression charac- where the exponent p is typically in the range 2.5–4 and F0 teristics of the piano hammers, it is necessary to use separate is an overall amplitude. This power law form seems to be at (and independent) experiments. least qualitatively consistent with many experiments and we This brings us to a third element of our modeling therefore used (1) in our initial modeling calculations. strategy—the problem of how to test the calculations. The While (1) has been widely used to analyze and inter- final output is the sound at the listener, so one could “test” pret experiments, and also in previous modeling work, it the model by simply evaluating the sounds via listening tests.