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Session 08 - Physics-based modelling

Physics-based modeling of musical instruments
V Välimäki
Helsinki University of Technology, Laboratory of Acoustics and Audio Signal Processing, Espoo, Finland

Physical models are being developed for musical instruments for two main purposes: research of acoustical properties and sound synthesis. This paper presents an overview and a short history of physical modeling of musical instruments. It also proposes a classification of physics-based methods. These include the source-filter, the finite-difference time-domain, the digital waveguide, the functional transformation, and the modal synthesis methods and some of their modifications and extensions. The methods are closely related to digital signal processing. Some commonly used signal processing techniques are presented. Tuning of model parameters and automatic parameter estimation are considered. A brief discussion on the control of physical modeling synthesis techniques is also included. Examples of simple models for vibrating strings generated with these methods are presented. The main focus of the paper is on music synthesis. The modeling and sound synthesis of the acoustic guitar and some wind and keyboard instruments are briefly covered. It is shown how the current physics-based methods are related to various traditional algorithms for synthesizing musical sounds. Current trends and future directions in physical modeling of musical instruments are discussed. The presentation will include sound examples of single tones and music synthesized with physical modeling techniques.

A power normalized non-linear lossy piano hammer
J Bensa¹, S D Bilbao², R Kronland-Martinet¹, J O Smith III³
¹LMA - CNRS, S2M, Marseille, France; ²Queen's University, SARC, Belfast, United Kingdom; ³Stanford University, Dept. of Music, Stanford, United States

For sound synthesis purposes, the vibration of a piano string may be simply modeled using digital waveguides which transport traveling wave-like signals in both directions. Such a digital wave-type formulation, in addition to yielding a particularly computationally efficient simulation routine, also possesses other important advantages. In particular, it is possible to couple the delay lines to a nonlinear exciting mechanism (the hammer) without compromising stability; in fact, if the hammer and string are passive, their digital counterparts will be exactly passive as well. The key to this good property (which can be carried over to other nonlinear elements in musical systems) is that all operations are framed in terms of the passive scattering of discrete signals in the network, the sum-of-squares of which serves as a discrete-time Lyapunov function for the system as a whole. Starting from a non-linear and lossy hammer model (proposed by Hunt and Crossley), we design a power-normalized digital hammer using the wave digital filters formalism. By coupling this hammer to a digital waveguide model, we simulate both the non-linear and hysteretic behavior of the hammer felt and the dissipation and dispersion phenomena involved in string wave propagation.

Sound synthesis for three-dimensional objects: dynamic contact between two arbitrary elastic bodies
J Bensoam, R Caussé, C Vergez, N Misdariis, N Ellis
Ircam, Acoustic Team, Paris, France

Modalys is a sound synthesis software developed at Ircam for research and musical applications. This software builds virtual instruments based on physical models to obtain the most entire range of expressive variations in the instrument in response to intuitive controls. A instrument, as a complex structure, is described by the mechanical/acoustical interaction of its components (strings, tubes, resonators, soundboard,...). This description relies on modal representation of vibrating objects. Modal data are obtained in two ways: in one hand, by analytic modeling for simple structures and on the other hand by experimental measurements for real structures .
Some new research have been done recently to extend the sound prediction to three-dimensional objects with the help of numerical methods. In particular, theoretical and numerical treatment of the unilateral and frictionless dynamic contact between two arbitrary elastic bodies was studied. In the context of infinitesimal deformation, a reciprocal formulation is used to reduce the well-posed problem to one involving Green functions defined only on contact surfaces. It is then often possible to approximate the system using considerably fewer unknowns than with finite difference algorithms. The ability of the method to predict the contact interaction between two elastic bodies, irrespective of the material constitution and geometry, is highlighted by numerical simulations implemented in Modalys.

