EURASIP Journal on Applied Signal Processing 2003:10, 941–952 c ○ 2003 Hindawi Publishing Corporation Physically Informed Signal Processing Methods for Piano Sound Synthesis: A Research Overview
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paper_title: Commuted Piano Synthesis

paper_content:

The \commuted piano synthesis" algorithm is described, based on a simpli ed acoustic model of the piano. The model includes multiple coupled strings, a nonlinear hammer, and an arbitrarily large soundboard and enclosure. Simpli cations are employed which greatly reduce computational complexity. Most of the simpli cations are made possible by the commutativity of linear, time-invariant systems. Special care is given to the felt-covered hammer which is highly nonlinear and therefore does not commute with other components. In its present form, a complete, two-key piano can be synthesized in real time on a single 25MHz Motorola DSP56001 signal processing chip.
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paper_title: Structured audio and effects processing in the MPEG-4 multimedia standard

paper_content:

While previous generations of the MPEG multimedia standard have focused primarily on coding and transmission of content digitally sampled from the real world,MPEG-4 contains extensive support for structured, syntheticand synthetic/natural hybrid coding methods. An overviewis presented of the "Structured Audio" and "AudioBIFS"components of MPEG-4, which enable the description ofsynthetic soundtracks, musical scores, and effects algorithmsand the compositing, manipulation, and synchronization ofreal and synthetic audio sources. A discussion of the separation of functionality between the systems layer and the audiotoolset of MPEG-4 is presented, and prospects for efficientDSP-based implementations are discussed.
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paper_title: Structured Audio: Creation, Transmission, and Rendering of Parametric Sound Representations

paper_content:

Structured audio representations are semantic and symbolic descriptions that are useful for ultralow-bit-rate transmission, flexible synthesis, and perceptually based manipulation and retrieval of sound. We present an overview of techniques for transmitting and synthesizing sound represented in structured format, and for creating structured representations from audio waveforms. We discuss applications for structured audio in virtual environments, music synthesis, gaming, content-based retrieval, interactive broadcast, and other multimedia contexts.
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paper_title: A Tutorial on Digital Sound Synthesis Techniques

paper_content:

Progress in electronics and computer technology has led to an ever-increasing utilization of digital techniques for musical sound production. Some of these are the digital equivalents of techniques employed in analog synthesizers and in other fields of electrical engineering. Other techniques have been specifically developed for digital music devices and are peculiar to these. This paper introduces the fundamentals of the main digital synthesis techniques. Mathematical developments have been restricted in the exposition and can be found in the papers listed in the references. To simplify the discussion, whenever possible, the techniques are presented with reference to continuous signals. Sound synthesis is a procedure used to produce a sound without the help of acoustic instruments. In digital synthesis, a sound is represented by a sequence of numbers (samples). Hence, a digital synthesis technique consists of a computing procedure or mathematical formula, which computes each sample value. Normally, the synthesis formula depends on some values, that is, parameters. Frequency and amplitude are examples of such parameters. Parameters can be constant or slowly time variant during the sound. Time-variant parameters are also called control functions. Synthesis techniques can be classified as (1) generation techniques (Fig. la), which directly produce the signal from given data, and (2) transformation techniques (Fig. Ib), which can be divided into two stages, the generation of one or more simple signals and their modification. Often, more or less elaborate combinations of these techniques are employed. Fixed-Waveform Synthesis
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paper_title: Understanding Musical Sound with Forward Models and Physical Models

paper_content:

This research report describes an approach to parameter estimation for physical models of sound-generating systems using distal teachers and foward models (Jordan & Rumelhart, 1992; Jordan, 1990). The general problem is to find an inverse model of a sound-generating system that transforms sounds to action parameters; these parameters constitute a model-base description of the sound. We first show that a two-layer feedforward model is capable of performing inverse mappings for a simple physical model of a violin string. We refer to this learning strategy as direct inverse modeling; it requires an explicit teacher and it is only suitable for convex regions of the parameter space. A model of two strings was implemented that had non-convex regions in its parameter space. We show how the direct modeling strategy failed at the task of learning the inverse model in this case and that forward models can be used, in conjunction with distal teachers, to bias the learning of an inverse model so that non-convex regions are mapped to single-point solutions in the parameter space. Our results show that forward models are appropriate for learning to map sounds to parametric representations
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paper_title: Synchronizing Computer Graphics Animation and Audio

paper_content:

