|Vol. 31 Issue 4 Reviews||Reviews > Publications >|
Gareth Loy: Musimathics: The Mathematical Foundations of Music, Volume One
Hardcover, The MIT Press, June 2006, ISBN-10 0-262-12282-0, ISBN-13 978-0-262-12282-5, 608 pages, 263 illustrations, forward (by Max Mathews), two appendices, glossary, notes, references, equation index, subject index, US$ 50.00/UK£ 32.95; The MIT Press, 55 Hayward Street, Cambridge, Massachusetts 02142-1493, USA; telephone (+1) 617-253-5646; fax (+1) 617-258-6779; Web mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=10916 or www.musimathics.com/.
Reviewed by Mark Ballora
With Musimathics, Gareth Loy solves a problem I have faced for a number of years now. As an instructor of a university Science of Music course, I have never found any single text that explains all necessary topics with equal depth and clarity, and thus have had to rely on course packs of photocopied compilations. In the future, this chore will be unnecessary, thanks to this book. Musimathics doesn’t have absolutely everything, but it comes so very close, and then provides a wealth of bonuses. The book is singularly precise, thorough, and often very funny.
The first chapters are preliminary, providing basic vocabulary for concepts that are treated in greater depth later. Chapter 1, “Music and Sound,” introduces air pressure, waves, and simple harmonic motion. Not a word is wasted anywhere. Students will likely be comforted by the low page count.
Chapter Two, “Representing Music,” gives an overview of how music is translated into pictorial symbols, with explanations of what is being represented at each step. Pitch is explained as both a matter of frequency and of interval ratios, and is supported by time-domain graphs of air pressure changes, the amplitude envelope, staff notation, and how pitch subsets form various scales. Duration and loudness are covered in terms of time, tempo, musical dynamic notation, and time signature. Timbre is discussed in terms of spectral changes and time. Graphs illustrate a string’s vibrational modes, plots of complex waves composed of harmonic partials, dynamic spectra, and sonograms.
Things take off with Chapter 3, “Musical Scales, Tuning, and Intonation.” Here is where everything gets treated down to the fine details. The nature of scales and ratios is discussed as they pertain to equal temperament, just intonation, Pythagorean intonation (and the problems introduced by the syntonic comma), meantone temperament, well temperament, various ethnic and microtonal scales (Hindustani and Partch, to name just two), and fret calculations used by guitar makers. Anything left out of a reader’s tuning background is likely to be covered here.
Chapter 4, “Physical Basis of Sound,” is approximately two-thirds “physical basis” and one-third “sound.” The majority of the chapter provides background in general physics, leading to musical considerations after covering underlying topics such as dimension, mass, density, velocity, Newton’s laws of motion, work, and conservative versus nonconservative forces. There is a good balance of equations and text, providing both a conceptual and a quantitative context for all topics (I’ve found that most other books emphasize one or the other, but rarely both). Here the equations get denser, as more symbols are applied to more terms. This is the chapter where novice students will learn to slow down and absorb material, symbol by symbol. Mr. Loy is well aware of the intimidation factor that can present itself in material like this, as he makes clear in his Preface: “I know what it’s like not to comprehend mathematics easily, and I also know what it’s like not to give up” (p. xvii). This chapter is where the non-mathematically inclined are likely to face a test of will. But, as he describes, the answers will come to those who understand that learning may require some effort on their part. And they will appreciate that this treatment is probably the most unambiguous description they’ll find anywhere.
Chapter 5, “Geometrical Basis of Sound,” tidily covers the relationship of sinusoids to circles and rotating vectors. This material is straightforward, and gives readers what they need to learn concepts like radians and angles by rote, just as musicians learn scales.
Chapter 6, “Psychophysical Basis of Sound,” goes inside of the human head, covering the auditory system and psychoacoustics. Here we get a thorough treatment of the auditory system as well as the differences between what can be objectively measured (F variables) and what is interpreted and perceived by the auditory apparatus (Y variables). This leads us through a number of peripheral topics including Shepard tones, the MP3 audio compression format, consonance and dissonance, localization, and timbre space.
Chapter 7, “Introduction to Acoustics,” picks up in many ways where Chapter 4 left off in its introduction to physics. This chapter focuses on wave behavior in strings and air, and culminates with reverberation and the musical character of rooms.
Chapter 8, “Vibrating Systems,” discusses the principles of musical instruments. What distinguishes this book from most other musical acoustics books is that the mechanics of the various instruments are not treated. Since the purview of this book is mathematics, math is what is presented. Instrument types are mentioned insofar as mathematics may describe the vibratory nature of strings, air columns, and membranes. But there is no discussion of the fipple mouthpiece, how the Bronx cheer forms the basis of brass embouchure, or the role of the sound post in the violin family. On the other hand, not every musical acoustics book presents Bessel functions to describe membrane vibrations, as the author does here.