Musical applications of multichannel generalized digital waveguides
C Burns¹, S Serafin², M Burtner³
¹CCRMA, Music, Stanford, CA, United States; ²CCRMA, Department of Music, Stanford, United States; ³VCCM, Music, University of Virginia, United States

One dimensional digital waveguides are a synthesis technique widely used in the computer music community to efficiently model waves propagating along different media, such as strings and tubes. In this paper we propose a generalized digital waveguide with the intent of creating an extended physical model. We built a network of eight independent digital waveguides. Each waveguide can be excited in a sustained or transient mode by different mechanisms. We implemented traditional excitations such as bowing, blowing or plucking. We also allowed the waveguide to be excited by different nonlinear functions that do not necessarily relate to traditional instruments, but noneless are interesting from a compositional perspective. Additionally, we experimented with unusual network topologies, including connecting the eight waveguides in a circular loop. In this situation, issues of stability of the system need to be taken into account; we demonstrate the use of nonlinear functions for gain control, alongside traditional coefficient-balancing procedures. In order to create an immersive environment, each waveguide is mapped to one channel in an eight channel configuration. Different musical applications of the generalized digital waveguide are proposed.

Synthesis of voiced sounds by means of waveform adaptive physical models
C Drioli
University of Padova, Information Engineering, Padova, Italy

The reproduction of sampled voice sounds by physical modeling is addressed. A major focus is put on the possibility of fitting a physically constrained model to real voice samples. The voice source model relies on a lumped mechano aerodynamic scheme inspired by the mass-spring paradigm. The vocal folds are represented by a mechanical resonator plus a delay line which takes into account the vertical phase differences. The vocal fold displacement is coupled to the glottal flow by means of a nonlinear subsystem, based on a general parametric nonlinear model, i.e., a radial basis function (RBF) network. The principal characteristics of the flow-induced oscillations are retained, and the overall model is suited for an identification approach where real (inverse filtered) glottal flow signals are to be reproduced. A data-driven identification procedure is outlined, where the parameters of the model are tuned in order to accurately match the target waveform. A nonlinear regression algorithm is used to train the nonlinear part. The vocal tract is modeled by a linear filter. The simultaneous optimization of the source and the vocal tract parameters is discussed. An algorithm based on the Kalman filtering approach is proposed and evaluated. A set of voice samples is used to train the model. Samples from different speakers, and with different voice qualities, are reproduced in order to asses the waveform adaptiveness of the model.

The development of a modular software paradigm for the physical modelling of musical instruments
I A Drumm
University of Salford, The School of Acoustics and Electronic Engineering, Manchester, United Kingdom

The paper will describe ongoing research by the author to investigate, develop and evaluate a modular and extensible paradigm for the software based construction of physical models of real musical instruments. The software makes full use of object orientated programming techniques to construct a user friendly application for the interconnecting of key components of musical instruments such as excitation mechanisms, wave-guides, resonators and environments by using a flexible exchange of parameters beyond the processing of an audio stream. The project aims to consolidate previous research findings of the wider community and further investigate the relative influence and interconnection of physical components.

The principle of closed wavetrains, resonance and efficiency: past, present and future
G Essl¹, P R Cook²
¹University of Florida, Computer & Information Science & Engineering, Gainesville, FL, United States; ²Princeton University, Computer Science & Music, Princeton, NJ, United States

The principle of closed wavetrains asserts the equivalence of the condition of a traveling wave closing onto itself in phase to the occurrence of a mode. This principle provides a direct conceptual link between spectral descriptions of dynamic responses and a path-based dynamic description. In this paper we present the history and development of the idea since d'Alembert first proposed traveling functional forms to solve the string equation, through the subsequent argument between Daniel Bernoulli, Euler and him which led to the development of Fourier analysis and contemporary theories of partial differential equations. Related are also the development of chaos theory connected to Poincare and Birkhoff, asymptotic solutions associated with Rayleigh, Wenzel, Kramers, Brillouin, Maslov and Keller, and Kac's famous isospectral problem. Then we discuss how the traveling functions have been utilized in the numerical simulation of musical instruments through work by Julius Smith, Karplus, Strong and others. This work has recently been extended to additional instrument types, in particular ideophones, yielding models that are not only more efficient than finite element based simulations, but have desirable properties of stability and ease of interpretation under perturbations. We conclude with outlining possible research based on the advantages and drawbacks of the method.

