sign in sign up
<<
Home , Gamelan notation, Musical prefix, Pyknon, Setar, Thomas Young (scientist), Tonality diamond

Musical Mathematics on the art and science of acoustic instruments

Cris Forster

MUSICAL MATHEMATICS

ON THE ART AND SCIENCE OF ACOUSTIC INSTRUMENTS

MUSICAL MATHEMATICS

ON THE ART AND SCIENCE OF ACOUSTIC INSTRUMENTS

Text and Illustrations

by Cris Forster Copyright © 2010 by Cristiano M.L. Forster

All Rights Reserved. No part of this book may be reproduced in any form without written permission from the publisher.

Library of Congress Cataloging-in-Publication Data available.

ISBN: 978-0-8118-7407-6

Manufactured in the United States.

All royalties from the sale of this book go directly to the Chrysalis Foundation, a public 501(c)3 nonprofit arts and education foundation. https://chrysalis-foundation.org

Photo Credits: Will Gullette, Plates 1–12, 14–16. Norman Seeff, Plate 13.

10 9 8 7 6 5 4 3 2 1

Chronicle Books LLC 680 Second Street San Francisco, California 94107 www.chroniclebooks.com In Memory of Page Smith

my enduring teacher

And to Douglas Monsour

our constant friend

I would like to thank the following individuals and foundations for their generous contributions in support of the writing, designing, and typesetting of this work:

Peter Boyer and Terry Gamble-Boyer The family of Jackson Vanfleet Brown Thomas Driscoll and Nancy Quinn Marie-Louise Forster David Holloway Jack Jensen and Cathleen O’Brien James and Deborah Knapp Ariano Lembi, Aidan and Yuko Fruth-Lembi Douglas and Jeanne Monsour Tim O’Shea and Peggy Arent Fay and Edith Strange Charles and Helene Wright

Ayrshire Foundation Chrysalis Foundation

The jewel that we find, we stoop and take’t, Because we see it; but what we do not see We tread upon, and never think of it.

W. Shakespeare For more information about Musical Mathematics: On the Art and Science of Acoustic Instruments please visit: https://chrysalis-foundation.org https://www.amazon.com CONTENTS

Foreword by David R. Canright v Introduction and Acknowledgments vii Tone Notation ix List of Symbols xi

Chapter 1 Mica Mass 1 Part I Principles of force, mass, and acceleration 1 Part II Mica mass definitions, mica unit derivations, and sample calculations 14 Notes 24

Chapter 2 Plain String and Wound String Calculations 27 Part I Plain strings 27 Part II Wound strings 36 Notes 41

Chapter 3 Flexible Strings 44 Part I Transverse traveling and standing , and simple motion in strings 44 Part II Period and equations of waves in strings 54 Part III Length, frequency, and interval ratios of the harmonic series on canon strings 59 Part IV Length, frequency, and interval ratios of non-harmonic tones on canon strings 69 Part V Musical, mathematical, and linguistic origins of length ratios 79 Notes 94

Chapter 4 Inharmonic Strings 98 Part I Detailed equations for stiffness in plain strings 98 Part II Equations for coefficients of in cents 108 Part III General equations for stiffness in wound strings 113 Notes 115

Chapter 5 Piano Strings vs. Canon Strings 118 Part I Transmission and reflection of mechanical and acoustic 118 Part II Mechanical impedance and soundboard-to-string impedance ratios 120 Part III Radiation impedance and air-to-soundboard impedance ratios 126 Part IV Dispersion, the speed of bending waves, and critical in soundboards 130 Part V Methods for tuning piano intervals to rates of coincident string 135 Part VI Musical advantages of thin strings and thin soundboards 141 Notes 143 ii Contents

Chapter 6 Bars, Rods, and Tubes 147 Part I Frequency equations, mode shapes, and restoring forces of free-free bars 147 Part II Free-free bar tuning techniques 160 Part III Frequency equations, mode shapes, and restoring forces of clamped-free bars 174 Part IV Clamped-free bar tuning techniques 176 Notes 178

Chapter 7 Acoustic Resonators 182 Part I Simple harmonic motion of longitudinal traveling waves in air 182 Part II Equations for the speed of longitudinal waves in solids, liquids, and gases 186 Part III Reflections of longitudinal traveling waves at the closed and open ends of tubes 189 Part IV Acoustic impedance and tube-to-room impedance ratio 196 Part V Longitudinal pressure and displacement standing waves in tubes 200 Part VI Length and frequency equations of tube resonators 203 Part VII Theory of cavity resonators 212 Part VIII Cavity resonator tuning techniques 219 Notes 223

Chapter 8 Simple Flutes 227 Part I Equations for the placement of tone holes on concert flutes and simple flutes 227 Part II Equations for analyzing the tunings of existing flutes 242 Part III Suggestions for making inexpensive yet highly accurate simple flutes 246 Notes 248

Chapter 9 The Geometric Progression, Logarithms, and Cents 253 Part I Human perception of the harmonic series as a geometric progression 253 Part II Logarithmic processes in mathematics and human hearing 257 Part III Derivations and applications of cent calculations 265 Part IV Logarithmic equations for guitar and musical rules 271 Notes 276

Chapter 10 Western Tuning Theory and Practice 280 Part I Definitions of prime, composite, rational, and irrational numbers 281 Part II Greek classifications of ratios, , scales, and modes 284 Part III Arithmetic and geometric divisions on canon strings 291 Part IV Philolaus, Euclid, , and 299 Part V Meantone temperaments, well-temperaments, and equal temperaments 334 Part VI 365 Notes 460

Chapter 11 World Tunings 485

Part I Chinese Music 485 Notes 504 Contents iii

Part II Indonesian Music: Java 508 Bali 522 Notes 535

Part III Indian Music: Ancient Beginnings 540 South India 564 North India 587 Notes 600

Part IV Arabian, Persian, and Turkish Music 610 Notes 774

Chapter 12 Original Instruments 788 Stringed Instruments: Chrysalis 788 Harmonic/Melodic Canon 790 Bass Canon 800 Just Keys 808 Percussion Instruments: Diamond Marimba 824 Bass Marimba 826 Friction Instrument: Glassdance 828

Wind Instruments: Simple Flutes 833

Chapter 13 Building a Little Canon 834 Parts, materials, labor, and detailed dimensions 834

Epilog by Heidi Forster 839

Plate 1: Chrysalis 845 Plate 2: Harmonic/Melodic Canon 846 Plate 3: Bass Canon 847 Plate 4: String Winder (machine) 848 Plate 5: String Winder (detail) 849 Plate 6: Just Keys 850 Plate 7: Diamond Marimba 851 Plate 8: Bass Marimba 852 Plate 9: Glassdance 853 Plate 10: Glassdance (back) 854 Plate 11: Simple Flutes 855 Plate 12: Little Canon 856 iv Contents

Plate 13: Cris Forster with Chrysalis 857 Plate 14: Heidi Forster playing Glassdance 858 Plate 15: David Canright, Heidi Forster, and Cris Forster 859 Plate 16: Chrysalis Foundation Workshop 860

Bibliography for Chapters 1–9 861 Bibliography for Chapter 10 866 Bibliography for Chapter 11 871 Bibliography for Chapter 12 877

Appendix A: Frequencies of Eight of 12-Tone 879 Appendix B: Conversion Factors 880 Appendix C: Properties of String Making Materials 882 Appendix D: Spring Steel Music Wire Tensile Strength and Break Strength Values 884 Appendix E: Properties of Bar Making Materials 885 Appendix F: Properties of Solids 888 Appendix G: Properties of Liquids 890 Appendix H: Properties of Gases 892

Index 895 Foreword

I met Cris Forster more than thirty years ago. Shortly thereafter, I saw him perform Song of Myself, his setting of Walt Whitman poems from Leaves of Grass. His delivery was moving and effective. Several of the poems were accompanied by his playing on unique instruments — one an elegant box with many steel strings and moveable bridges, a bit like a koto in concept; the other had a big wheel with strings like spokes from offset hubs, and he rotated the wheel as he played and intoned the poetry. I was fascinated. Since that time, Cris has built several more instruments of his own design. Each shows exquisite care in conception and impeccable craftsmanship in execution. And of course, they are a delight to hear. Part of what makes them so good is his deep understanding of how acoustic musical instruments work, and part is due to his skill in working the materials to his exacting standards. But another important aspect of their sound, and indeed one of the main reasons Cris could not settle for standard instruments, is that his music uses scales and that are not found in the standard Western system of intonation (with each divided into twelve equal , called equal temperament). Rather, his music employs older notions of consonance, which reach back as far as ancient Greek music and to other cultures across the globe, based on what is called just intonation. Here, the musical intervals that make up the scales and chords are those that occur naturally in the harmonic series of , in stretched flexible strings, and in organ pipes, for example. In just intonation, the octave is necessarily divided into unequal parts. In comparison to equal temperament, the harmonies of just intonation have been described as smoother, sweeter, and/or more powerful. Many theorists consider just intonation to be the standard of comparison for con- sonant intervals. There has been a resurgence of interest in just intonation since the latter part of the twentieth century, spurred by such pioneers as and . Even so, the community of just intonation composers remains comparatively quite small, and the subset of those who employ only acoustic instruments is much smaller still. I know of no other living composer who has created such a large and varied ensemble of high-quality just intoned acoustical instruments, and a body of music for them, as Cris Forster. Doing what he has done is not easy, far from it. The long process of developing his instruments has required endless experimentation and careful measurement, as well as intense study of the liter- ature on of musical instruments. In this way Cris has developed deep and rich knowledge of how to design and build instruments that really work. Also, in the service of his composing, Cris has studied the history of intonation practices, not only in the Western tradition, but around the world. This book is his generous offering of all that hard-earned knowledge, presented as clearly as he can make it, for all of you who have an interest in acoustic design and/or musi- cal scales over time and space. The unifying theme is how mathematics applies to music, in both the acoustics of resonant instruments and the analysis of musical scales. The emphasis throughout is to show how to use these mathematical tools, without requiring any background in higher mathemat- ics; all that is required is the ability to do arithmetic on a pocket calculator, and to follow Cris’ clear step-by-step instructions and examples. Any more advanced mathematical tools required, such as logarithms, are carefully explained with many illustrative examples. The first part of the book contains practical information on how to design and build musical instruments, starting from first principles of vibrating sound sources of various kinds. The ideas are explained clearly and thoroughly. Many beautiful figures have been carefully conceived to illumi- nate the concepts. And when Cris gives, say, formulas for designing flutes, it’s not just something he read in a book somewhere (though he has carefully studied many books); rather, you can be

