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Publisher:

MIT Press

Publication Date:

2007

Number of Pages:

562

Format:

Hardcover

Price:

50.00

ISBN:

9780262122856

Category:

Monograph

[Reviewed by , on ]

Sarah Boslaugh

09/1/2007

*Musimathics: the mathematical foundations of music* vol. 2 is a continuation of volume 1 of the same title, focusing in even greater depth on the mathematics behind music and sound, with particular emphasis on digital and computer-based music. Although volume 1 is intended as preparation for volume 2, either can be read independently.

This is not a book for the mathematically faint-of-heart, but interested musicians (or mathematicians interested in music) will find it rewarding if they are willing to put forth the effort to work through the explanations provided.

Loy writes with the assumption that, as he states in the preface, “enlightened common sense and inference are the whole of mathematics” and that “the cure for any lack of mathematical preparation on the reader’s part is simply to focus on what makes the most sense, and the rest will follow.” Fair enough: Loy is a remarkably clear writer and if you don’t understand, say, sampling and aliasing after diligent study of his text, perhaps you never will. Additionally, it’s not necessary to read *Musimathics* from cover to cover: individual chapters can be read as the interest and needs of the reader dictates, and each chapter includes the necessary mathematics to understand the content of that chapter.

Still, this is a very technical text which demands the reader be willing to put forth a fair amount of intellectual effort to get any benefit from reading it. Readers, particularly those working in electronic music, who are willing to put in the necessary time will be amply rewarded with increased understanding and appreciation of both music and mathematics. Additional material related to *Musimathics* vols 1 and 2 is available from http://www.musimathics.com/.

Gareth Loy is a performing musician, composer, software architect and digital audio systems engineer. He received his DMA in composition from Stanford University in 1980. Further information about his musical and engineering activities is available from Loy’s personal web page, http://www.garethloy.com/ and the web page for his consulting company, http://www.garethinc.com/.

Sarah Boslaugh (seb5632@bjc.org) is a Performance Analyst for BJC HealthCare and an adjunct professor at the Washington University School of Medicine in St. Louis, Missouri. She has written two books, *An Intermediate Guide to SPSS Programming: Using Syntax for Data Management* (SAGE, 2005) and *Secondary Data Sources for Public Health: A Practical Guide* (Cambridge University Press, forthcoming May 2007) and is editor-in-chief of *The Encyclopedia of Epidemiology* (forthcoming from Sage, November 2007).

