on the art and
science of acoustic instruments
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Also available from the publisher at
Chronicle Books, San Francisco.
Cristiano M.L. Forster
All rights reserved.
Introduction and Acknowledgment
In simplest terms, human
beings identify musical instruments by two aural characteristics: a
particular kind of sound or timbre, and a particular kind of scale or
tuning. To most listeners, these two aspects of musical sound do not
vary. However, unlike the constants of nature — such as gravitational
acceleration on earth, or the speed of sound in air — which we cannot
change, the constants of music — such as string, percussion, and wind
instruments — are subject to change. A creative investigation into
musical sound inevitably leads to the subject of musical mathematics,
and to a reexamination of the meaning of variables.
The first chapter entitled
"Mica Mass" addresses an exceptionally thorny subject, namely, the
derivation of a unit of mass based on an inch constant for
acceleration. This unit is intended for builders who measure wood,
metal, and synthetic materials in inches. For example, with the mica
unit, builders of string instruments can calculate tension in
pounds-force, or lbf, without first converting the diameter of a
string from inches to feet. Similarly, builders of tuned bar
percussion instruments who know the modulus of elasticity of a given
material in pounds-force per square inch, or lbf/in2,
need only the mass density in mica/in3
to calculate the speed of sound in the material in inches per second;
a simple substitution of this value into another equation gives the
mode frequencies of uncut bars.
Chapters 2-4 explore many physical, mathematical, and
musical aspects of strings. In Chapter 3, I distinguish between four
different types of ratios: ancient length ratios, modern length
ratios, frequency ratios, and interval ratios. Knowledge of these
ratios is essential to Chapters 10 and 11. Many writers are unaware of
the crucial distinction between ancient length ratios and frequency
ratios. Consequently, when they attempt to define arithmetic and
harmonic divisions of musical intervals based on frequency ratios, the
results are diametrically opposed to those based on ancient length
ratios. Such confusion leads to anachronisms, and renders the works of
theorists like Ptolemy, Al-Farabi, Ibn Sina, and Zarlino
Chapter 5 investigates the mechanical interactions
between piano strings and soundboards, and explains why the large
physical dimensions of modern pianos are not conducive to explorations
of alternate tuning systems.
Chapters 6 and 7 discuss the theory and practice of
tuning marimba bars and resonators. The latter chapter is essential to
Chapter 8, which examines a sequence of equations for the placement of
tone holes on concert flutes and simple flutes.
Chapter 9 covers logarithms, and the modern cent unit.
This chapter serves as an introduction to calculating scales and
tunings discussed in Chapters 10 and 11.
In summary, this book is divided into three parts. (1) In
Chapters 1-9, I primarily examine various vibrating systems found in
musical instruments; I also focus on how builders can customize their
work by understanding the functions of variables in mathematical
equations. (2) In Chapter 10, I discuss scale theories and tuning
practices in ancient Greece, and during the Renaissance and
Enlightenment in Europe. Some modern interpretations of these theories
are explained as well. In Chapter 11, I describe scale theories and
tuning practices in Chinese, Indonesian, and Indian music, and in
Arabian, Persian, and Turkish music. For Chapters 10 and 11, I
consistently studied original texts in modern translations. I also
translated passages in treatises by Ptolemy, Al-Kindi, the Ikhwan
al-Safa, Ibn Sina, Stifel, and Zarlino from German into English; and
in collaboration with two contributors, I participated in translating
portions of works by Al-Farabi, Ibn Sina, Safi Al-Din, and Al-Jurjani
from French into English. These translations reveal that all the
above-mentioned theorists employ the language of ancient length
ratios. (3) Finally, Chapters 12 and 13 recount musical instruments I
have built and rebuilt since 1975.
I would like to acknowledge the assistance and
encouragement I received from Dr. David R. Canright, associate
professor of mathematics at the Naval Postgraduate School in Monterey,
California. David’s unique understanding of mathematics, physics, and
music provided the foundation for many conversations throughout the
ten years I spent writing this book. His mastery of differential
equations enabled me to better understand dispersion in strings, and
simple harmonic motion of air particles in resonators. In Chapter 4,
Section 6, David’s equation for the effective length of stiff strings
is central to the study of inharmonicity; and in Chapter 6, Section 7,
David’s figure, which shows the effects of two restoring forces on the
geometry of bar elements, sheds new light on the physics of vibrating
bars. Furthermore, David’s plots of compression and rarefaction pulses
inspired numerous figures in Chapter 7. Finally, we also had extensive
discussions on Newton’s laws. I am very grateful to David for his
patience and contributions.
Heartfelt thanks go to my wife, Heidi Forster. Heidi
studied, corrected, and edited myriad versions of the manuscript.
Also, in partnership with the highly competent assistance of
professional translator Cheryl M. Buskirk, Heidi did most of the work
translating extensive passages from La Musique Arabe into
English. To achieve this accomplishment, Heidi mastered the often
intricate verbal language of ratios. Heidi also assisted me in
transcribing the Indonesian and Persian musical scores in Chapter 11,
and transposed the traditional piano score of “The Letter” in Chapter
12. Furthermore, she rendered invaluable services during all phases of
book production by acting as my liaison with the editorial staff at
Chronicle Books. Finally, when the writing became formidable, she
became my sparring partner and helped me through the difficult process
of restoring my focus. I am very thankful to Heidi for all her love,
friendship, and support.
I would also like to express my appreciation to Dr. John
H. Chalmers. Since 1976, John has generously shared his vast knowledge
of scale theory with me. His mathematical methods and techniques have
enabled me to better understand many historical texts, especially
those of the ancient Greeks. And John’s scholarly book Divisions of
the Tetrachord has furthered my appreciation for world tunings.
I am very grateful to Lawrence Saunders, M.A. in
ethnomusicology, for reading Chapters 3, 9, 10, and 11, and for
suggesting several technical improvements.
Finally, I would like to thank Will Gullette for his
twelve masterful color photographs of the Original Instruments and
String Winder. Will’s skill and tenacity have illuminated this book in
ways that words cannot convey.
From left to right:
Heidi Forster, and Cris
Photo by Will Gullette