v vi Foreword sure it is something he has tried out: he knows it works from direct experience. While some of this information can be found (albeit in a less accessible form) in other books on , other information appears nowhere else. For example, Cris developed a method for tuning the over- tones of marimba bars that results in a powerful, unique tone not found in commercial instruments. Step-by-step instructions are given for applying this technique (see Chapter 6). Another innovation is Cris’ introduction of a new unit of mass, the “mica,” that greatly simplifies calculations using lengths measured in inches. And throughout Cris gives careful explanations, in terms of physical principles, that make sense based on one’s physical intuition and experience. The latter part of the book surveys the development of musical notions of consonance and scale construction. Chapter 10 traces Western ideas about intonation, from Pythagoras finding number in , through “meantone” and then “well-temperament” in the time of J.S. Bach, up to modern equal temperament. The changing notions of which intervals were considered consonant when, and by whom, make a fascinating story. Chapter 11 looks at the largely independent (though sometimes parallel) development of musical scales and tunings in various Eastern cultures, including China, India, and , as well as Persian, Arabian, and Turkish musical traditions. As far as possible, Cris relies on original sources, to which he brings his own analysis and explication. To find all of these varied scales compared and contrasted in a single work is unique in my experience. The book concludes with two short chapters on specific original instruments. One introduces the innovative instruments Cris has designed and built for his music. Included are many details of construction and materials, and also scores of his work that demonstrate his notation for the instruments. The last chapter encourages the reader (with explicit plans) to build a simple stringed instrument (a “canon”) with completely adjustable tuning, to directly explore the tunings discussed in the book. In this way, the reader can follow in the tradition of Ptolemy, of learning about music through direct experimentation, as has Cris Forster.

David R. Canright, Ph.D. Del Rey Oaks, California January 2010 Introduction and Acknowledgments

In simplest terms, human beings identify musical instruments by two aural characteristics: a par- ticular 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 gravita- tional 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: 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 diam- eter 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- musi cal 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-Frb, Ibn Sn, 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-Kind, the Ikhwn al-a, Ibn Sn, Stifel, and Zarlino from German into English; and in collaboration with two contributors, I participated in translating portions of works by Al-Frb, Ibn Sn, a Al-Dn, and Al-Jurjn from French into English. These translations reveal that all the above- mentioned theorists employ the 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,

vii viii Introduction and Acknowledgments

California. David’s unique understanding of mathematics, , and music provided the foun- dation 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 Section 4.5, David’s equation for the effective length of stiff strings is central to the study of inharmonicity; and in Section 6.6, David’s figure, which shows the effects of two restoring forces on the geometry of bar elements, sheds new 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 ver- sions 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, she mastered the often intricate ver- bal 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 tech- niques have enabled me to better understand many historical texts, especially those of the ancient Greeks. And John’s scholarly book Divisions of the has furthered my appreciation for world tunings. I am very grateful to Lawrence Saunders, M.A. in , 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 plates of the Original Instruments and String Winder, plus three additional plates. Will’s skill and tenacity have illumi- nated this book in ways that words cannot convey.

Cris Forster San Francisco, California January 2010 TONE NOTATION

32' 16' 8' 4' 2' 1' Z\x' Z\v' Z\,'

           

1. C0 C1 C2 C3 C4 C5 C6 C7 C8 2. C C C cc c c c cV   0 1 2 3 4 5 3. C2 C1 C0 c c c c c c

1. American System, used throughout this text.

2. Helmholtz System.

3. German System.

ix

LIST OF SYMBOLS

Latin

12-TET 12-tone equal temperament a Acceleration; in/s2 a.l.r. Ancient length ratio; dimensionless B Bending stiffness of bar; lbfin2, or micain3/s2 B¢ Bending stiffness of plate; lbfin, or micain2/s2 2 2 BA Adiabatic bulk modulus; psi, lbf/in , or mica/(ins ) 2 2 BI Isothermal bulk modulus; psi, lbf/in , or mica/(ins ) b Width; in ¢ Cent, 1/100 of a “,” or 1/1200 of an “octave”; dimensionless ¢ Coefficient of inharmonicity of string; cent cB Bending speed; in/s cL Longitudinal wave speed, or speed of sound; in/s cT Transverse wave speed; in/s c.d. Common difference of an arithmetic progression; dimensionless c.r. Common ratio of a geometric progression; dimensionless cps Cycle per second; 1/s D Outside diameter; in

Di Inside diameter of wound string; in Dm Middle diameter of wound string; in Do Outside diameter of wound string; in Dw Wrap wire diameter of wound string; in d Inside diameter, or distance; in E Young’s modulus of elasticity; psi, lbf/in2, or mica/(ins2) F Frequency; cps

Fc Critical frequency; cps Fn Resonant frequency; cps Fn Inharmonic mode frequency of string; cps f Force; lbf, or micain/s2 f.r. Frequency ratio; dimensionless g Gravitational acceleration; 386.0886 in/s2 h Height, or thickness; in I Area moment of inertia; in4 i.r. Interval ratio; dimensionless J Stiffness parameter of string; dimensionless K Radius of gyration; in k Spring constant; lbf/in, or mica/s2 L Length; in, cm, or mm

M Multiple loop length of string; in S Single loop length of string; in l.r. Length ratio; dimensionless lbf Pounds-force; micain/s2 lbm Pounds-mass; 0.00259008 mica

xi xii List of Symbols

M/u.a. Mass per unit area; mica/in2, or lbfs2/in3 M/u.l. Mass per unit length; mica/in, or lbfs2/in2 m Mass; mica, or lbfs2/in n Mode number, or harmonic number; any positive integer P Pressure; psi, lbf/in2, or mica/(ins2) p Excess acoustic pressure; psi, lbf/in2, or mica/(ins2) psi Pounds-force per square inch; lbf/in2, or mica/(ins2) q Bar parameter; dimensionless R Ideal gas constant; inlbf/(mica°R), or in2/(s2°R) r Radius; in S Surface area; in2 SHM Simple harmonic motion T Tension; lbf, or micain/s2

TA Absolute temperature; dimensionless t Time; s U Volume velocity; in3/s u Particle velocity; in/s V Volume; in3 v Phase velocity; in/s W Weight density, or weight per unit volume; lbf/in3, or mica/(in2s2) w Weight; lbf, or micain/s2 4 YA Acoustic admittance; in s/mica 4 ZA Acoustic impedance; mica/(in s) 4 Zr Acoustic impedance of room; mica/(in s) 4 Zt Acoustic impedance of tube; mica/(in s) ZM Mechanical impedance; mica/s Zb Mechanical impedance of soundboard; mica/s Zp Mechanical impedance of plate; mica/s Zs Mechanical impedance of string; mica/s ZR Radiation impedance; mica/s Za Radiation impedance of air; mica/s z Specific acoustic impedance; mica/(in2s) 2 za Characteristic impedance of air; 0.00153 mica/(in s)

Greek

 Correction coefficient, or end correction coefficient; dimensionless  Correction, or end correction; in, cm, or mm  Departure of tempered ratio from just ratio; cent  Ratio of specific heat; dimensionless  Angle; degree  Conductivity; in  Bridged canon string length; in A Arithmetic mean string length; in G Geometric mean string length; in H Harmonic mean string length; in List of Symbols xiii

; in B Bending wavelength; in L Longitudinal wavelength; in T Transverse wavelength; in  Poisson’s ratio; dimensionless  Fretted guitar string length; mm  Pi; » 3.1416  Mass density, or mass per unit volume; mica/in3, or lbfs2/in4  Period, or second per cycle; s

Musical Mathematics: On the Art and Science of Acoustic Instruments © 2000–2010 Cristiano M.L. Forster All rights reserved. www.chrysalis-foundation.org

BIBLIOGRAPHY

Chapters 1–9

Askenfelt, A., Editor (1990). Five Lectures on The Acoustics of the Piano. Royal Swedish Academy of Music, No. 64, Stockholm, Sweden.