Foreword John Chowning |
xiii | |

Preface | xv | |

Acknowledgments | xvi | |

1 |
Digital Signals and Sampling |
1 |

1.1 | Measuring the Ephemeral | 1 |

1.2 | Analog-to-Digital Conversion | 9 |

1.3 | Aliasing | 11 |

1.4 | Digital-to-Analog Conversion | 20 |

1.5 | Binary Numbers | 22 |

1.6 | Synchronization | 28 |

1.7 | Discretization | 28 |

1.8 | Precision and Accuracy | 29 |

1.9 | Quantization | 29 |

1.10 | Noise and Distortion | 33 |

1.11 | Information Density of Digital Audio | 38 |

1.12 | Codecs | 40 |

1.13 | Further Refinements | 42 |

1.14 | Cultural Impact of Digital Audio | 46 |

Summary | 47 | |

2 |
Musical Signals |
49 |

2.1 | Why Imaginary Numbers? | 49 |

2.2 | Operating with Imaginary Numbers | 51 |

2.3 | Complex Numbers | 52 |

2.4 | de Moivre's Theorem | 62 |

2.5 | Euler's Formula | 64 |

2.6 | Phasors | 68 |

2.7 | Graphing Comlpex Signals | 86 |

2.8 | Spectra of Complex Sampled Signals | 87 |

2.9 | Multiplying Phasors | 89 |

2.10 | Graphing Complex Spectra | 92 |

2.11 | Analytic Signals | 95 |

Summary | 100 | |

3 |
Spectral Analysis and Synthesis |
103 |

3.1 | Introduction to the Fourier Transform | 103 |

3.2 | Discrete Fourier Transform | 103 |

3.3 | Discrete Fourier Transform in Action | 125 |

3.4 | Inverse Discrete Fourier Transform | 134 |

3.5 | Analyzing Real-World Signals | 138 |

3.6 | Windowing | 141 |

3.7 | Fast Fourier Transform | 145 |

3.8 | Properties of the Discrete Fourier Transform | 147 |

3.9 | A Practical Hilbert Transform | 154 |

Summary | 156 | |

4 |
Convolution |
159 |

4.1 | Rolling Shutter Camera | 159 |

4.2 | Defining Convolution | 161 |

4.3 | Numerical Examples of Convolution | 163 |

4.4 | Convolving Spectra | 168 |

4.5 | Convolving Sigals | 172 |

4.6 | Convolution and the Fourier Transform | 180 |

4.7 | Domain Symmetry between Signals and Spectra | 180 |

4.8 | Convolution and Sampling Theory | 187 |

4.9 | Convolution and Windowing | 187 |

4.10 | Correlation Functions | 191 |

Summary | 193 | |

Suggested Reading | 194 | |

5 |
Filtering |
195 |

5.1 | Tape Recorder as a Model of Filtering | 195 |

5.2 | Introduction to Filtering | 199 |

5.3 | A Sample Filter | 201 |

5.4 | Finding the Frequency Response | 203 |

5.5 | Linearity and Time Invariance of Filters | 217 |

5.6 | FIR Filters | 218 |

5.7 | IIR Filters | 218 |

5.8 | Canonical Filter | 219 |

5.9 | Time Domain Behavior of Filters | 219 |

5.10 | Filtering as Convolution | 222 |

5.11 | Z Transform | 224 |

5.12 | Z Transform of the General Difference Equation | 232 |

5.13 | Filter Families | 244 |

Summary | 261 | |

6 |
Resonance |
263 |

6.1 | The Derivative | 263 |

6.2 | Differential Equations | 276 |

6.3 | Mathematics of Resonance | 280 |

Summary | 297 | |

7 |
The Wave Equation |
299 |

7.1 | One-Dimensional Wave Equation and String Motion | 299 |

7.2 | An Example | 307 |

7.3 | Modeling Vibration with Finite Difference Equations | 310 |

7.4 | Striking Points, Plucking Points, and Spectra | 319 |

Summary | 324 | |

8 |
Acoustical Systems |
325 |

8.1 | 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 |

8.10 | Resistance and Alternating Current | 337 |

8.11 | Capacitance and Alternating Current | 337 |

8.12 | Acoustical Impedance | 339 |

8.13 | Sound Propagation and Sound Transmission | 344 |

8.14 | Input Impedance: Fingerprinting a Resonant System | 351 |

8.15 | Scattering Junctions | 357 |

Summary | 360 | |

Suggested Reading | 362 | |

9 |
Sound Synthesis |
363 |

9.1 | 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 |

Summary | 450 | |

10 |
Dynamic Spectra |
453 |

10.1 | Gabor's Elementary Signal | 454 |

10.2 | The Short-Time Fourier Transform | 459 |

10.3 | Phase Vocoder | 486 |

10.4 | Improving on the Fourier Transform | 496 |

10.5 | Psychoacoustic Audio Encoding | 502 |

Summary | 507 | |

Suggested Reading | 509 | |

Epilogue | 511 | |

Appendix |
513 | |

A.1 | About Algebra | 513 |

A.2 | About Trigonometry | 514 |

A.3 | Series and Summations | 517 |

A.4 | Trigonometric Identities | 518 |

A.5 | Modulo Arithmetic and Congruence | 522 |

A.6 | Finite Difference Approximations | 523 |

A.7 | Walsh-Hadamard Transform | 525 |

A.8 | Sampling, Reconstruction, and Sinc Function | 526 |

A.9 | Fourier Shift Theorem | 528 |

A.10 | Spectral Effects of Ring Modulation | 529 |

A.11 | Derivation of the Reflection Coefficient | 530 |

Notes | 533 | |

Glossary | 539 | |

References | 543 | |

Equation Index | 547 | |

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