# Musimathics: The Mathematical Foundations of Music

Gareth Loy
Foreword by John Chowning
Pages: 584
https://www.jstor.org/stable/j.ctt5hhm8g

1. Front Matter
(pp. i-vi)
(pp. vii-x)
3. Foreword
(pp. xi-xii)
John Chowning

During the 1960s and 1970s the Artificial Intelligence Laboratory at Stanford University was a multidisciplinary facility populated by enormously gifted and dedicated workers trained in the sciences, engineering sciences, social sciences, and music. The musicians were part of the Center for Computer Research in Music and Acoustics (CCRMA) and shared the use of the A. I. Lab’s computer. While the scientists had no professional interest in music, their scientific and technical knowledge was of critical importance to the musicians in learning to make effective use of the ever-evolving hardware and software. We had been seduced by Max Mathews’s now famous...

4. Preface
(pp. xiii-xiv)
5. Acknowledgments
(pp. xiv-xvi)
Gareth Loy
6. 1 Digital Signals and Sampling
(pp. 1-48)

Digital audio has fundamentally changed the way music is made, distributed, and shared. It is now so pervasive that most of the music we hear is digitally stored and processed. A good deal of it is also created digitally.

The public has benefited enormously from the technological advances of digital audio, but at a price. Legal efforts to limit and regulate music copying have come about largely because of digital audio’s ability to make perfect copies of recordings. Digital audio has become the proverbial lion in the pathway that our society must address in order to restore balance between artistic,...

7. 2 Musical Signals
(pp. 49-102)

Mathematicians aren’t above imagining new kinds of numbers when circumstances warrant. The natural numbers—whole numbers greater than zero—are probably as old as civilization. But when the natural numbers failed to solve equations such asc=a-bfor all possible natural numbers a and b, mathematicians invented negative numbers. The result was the birth of the integers. Rational numbers were developed when integers failed to solve equations such asc=a÷bfor all possible integersaandb. When a careful look at irrational numbers such asπshowed the limitations of rational...

8. 3 Spectral Analysis and Synthesis
(pp. 103-158)

Joseph Fourier¹ contributed a penetrating insight to our knowledge of waveforms in general and music in particular.

Any periodic vibration, no matter how complicated it seems, can be built up from sinusoids whose frequencies are integer multiples of a fundamental frequency, by choosing the proper amplitudes and phases.

Signals whose frequencies are integer multiples of some fundamental are theharmonicsof that frequency. The process of building up a compound signal from simple sinusoids is known asFourier synthesis, orspectral synthesis.

Fourier synthesis allows us to create a waveform from a specification of the strengths of its various harmonics....

9. 4 Convolution
(pp. 159-194)

Convolution lies at the heart of modem digital audio. The quickest and most intuitive introduction to convolution that I’ve found comes by way of an antique rolling shutter camera. When photography was first being developed early in the twentieth century, some cameras used a rolling shutter instead of an iris to control exposure of the film. A narrow slit was cut in a roll of opaque paper attached between two spring-loaded rollers that when released by the camera’s trigger caused the slit to scroll quickly across the film plate between the lens and the film, exposing it to light and...

10. 5 Filtering
(pp. 195-262)

Just as an antique camera technology provided an intuitive introduction for convolution (chapter 4), analog tape recorders provide a very effective model for understanding filtering.

Analog tape recorders operate by dragging a magnetic tape past erase, record, and playback heads at a constant rate (figure 5.1). The tape is pinched between the motor-driven capstan and the pinch roller to pull it along. The reel on the left supplies tape at the rate determined by the angular velocity of the capstan; the motor-driven reel on the right takes up the tape after it passes the capstan.

When recording, the tape first...

11. 6 Resonance
(pp. 263-298)

The study of resonance unlocks the deepest understanding of musical instrument sounds. It explains how sound propagates in rooms, why microphones and loudspeakers sound the way they do, and how our ears work. The subject can be boiled down to just three characteristics of vibration: displacement, velocity, and acceleration. Furthermore, these three qualities can be seen as just different aspects of the same underlying phenomenon. The unifying perspective is called thederivative. It provides a simpler and more coherent way to understand resonance than was given in volume 1, section 8.9, and it unifies a broad range of musical phenomena....

12. 7 The Wave Equation
(pp. 299-324)

The characteristic movement of simple harmonic motion produces the shape of a sinusoid when its displacement is plotted against time (volume 1, chapter 8). What about the vibrating patterns of higher-dimensional shapes, such strings, membranes, and air? While it may seem daunting to consider more complex systems, it needn’t be so. Thewave equationprovides a unified perspective for all forms of physical vibration in terms not much more complex than simple harmonic motion. It describes, for some point on an object, how its displacement from equilibrium changes from moment to moment based on the forces in its immediate neighborhood....

13. 8 Accoustical Systems
(pp. 325-362)

We have seen how conservative forces sustain vibration within strings, pipes, membranes, and bars, holding energy in a vibrating system (volume 1, chapter 8). This chapter focuses on the way energy flows between coupled acoustical systems. The study of acoustics is greatly simplified by understanding the circumstances governing the flow of sound energy because instruments, ears, and rooms can all be viewed as networks of interconnected vibrating elements.

For simplicity, the focus here is on one-dimensional systems such as musical instrument tubes, the subject of classical hydrodynamics. With this approach we can look at wave motion in the bore of...

14. 9 Sound Synthesis
(pp. 363-452)

Fourier synthesis can be used to create any periodic vibration (see chapter 3). But Fourier methods are only one of an essentially limitless number of techniques that can be used to synthesize sounds.

Linear synthesis techniques can generally be used to reproduce a sound that is identical to the original. The Fourier transform, for example, can be used to reproduce any periodic waveform. This is part of the reason we call it a transform: we can analyze a signal into its transformed state and then use its inverse transform to reproduce the original. Nonlinear techniques generally provide no way to...

15. 10 Dynamic Spectra
(pp. 453-510)

Because musical signals demonstrate highly complex interactions between time and frequency, music notation systems throughout the world are designed to convey information that is local to both time and frequency. But up to this point in this book, the time and frequency aspects of sound have been held strictly separate. In this chapter, we examine ways to treat both in one unified representation.

In the opening quotation Gabor characterizes Fourier analysis as “a timeless description.” While this is true, there are shades of meaning. Gabor is reminding us that the Fourier transform (defined in equation 3.5) must be evaluated for...

16. Epilogue
(pp. 511-512)

Our culture is what we have learned to our advantage about the ways of the world. There are as many methods of cultural transmission as there are kinds of knowledge.

It might be as simple as telling tales at night around a camp fire. Polynesian sailors navigated among the thousands of South Pacific islands by reckoning with symbolic maps they made from slats of bamboo. The Australian aborigines combine place and time, and link themselves to their ancestors in the dreamtime via songlines. The druids at Stonehenge encoded the grinding of the equinoctal plane against the ecliptic plane by a...

17. Appendix
(pp. 513-532)
18. Notes
(pp. 533-538)
19. Glossary
(pp. 539-542)
20. References
(pp. 543-546)
21. Index of Equations and Mathematical Formulas
(pp. 547-550)
22. Subject Index
(pp. 551-562)