Musical Mathematics


on the art and science of acoustic instruments

 

Table of Contents

 

 

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Also available from the publisher at Chronicle Books, San Francisco.

 

© 2000–2017 Cristiano M.L. Forster
All rights reserved.

 

www.chrysalis-foundation.org

 

 

Introduction and Acknowledgments

          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 incomprehensible.

          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:  David Canright,

Heidi Forster, and Cris Forster

Photo by Will Gullette