音乐数学探索：第一卷——音乐元素的数学基础

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《Musimathics：音乐元素的数学基础》（Volume I）是Gareth Loy编著的一本深入探讨音乐与数学之间联系的专业书籍，由麻省理工学院出版社出版。这本书旨在为读者提供一套系统性的指导，揭示音乐中的数学原理，尤其是对于音乐理论、旋律、和声、节奏等基本元素背后的数学结构进行深入剖析。在《Musimathics Volume I》中，Loy教授从数学的角度出发，探讨了音乐的各个方面，如音高系统的对数比例（如十二平均律），音乐中的比例（如黄金分割和三分损益法在音阶构建中的应用），以及音乐中的周期性和频率。他展示了如何通过数学模型来解释音程的和谐、音调的连续性，以及音乐的结构性元素，如节奏模式和节拍的划分。本书的重要章节可能包括： 1. **数学基础介绍**：章节阐述了数学语言在音乐中的应用，比如集合论、函数、微积分等工具如何帮助理解音乐规律。 2. **音高与频率**：这部分深入讨论了赫兹（Hz）与音乐音阶之间的关系，以及对数频率系统的实用性，如半音阶的十二个等距关系。 3. **音程与和弦**：通过数学模型解析不同音程的数学构成，如大调、小调、和弦的结构及其变化规律。 4. **节奏与时间结构**：探讨了分数与比率在音乐节奏中的作用，如二分音符、四分音符、八分音符等以及它们的时间值。 5. **音乐的周期性和模式**：展示了循环结构（如重复和周期性节奏）的数学表现，以及音乐形式的发展规则。 6. **音乐的数学美学**：分析了数学在创造音乐美感中的潜在作用，如数学比例与音乐美感的关联。 7. **案例研究与应用**：通过具体的音乐作品分析，展示数学理论如何影响作曲和演奏实践。 8. **索引与参考文献**：提供了丰富的资源，供读者进一步探索音乐数学的广阔领域。《Musimathics Volume I》不仅是一本理论著作，也是一本实用的工具书，适合音乐理论爱好者、作曲家、音乐教育者以及对数学与艺术交叉学科感兴趣的读者。通过这本书，读者可以拓宽视野，认识到音乐与数学并非孤立的概念，而是相互交织、相辅相成的。

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Preface

To start a great enterprise requires at the beginning only the first step.

Mathematics can be as effortless as humming a tune, if you know the tune. But our culture does

not prepare us for appreciation of mathematics as it does for appreciation of music. Though we

start hearing music very early in life, the same cannot be said of mathematics, even though the two

subjects are twins. This is a shame; to know music without knowing its mathematics is like hearing

a melody without its accompaniment.

If you are drawn to mathematics because of your love of music, then this book is for you. It pro-

vides a commonsense, self-contained, self-consistent, self-referential introduction to these sub-

jects for nonspecialist readers. It is designed for musicians who find their art increasingly

mediated by technology, and it is written for anyone who desires to understand the intersection

between art and science.

It has been my experience that there are many who want a deeper understanding of the math-

ematics of music if the subject could be presented in a manner accessible to them. This book aims

to meet that need. My goal is always to sustain readers’ motivation while competence is gradually

built up in mathematical fundamentals.

Readers will need only average experience with mathematics and music—advanced high

school math or college freshman algebra and some basic music theory. No knowledge of the cal-

culus, apart from a small amount supplied in volume 2, is required. Some physics background is

helpful, but the text supplies almost everything necessary for understanding.

Virtually all of this book is focused on the mathematics of music:

■

The topics are all subjects that contemporary composers, musicians, and music engineers have

found to be important.

■

The examples are all practical problems in music and audio.

■

Even the fundamentals are cast in terms of the goal: I try to make it clear up front why a foun-

dation is relevant and what readers will be able to do with it once it is mastered.

This is not a book for the mathematically inexperienced, nor is it for experts. My aim is balance.

I travel at a somewhat leisurely pace through this very remarkable material, examining not just

its mathematical content but its aesthetic and philosophical qualities as well.

loy79076_fm.fm Page xv Wednesday, April 26, 2006 11:53 AM

xvi Preface

Musimathics

presents the story of music engineering by examining its mathematics. Since engi-

neering is basically about applying human values to nature, readers will discover a lot about them-

selves, about the world of sound and music, and about what human cultures have valued. However,

because I approach these values from an abstract perspective, they can be seen objectively, giving

a better vantage point from which readers can make their own choices.

There are three main directions of inquiry in volume 1:

■

The materials of music: notes, intervals, scales

■

The physical properties of music: frequency, amplitude, duration, timbre

■

The perception of music and sound: how we hear

■

Music composition

Volume 2 presents a deeper cut into the underlying mathematics of music and sound, including

■

Digital audio, sampling, binary numbers

■

Complex numbers and how they simplify representation of musical signals

■

Fourier transform, convolution, and filtering

■

Resonance, the wave equation, and the behavior of acoustical systems

■

Sound synthesis

■

The short-time Fourier transform, phase vocoder, and the wavelet transform

The Web site, http://www.musimathics.com/,

contains additional source material, animations,

figures, and sources for other program examples in this book. Also, try saying “Musimathics” to

your favorite Web browser and see what happens.

