Music by the Numbers: From Pythagoras to Schoenberg

176 Pages·2018·25.88 MB·English

by  Eli Maor

MUSIC BY THE NUMBERS

Music
by the
Numbers
FROM PYTHAGORAS
TO SCHOENBERG
Eli Maor
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
Copyright © 2018 by Princeton University Press
Published by Princeton University Press
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In the United Kingdom: Princeton University Press
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Visit chenalexander.com/harmonics for an interactive
diagram of the harmonic series by Alexander Chen
All Rights Reserved
Library of Congress Cataloging- in- Publication Data
Names: Maor, Eli, author.
Title: Music by the numbers : from Pythagoras to Schoenberg / Eli Maor.
Description: Princeton : Princeton University Press, [2018] | Includes
bibliographical references and index.
Identifiers: LCCN 2017031417 | ISBN 9780691176901 (hardcover : alk
paper)
Subjects: LCSH: Music—Mathematics. | Music—Acoustics and physics.
Classification: LCC ML3805 .M3 2018 | DDC 781 .2—dc23
LC record available at https:// lccn .loc .gov /2017031417
Editorial: Vickie Kearn and Lauren Bucca
Production Editorial: Deborah Tegarden
Text and Jacket Design: Chris Ferrante
Production: Jacquie Poirier
Publicity: Sarah Henning-Stout and Lucy Zhou
Copyeditor: Carole Schwager
British Library Cataloging- in- Publication Data is available
This book has been composed in New Century Schoolbook and Gotham
Printed on acied- free paper. ∞
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
In memory of my grandfather,
Karl Stiefel (1881–1947),
who instilled in me an interest
in science and love of music.

CONTENTS
Preface ix
1 Prologue: A World in Crisis 1
2 String Theory, 500 BCE 13
SIDEBAR A. It’s All about Nomenclature 22
3 Enlightenment 24
4 The Great String Debate, 1730–1780 38
SIDEBAR B. The Slinky 52
5 A Most Precious Gift 55
6 Musical Temperament 70
SIDEBAR C. Music for the Record Books: The
Lowest, the Longest, the Oldest, and the Weirdest 78
7 Musical Gadgets: The Tuning Fork and
the Metronome 82
8 Rhythm, Meter, and Metric 91
9 Frames of Reference: Where Am I? 99
SIDEBAR D. Musical Hierarchies 116
10 Relativistic Music 120
11 Aftermath 130
SIDEBAR E. The Bernoulli 138
12 The Last Pythagoreans 142
Bibliography 147
Illustration Credits 149
Index 151

PREFACE
THE GREAT COMPOSER Igor Stravinsky once said, “ Musical
form is close to mathematics—not perhaps to m athe matics
itself, but certainly to something like mathematical
thinking and relationships.” Indeed, numerous writers
have commented on the supposed affinity between math-
ematics and music, citing the fact that many scientists
enjoy listening to music or actually practice it; Albert
Einstein and his iconic violin immediately come to mind,
but there were many, many others.
This may be true, but the relations between the two
disciplines were never truly symmetric. Yes, there are
many similarities between the two. For example, math-
ematics and music both depend on an efficient system of
notation—a set of written symbols that convey a precise,
unambiguous meaning to its practitioners (although in
music this is augmented by a large assortment of verbal
terms to indicate the more emotional aspects of playing).
It is also interesting to note that the two systems started
to evolve roughly around the same time, beginning about
1000 CE, although in mathematics this system of nota-
tion continues to evolve even today as new branches of the
discipline come into being.
Mathematics and music also share many terms. Take,
for example, the word harmonic. As an adjective it means
“pleasing to the ear”; as a noun, it refers to the series of
higher overtones that accompany the sound of practically
all musical instruments. Now this word is almost as com-
mon in mathematics as it is in music; the two- volume En-
cyclopedic Dictionary of Mathematics lists no fewer than
twenty usages of the word, including harmonic mean,