Askill, J. (1979). Physics of Musical Sound. D. Van Nostrand Company, New York.

Baines, A. (1967). Woodwind Instruments and Their History. Dover Publications, Inc., New York, 1991.

Barbera, A., Translator (1991). The Euclidean Division of the Canon: Greek and Latin Sources. University of Nebraska Press, Lincoln, Nebraska.

Barker, A., Translator (1989). Greek Musical Writings. Two Volumes. Cambridge University Press, Cam- , Massachusetts.

Bell, A.J., and Firth, I.M. (1986). The physical properties of gut musical instrument strings. Acustica 60, No. 1, pp. 87–89.

Benade, A.H., and French, J.W. (1965). Analysis of the flute head joint. Journal of the Acoustical Society of America 37, No. 4, pp. 679–691.

Benade, A.H. (1967). Measured end corrections for woodwind toneholes. Journal of the Acoustical Society of America 41, No. 6, p. 1609.

Benade, A.H. (1976). Fundamentals of Musical Acoustics. Dover Publications, Inc., New York, 1990.

Berliner, P.F. (1978). The Soul of Mbira. University of California Press, Berkeley, California, 1981.

Blevins, R.D. (1979). Formulas for Natural Frequency and Mode Shape, Reprint. Krieger Publishing Com- pany, Malabar, Florida, 1993.

Boehm, T. (1847). On the Construction of Flutes, Über den Flötenbau. Frits Knuf Buren, Amsterdam, Neth- erlands, 1982.

Boehm, T. (1871). The Flute and Flute-Playing. Dover Publications, Inc., New York, 1964.

Boyer, H.E., and Gall, T.L., Editors (1984). Metals Handbook, Desk Edition. American Society for Metals, Metals Park, Ohio, 1989.

Bray, A., Barbato, G., and Levi, R. (1990). Theory and Practice of Force Measurement. Academic Press, San Diego, California.

Burkert, W. (1962). Lore and Science in Ancient Pythagoreanism. Translated by E.L. Minar, Jr. Harvard University Press, Cambridge, Massachusetts, 1972.

Cadillac Plastic Buyer’s Guide. Cadillac Plastic and Chemical Company, Troy, Michigan, 1986.

Campbell, M., and Greated, C. (1987). The Musician’s Guide to Acoustics. Schirmer Books, New York, 1988.

Capstick, J.W. (1913). Sound. Cambridge University Press, London, , 1932.

Chapman, R.E., Translator (1957). Harmonie universelle: The Books on Instruments, by . Martinus Nijhoff, The Hague, Netherlands.

861 862 Bibliography

Cohen, H.F. (1984). Quantifying Music. D. Reidel Publishing Company, Dordrecht, Netherlands.

Coltman, J.W. (1979). Acoustical analysis of the Boehm flute. Journal of the Acoustical Society of America 65, No. 2, pp. 499–506.

Cremer, L., Heckl, M., and Ungar, E.E. (1973). Structure-Borne Sound, 2nd ed. Springer-Verlag, Berlin and New York, 1988.

Cremer, L. (1981). The Physics of the Violin, 2nd ed. The MIT Press, Cambridge, Massachusetts, 1984.

Crew, H., and De Salvio, A., Translators (1914). Dialogues Concerning Two New Sciences, by Galileo Galilei. Dover Publications, Inc., New York.

D’Addario Brochure (2007). “Catalog Supplement/String Tension Specifications.” Online publication, pp. 1–14. J. D’Addario & Company, Inc., Farmingdale, New York.

Den Hartog, J.P. (1934). Mechanical Vibrations. Dover Publications, Inc., New York, 1985.

Den Hartog, J.P. (1948). Mechanics. Dover Publications, Inc., New York, 1984.

D’Erlanger, R., Bakkouch, .., and Al-Sans, M., Translators (Vol. 1, 1930; Vol. 2, 1935; Vol. 3, 1938; Vol. 4, 1939; Vol. 5, 1949; Vol. 6, 1959). La Musique Arabe. Librairie Orientaliste Paul Geuthner, Paris, France.

Diels, H. (1903). Die Fragmente der Vorsokratiker, Griechisch und Deutsch. Three Volumes. Weidmannsche Verlagsbuchhandlung, Berlin, Germany, 1951.

D’Ooge, M.L., Translator (1926). Nicomachus of Gerasa: Introduction to Arithmetic. The Macmillan Com- pany, New York.

Dunlop, J.I. (1981). Testing of poles by using acoustic pulse method. Wood Science and Technology 15, pp. 301–310.

Du Pont Bulletin: “Tynex 612 Nylon Filament.” Du Pont Company, Wilmington, Delaware.

Düring, I., Translator (1934). Ptolemaios und Porphyrios über die Musik. Georg Olms Verlag, Hildesheim, Germany, 1987.

Einarson, B., Translator (1967). On Music, by Plutarch. In Plutarch’s Moralia, Volume 14. Harvard Univer- sity Press, Cambridge, Massachusetts.

Elmore, W.C., and Heald, M.A. (1969). Physics of Waves. Dover Publications, Inc. New York, 1985.

Fenner, K., On the Calculation of the Tension of Wound Strings, 2nd ed. Verlag Das Musikinstrument, Frank- furt, Germany, 1976.

Fishbane, P.M., Gasiorowicz, S., and Thornton, S.T. (1993). Physics for and Engineers. Prentice- Hall, Englewood Cliffs, New Jersey.

Fletcher, H., Blackham, E.D., and Stratton, R.S. (1962). Quality of piano tones. Journal of the Acoustical Society of America 34, No. 6, pp. 749–761.

Fletcher, H. (1964). Normal vibration frequencies of a stiff piano string. Journal of the Acoustical Society of America 36, No. 1, pp. 203–209.

Fletcher, N.H., and Rossing, T.D. (1991). The Physics of Musical Instruments, 2nd ed. Springer-Verlag, Ber- lin and New York, 1998. Bibliography: Chapters 1–9 863

Fogiel, M., Editor (1980). The Strength of Materials & Mechanics of Solids Problem Solver. Research and Education Association, Piscataway, New Jersey, 1990.

Goodway, M., and Odell, J.S. (1987). The Historical Harpsichord, Volume Two: The Metallurgy of 17th– and 18th–Century Music Wire. Pendragon Press, Stuyvesant, New York.

Gray, D.E., Editor (1957). American Institute of Physics Handbook, 3rd ed. McGraw-Hill Book Company, New York, 1972.

Halliday, D., and Resnick, R. (1970). Fundamentals of Physics, 2nd ed. John Wiley & Sons, New York, 1981.

Hamilton, E., and Cairns, H., Editors (1963). The Collected Dialogues of Plato. Random House, Inc., New York, 1966.

Helmholtz, H.L.F., and Ellis A.J., Translator (1885). On the . Dover Publications, Inc., New York, 1954.

Hoadley, R.B. (1980). Understanding Wood. The Taunton Press, Newtown, Connecticut, 1981.

Hubbard, F. (1965). Three Centuries of Harpsichord Making, 4th ed. Harvard University Press, Cambridge, Massachusetts, 1972.

Ingard, U. (1953). On the theory and design of acoustic resonators. Journal of the Acoustical Society of America 25, No. 6, pp. 1037–1061.

Ingard, K.U. (1988). Fundamentals of Waves and Oscillations. Cambridge University Press, Cambridge, Mas- sachusetts, 1990.

Jan, K. von, Editor (1895). Musici Scriptores Graeci. Lipsiae, in aedibus B.G. Teubneri.

Jerrard, H.G., and McNeill, D.B. (1963). Dictionary of Scientific Units, 6th ed. Chapman and Hall, London, England, 1992.

Jones, A.T. (1941). End corrections of organ pipes. Journal of the Acoustical Society of America 12, pp. 387–394.

Kinsler, L.E., and Frey, A.R. (1950). Fundamentals of Acoustics, 2nd ed. John Wiley & Sons, Inc., New York, 1962.

Klein, H.A. (1974). The Science of Measurement. Dover Publications, Inc., New York, 1988.

Land, F. (1960). The Language of Mathematics. Doubleday & Company, Inc., Garden City, New York.

Lemon, H.B., and Ference, M., Jr. (1943). Analytical . The University of Chicago Press, Chicago, Illinois.

Levin, F.R., Translator (1994). The Manual of Harmonics, of Nicomachus the Pythagorean. Phanes Press, Grand Rapids, Michigan.

Liddell, H.G., and Scott, R. (1843). A Greek-English Lexicon. The Clarendon Press, Oxford, England, 1992.

Lide, D.R., Editor (1918). CRC Handbook of Chemistry and Physics, 73rd ed. CRC Press, Boca Raton, Florida, 1992.

Lindeburg, M.R. (1988). Engineering Unit Conversions, 2nd ed. Professional Publications, Inc., Belmont, California, 1990. 864 Bibliography

Lindeburg, M.R. (1990). Engineer-in-Training Reference Manual, 8th ed. Professional Publications, Inc., Belmont, California, 1992.

Lindley, M. (1987). “Stimmung und Temperatur.” In Geschichte der Musiktheorie, Volume 6, F. Zaminer, Editor. Wissenschaftliche Buchgesellschaft, Darmstadt, Germany.

McLeish, J. (1991). Number. Bloomsbury Publishing Limited, London, England.