About the Author

This section is here to give readers a sense of comfort that they are in good hands. I received my

Doctor of Musical Arts (DMA) degree from Stanford University in 1980 in composition of com-

puter music. I did my graduate work at the Stanford Center for Computer Research in Music and

Acoustics (CCRMA), one of the premier institutions for the study of this subject, then housed in

the Stanford Artificial Intelligence Laboratory. I have been a performing musician all my life (vio-

lin, guitar, lute, sitar, and voice) and am an award-winning composer (Bourges prize) and a

National Endowment for the Arts grant recipient. I spent over a decade conducting research and

teaching computer music, electronic music, and musical acoustics at the University of California,

San Diego, as Director of Research at the Center for Music Experiment. More recently I’ve been

a computer programmer, software architect, and digital audio systems engineer in various companies

in Silicon Valley. I am president of a (very) small corporation, http://www.GarethInc.com/, which

provides engineering consulting services internationally.

loy79076_fm.fm Page xvi Wednesday, April 26, 2006 11:53 AM

Preface xvii

But there’s more about me that you should know. Mathematics has never been an easy subject

for me; I am a composer by training, not a mathematician. My academic career suffered badly in

proportion to the amount of mathematics included in the syllabus. The aim of confessing this is

paradoxically to give readers confidence. I know what it’s like not to comprehend mathematics

easily, and I also know what it’s like not to give up.

Notwithstanding my inability to add a column of figures and come up with the same answer

twice, I found that mathematics was the lion in my path, the invariant obstacle to the realization

of my artistic visions. So it was more out of necessity than facility that I came to study mathe-

matics. The composer Harry Partch constructed an entire orchestra of novel instruments to realize

his artistic vision and once called himself “a composer seduced into carpentry.” By analogy, I

suppose I’m a composer seduced into mathematics.

I considered subtitling this book, “Everything I wanted to know about music when I was

eleven.” At that age I prowled the stacks of a nearby university library in search of answers to my

burning questions, only to discover that they were out of reach because I didn’t understand the jar-

gon in which the answers were written. At that age we are still intellectually fearless. In my expe-

rience as a child and as a father and teacher, I’ve come to believe that there is nothing an

eleven-year-old can’t understand given the right explanation. But by the time most of us have

reached adulthood, this inquisitive quality is in eclipse, in large part because the right explanations

are very hard to come by. This book is my gift to myself all those years ago, of all the best expla-

nations I’ve been able to find or invent for many of the questions I had. And this book is my gift

to you; may it help throw open the doors to the mathematics of music, one of the crown jewels

of our civilization.

C. G. Jung (1962) wrote, “The decisive question for man is: Is he related to something infinite

or not? In the final analysis, we count for something only because of the essential we embody, and

if we do not embody that, life is wasted.”

In the storm called life, mathematics and music are two sure guides to that essential that we all

embody.

Acknowledgments

This work was supported in part by a generous grant of love and encouragement from my wife,

Lisa, and my children, Morgan, Greta, and Tutti.

Thanks to all those whose passion for the subject has helped inflame my own, including my

teachers Herbert Bielawa, John Chowning, Andy Moorer, John Grey, Loren Rush, Leonard Ratner,

and Leland Smith. Thanks to those who have helped keep the dream in focus: Connie Strohbehn,

Shari Carlson, Linda Grahm.

I am grateful to all whose scholarship and research have fed into the rich stream of knowledge

that this book can at best sample and summarize. The enormous list of these individuals begins

with the bibliography of this book and extends recursively through all the influences they cite. If

loy79076_fm.fm Page xvii Wednesday, April 26, 2006 11:53 AM

Music and Sound

“How did you know how to do that?” he asks.

“You just have to figure it out.”

“I wouldn’t know where to start,” he says.

I think to myself, That’s the problem, all right, where to start. To reach him you have to back up and back up,

and the further back you go, the further back you see you have to go, until what looked like a small problem

of communication turns into a major philosophic enquiry.”

—Robert M. Pirsig, Zen and the Art of Motorcycle Maintenance

The problem of finding the right place to begin an explanation is rather like finding the right fulcrum

point to move a stone with a lever. Putting the fulcrum point too close to the stone provides great lever-

age but little range of movement (figure 1.1a). Putting it too far from the stone provides great range

of movement but no leverage (figure 1.1b). The fulcrum point of an explanation is the knowledge and

assumptions the reader must already have in order to make sense of the explanation. The assumptions

are like the axioms in geometry: a short list of simple, self-evident facts from which the entire subject

can ultimately be derived.

This chapter is such a fulcrum for the rest of this book, and it therefore runs the greatest risk of

overwhelm or underwhelm. Given the choice, I’ve decided to err on the side of underwhelm. The

rest of this chapter introduces some basic properties of sound that will become immediately useful

in chapter 2. If it looks like there are no surprises here, skip this chapter.

And if this subject is new to you, I have a suggestion: if any of the material seems beyond you at

times, just read it like a mystery novel. Seriously! I recommend this approach based on years of per-

sonal experience reading things I didn’t at first understand. You don’t have to speak fluent French

in order to enjoy Paris, but you’ll certainly get more out of it if you pick some up along the way.

1.1 Basic Properties of Sound

If you were to strike a tuning fork and hold it next to your ear, you would hear one of nature’s purest,

simplest sounds. What you hear is a result of the periodic changes in air pressure at your ear drum

caused by the vibration of the air set in motion by the tines of the fork (figure 1.2a). Figure 1.2b is

a representation of the air molecules in the vicinity of the fork, showing areas of greater and lesser

loy79076_ch01.fm Page 1 Wednesday, April 26, 2006 11:38 AM

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