Real-time synthesis models of wind instruments based on physical models
P Guillemain, J Kergomard, T Voinier
CNRS-LMA, equipe S2M, Marseille, France

A real-time synthesis model of wind instruments based on a simple physical model is presented. The model includes a main linear part, that model directly the input impedance of the resonator of the instrument, avoiding the (p+, p-) decomposition, and made of combinations of elementary impedances of both local and propagative elements. In particular, the tangent function which constitutes the input impedance of a simple cylindical bore is written as a combination of two identical delay+filter functions. The second linear part models the reed/lips displacement. The non linear part, coupled with the linear parts, models the non linear interaction between the flow and the pressure at the embouchure level. The sampled version of the model uses digital filters the coefficients of which are expressed analytically as functions of the geometrical parameters of the resonator. Moreover, a suitable discretization of the reed/lips displacement makes possible an explicit resolution scheme of the coupled non linear system, avoiding the use of iterative schemes. Since the model makes an explicit use of the physical parameters, its real-time control is easy and natural. The application of these approaches to the synthesis of clarinet-like tones will be detailed. During the presentation, sound examples recorded during live performances will be played.

Time-domain physical modeling and real-time synthesis using mixed modeling paradigms
M Karjalainen
Helsinki University of Technology, Acoustics Laboratory, Espoo, Finland

Three approaches, often used in discrete-time modeling of physical systems, are digital waveguides (DWG), finite difference time-domain schemes (FDTD), and wave digital filters (WDF). However, they have not been systematically applied in hybrid modeling by mixing the approaches. We have shown this feasible by formulating a modeling framework, utilizing the best features of each of them. Based on the theoretical framework we have developed a software tool, called the BlockCompiler, which is used for high-level description of model structures, automatically compiled to efficient simulation and real-time synthesis. Physical models are specified as K-type (with Kirchhoff variables: FDTDs) and W-type (wave variables: DWGs and WDFs) blocks that have interconnection ports of two-directional interaction. The ports can be connected in series or parallel. This allows for simulation of arbitrary spatially distributed 1-, 2-, or 3-D systems as well as lumped element models. Simulation and sound synthesis is made efficiently in the time domain. We will describe the theoretical approach, the BlockCompiler, and some basic applications of the approach to different instrument families: plucked string instruments and wind instruments as 1-D systems, as well as drums membranes as 2-D systems. Real-time synthesis demonstrations will be included in the final presentation.

A new method for the calculation of self-sustained oscillations: the perturbation of the Helmholtz motion
J Kergomard, S Farner
CNRS, Laboratoire de Mécanique et d'Acoustique, Marseille, France

When losses are ignored, elementary solutions for the classical models of self sustained instruments, such as reed or bowed string instruments, are pure square or "rectangular" signals, called Helmholtz motion. When losses are introduced, round corner signals are obtained, and the calculation becomes delicate. Ab initio calculation is possible, but methods limited to the steady-state regime make it easier to study the influence of the parameters on the spectrum and the playing frequency: the harmonic balance method is well know, but, because losses are small, another, iterative technique is suggested. Considering e.g. reed instruments, the Fourier components of the input pressure signal can be divided into two parts, the components with high input impedance, and those with low input impedance (corresponding to the missing harmonics of the rectangular signal), and a perturbation method can be obtained by starting from infinite and zero impedances, respectively. A key point is that at each step, the frequency is fixed in order to calculate the perturbation, then a new value is calculated using any equation of the harmonic balance system, an excellent candidate being the reactive power defined by Boutillon. Results are compared to these of the harmonic balance method, and are very interesting, especially far from the oscillation thresholds. Taking into account discontinuities in the nonlinear characteristic is discussed.