Moore, J.L. (1971). Acoustics of Bar Percussion Instruments. Ph.D. dissertation printed and distributed by University Microfilms, Inc., Ann Arbor, Michigan.

Morse, P.M., and Ingard, K.U. (1968). Theoretical Acoustics. Princeton University Press, Princeton, New Jersey, 1986.

Nash, W.A. (1957). Strength of Materials, 3rd ed. Schaum’s Outline Series, McGraw-Hill, Inc., New York, 1994.

Nederveen, C.J. (1969). Acoustical Aspects of Woodwind Instruments. Frits Knuf, Amsterdam, Netherlands.

Nederveen, C.J. (1973). Blown, passive and calculated resonance frequencies of the flute. Acustica 28, pp. 12–23.

Newton, R.E.I. (1990). Wave Physics. Edward Arnold, a division of Hodder & Stoughton, London, Eng- land.

Norton, M.P. (1989). Fundamentals of Noise and Vibration Analysis for Engineers. Cambridge University Press, Cambridge, Massachusetts.

Oberg, E., Jones, F.D., Horton, H.L., and Ryffel, H.H. (1914). Machinery’s Handbook, 24th ed. Industrial Press Inc., New York, 1992.

Olson, H.F. (1952). Music, Physics and Engineering, 2nd ed. Dover Publications, Inc., New York, 1967.

Pierce, A.D. (1981). Acoustics. Acoustical Society of America, Woodbury, New York. 1991.

Pierce, J.R. (1983). The Science of Musical Sound. Scientific American Books, W.H. Freeman and Company, New York.

Pikler, A.G. (1966). Logarithmic frequency systems. Journal of the Acoustical Society of America 39, No. 6, pp. 1102–1110.

Rao, S.S. (1986). Mechanical Vibrations, 2nd ed. Addison-Wesley Publishing Company, Reading, Massachu- setts, 1990.

Richardson, E.G. (1929). The Acoustics of Orchestral Instruments and of the Organ. Edward Arnold & Co., London, England.

Rossing, T.D. (1989). The Science of Sound, 2nd ed. Addison-Wesley Publishing Co., Inc., Reading, Mas- sachusetts, 1990.

Sadie, S., Editor (1984). The New Grove Dictionary of Musical Instruments. Macmillan Press Limited, Lon- don, England.

Schlesinger, K. (1939). The Greek Aulos. Methuen & Co. Ltd., London, England.

Schuck, O.H., and Young, R.W. (1943). Observations on the vibrations of piano strings. Journal of the Acous- tical Society of America 15, No. 1, pp. 1–11. Bibliography: Chapters 1–9 865

Sears, F.W., Zemansky, M.W., and Young, H.D., University Physics, 7th ed. Addison-Wesley Publishing Company, Reading, Massachusetts, 1988.

Skudrzyk, E. (1968). Simple and Complex Vibratory Systems. Pennsylvania State University Press, University Park, Pennsylvania, 1981.

Smith, D.E. (1925). History of Mathematics. Two Volumes. Dover Publications, Inc., New York, 1958.

Standards Handbook, Part 2 — Alloy Data, Wrought Copper and Copper Alloy Mill Products, Eighth Edition, Copper Development Association, Inc., Greenwich, Connecticut, 1985.

Stauss, H.E., Martin, F.E., and Billington, D.S. (1951). A piezoelectric method for determining Young’s modulus and its temperature dependence. Journal of the Acoustical Society of America 23, No. 6, pp. 695–696.

Steinkopf, O. (1983). Zur Akustik der Blasinstrumente. Moeck Verlag, Celle, Germany.

Stiller, A. (1985). Handbook of Instrumentation. University of California Press, Berkeley, California.

Suzuki, H. (1986). Vibration and sound radiation of a piano soundboard. Journal of the Acoustical Society of America 80, No. 6, pp. 1573–1582.

Thompson, S.P. (1910). Calculus Made Easy, 3rd ed. St. Martin’s Press, New York, 1984.

Timoshenko, S., and Woinowsky-Krieger, S. (1940). Theory of Plates and Shells, 2nd ed., McGraw-Hill Book Company, New York, 1959.

Timoshenko, S.P. (1953). History of Strength of Materials. Dover Publications, Inc., New York, 1983.

Towne, D.H. (1967). Wave Phenomena. Dover Publications, Inc., New York, 1988.

Tropfke, J. (1921). Geschichte der Elementar-Mathematik. Seven Volumes. Vereinigung Wissenschaftlicher Verleger, Walter de Gruyter & Co., Berlin and Leipzig, Germany.

U.S. Business and Defense Services Administration (1956). Materials Survey: Aluminum. Department of Commerce, Washington, D.C.

Weaver, W., Jr., Timoshenko, S.P., and Young, D.H., Vibration Problems in Engineering, 5th ed. John Wiley and Sons, New York, 1990.

White, W.B. (1917). Piano Tuning and Allied Arts, 5th ed. Tuners Supply Company, Boston, Massachusetts, 1972.

Wogram, K. (1981). Akustische Untersuchungen an Klavieren. Teil I: Schwingungseigenschaften des Reso- nanzbodens. Das Musikinstrument 24, pp. 694–702, 776–782, 872–879. English translation: Acoustical research on pianos. Part I: Vibrational characteristics of the soundboard. In Musical Acoustics: Selected Reprints, T.D. Rossing, Editor, pp. 85–98. American Association of Physics Teachers, College Park, Maryland, 1988.

Wolfenden, S. (1916). A Treatise on the Art of Pianoforte Construction. The British Piano Museum Chari- table Trust, Brentford, Middlesex, England, 1975.

Wood, A.B. (1930). A Textbook of Sound. The Macmillan Company, New York, 1937.

Wood, A. (1940). Acoustics. Dover Publications, Inc., New York, 1966. 866 Bibliography

Young, R.W. (1952). Inharmonicity of plain wire piano strings. Journal of the Acoustical Society of America 24, No. 3, pp. 267–272.

Zanoncelli, L., Translator (1990). La Manualistica Musicale Greca. Angelo Guerini e Associati, Milan, Italy.

Zebrowski, E., Jr. (1979). Fundamentals of Physical Measurement. Duxbury Press, Belmont, California.

Chapter 10

Adkins, C.D. (1963). The Theory and Practice of the . Ph.D. dissertation printed and distributed by University Microfilms, Inc., Ann Arbor, Michigan.

Al-Faruqi, L.I. (1974). The Nature of the Musical Art of Islamic Culture: A Theoretical and Empirical Study of Arabian Music. Ph.D. dissertation printed and distributed by University Microfilms, Inc., Ann Arbor, Michigan.

Asselin, P. (1985). Musique et Tempérament. Éditions Costallat, Paris, France.

Barbera, C.A. (1977). Arithmetic and geometric divisions of the tetrachord. Journal of 21, No. 2, pp. 294–323.

Barbera, A., Translator (1991). The Euclidean Division of the Canon: Greek and Latin Sources. University of Nebraska Press, Lincoln, Nebraska.

Barbour, J.M. (1933). The persistence of the system. Scripta Mathematica, Vol. 1, pp. 286–304.

Barbour, J.M. (1951). Tuning and Temperament. Press, New York, 1972.

Barker, A., Translator (1989). Greek Musical Writings. Two Volumes. Cambridge University Press, Cam- bridge, England.

Barnes, J. (1979). Bach’s keyboard temperament. Early Music 7, No. 2, pp. 236–249.

Beck, C., Translator (1868). Flores musice omnis cantus Gregoriani, by Hugo Spechtshart [von Reutlingen]. Bibliothek des Litterarischen Vereins, Stuttgart, Germany.

Bower, C.M., Translator (1989). Fundamentals of Music, by A.M.S. Boethius. Yale University Press, New Haven, Connecticut.

Briscoe, R.L., Translator (1975). Rameau’s “Démonstration du principe de l’harmonie” and “Nouvelles reflec- tions de M. Rameau sur sa démonstration du principe de l’harmonie:” An Annotated Translation of Two Treatises by Jean-Philippe Rameau. Ph.D. dissertation printed and distributed by University Microfilms, Inc., Ann Arbor, Michigan.

Brun, V. (1964). Euclidean algorithms and musical theory. L’Enseignement Mathématique X, pp. 125–137.

Burkert, W. (1962). Lore and Science in Ancient Pythagoreanism. Translated by E.L. Minar, Jr. Harvard University Press, Cambridge, Massachusetts, 1972.

Chalmers, J.H., Jr. (1993). Divisions of the Tetrachord. Frog Peak Music, Hanover, New Hampshire.

Chandler, B.G., Translator (1975). Rameau’s “Nouveau système de musique théorique:” An Annotated Trans- lation with Commentary. Ph.D. dissertation printed and distributed by University Microfilms, Inc., Ann Arbor, Michigan. Bibliography: Chapter 10 867

Chapman, R.E., Translator (1957). Harmonie universelle: The Books on Instruments, by Marin Mersenne. Martinus Nijhoff, The Hague, Netherlands.

Coelho, V., Editor (1992). Music and Science in the Age of Galileo. Kluwer Academic Publishers, Dordrecht, Netherlands.

Cohen, H.F. (1984). Quantifying Music. D. Reidel Publishing Company, Dordrecht, Netherlands.

