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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
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