Musimathics The Mathematical Foundations of Music Volume 2 Gareth Loy Foreword by John Chowning The MIT Press Cambridge, Massachusetts London, England
..2.3.4.5.6.7.8.9.0..2.3.4 2 2. 2.2 2.3 2.4 2.5 2.6 Foreword by John Chowning Preface Acknowledgments Digital Signals and Sampling Measuring the Ephemeral Analog-to-Digital Conversion Aliasing Digital-to-Analog Conversion Binary Numbers Synchronization Discretization Precision and Accuracy Quantization Noise and Distortion Information Density of Digital Audio Codecs Further Refinements Cultural Impact of Digital Audio Musical Signals Why Imaginary Numbers? Operating with Imaginary Numbers Complex Numbers de Moivre's Theorem Euler's Formula Phasors XI xiii xiv 9 20 22 28 28 29 29 33 38 40 42 46 47 49 49 5 52 62 64 68
2.7 2.8 2.9 2.0 2. 3 3. 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4 4. 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.0 5 5. 5.2 5.3 5.4 5.5 5.6 Graphing Complex Signals Spectra of Complex Sampled Signals Multiplying Phasors Graphing Complex Spectra Analytic Signals Spectral Analysis and Synthesis Introduction to the Fourier Transform Discrete Fourier Transform Discrete Fourier Transfom in Action Inverse Discrete Fourier Transform Analyzing Real-World Signals Windowing Fast Fourier Transform Properties of the Discrete Fourier Transform A Practical Hilbert Transform Convolution Rolling Shutter Camera Defining Convolution Numerical Examples of Convolution Convolving Spectra Convolving Signals Convolution and the Fourier Transform Domain Symmetry between Signals and Spectra Convolution and Sampling Theory Convolution and Windowing Correlation Functions Suggested Reading Filtering Tape Recorder as a Model of Filtering Introduction to Filtering A Simple Filter Finding the Frequency Response Linearity and Time Invariance of Filters FIR Filters 86 87 89 92 95 00 03 03 4 25 34 38 4 45 47 54 56 59 59 6 63 68 72 80 80 85 87 9 93 94 95 95 99 20 203 27 28
ix 5.7 IIR Filters 28 5.8 Canonical Filter 29 5.9 Time Domain Behavior of Filters 29 5.0 Filtering as Convolution 222 5. Z Transform 224 5.2 Z Transform of the General Difference Equation 232 5.3 Filter Families 244 26 6 Resonance 263 6. The Derivative 263 6.2 Differential Equations 276 6.3 Mathematics of Resonance 280 297 7 The Wave Equation 299 7. One-Dimensional Wave Equation and String Motion 299 7.2 An Example 307 7.3 Modeling Vibration with Finite Difference Equations 30 7.4 Striking Points, Plucking Points, and Spectra 39 324 8 Acoustical Systems 325 8. Dissipation and Radiation 325 8.2 Acoustical Current 326 8.3 Linearity of Frictional Force 329 8.4 Inertance, Inductive Reactance 332 8.5 Compliance, Capacitive Reactance 333 8.6 Reactance and Alternating Current 334 8.7 Capacitive Reactance and Frequency 335 8.8 Inductive Reactance and Frequency 336 8.9 Combining Resistance, Reactance, and Alternating Current 336 8.0 Resistance and Alternating Current 337 8. Capacitance and Alternating Current 337 8.2 Acoustical Impedance 340 8.3 Sound Propagation and Sound Transmission 344 8.4 Input Impedance: Fingerprinting a Resonant System 35 8.5 Scattering Junctions 357 360 Suggested Reading 362
x Contents 9 Sound Synthesis 363 9. Forms of Synthesis 363 9.2 A Graphical Patch Language for Synthesis 365 9.3 Amplitude Modulation 384 9.4 Frequency Modulation 389 9.5 Vocal Synthesis 409 9.6 Synthesizing Concert Hall Acoustics 425 9.7 Physical Modeling 433 9.8 Source Models and Receiver Models 449 450 0 Dynamic Spectra 453 0. Gabor's Elementary Signal 454 0.2 Short-Time Fourier Transform 459 0.3 Phase Vocoder 486 0.4 Improving on the Fourier Transform 496 0.5 Psychoacoustic Audio Encoding 502 507 Suggested Reading 509 Epilogue 5 Appendix 53 A.I About Algebra 53 A.2 About Trigonometry 54 A.3 Series and Summations 57 A.4 Trigonometric Identities 58 A.5 Modulo Arithmetic and Congruence 522 A.6 Finite Difference Approximations 523 A.7 Walsh-Hadamard Transform 525 A.8 Sampling, Reconstruction, and the Sine Function 526 A.9 Fourier Shift Theorem 528 A. 0 Spectral Effects of Ring Modulation 529 A. Derivation of the Reflection Coefficient 530 Notes 533 Glossary 539 References 543 Index of Equations and Mathematical Formulas 547 Subject Index 55