Compact Edition of the Oxford English Dictionary. Oxford University Press, Oxford, England, 1974.

Crew, H., and De Salvio, A., Translators (1914). Dialogues Concerning Two New Sciences, by Galileo Galilei. Dover Publications, Inc., New York.

Crocker, R.L. (1963). Pythagorean mathematics and music. The Journal of Aesthetics and Art Criticism XXII, No. 2, Part I: pp. 189–198, and No. 3, Part II: pp. 325–335.

Crocker, R.L. (1966). “Aristoxenus and Greek Mathematics.” In Aspects of Medieval and : A Birthday Offering to Gustave Reese, J. LaRue, Editor. Pendragon Press, New York.

Crone, E., Editor; Fokker, A.D., Music Editor; Dikshoorn, C., Translator (1966). The Principal Works of Simon Stevin. Five Volumes. C.V. Swets & Zeitlinger, Amsterdam.

Crookes, D.Z., Translator (1986). Syntagma musicum II: De organographia, Parts I and II, by Michael Prae- torius. The Clarendon Press, Oxford, England.

Daniels, A.M. (1962). The De musica libri VII of Francisco de Salinas. Ph.D. dissertation printed and distrib- uted by University Microfilms, Inc., Ann Arbor, Michigan.

De Haan, D.B., Publisher (1884). Vande Spiegeling der Singconst, by Simon Stevin. Amsterdam.

D’Erlanger, R., Bakkouch, .., and Al-Sans, M., Translators (Vol. 1, 1930; Vol. 2, 1935; Vol. 3, 1938; Vol. 4, 1939; Vol. 5, 1949; Vol. 6, 1959). La Musique Arabe, Librairie Orientaliste Paul Geuthner, Paris, France.

Diels, H. (1903). Die Fragmente der Vorsokratiker, Griechisch und Deutsch. Three Volumes. Weidmannsche Verlagsbuchhandlung, Berlin, Germany, 1951.

D’Ooge, M.L., Translator (1926). Nicomachus of Gerasa: Introduction to Arithmetic. The Macmillan Com- pany, New York.

Dupont, W. (1935). Geschichte der musikalischen Temperatur. C.H. Beck’sche Buchdruckerei, Nördlingen, Germany.

Düring, I., Editor (1930). Die Harmonielehre des Klaudios Ptolemaios. Original Greek text of Ptolemy’s Harmonics. Wettergren & Kerbers Förlag, Göteborg, Sweden.

Düring, I., Translator (1934). Ptolemaios und Porphyrios über die Musik. Georg Olms Verlag, Hildesheim, Germany, 1987.

Farmer, H.G. (1965). The Sources of Arabian Music. E.J. Brill, Leiden, Netherlands.

Farmer, H.G., Translator (1965). Al-Farabi’s Arabic-Latin Writings on Music. Hinrichsen Edition Ltd., New York.

Fend, M., Translator (1989). Theorie des Tonsystems: Das erste und zweite Buch der Istitutioni harmoniche (1573), von Gioseffo Zarlino. Peter Lang, Frankfurt am Main, Germany.

Fernandez de la Cuesta, I., Translator (1983). Siete libros sobre la musica, by Francisco Salinas. Editorial Alpuerto, Madrid, Spain. 868 Bibliography

Flegg, G., Hay, C., and Moss, B., Translators (1985). Nicolas Chuquet, Renaissance Mathematician. D. Reidel Publishing Company, Dordrecht, Holland.

Forster, C. (2015). “The Partch Hoax Doctrines.” Online article, pp. 1–22. The Chrysalis Foundation, San Francisco, California.

Gossett, P., Translator (1971). Traité de l’harmonie [Treatise on Harmony], by Jean-Philippe Rameau. Dover Publications, Inc., New York.

Green, B.L. (1969). The Harmonic Series From Mersenne to Rameau: An Historical Study of Circumstances Leading to Its Recognition and Application to Music. Ph.D. dissertation printed and distributed by Uni- versity Microfilms, Inc., Ann Arbor, Michigan.

Guthrie, K.S., Translator (1987). The Pythagorean Sourcebook and Library. Phanes Press, Grand Rapids, Michigan.

Hamilton, E., and Cairns, H., Editors (1966). The Collected Dialogues of Plato. Random House, Inc., New York.

Hawkins, J. (1853). A General History of the Science and Practice of Music. Dover Publications, Inc., New York, 1963.

Hayes, D., Translator (1968). Rameau’s Theory of Harmonic Generation; An Annotated Translation and Commentary of “Génération harmonique” by Jean-Philippe Rameau. Ph.D. dissertation printed and dis- tributed by University Microfilms, Inc., Ann Arbor, Michigan.

Heath, T.L., Translator (1908). Euclid’s Elements. Dover Publications, Inc., New York, 1956.

Heath, T. (1921). A History of Greek Mathematics. Dover Publications, Inc., New York, 1981.

Hitti, P.K. (1937). History of the Arabs. Macmillan and Co. Ltd., London, England, 1956.

Hubbard, F. (1965). Three Centuries of Harpsichord Making, 4th ed. Harvard University Press, Cambridge, Massachusetts, 1972.

Hyde, F.B. (1954). The Position of Marin Mersenne in the History of Music. Two Volumes. Ph.D. disserta- tion printed and distributed by University Microfilms, Inc., Ann Arbor, Michigan.

Ibn Sn (Avicenna): Auicene perhypatetici philosophi: ac medicorum facile primi opera in luce redacta... This Latin translation was published in 1508. Facsimile Edition: Minerva, Frankfurt am Main, Germany, 1961.

Jacobi, E.R., Editor (1968). Jean-Philippe Rameau (1683–1764): Complete Theoretical Writings. American Institute of Musicology, [Rome, Italy].

James, G., and James, R.C. (1976). Mathematics Dictionary, 4th ed. Van Nostrand Reinhold, New York.

Jorgensen, O. (1977). Tuning the Historical Temperaments by Ear. The Northern Michigan University Press, Marquette, Michigan.

Jorgenson, D.A. (1957). A History of Theories of the Minor Triad. Ph.D. dissertation printed and distributed by University Microfilms, Inc., Ann Arbor, Michigan.

Jorgenson, D.A. (1963). A résumé of harmonic dualism. Music and Letters XLIV, No. 1, pp. 31–42.

Kastner, M.S., Editor (1958). De musica libri VII, by Francisco Salinas. Facsimile Edition. Bärenreiter- Verlag, Kassel, Germany.

Kelleher, J.E. (1993). Zarlino’s “Dimostrationi harmoniche” and Demonstrative Methodologies in the Six- teenth Century. Ph.D. dissertation printed and distributed by University Microfilms, Inc., Ann Arbor, Michigan. Bibliography: Chapter 10 869

Lawlor, R. and D., Translators (1978). Mathematics Useful for Understanding Plato, by Theon of Smyrna. Wizards Bookshelf, San Diego, California, 1979.

Levin, F.R., Translator (1994). The Manual of Harmonics, of Nicomachus the Pythagorean. Phanes Press, Grand Rapids, Michigan.

Lindley, M. (1984). , Viols and Temperaments. Cambridge University Press, Cambridge, England.

Litchfield, M. (1988). Aristoxenus and empiricism: A reevaluation based on his theories. Journal of Music Theory 32, No. 1, pp. 51–73.

Mackenzie, D.C., Translator (1950). Harmonic Introduction, by Cleonides. In Source Readings in Music His- tory, O. Strunk, Editor. W. W. Norton & Company, Inc., New York.

Macran, H.S., Translator (1902). The Harmonics of Aristoxenus. Georg Olms Verlag, Hildesheim, Germany, 1990.

Marcuse, S. (1964). Musical Instruments: A Comprehensive Dictionary. W. W. Norton & Company, Inc., New York, 1975.

Maxham, R.E., Translator (1976). The Contributions of to Acoustics. Two Volumes. Ph.D. dissertation printed and distributed by University Microfilms, Inc., Ann Arbor, Michigan.

Mersenne, M. (1636–37). Harmonie universelle contenant la théorie et la pratique de la musique. Three Vol- umes. Facsimile Edition. Éditions du Centre National de la Recherche Scientifique, Paris, France, 1963.

Meyer, M.F. (1929). The Musician’s Arithmetic. Oliver Ditson Company, Boston, Massachusetts.

Miller, C.A., Translator (1993). Musica practica, by Bartolomeo Ramis de Pareia. Hänssler-Verlag, Neuhausen- Stuttgart, Germany.

Niven, I. (1961). Numbers: Rational and Irrational. Random House, New York.

Palisca, C.V. (1961). “Scientific Empiricism in Musical Thought.” In Seventeenth Century Science and the Arts, H.H. Rhys, Editor. Princeton University Press, Princeton, New Jersey.

Palisca, C.V. (1985). Humanism in Italian Renaissance Musical Thought. Yale University Press, New Haven, Connecticut.

Palisca, C.V., Translator (2003). Dialogue on Ancient and Modern Music, by Vincenzo Galilei. Yale Univer- sity Press, New Haven, Connecticut.

Partch, H. (1949). , 2nd ed. Da Capo Press, New York, 1974.

Rameau, J.P. (1722). Traité de l’harmonie reduite à ses principes naturels. Facsimile Edition. Biblioteca Nacional de Madrid, Spain, 1984.