The final chapter, “Composition and Methodology,” will be regarded by many as the book’s tour de force. At 123 pages, it is longer than any other chapter, and comprises some 25 percent of the book’s length. The chapter is striking in its integration of both the philosophies and the nuts and bolts of objective compositional methodologies dating from Guido d’Arezzo to the present. The list includes set theory, stochastics, probability, information theory, neural nets, and finally an exploration of the nature of intelligence itself. The scope of Mr. Loy’s discussion is broad and noble, indeed. Examples of all methodologies are realized with a programming language, Musimat, that the author created for this book. Readers are encouraged to download the code and create their own examples.
The Appendices provide background, trivia, and extra working tools. Appendix A provides a review of relevant mathematics, plus an enticing set of extra features that includes Xeno’s Paradox and excerpts from Pope John XXII’s fourteenth century Bull advising composers on how to write Godly music. A good example of Loy’s apt humor is evident in his justification for presenting the Greek alphabet as being “useful not only for the study of mathematics but also for students being rushed for fraternities. It may also come in handy when eating alphabet soup in Greece” (p. 419).
The second Appendix presents Musimat, the procedural programming language Mr. Loy created that specializes in realizing the musical operations presented in Chapter 9. While this program is handy for demonstrating procedures for musical algorithms, it is an illustration of principles only, as it does not synthesize audio. But as a generator of symbolic examples, it is certainly a well-conceived program, and no doubt some will find it highly effective.
Mr. Loy provides a new organization and integration of these subjects that is beautifully conceived and clearly relevant to musicians, using humor to skip elegantly to and fro in areas of mathematics, philosophy, and cultural commentary. The ordering of topics differs from book to book and class to class. Some instructors will want to follow the author’s ordering, others won’t. But given the tidy outlining and segmentation that characterizes the book (and many other books from MIT Press), it will be an easy matter for instructors to create a reading sequence of their choosing by simply re-ordering the well-defined sections. Often these topics are covered far more thoroughly than will be necessary for an introductory course, which means that those students wishing to go more in depth will have ample resources available to them in the text.
While the book is amply illustrated, it is visually “lo-fi.” All images are simply designed and none are in color. This probably accounts for the book’s low cost, and will be a welcome tradeoff for most purchasers. Most are entirely sufficient, although in some places more detailed illustrations would be helpful. The section on the human auditory system, for example, would be well served by some more detailed anatomical images. Such images are easy enough to find elsewhere, but those who want them will have to take the extra step of consulting other sources.
The difficulty in presenting this material is the interdependence of many of its topics, and it can be confounding to navigate through this web of dependencies in the one-dimensional book format. Mr. Loy tackles this challenge admirably for the most part, although in some cases terms are introduced before they have been defined. Standing waves, for example, are mentioned on p. 193, 62 pages before they are actually defined and described in the context of resonance.
This small breach of information ordering may be handled easily enough by a competent instructor, but it does bring about a larger point. Those familiar with this material will welcome the book's precise, concise, and modular format. But it is not a casual read for initiates. When it gets dense, many readers will likely benefit from an instructor who is able to expand upon it for them. (Don’t read this at home alone, kids—consult a professional!)
Sometimes descriptions are a little too concise. For example, the nature of nodes and antinodes in pipes is explained in its entirety as follows: “At the open end of a pipe, there is a displacement antinode because the air inside is free to move in and out of the tube. A the closed end of a pipe, there is a displacement node because the air can't move longitudinally (the closed end prevents it)” (p. 263). There's a bit more to it than this, and a few more paragraphs could clarify things considerably. While this might be a good reminder for someone already familiar with vibrational modes of pipes, a novice is likely to need a bit more explanation.
Given the thorough treatment of most topics, it is surprising when some areas get comparatively thin coverage. The definition of the Golden Mean is simply “about 2/3.” I personally found this a bit disappointing, considering the interesting geometrical explanation of this ratio, and its rich underlying role in nature and culture. But given the wealth of topics covered in the final chapter, understandably the line had to be drawn somewhere.
Mr. Loy’s definition of chaos theory is also not entirely complete—while emphasizing the deterministic nature of chaotic systems and their sensitivity to initial conditions, he does not discuss their aperiodicity or boundedness, which would have rounded out the four essential characteristics of chaos. But even though one might nitpick this issue, such a position risks losing sight of the larger issue, which is explained in the author’s unusually sensitive and insightful summation of nonlinear dynamics’ relevance to musical creation:
These characteristics of self-regulation are cornerstones of healthy responsiveness to life and mental well-being. How appropriate that stability, adaptability, and flexibility are also hallmarks of successful music… Here is the foundation for a music theory that weaves together information theory, chaos theory, complexity theory, cognitive psychology, and nonlinear dynamics in a way that honors music’s therapeutic capacities. (p. 306)
This book distinguishes itself from the vast majority of acoustics texts with its thorough treatment of the subject matter and its frequent reinforcement with philosophical underpinnings. Mr. Loy has, overall, heeded Albert Einstein’s advice to keep things as simple as possible, but no simpler. Volume Two, which explores digital audio, is scheduled to appear June 2007, after this review has been submitted. Based on Volume One, I eagerly await Mr. Loy’s treatment of digital audio and digital signal processing theory.