Digital Sound Synthesis by Physical Modeling Using the Functional Transformation Method

Abstract

1. Introduction.- 2. Sound-Based Synthesis Methods.- 1 Wavetable synthesis.- 1.1 Looping.- 1.2 Pitch shifting.- 1.3 Enveloping.- 1.4 Filtering.- 2 Granular synthesis.- 2.1 Asynchronous granular synthesis.- 2.2 Pitch-synchronous granular synthesis.- 3 Additive synthesis.- 4 Subtractive synthesis.- 5 FM synthesis.- 6 Combinations of sound-based synthesis methods.- 3. Physical Description of Musical instruments.- 1 General notation.- 2 Subdivision of a musical instrument into vibration generators and a resonant body.- 2.1 Division of stringed instruments into single strings and the resonant body.- 2.1.1 Construction of stringed instruments.- 2.1.2 Fixed strings filtered with the resonant body.- 2.1.3 Strings terminated with independent impedances.- 2.1.4 Strings terminated with an impedance network.- 2.2 Division of a kettle drum into a membrane and the kettle.- 2.2.1 Construction of drums.- 2.2.2 Drum body simulation by modifying the physical parameters of the membrane.- 2.2.3 Drum body simulation by room acoustic simulation with the membrane as vibrating boundary.- 3 Physical description of string vibrations.- 3.1 Longitudinal string vibrations.- 3.2 Torsional string vibrations.- 3.3 Transversal string vibrations.- 3.3.1 Basic linear model.- 3.3.2 Nonlinear excitation functions.- 3.3.3 Nonlinear PDE with solution-dependent coefficients.- 4 Physical description of membrane vibrations.- 4.1 Bending membrane vibrations.- 5 Physical description of resonant bodies.- 6 Chapter summary.- 4. Classical Synthesis Methods Based on Physical Models.- 1 Finite difference method.- 1.1 FDM applied to scalar PDEs.- 1.2 FDM applied to vector PDEs.- 2 Digital waveguide method.- 2.1 Digital waveguides simulating string vibrations.- 2.2 Digital waveguide meshes simulating membrane vibrations.- 3 Modal synthesis.- 4 Chapter summary.- 5. Functional Transformation Method.- 1 Fundamental principles of the FTM.- 1.1 FTM applied to scalar PDEs.- 1.1.1 Laplace transformation.- 1.1.2 Sturm-Liouville transformation.- 1.1.3 Transfer function model.- 1.1.4 Discretization of the MD TFM.- 1.1.5 Inverse Sturm-Liouville transformation.- 1.1.6 Inverse z-transformation.- 1.2 FTM applied to vector PDEs.- 1.2.1 Laplace transformation.- 1.2.2 Sturm-Liouville transformation.- 1.2.3 Transfer function model.- 1.2.4 Discretization of the MD TFM.- 1.2.5 Inverse Sturm-Liouville transformation.- 1.2.6 Inverse z-transformation.- 1.3 FTM applied to PDEs with nonlinear excitation functions.- 1.4 FTM applied to PDEs with solution-dependent coefficients.- 1.5 Stability and simulation accuracy of the FTM.- 1.6 Section summary.- 2 Application of the FTM to vibrating strings.- 2.1 Transversal string vibrations described by a scalar PDE.- 2.2 Longitudinal string vibrations described by vector PDEs.- 2.2.1 Boundary conditions of second kind.- 2.2.2 Boundary conditions of third kind.- 2.2.3 Two interconnected strings.- 2.3 Transversal string vibrations with nonlinear excitation functions.- 2.3.1 Piano hammer excitation.- 2.3.2 Slapped bass.- 2.4 Transversal string vibrations with tension-modulated nonlinearities.- 3 Application of the FTM to vibrating membranes.- 3.1 Rectangular reverberation plate.- 3.2 Circular drum heads.- 4 Application of the FTM to resonant bodies.- 5 Chapter summary.- 6. Comparison of the Ftm with the Classical Physical Modeling Methods.- 1 Comparison of the FTM with the FDM.- 2 Comparison and combination of the FTM with the DWG.- 2.1 Comparison of the FTM with the DWG.- 2.2 Combination of the DWG with the FTM.- 2.2.1 Designing the loss filter.- 2.2.2 Designing the dispersion filter.- 2.2.3 Designing the fractional delay filter.- 2.2.4 Adjusting the excitation function.- 2.3 Limits of the combination.- 3 Comparison of the FTM with the MS.- 4 Chapter conclusions.- 7. Summary, Conclusions, and Outlook.