Rasch, R., Editor (1983). Musicalische Temperatur, by Andreas Werckmeister. The Diapason Press, Utrecht, Netherlands.

Rasch, R., Editor (1984). Collected Writings on Musical Acoustics, by Joseph Sauveur. The Diapason Press, Utrecht, Netherlands.

Rasch, R., Editor (1986). Le cycle harmonique (1691), Novus cyclus harmonicus (1724), by Christiaan Huy- gens. The Diapason Press, Utrecht, Netherlands.

Reichenbach, H. (1951). The Rise of Scientific Philosophy. The University of California Press, Berkeley and Los Angeles, California, 1958. 870 Bibliography

Roberts, F. (1692). A discourse concerning the musical notes of the trumpet, and the trumpet-marine, and of the defects of the same. Philosophical Transactions of the Royal Society of London XVII, pp. 559–563.

Rossing, T.D. (1989). The Science of Sound, 2nd ed. Addison-Wesley Publishing Co., Inc., Reading, Mas- sachusetts, 1990.

Sadie, S., Editor (1980). The New Grove Dictionary of Music and Musicians. Macmillan Publishers Limited, London, England, 1995.

Shirlaw, M. (1917). The Theory of Harmony. Da Capo Press Reprint Edition. Da Capo Press, New York, 1969.

Solomon, J., Translator (2000). Ptolemy Harmonics. Brill, Leiden, Netherlands.

Soukhanov, A.H., Executive Editor (1992). The American Heritage Dictionary of the English Language, 3rd ed. Houghton Mifflin Company, Boston, Massachusetts.

Stephan, B. (1991). Geometry: Plane and Practical. Harcourt Brace Jovanovich, Publishers, San Diego, Cali- fornia.

Truesdell, C. (1960). The Rational Mechanics of Flexible or Elastic Bodies: 1638–1788. Orell Füssli, Zürich, Switzerland.

Wallis, J. (1677). Dr. Wallis’ letter to the publisher, concerning a new musical discovery. Philosophical Trans- actions of the Royal Society of London XII, pp. 839–842.

West, M.L. (1992). Ancient Greek Music. The Clarendon Press, Oxford, England, 1994.

White, W.B. (1917). Piano Tuning and Allied Arts, 5th ed. Tuners Supply Company, Boston, Massachusetts, 1972.

Wienpahl, R.W. (1959). Zarlino, the Senario, and . Journal of the American Musicological Society XII, No. 1, pp. 27–41.

Williams, R.F., Translator (1972). Marin Mersenne: An Edited Translation of the Fourth Treatise of the “Har- monie universelle.” Three Volumes. Ph.D. dissertation printed and distributed by University Microfilms, Inc., Ann Arbor, Michigan.

Williamson, C. (1938). The frequency ratios of the tempered scale. Journal of the Acoustical Society of America 10, pp. 135–136.

Winnington-Ingram, R.P. (1932). Aristoxenus and the intervals of Greek music. The Classical Quarterly XXVI, Nos. 3–4, pp. 195–208.

Winnington-Ingram, R.P. (1936). Mode in Ancient Greek Music. Cambridge University Press, London, Eng- land.

Winnington-Ingram, R.P. (1954). “Greek Music (Ancient).” In Grove’s Dictionary of Music and Musicians, Volume 3, 5th ed., E. Blom, Editor. St. Martin’s Press, Inc., New York, 1970.

Zarlino, R.M.G. (1571). Dimostrationi harmoniche. Facsimile Edition, The Gregg Press Incorporated, Ridge- wood, New Jersey, 1966.

Zarlino, R.M.G. (1573). Istitutioni harmoniche. Facsimile Edition, The Gregg Press Limited, Farnborough, Hants., England, 1966. Bibliography: Chapter 11 871

Chapter 11

Chinese Music

Apel, W., Editor (1944). Harvard Dictionary of Music, 2nd ed. Harvard University Press, Cambridge, Mas- sachusetts, 1972.

Gulik, R.H., Translator (1941). Poetical Essay on the , by Hsi K’ang. In Gulik’s Hsi K’ang and His Poeti- cal Essay on the Lute, Sophia University, Tokyo, Japan.

Kaufmann, W. (1967). Musical Notations of the Orient. Indiana University Press, Bloomington, Indiana.

Kaufmann, W. (1976). Musical References in the Chinese Classics. Detroit Monographs in Musicology, De- troit, Michigan.

Kuttner, F.A. (1965). A musicological interpretation of the twelve lüs in China’s traditional tone system. Journal of the Society for Ethnomusicology IX, No. 1, pp. 22–38.

Lieberman, F., Translator (1977). The Mei-an Ch’in-p’u, by Hsü Li-sun. In Lieberman’s The Chinese Long Zither Ch’in: A Study Based on the Mei-an Ch’in-p’u. Ph.D. dissertation printed and distributed by University Microfilms, Inc., Ann Arbor, Michigan.

Lieberman, F., Translator (1983). The Mei-an Ch’in-p’u, by Hsü Li-sun. In Lieberman’s A Chinese Zither Tutor. University of Washington Press, Seattle, Washington.

Lui, T. (1968). A short guide to ch’in. Selected Reports I, No. 2, pp. 180–201. Publication of the Institute of Ethnomusicology of the University of California at Los Angeles.

Needham, J. (1962). Science and Civilization in China, Volume 4, Part I. Cambridge University Press, Cam- bridge, England.

Reinhard, K. (1956). Chinesische Musik, 2nd ed. Im Erich Röth-Verlag, Kassel, Germany.

Robinson, K. (1980). A Critical Study of Chu Tsai-yü’s Contribution to the Theory of Equal Temperament in Chinese Music. Franz Steiner Verlag GmbH, Wiesbaden, Germany.

Sachs, C. (1940). The History of Musical Instruments. W. W. Norton & Company, Inc., New York.

Sadie, S., Editor (1980). The New Grove Dictionary of Music and Musicians. Macmillan Publishers Limited, London, England, 1995.

Wang, L., and Needham, J. (1955). Horner’s Method in Chinese Mathematics: Its origins in the root-extraction procedures of the Han Dynasty. T’oung Pao Archives XLIII, No. 5, pp. 345–401. Leiden, Netherlands.

Wang, K. (1956). Chung-kuo yin yueh shih. Taipei, Formosa: Chung hua shu chu.

Indonesian Music

Apel, W., Editor (1944). Harvard Dictionary of Music, 2nd ed. Harvard University Press, Cambridge, Mas- sachusetts, 1972.

Blom, E., Editor (1954). Grove’s Dictionary of Music and Musicians, 5th ed. St. Martin’s Press, Inc., New York, 1970.

Hood, M. (1954). The Nuclear Theme as a Determinant of Patet in Javanese Music. Da Capo Press, Inc., New York, 1977. 872 Bibliography

Hood, M. (1966). Sléndro and pélog redefined. Selected Reports I, No. 1, pp. 28–48. Publication of the Insti- tute of Ethnomusicology of the University of California at Los Angeles.

Lentz, D.A. (1965). The and Bali. University of Nebraska Press, Lincoln, Nebraska.

Martopangrawit, R.L. (1972). Catatan-Catatan Pengetahuan Karawitan [Notes on Knowledge of Gamelan Music], translated by M.F. Hatch. In Karawitan, Volume 1, J. Becker and A.H. Feinstein, Editors. Center for South and Southeast Asian Studies, The University of Michigan, 1984.

McDermott, V., and (1975). Central Javanese music: The patet of laras sléndro and the barung. Journal of the Society for Ethnomusicology XIX, No. 2, pp. 233–244.

McPhee, C. (1966). Music in Bali: A Study in Form and Instrumental Organization in Balinese Orchestral Music. Yale University Press, New Haven, Connecticut.

Ornstein, R.S. (1971). Gamelan Kebjar: The Development of a Balinese Musical Tradition. Ph.D. dis- sertation printed and distributed by University Microfilms, Inc., Ann Arbor, Michigan.

Poerbapangrawit, R.M.K. (1955). Gendhing Jawa [Javanese Gamelan Music], translated by J. Becker. In Karawitan, Volume 1, J. Becker and A.H. Feinstein, Editors. Center for South and Southeast Asian Stud- ies, The University of Michigan, 1984.

Rai, I, W. (1996). Balinese Gamelan Semar Pagulingan Saih Pitu: The Modal System. Ph.D. dissertation printed and distributed by University Microfilms, Inc., Ann Arbor, Michigan.

Randel, D.M., Editor (1986). The New Harvard Dictionary of Music, 6th ed. The Belknap Press of Harvard University Press, Cambridge, Massachusetts, 1993.

Sadie, S., Editor (1984). The New Grove Dictionary of Musical Instruments. Macmillan Press Limited, Lon- don, England.

Schaareman, D., Editor (1992). Balinese Music in Context: A Sixty-fifth Birthday Tribute to Hans Oesch. Amadeus Verlag, Winterthur, Switzerland.

Sindoesawarno, K. (1955). Ilmu Karawitan [Knowledge About Gamelan Music], translated by M.F. Hatch. In Karawitan, Volume 2, J. Becker and A.H. Feinstein, Editors. Center for South and Southeast Asian Studies, The University of Michigan, 1987.