Simulating the mechanism of sound generation in flutes using the lattice Boltzmann method
H Kühnelt
University of Music and Performing Arts Vienna, Institute of Musical Acoustics, Vienna, Austria

The sound generation in flute like instruments, as flutes, recorders and flue organ pipes, is determined by non linear acoustical and fluid dynamical processes, which interact closely. Because of different scales the coupling is rather weak. In addition to that, geometrical parameters of the mouth of the flute as the shape of the labium or the flue exit, where the jet is formed, influence the amplitude and the spectrum of the generated sound considerably. A model for simulating self sustained oscillations in flutes should be able to include both, weak coupling and strong dependence on boundary conditions.
The lattice Boltzmann method is a rather new method for simulating fluid flow, and as a special discretization of the Boltzmann equation it incorporates fluid dynamics and acoustics intrinsically. In addition to that complex boundary conditions can be modeled very easily. The advantages and shortcomings of the lattice Boltzmann method will be discussed. First results of simulations of a small organ pipe with recorder like dimensions will be presented. They show, that the lattice Boltzmann method is able to model self sustained oscillations in flutes.

Methods for synthesizing very high Q parametrically well behaved two pole filters
M V Mathews, J O Smith
Stanford University, CCRMA-Music Department, Stanford California, United States

Techniques for synthesizing two pole filters are well known. A number of techniques introduce unpleasant sounding transients in the filter response when the frequency or damping of the filter is rapidly changed. We will demonstrate a difference equation for a digital filter in which both the frequency and the damping can be changed without producing discontinuities in the filter output. The technique is based on the well known property of the product of complex numbers. In polar form, the magnitude of the product of two numbers is the product of their magnitudes and the angle of the product is the sum of their angles. Successive multiplies can produce a rotating vector whose real or imaginary parts are samples of constant amplitude sine waves or of exponentially damped sine waves. The frequency and damping of the resulting waves can be changed without changing the amplitude of the waves. These properties can be used to make a digital filter whose input, frequency, and damping can all be functions of time. A program to demonstrate some musical applications of these filters will be shown.

Modeling and real-time synthesis of the kantele using distributed tension modulation
J Pakarinen, M Karjalainen, V Valimaki
Helsinki University of Technology, Laboratory of Acoustics and Audio Signal Processing, Espoo, Finland

Nonlinear behavior of a vibrating string is responsible for acoustical features in some plucked-string instruments, resulting in a characteristic and easily-recognizable tone. That is also the case for the Finnish kantele, a traditional plucked-string folk music instrument. Earlier we have analyzed the general acoustic properties of the kantele and discussed related sound synthesis techniques. In this study, a novel modeling and sound synthesis method for nonlinear string vibrations with spatially distributed tension modulation is presented. The modeling is conducted through a digital waveguide approach, using controllable fractional delay elements in implementing the distributed tension modulation nonlinearity. The elongation of the vibrating string is estimated and the result is used in tuning the fractional delay values accordingly. Because of the spatially distributed nature of the approach, control of the string model parameters and observation of its behavior can be implemented at any point along it, in contrast to prior digital waveguide string models. This new approach is applied in constructing a physical model of a five-string kantele. Real-time sound synthesis is implemented using an efficient, block-based modeling tool, the BlockCompiler.

The estimation of birdsong control parameters using maximum likelihood and minimum action
T Smyth¹, J S Abel², J O Smith¹
¹Stanford University, Music, Stanford, United States; ²Universal Audio, Inc, Santa Cruz, United States

The bird's airway consists of a trachea which divides into left and right bronchi, and a membrane forming a valve at the top of each bronchial lumen. A physical model of the bird's vocal tract was developed using waveguide synthesis techniques for the bronchi and trachea tubes and finite difference methods for the nonlinear vibrating syringeal membranes. Here, a method is presented for extracting two important model control parameters from recorded birdsong: lung pressure and membrane tension. A look-up table pairs combinations of pressure and tension with the model's corresponding output power spectra. At each time frame, a generalized likelihood ratio fills a pressure-tension matrix indicating similarity between the birdsong power spectrum and the tabulated spectra. Successive pressure-tension matrices are stacked, and points exhibiting a good fit to the data align to form trajectories corresponding to changes in pressure and tension over time. In the event a range of trajectories matches the data well, selection is taken to be that of least action. This research serves two purposes: 1) to judge the model's ability to produce realistic birdsong by calibrating it to recorded birdsong and 2) to restrict and scale the control parameter space so as to improve the user's ability to interact with the model, e.g., by having a controller that follows predetermined trajectories through the matrix stack.

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