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Musimathics, Volume 2 The Mathematical Foundations of Music

ISBN-10: 0262122855

ISBN-13: 9780262122856

Edition: 2007

Authors: Gareth Loy, John Chowning

List price: $55.00
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Description:

"Mathematics can be as effortless as humming a tune, if you know the tune," writes Gareth Loy. In Musimathics,Loy teaches us the tune, providing a friendly and spirited tour of the mathematics of music--a commonsense, self-contained introduction for the nonspecialist reader. Volume 2 of Musimathicscontinues the story of music engineering begun in volume 1, focusing on the digital and computational domain. Loy goes deeper into the mathematics of music and sound, beginning with digital audio, sampling, and binary numbers, as well as complex numbers and how they simplify representation of musical signals. Chapters cover the Fourier transform, convolution, filtering, resonance, the wave equation, acoustical systems, sound synthesis, the short time Fourier transform, and the wavelet transform. These subjects provide the theoretical underpinnings of today's music technology. The material in volume 1 is all preparatory to the subjects presented in this volume, although either volume can be read independently. Cross-references to volume 1 are provided for concepts introduced in the earlier volume, and additional mathematical orientation is offered where necessary. The topics are all subjects that contemporary composers, musicians, and music engineers have found to be important. The examples given are all practical problems in music and audio. The level of scholarship and the pedagogical approach also make Musimathicsideal for classroom use. Additional material can be found at a companion web site.
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Book details

List price: $55.00
Copyright year: 2007
Publisher: MIT Press
Publication date: 5/11/2007
Binding: Hardcover
Pages: 584
Size: 7.25" wide x 9.25" long x 1.50" tall
Weight: 2.244

Foreword
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
Summary
Musical Signals
Why Imaginary Numbers?
Operating with Imaginary Numbers
Complex Numbers
de Moivre's Theorem
Euler's Formula
Phasors
Graphing Comlpex Signals
Spectra of Complex Sampled Signals
Multiplying Phasors
Graphing Complex Spectra
Analytic Signals
Summary
Spectral Analysis and Synthesis
Introduction to the Fourier Transform
Discrete Fourier Transform
Discrete Fourier Transform in Action
Inverse Discrete Fourier Transform
Analyzing Real-World Signals
Windowing
Fast Fourier Transform
Properties of the Discrete Fourier Transform
A Practical Hilbert Transform
Summary
Convolution
Rolling Shutter Camera
Defining Convolution
Numerical Examples of Convolution
Convolving Spectra
Convolving Sigals
Convolution and the Fourier Transform
Domain Symmetry between Signals and Spectra
Convolution and Sampling Theory
Convolution and Windowing
Correlation Functions
Summary
Suggested Reading
Filtering
Tape Recorder as a Model of Filtering
Introduction to Filtering
A Sample Filter
Finding the Frequency Response
Linearity and Time Invariance of Filters
FIR Filters
IIR Filters
Canonical Filter
Time Domain Behavior of Filters
Filtering as Convolution
Z Transform
Z Transform of the General Difference Equation
Filter Families
Summary
Resonance
The Derivative
Differential Equations
Mathematics of Resonance
Summary
The Wave Equation
One-Dimensional Wave Equation and String Motion
An Example
Modeling Vibration with Finite Difference Equations
Striking Points, Plucking Points, and Spectra
Summary
Acoustical Systems
Dissipation and Radiation
Acoustical Current
Linearity of Frictional Force
Inertance, Inductive Reactance
Compliance, Capacitive Reactance
Reactance and Alternating Current
Capacitive Reactance and Frequency
Inductive Reactance and Frequency
Combining Resistance, Reactance, and Alternating Current
Resistance and Alternating Current
Capacitance and Alternating Current
Acoustical Impedance
Sound Propagation and Sound Transmission
Input Impedance: Fingerprinting a Resonant System
Scattering Junctions
Summary
Suggested Reading
Sound Synthesis
Forms of Synthesis
A Graphical Patch Language for Synthesis
Amplitude Modulation
Frequency Modulation
Vocal Synthesis
Synthesizing Concert Hall Acoustics
Physical Modeling
Source Models and Receiver Models
Summary
Dynamic Spectra
Gabor's Elementary Signal
The Short-Time Fourier Transform
Phase Vocoder
Improving on the Fourier Transform
Psychoacoustic Audio Encoding
Summary
Suggested Reading
Epilogue
Appendix
About Algebra
About Trigonometry
Series and Summations
Trigonometric Identities
Modulo Arithmetic and Congruence
Finite Difference Approximations
Walsh-Hadamard Transform
Sampling, Reconstruction, and Sinc Function
Fourier Shift Theorem
Spectral Effects of Ring Modulation
Derivation of the Reflection Coefficient
Notes
Glossary
References
Equation Index