Sumarsam (1975). Gendèr barung, its technique and function in the context of Javanese gamelan. Indonesia 20, pp. 161–172.

Sumarsam (1992). Gamelan: Cultural Interaction and Musical Development in Central Java. The University of Chicago Press, Chicago, Illinois, 1995.

Surjodiningrat, W., Sudarjana, P.J., and Susanto, A. (1972). Tone Measurements of Outstanding Javanese in Jogjakarta and Surakarta, 2nd ed. Gadjah Mada University Press, Jogjakarta, Indonesia.

Tenzer, M. (2000). : The Art of Twentieth-Century Balinese Music. The University of Chicago Press, Chicago, Illinois.

Toth, A.F. (1975). The Gamelan Luang of Tangkas, Bali. Selected Reports II, No. 2, pp. 65–79. Publication of the Institute of Ethnomusicology of the University of California at Los Angeles.

Toth, A.F. (1993). “Selera Yang Selaras: Pepatutan Gong Ditinjau Dari Segi Akustika dan Estetika.” Mudra: Jurnal Seni Budaya, pp. 92–117. Edisi Khusus. Bibliography: Chapter 11 873

Indian Music

Aiyar, M.S.R., Translator (1932). Svaramelakalanidhi, by Rmmtya. The Annamalai University, India.

Ayyar, C.S. (1939). The Grammar of South Indian (Karnatic) Music. Smt. Vidya Shankar, Madras, India, 1976.

Bhandarkar, R.S.P.R. (1912). Contribution to the study of ancient Hindu music. The Indian Antiquary XLI, pp. 157–164, pp. 185–195, pp. 254–265.

Bhandarkar, R.S.P.R. (1913–1914). Kuimiymalai inscription on music. Epigraphia Indica XII, No. 28, pp. 226–237.

Bhatkhande, V.N. (1930). A Comparative Study of Some of the Leading Music Systems of the 15th, 16th, 17th, and 18th Centuries. Indian Musicological Society, Baroda, India, 1972.

Bhise, U.R., Translator (1986). Naradiya Siksa, by Nrada. Bhandarkar Institute Press, Poona, India.

Coomaraswamy, A.K. (1930). The parts of a vina. Journal of the American Oriental Society 50, No. 3, pp. 244–253.

Gangoly, O.C. (1935). Ragas and Raginis. Nalanda Publications, Bombay, India, 1948.

Ghosh, M., Translator (Vol. 1, Ch. 1–27, 1950; Vol. 2, Ch. 28–36, 1961). The Natyasastra, by Bharata. Bib- liotheca Indica, The Asiatic Society, Calcutta, India.

Iyer, T.L.V. (1940). The scheme of 72 melas in Carnatic Music. The Journal of the Music Academy XI, Parts I–IV, pp. 80–86, Madras, India.

Jairazbhoy, N.A. (1971). The Rags of North Indian Music. Wesleyan University Press, Middletown, Con- necticut.

Jairazbhoy, N.A. (1975). An interpretation of the 22 srutis. Asian Music VI, Nos. 1–2, pp. 38–59.

Junius, M.M. (1974). The . Heinrichshofen’s Verlag, Wilhelmshaven, Germany.

Kaufmann, W. (1968). The Ragas of North India. Indiana University Press, Bloomington, Indiana.

Kaufmann, W. (1976). The Ragas of South India. Indiana University Press, Bloomington, Indiana.

Krishnaswamy, A. (1981). Melakarta and Janya Raga Chart. Sakthi Priya Publication, Madras, India.

Lath, M., Translator (1978). A Study of Dattilam: A Treatise on the Sacred Music of Ancient India. Impex India, New Delhi, India.

Marcotty, T. (1974). Djovari: Giving Life to the Sitar. This essay on how to make a parabolic sitar bridge is in The Sitar, by Manfred M. Junius. Heinrichshofen’s Verlag, Wilhelmshaven, Germany.

Nijenhuis, E.W., Translator (1970). Dattilam: A Compendium of Ancient Indian Music. E. J. Brill, Leiden, Netherlands.

Powers, H.S. (1958). The Background of the South Indian Raga-System. Ph.D. dissertation printed and dis- tributed by University Microfilms, Inc., Ann Arbor, Michigan.

Ramachandran, K.V. (1938). The melakarta — a critique. The Journal of the Music Academy IX, Parts I–IV, pp. 31–33, Madras, India.

Ramachandran, K.V. (1950). Carnatic ragas from a new angle — Sankarabharana. The Journal of the Music Academy XXI, Parts I–IV, pp. 88–99, Madras, India. 874 Bibliography

Ramachandran, K.V. (1950). Carnatic ragas and the textual tradition. The Journal of the Music Academy XXI, Parts I–IV, pp. 99–106, Madras, India.

Ramachandran, K.V. (1950). Apurva ragas of Tygarja’s songs. The Journal of the Music Academy XXI, Parts I–IV, pp. 107–109, Madras, India.

Ramachandran, N.S. (1938). The Ragas of Karnatic Music. University of Madras, Madras, India.

Rao, T.V.S. (1945 and 1946). The ragas of the Sangita Saramrita. The Journal of the Music Academy XVI, Parts I–IV, pp. 45–64, and XVII, Parts I–IV, pp. 104–134, Madras, India.

Rowell, L. (1981). Early Indian musical speculation and the theory of melody. Journal of Music Theory 25.2, pp. 217–244.

Roy, H.L. (1937). Problems of Hindustani Music. Bharati Bhavan, Calcutta, India.

Sachs, C. (1940). The History of Musical Instruments. W. W. Norton & Company, Inc., New York.

Sadie, S., Editor (1980). The New Grove Dictionary of Music and Musicians. Macmillan Publishers Limited, London, England, 1995.

Sadie, S., Editor (1984). The New Grove Dictionary of Musical Instruments. Macmillan Press Limited, Lon- don, England.

Sambamoorthy, P. (1960). History of Indian Music. The Indian Music Publishing House, Madras, India.

Sambamoorthy, P. (Vol. 1, A–F, 1952; Vol. 2, G–K, 1959; Vol. 3, L–N, 1971). A Dictionary of South Indian Music and Musicians. The Indian Music Publishing House, Madras, India.

Sastri, S.S. (1931). Venkaamakhi and his twelve notes. The Journal of the Music Academy II, No. 1, pp. 22–23, Madras, India.

Sathyanarayana, R., Editor (1957). Kudimiyamalai Inscription on Music. ri Varalakshmi Academies of Fine Arts, Parimala Press, Mysore, India.

Shankar, R. (1968). My Music, My Life. Simon and Schuster, New York, New York.

Sharma, P.L., Editor and Translator (Vol. 1, Ch. 1, 1992; Vol. 2, Ch. 2–6, 1994). Brhaddesi of ri Mataga Muni. Indira Gandhi National Centre for the Arts in association with Motilal Banarsidass Publishers, Delhi, India.

Shringy, R.K., and Sharma, P.L., Translators (Vol. 1, Ch. 1, 1978; Vol. 2, Ch. 2–4, 1989). Sangitaratnakara, by Srgadeva. Volume 1, Motilal Banarsidass, Delhi, India; Volume 2, Munshiram Manoharlal, New Delhi, India.

Sorrell, N., and Narayan, R. (1980). Indian Music in Performance. New York University Press, New York, New York. This book comes boxed with an excellent cassette tape recording.

Widdess, R. (1995). The Ragas of Early Indian Music. The Clarendon Press, Oxford, England.

Arabian, Persian, and Turkish Music

Al-Faruqi, L.I. (1974). The Nature of the Musical Art of Islamic Culture: A Theoretical and Empirical Study of Arabian Music. Ph.D. dissertation printed and distributed by University Microfilms, Inc., Ann Arbor, Michigan.

Al-Faruqi, L.I. (1981). An Annotated Glossary of Arabic Musical Terms. Greenwood Press, Westport, Con- necticut. Bibliography: Chapter 11 875

Avenary, H., Translator (1974). The Hebrew version of Ab l-alt’s treatise on music. Yuval III, pp. 7–82.

Barker, A., Translator (1989). Greek Musical Writings. Two Volumes. Cambridge University Press, Cam- bridge, England.

Cowl, C., Translator (1966). Al-Kind’s essay on the composition of melodies. The Consort, No. 23, pp. 129–159.

Crookes, D.Z., Translator (1986). Syntagma musicum II: De organographia, Parts I and II, by Michael Prae- torius. The Clarendon Press, Oxford, England. The first edition of this work was published in 1618.

D’Erlanger, R., Bakkouch, .., and Al-Sans, M., Translators (Vol. 1, 1930; Vol. 2, 1935; Vol. 3, 1938; Vol. 4, 1939; Vol. 5, 1949; Vol. 6, 1959). La Musique Arabe, Librairie Orientaliste Paul Geuthner, Paris, France.

Dieterici, F., Translator (1858–1890; 16 Volumes). Die Philosophie der Araber im IX. und X. Jahrhundert n. Chr. aus der Theologie des Aristoteles, den Abhandlungen Alfarabis und den Schriften der Lautern Brüder. The quoted passage on music is in Volume 6, entitled: Die Propaedeutik der Araber im zehnten Jahrhundert. E.S. Mittler und Sohn, Berlin, Germany, 1865.

