1. Introduction -- 2. Sound-Based Synthesis Methods -- 1 Wavetable synthesis -- 2 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 -- 3 Physical description of string vibrations -- 4 Physical description of membrane vibrations -- 5 Physical description of resonant bodies -- 6 Chapter summary -- 4. Classical Synthesis Methods Based on Physical Models -- 1 Finite difference method -- 2 Digital waveguide method -- 3 Modal synthesis -- 4 Chapter summary -- 5. Functional Transformation Method -- 1 Fundamental principles of the FTM -- 2 Application of the FTM to vibrating strings -- 3 Application of the FTM to vibrating membranes -- 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 -- 3 Comparison of the FTM with the MS -- 4 Chapter conclusions -- 7. Summary, Conclusions, and Outlook.