Our prototype authoring tool synchronizes computer graphics animation, narration and background music. It lets users visualize the contents of a background music track and narration along a single time line and displays an animation's scenario sharing the same time line. A graphical user interface supports interactive editing of both background music and narration without any loss in quality.
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paper_title: Principles of Digital Waveguide Models of Musical Instruments

paper_content:

Basic principles of digital waveguide modeling of musical instruments are presented in a tutorial introduction intended for graduate students in electrical engineering with a solid background in signal processing and acoustics. The vibrating string is taken as the principal illustrative example, but the formulation is unified with that for acoustic tubes. Modeling lossy stiff strings using delay lines and relatively low-order digital filters is described. Various choices of wave variables are discussed, including velocity waves, force waves, and root-power waves. Signal scattering at an impedance discontinuity is derived for an arbitrary number of waveguides intersecting at a junction. Various computational forms are discussed, including the Kelly-Lochbaum, one-multiply, and normalized scattering junctions. A relatively new three-multiply normalized scattering junction is derived using a two-multiply transformer to normalize a one-multiply scattering junction. Conditions for strict passivity of the model are discussed. Use of commutativity of linear, time-invariant elements to greatly reduce computational cost is described. Applications are summarized, and models of the clarinet and bowed-string are described in some detail. The reed-bore and bow-string interactions are modeled as nonlinear scattering junctions attached to the bore/string acoustic waveguide.
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paper_title: Hysteretic model of the grand piano hammer felt

paper_content:

The experimental relationships of dynamic force versus grand piano hammer felt deformation show the significant influence of hysteresis characteristics. To explain this phenomenon, a new mathematical model of the hammer felt is proposed. In this model the hammer felt is considered as a nonlinear history‐dependent (hysteretic) material with an exponential kernel function. The numerical simulation of interaction of the hammer with a fixed target was used to identify the nonlinear and hysteresis parameters of the felt, and good agreement with experiments was achieved. Also, this model is used here for the analysis of interaction of the hammer with a real grand piano string.
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paper_title: Numerical simulations of piano strings. II. Comparisons with measurements and systematic exploration of some hammer‐string parameters

paper_content:

A physical model of the piano string, using finite difference methods, has recently been developed. [Chaigne and Askenfelt, J. Acoust. Soc. Am. 95, 1112–1118 (1994)]. The model is based on the fundamental equations of a damped, stiff string interacting with a nonlinear hammer, from which a numerical finite difference scheme is derived. In the present study, the performance of the model is evaluated by systematic comparisons between measured and simulated piano tones. After a verification of the accuracy of the method, the model is used as a tool for systematically exploring the influence of string stiffness, relative striking position, and hammer‐string mass ratio on string waveforms and spectra.
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paper_title: Piano string excitation. VI: Nonlinear modeling

paper_content:

Accurate modeling of the piano string–hammer interaction requires that the nonlinearity of the force‐displacement relation for the hammer be recognized and included. Two models that accomplish this with numerical integration are described, and sample results are presented. The first model is valid for a perfectly flexible string, where nondispersive traveling waves can be conveniently followed with a digital delay line. The second model uses standing waves instead to describe the vibrations of a stiff string, with the stiffness itself justifying truncation of what would otherwise be a poorly converging normal‐mode summation. Predictions with these models give significantly better agreement with data than did previous calculations with completely linear models.
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paper_title: Numerical simulations of piano strings. I. A physical model for a struck string using finite difference methods

paper_content:

The first attempt to generate musical sounds by solving the equations of vibrating strings by means of finite difference methods (FDM) was made by Hiller and Ruiz [J. Audio Eng. Soc. 19, 462–472 (1971)]. It is shown here how this numerical approach and the underlying physical model can be improved in order to simulate the motion of the piano string with a high degree of realism. Starting from the fundamental equations of a damped, stiff string interacting with a nonlinear hammer, a numerical finite difference scheme is derived, from which the time histories of string displacement and velocity for each point of the string are computed in the time domain. The interacting force between hammer and string, as well as the force acting on the bridge, are given by the same scheme. The performance of the model is illustrated by a few examples of simulated string waveforms. A brief discussion of the aspects of numerical stability and dispersion with reference to the proper choice of sampling parameters is also incl...
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paper_title: Traveling Wave Implementation of a Lossless Mode-coupling Filter and the Wave Digital Hammer

paper_content:

Monochloro-substituted saturated compounds may be prepared by condensing a saturated compound such as cycloalkyl hydrocarbon with a chloromonoolefin possessing not more than 4 carbon atoms and having the chlorine atom attached to one of the doubly-bonded carbon atoms in the presence of a free radical-generating catalyst and a promoter comprising a hydrogen chloride compound.
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paper_title: Elimination of delay-free loops in discrete-time models of nonlinear acoustic systems

paper_content:

Nonlinear acoustic systems are often described by means of nonlinear maps acting as instantaneous constraints on the solutions of a system of linear differential equations. This description leads to discrete-time models exhibiting noncomputable loops. We present a solution to this computability problem by means of geometrical transformation of the nonlinearities and algebraic transformation of the time-dependent equations. The proposed solution leads to stable and accurate simulations even at relatively low sampling rates.
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paper_title: Numerical simulations of piano strings. II. Comparisons with measurements and systematic exploration of some hammer‐string parameters

paper_content:

A physical model of the piano string, using finite difference methods, has recently been developed. [Chaigne and Askenfelt, J. Acoust. Soc. Am. 95, 1112–1118 (1994)]. The model is based on the fundamental equations of a damped, stiff string interacting with a nonlinear hammer, from which a numerical finite difference scheme is derived. In the present study, the performance of the model is evaluated by systematic comparisons between measured and simulated piano tones. After a verification of the accuracy of the method, the model is used as a tool for systematically exploring the influence of string stiffness, relative striking position, and hammer‐string mass ratio on string waveforms and spectra.
---
paper_title: Numerical simulations of piano strings. I. A physical model for a struck string using finite difference methods

paper_content:

The first attempt to generate musical sounds by solving the equations of vibrating strings by means of finite difference methods (FDM) was made by Hiller and Ruiz [J. Audio Eng. Soc. 19, 462–472 (1971)]. It is shown here how this numerical approach and the underlying physical model can be improved in order to simulate the motion of the piano string with a high degree of realism. Starting from the fundamental equations of a damped, stiff string interacting with a nonlinear hammer, a numerical finite difference scheme is derived, from which the time histories of string displacement and velocity for each point of the string are computed in the time domain. The interacting force between hammer and string, as well as the force acting on the bridge, are given by the same scheme. The performance of the model is illustrated by a few examples of simulated string waveforms. A brief discussion of the aspects of numerical stability and dispersion with reference to the proper choice of sampling parameters is also incl...
---
paper_title: Commuted Piano Synthesis

paper_content:

The \commuted piano synthesis" algorithm is described, based on a simpli ed acoustic model of the piano. The model includes multiple coupled strings, a nonlinear hammer, and an arbitrarily large soundboard and enclosure. Simpli cations are employed which greatly reduce computational complexity. Most of the simpli cations are made possible by the commutativity of linear, time-invariant systems. Special care is given to the felt-covered hammer which is highly nonlinear and therefore does not commute with other components. In its present form, a complete, two-key piano can be synthesized in real time on a single 25MHz Motorola DSP56001 signal processing chip.
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paper_title: Physical Modeling of Plucked String Instruments with Application to Real-Time Sound Synthesis

paper_content:

An efficient approach for real-time synthesis of plucked string instruments using physical modeling and DSP techniques is presented. Results of model-based resynthesis are illustrated to demonstrate that high-quality synthetic sounds of several string instruments can be generated using the proposed modeling principles. Real-time implementation using a signal processor is described, and several aspects of controlling physical models of plucked string instruments are studied.
---
paper_title: Principles of Digital Waveguide Models of Musical Instruments

paper_content:

Basic principles of digital waveguide modeling of musical instruments are presented in a tutorial introduction intended for graduate students in electrical engineering with a solid background in signal processing and acoustics. The vibrating string is taken as the principal illustrative example, but the formulation is unified with that for acoustic tubes. Modeling lossy stiff strings using delay lines and relatively low-order digital filters is described. Various choices of wave variables are discussed, including velocity waves, force waves, and root-power waves. Signal scattering at an impedance discontinuity is derived for an arbitrary number of waveguides intersecting at a junction. Various computational forms are discussed, including the Kelly-Lochbaum, one-multiply, and normalized scattering junctions. A relatively new three-multiply normalized scattering junction is derived using a two-multiply transformer to normalize a one-multiply scattering junction. Conditions for strict passivity of the model are discussed. Use of commutativity of linear, time-invariant elements to greatly reduce computational cost is described. Applications are summarized, and models of the clarinet and bowed-string are described in some detail. The reed-bore and bow-string interactions are modeled as nonlinear scattering junctions attached to the bore/string acoustic waveguide.
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paper_title: Development and Calibration of a Guitar Synthesizer

paper_content:

A specific efficient implementation for the synthesis of acoustic guitar tones is described which is based on recent digital waveguide synthesis techniques. Enhanced analysis methods are used for calibrating the synthesis model based on digitized tones. The lowest resonances of the guitar body are implemented with separate parametric second-order resonators that run at a lower sampling rate than the string model.
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paper_title: Perceptual study of decay parameters in plucked string synthesis

paper_content:

A listening experiment was conducted to study the audibility of variation of decay parameters in plucked string synthesis. A digital commuted-waveguide-synthesis model was used to generate the test sounds. The decay of each tone was parameterized with an overall and a frequency-dependent decay parameter. Two different fundamental frequencies, tone durations, and types of excitation signals were used totalling in eight test sets for both parameters. The results indicate that variations between 25% and 40% in the time constant of decay are inaudible. This suggests that large deviations in decay parameters can be allowed from a perceptual viewpoint. The results are applied in model-based audio processing.
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paper_title: Robust loss filter design for digital waveguide synthesis of string tones

paper_content:

A robust loss filter design method is presented for digital waveguide string models, which can be used with high filter orders. The method aims at minimizing the decay time error in partials of the synthetic tone. This is achieved by a new weighting function based on the first-order Taylor series approximation of the decay time errors. Smoothing of decay time data and requiring the design to be minimum-phase are also proposed to facilitate the stability of the design. The new method is applicable to analysis-based sound synthesis of piano and guitar tones, for example.
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paper_title: Physical Modeling of Plucked String Instruments with Application to Real-Time Sound Synthesis

paper_content:

An efficient approach for real-time synthesis of plucked string instruments using physical modeling and DSP techniques is presented. Results of model-based resynthesis are illustrated to demonstrate that high-quality synthetic sounds of several string instruments can be generated using the proposed modeling principles. Real-time implementation using a signal processor is described, and several aspects of controlling physical models of plucked string instruments are studied.
---
paper_title: Synchronizing Computer Graphics Animation and Audio

paper_content:

Our prototype authoring tool synchronizes computer graphics animation, narration and background music. It lets users visualize the contents of a background music track and narration along a single time line and displays an animation's scenario sharing the same time line. A graphical user interface supports interactive editing of both background music and narration without any loss in quality.
---
paper_title: Simple and robust method for the design of allpass filters using least-squares phase error criterion

paper_content:

We consider a simple scheme for the design of allpass filters for approximation (or equalization) of a given phase function using a least-squares error criterion. Assuming that the desired phase response is prescribed at a discrete set of frequency points, we formulate a general least-squares equation-error solution with a possible weight function. Based on the general formulation and detailed analysis of the introduced error, we construct a new algorithm for phase approximation. In addition to iterative weighting of the equation error, the nominal value of the desired group delay is also adjusted iteratively to minimize the total phase error measure in equalizer applications. This new feature essentially eliminates the difficult choice of the nominal group delay which is known to have a profound effect on the stability of the designed allpass filter. The proposed method can be used for highpass and bandpass equalization as well, where the total phase error can be further reduced by introducing an adjustable-phase offset in the optimization. The performance of the algorithm is analyzed in detail with examples. First we examine the approximation of a given phase function. Then we study the equalization of the nonlinear phase of various lowpass filters. Also, a bandpass example is included. Finally we demonstrate the use of the algorithm for the design of approximately linear-phase recursive filters as a parallel connection of a delay line and an allpass filter. >
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paper_title: Quality of Piano Tones

paper_content:

A synthesizer was constructed to produce simultaneously 100 pure tones with means for controlling the intensity and frequency of each one of them. The piano tones were analyzed by conventional apparatus and methods and the analysis set into the synthesizer. The analysis was considered correct only when a jury of eight listeners could not tell which were real and which were synthetic tones. Various kinds of synthetic tones were presented to the jury for comparison with real tones. A number of these were judged to have better quality than the real tones. According to these tests synthesized piano like tones were produced when the attack time was less than 0.01 sec. The decay can be as long as 20 sec for the lower notes and be less than 1 sec for the very high ones. The best quality is produced when the partials decrease in level at the rate of 2 db per 100 cps increase in the frequency of the partial. The partials below middle C must be inharmonic in frequency to be piano like.
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paper_title: Audibility of the timbral effects of inharmonicity in stringed instrument tones

paper_content:

Listening tests were conducted to find the audibility of inharmonicity in musical sounds produced by stringed instruments, such as the piano or the guitar. The audibility threshold of inharmonicity was measured at five fundamental frequencies. Results show that the detection of inharmonicity is strongly dependent on the fundamental frequency f0. A simple model is presented for estimating the threshold as a function of f0. The need to implement inharmonicity in digital sound synthesis is discussed.
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paper_title: Bandwidth of perceived inharmonicity for physical modeling of dispersive strings

paper_content:

The influence of accurate reproduction of inharmonicity on the perceived quality of piano tones is investigated. Acoustic piano tones were resynthesized by changing the bandwidth of correct positioning of partials. Cutoff frequencies of inharmonicity for different pitches have been established by psychoacoustic experimentation. The results are applicable to the design of dispersive resonators in sound synthesis by physical modeling.
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paper_title: Circulant and elliptic feedback delay networks for artificial reverberation

paper_content:

The feedback delay network (FDN) has been proposed for digital reverberation, The digital waveguide network (DWN) is also proposed with similar advantages. This paper notes that the commonly used FDN with an N/spl times/N orthogonal feedback matrix is isomorphic to a normalized digital waveguide network consisting of one scattering junction joining N reflectively terminated branches. Generalizations of FDNs and DWNs are discussed. The general case of a lossless FDN feedback matrix is shown to be any matrix having unit-modulus eigenvalues and linearly independent eigenvectors. A special class of FDNs using circulant matrices is proposed. These structures can be efficiently implemented and allow control of the time and frequency behavior. Applications of circulant feedback delay networks in audio signal processing are discussed.
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paper_title: Resynthesis of Coupled Piano String Vibrations Based on Physical Modeling

paper_content:

This paper presents a technique to resynthesize the sound generated by the vibrations of two piano strings tuned to a very close pitch and coupled at the bridge level. Such a mechanical system produces doublets of components, thus generating beats and double decays on the amplitudes of the partials. We design a digital waveguide model by coupling two elementary digital waveguides. This model is able to reproduce perceptually relevant sounds. The parameters of the model are estimated from the analysis of real signals collected directly on the strings by laser velocimetry. Sound transformations can be achieved by modifying relevant parameters in order to simulate various physical situations.
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paper_title: Pedal resonance effect simulation device for digital pianos

paper_content:

