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Contents
Contents
Index
Contents
Book Summary
As Teaching Aid
Other Reading
Sound Synthesis and Physical Modeling
Abstract Digital Sound Synthesis
Additive Synthesis
Subtractive Synthesis
Wavetable Synthesis
AM and FM Synthesis
Other Methods
Physical Modeling
Lumped Mass-Spring Networks
Modal Synthesis
Digital Waveguides
Hybrid Methods
Direct Numerical Simulation
Physical Modeling: A Larger View
Physical Models as Descended from Abstract Synthesis
Connections: Direct Simulation and Other Methods
Complexity of Musical Systems
Why?
Time Series and Difference Operators
Finite Differences
Time Series
Difference and Averaging Operators
Frequency Domain Interpretation
Problems
Programming Exercises
The Oscillator
The Simple Harmonic Oscillator
Sinusoidal Solution
Energy
As First-Order System
Coupled Systems of Oscillators
A Simple Scheme
As Recursion
Initialization
Frequency Domain Analysis
Accuracy
Energy Analysis
Computational Considerations
Other Schemes
Using Time-averaging Operators
A Second-order Family of Schemes
Hughes-Taylor and Newmark Methods
An Exact Solution
Further Methods
Lumped Mass-Spring Networks
Loss
Finite Difference Scheme
Numerical Decay Time
An Exact Solution
Problems
Programming Exercises
The Oscillator in Musical Acoustics
Nonlinear Oscillators of Series Type
Difference Schemes
The Hammer
The Reed
The Glottis
Lip Models
The Bow
General Collisions
Problems
Programming Exercises
Grid Functions and Finite Difference Operators
Grid Functions and Difference Operators in One Dimension
Grid Functions
Time Difference and Averaging Operators
Spatial Difference Operators
Mixed Spatial-Temporal Difference Operators
Interpolation Operators
Frequency Domain Interpretation
Matrix Interpretation of Difference Operators
Inner Products and Identities
Energetic Identities, Manipulations and Inequalities
Grid Functions and Difference Operators in Two Dimensions: Cartesian Coordinates
Stencil Width
Frequency Domain Analysis
Anisotropic Behaviour
An Inner Product and Energetic Manipulations
Grid Functions and Difference Operators in Two Dimensions: Radial Coordinates
Problems
Programming Exercises
The 1D Wave Equation
Definition
Non-dimensionalized Form
Initial Conditions
Dispersion Relation
Phase and Group Velocity
Energy Analysis
Boundary Conditions
Bounds on Solution Size
Modes
Characteristics and Travelling Wave Solutions
A Simple Finite Difference Scheme
Initialization
von Neumann Analysis
Numerical Dispersion
Energy Analysis
Numerical Boundary Conditions
Bounds on Solution Size and Numerical Stability
Matrix Form
Output and Interpolation
Digital Waveguide Interpretation
Other Schemes
A Stencil-width Five Scheme
A Compact Implicit Scheme
A Comparative Study: Physical Modeling Sound Synthesis Methods
Lumped Network Models
Modal Synthesis
Digital Waveguides
Problems
Programming Exercises
Bars and Linear Strings
The Ideal Bar
Initial and Boundary Conditions
Frequency Domain Analysis
Energy Analysis
Finite Difference Schemes
Stiff Strings
Frequency Domain Analysis
Energy Analysis
Finite Difference Schemes
Matrix Interpretation
Hammer Models and Strings
Multiple Strings
Prepared Strings
Bars of Variable Cross-section
Problems
Programming Exercises
Nonlinear Strings
The Kirchhoff-Carrier String Model
Pitch Glides
Energy Analysis
Finite Difference Schemes
A Quasi-modal Form
A Digital Waveguide Form
General Planar Nonlinear String Motion
Coupling Between Transverse and Longitudinal Modes and Phantom Partials
Finite Difference Schemes
Nonplanar String Motion
Finite Difference Schemes
Instability of Planar Motion and Whirling Behaviour
Problems
Programming Exercises
Acoustic Tubes
Cylindrical Tubes
Conical Tubes
Tonehole Models
Coupling with Reed Models
Webster's Equation and the Vocal Tract
The Kelly-Lochbaum Speech Synthesis Model
Coupling with a Glottis Model
Problems
Programming Exercises
Linear Membranes and Plates
The 2D Wave Equation
Finite Difference Schemes
Digital Waveguide Meshes
Lumped Networks
Modal Synthesis
The Kirchhoff Thin Plate Model
Finite Difference Schemes
Modal Synthesis
Realistic Plate Models
Finite Difference Schemes
Plate Reverberation
Soundboards
Problems
Programming Exercises
Nonlinear Plates
The Berger Plate Model
Definition
Energy Analysis
A Finite Difference Scheme
Pitch Glides
The von Karman Plate Model
The Operator
Energy Analysis
A Finite Difference Scheme
Cymbals and Gongs
Problems
Programming Exercises
Finite Element Methods
Variational Forms
Shape Functions
Grid Generation
The Stiff String Revisited
The Linear Plate Revisited
Musical Instrument Modeling
Problems
Programming Exercises
Spectral Methods
Basis Functions
Fourier Series
Polynomials
Collocation Methods
Galerkin Methods
Spectral Derivatives
As Limiting Case of Finite Difference Approximations
Chebyshev Grids
Fast Fourier Transforms
High Order Time Stepping Methods
Forward and Backward Euler
Adams-Bashforth Methods
Adams-Moulton Methods
Runge-Kutta Methods
The Stiff String Again Revisited
The Linear Plate Again Revisited
Problems
Programming Exercises
Concluding Remarks
Physical Modeling Sound Synthesis Methods: Strengths and Weaknesses
Modal Synthesis
Digital Waveguides
Lumped Mass Spring Networks
Direct Simulation
The Future
MATLAB Code Examples
The Simple Harmonic Oscillator
The 1D Wave Equation: Finite Difference Scheme
The 1D Wave Equation: Modal Synthesis
The 1D Wave Equation: Digital Waveguide Synthesis
The Stiff String
A Tension-modulated String
Vocal Tract Synthesis
The 2D Wave Equation
Plate Reverberation
Gong
Bibliography
Index
About this document ...
Stefan Bilbao 2006-11-15
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