The present invention relates to a pedal resonance effect simulation device for digital pianos consisting of a resonance effect circuit for the simulation of the resonance effects in the strings of a traditional mechanical piano, coupled with a reference model which varies the resonance contributions of the various strings, which are equivalent to those of a mechanical piano, by using delay lines with a method which considers the position of the resonance pedal pressed by the performer upon reproduction.
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paper_title: Coupled piano strings

paper_content:

The admittance of the piano bridge has a crucial effect on piano tone by coupling together the strings belonging to one note into a single dynamical system. In this paper, we first develop theoretical expressions that show how the rate of energy transmission to the bridge as a function of time (including the phenomena of beats and ’’aftersound’’) depends on bridge admittance, hammr irregularities, and the exact state in which the piano is tuned. We then present experimental data showing the effects of mutual string coupling on beats and aftersound, as well as the great importance of the two polarizations of the string motion. The function of the una corda pedal in controlling the aftersound is explained, and the stylistic possibilities of a split damper are pointed out. The way in which an excellent tuner can use fine tuning of the unisons to make the aftersound more uniform is discussed.
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paper_title: Accurate and efficient modeling of beating and two-stage decay for string instrument synthesis

paper_content:

In this paper a novel modeling method is presented for beating and two-stage decay. Here, one digital waveguide is used for each note and some resonators are run in parallel to simulate the beating and two-stage decay of those partials, where these phenomena are most prominent. The resonator bank is implemented by using the multi-rate approach, resulting in a decrease of computational cost by a factor of 10. By taking the advantage of the characteristics of the resonators, relatively simple upsampling and downsampling filters are used. Two different filtering approaches are presented and compared with respect to computational complexity. Examples are shown with the application to piano sound synthesis.
---
paper_title: Circulant and elliptic feedback delay networks for artificial reverberation

paper_content:

The feedback delay network (FDN) has been proposed for digital reverberation, The digital waveguide network (DWN) is also proposed with similar advantages. This paper notes that the commonly used FDN with an N/spl times/N orthogonal feedback matrix is isomorphic to a normalized digital waveguide network consisting of one scattering junction joining N reflectively terminated branches. Generalizations of FDNs and DWNs are discussed. The general case of a lossless FDN feedback matrix is shown to be any matrix having unit-modulus eigenvalues and linearly independent eigenvectors. A special class of FDNs using circulant matrices is proposed. These structures can be efficiently implemented and allow control of the time and frequency behavior. Applications of circulant feedback delay networks in audio signal processing are discussed.
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Title: Physically Informed Signal Processing Methods for Piano Sound Synthesis: A Research Overview

Section 1: INTRODUCTION
Description 1: Write an introduction to the paper, outlining the primary motivation, context, and objectives of physical modeling in piano sound synthesis.

Section 2: ACOUSTICS AND MODEL STRUCTURE
Description 2: Discuss the general aspects of piano acoustics and the structure of the physical model used for piano sound synthesis.

Section 3: THE HAMMER
Description 3: Delve into the physical and numerical modeling of the piano hammer, discussing hammer-string interaction and various implementation approaches.

Section 4: THE STRING
Description 4: Describe different string modeling techniques with a focus on digital waveguide methods, including loss filter design, dispersion simulation, and string coupling.

Section 5: RADIATION MODELING
Description 5: Address the modeling of the soundboard and the radiation properties, including different implementation strategies and computational considerations.

Section 6: CONCLUSIONS
Description 6: Summarize the main stages of development in physical modeling for the piano, highlighting key computational aspects and discussing the importance of perceptual and stability considerations.

Section 7: ACKNOWLEDGMENTS
Description 7: Provide acknowledgments for any contributing individuals or organizations and mention funding sources where applicable.