El-Hefny, M., Translator (1931). Ibn Sina’s Musiklehre [Ibn Sn’s teaching on music]. Ph.D. dissertation printed by Otto Hellwig, Berlin, Germany.

Farhat, H. (1965). The Concept in Persian Music. Ph.D. dissertation printed and distributed by University Microfilms, Inc., Ann Arbor, Michigan.

Farhat, H. (1990). The Dastgah Concept in Persian Music. Cambridge University Press, Cambridge, Eng- land.

Farmer, H.G. (1925). Clues for the Arabian influence on European musical theory. The Journal of the Royal Asiatic Society, First Quarter, pp. 61–80.

Farmer, H.G. (1928). Ibn Khurddhbih on musical instruments. The Journal of the Royal Asiatic Society, Third Quarter, pp. 509–518.

Farmer, H.G. (1929). A History of Arabian Music. Luzac Oriental, London, England, 1994.

Farmer, H.G. (1953–1954). The song captions in the Kitab al-aghani al-kabir. Transactions of the Glasgow University Oriental Society XV, pp. 1–10.

Farmer, H.G. (1954). “d.” In Grove’s Dictionary of Music and Musicians, Volume 8, 5th ed., E. Blom, Edi- tor. St. Martin’s Press, Inc., New York, 1970.

Farmer, H.G. (1957). “The Music of Islam.” In New Oxford History of Music, Volume 1: Ancient and Oriental Music, E. Wellesz, Editor. Oxford University Press, London, England, 1960.

Farmer, H.G. (1965). The Sources of Arabian Music. E.J. Brill, Leiden, Netherlands.

Farmer, H.G., Translator (1965). Al-Farabi’s Arabic-Latin Writings on Music. Hinrichsen Edition Ltd., New York.

Farmer, H.G. (1965). The old Arabian melodic modes. Journal of the Royal Asiatic Society, Parts 3 & 4, pp. 99–102.

Farmer, H.G. (1978). Studies in Oriental Musical Instruments, First and Second Series. Longwood Press Ltd., Tortola, British Virgin Islands.

Hitti, P.K. (1937). History of the Arabs. Macmillan and Co. Ltd., London, England, 1956. 876 Bibliography

Lachmann, R. and El-Hefni, M., Translators (1931). Risala  hubr t2lif al-alhan [Über die Komposition der Melodien], by Al-Kind. Fr. Kistner & C.F.W. Siegel, Leipzig, Germany.

Lewis, B., Editor (1976). Islam and the Arab World. Alfred A. Knopf, New York.

Maas, M. and Snyder, J.M. (1989). Stringed Instruments of Ancient Greece. Yale University Press, New Ha- ven, Connecticut.

Marcus, S.L. (1989). Arab Music Theory in the Modern Period. Ph.D. dissertation printed and distributed by University Microfilms, Inc., Ann Arbor, Michigan.

Marcus, S.L. (1992). Modulation in Arab music: Documenting oral concepts, performance rules and strate- gies. Ethnomusicology 36, No. 2, pp. 171–195.

Muallem, D. (2010). The Maqam Book: A Doorway to Arab Scales and Modes. OR-TAV Music Publications, Kfar Sava, Israel.

Racy, A.J. (1977). Musical Change and Commercial Recording in Egypt, 1904–1932. Ph.D. dissertation print- ed and distributed by University Microfilms, Inc., Ann Arbor, Michigan.

Racy, A.J. (1978). “Music.” In The Genius of Arab Civilization, 2nd ed., J.R. Hayes, Editor. MIT Press, Cambridge, Massachusetts, 1983.

Ribera, J. (1929). Music in Ancient Arabia and Spain. Translated and abridged from the Spanish by Eleanor Hague and Marion Leffingwell. Stanford University Press, Stanford University, California.

Ronzevalle, P.L., Translator (1913). Un Traité de Musique Arabe Moderne, by Mkhl Mashqah. Mélanges de la Faculté Orientale VI, pp. 1–120. Université Saint-Joseph, Beirut, Lebanon.

Sachs, C. (1940). The History of Musical Instruments. W. W. Norton & Company, Inc., New York.

Sadie, S., Editor (1984). The New Grove Dictionary of Musical Instruments. Macmillan Press Limited, Lon- don, England.

Shiloah, A. (1972). The simsimiyya: A stringed instrument of the Red Sea area. Asian Music IV, No. 1, pp. 15–26.

Shiloah, A., Translator (1978). The Epistle on Music of the Ikhwan Al-Sa. Tel-Aviv University, Tel-Aviv, Israel.

Shiloah, A. (1979). The Theory of Music in Arabic Writings (c. 900–1900). G. Henle Verlag, München, Germany.

Shiloah, A. (1981). The Arabic concept of mode. Journal of the American Musicological Society XXXIV, No. 1, pp. 19–42.

Signell, K.L. (1973). The Turkish ‘Makam’ System in Contemporary Theory and Practice. Ph.D. dissertation printed and distributed by University Microfilms, Inc., Ann Arbor, Michigan.

Signell, K.L. (1986). Makam: Modal Practice in Turkish Art Music. Da Capo Press, New York, New York.

Smith, E., Translator (1847). A Treatise on Arab Music, by Mkhl Mashqah. Journal of the American Oriental Society I, No. 3, pp. 173–217. Boston, Massachusetts.

Touma, H.H. (1996). The Music of the Arabs. Amadeus Press, Portland, Oregon.

Wright, O. (1966). Ibn al-Munajjim and the early Arabian modes. The Galpin Society Journal XIX, pp. 27–48. Bibliography: Chapter 12 877

Chapter 12

Writings on Cris Forster

Editors. “Cris Forster.” In Für Augen und Ohren, Magazine of the Berlin Music Festival. January–February, 1980, pp. 16–17. Dr. Ulrich Eckhardt, Publisher. Berlin, Germany.

Mahlke, S. “Barde mit zwei Saitenspielen.” Der Tagesspiegel. January 25, 1980. Berlin, Germany.

Garr, D. “The Endless Scale.” Omni. March, 1981, p. 48. New York, New York.

McDonald, R. “Cris Forster: Making Music.” San Diego Magazine. September, 1982, pp. 136–139, 198, 228. San Diego, California.

Fleischer, D. “ of Infinity.” Connoisseur. August, 1983, pp. 102, 105. New York, New York.

Levine, J. “Expanded Musical Palette Is Inventor’s Note-able Goal.” The Tribune. September 6, 1983. San Diego, California.

Brewster, T. “A Medley of New Instruments: A Wheel Like the Wind.” Life. November, 1983, p. 142. New York, New York.

Editors. “Klingen wie der Wind.” Stern. January, 1984, pp. 146–148. Hamburg, Germany.

Davies, H. (1984). “Microtonal Instruments,” pp. 656–657. In The New Grove Dictionary of Musical Instru- ments, Volume 2, S. Sadie, Editor. Macmillan Press Limited, London, England.

Drye, S.L. (1984). The History and Development of the Musical Glasses, pp. 46–49. Master’s thesis, North Texas State University.

Arnautoff, V. “Hill Musician Composes, Builds Own Instruments.”The Potrero View. November, 1985, p. 5. San Francisco, California.

Editors. “This Californian and His Bearings Are Making Beautiful Music Together.” Precisionist: A Publica- tion of the Torrington Company. Summer, 1987, pp. 14–15. Torrington, Connecticut.

Bowen, C. “Making Music from Scratch.” Métier. Fall, 1987, p. 7. San Francisco, California.

Snider, J. “Chrysalis: A Transformation in Music.” Magical Blend. April, 1989, pp. 98–102. San Francisco, California.

Hopkin, B. “Review.” Experimental Musical Instruments. April, 1990, p. 4. Nicasio, California.

Editors. “Forster, Cris.” PITCH for the International Microtonalist I, No. 4, Spring 1990, p. 142. Johnny Reinhard, Editor. New York, New York.

Canright, D. “Performance: Concert in Celebration of the Chrysalis New Music Studio — Instruments and Music by Cris Forster.” 1/1, the Journal of the Just Intonation Network 11, No. 4. Spring, 2004, pp. 19–20. San Francisco, California.

Seven extensive reviews of Musical Mathematics: On the Art and Science of Acoustic Instruments. https://chrysalis-foundation.org/musical-mathematics-extensive-reviews

Kaliss, J. “Cris Forster’s ‘Just’ Musical Menagerie.” San Francisco Classical Voice. July 23, 2013. San Fran- cisco, California. (See San Francisco Classical Voice Article.) 878 Bibliography

Spaulding, B. “How Math Is Helping One Man Push the Boundaries of Acoustic Music.” PTC Product Lifecycle Report. January 4, 2016. Needham, Massachusetts. (See PTC Mathcad Article.)

Forster, H. “Support the Evolution of Acoustic Music.” Kickstarter Project. November 29, 2016–January 15, 2017. Brooklyn, New York. (See Kickstarter Article.)

Videos on Cris Forster

Gaikowski, R., Videographer and Editor (1988). Musical Wood, Steel, and Glass: A video featuring Cris Forster, composer, musician, inventor. Production by One Way Films/Videos, San Francisco, Califor- nia.

Noyes, E., Videographer and Editor; Forster, H., Producer and Writer (2006). A Voyage in Music: A retro- spective of Cris Forster’s work over the past thirty years. Production by Alligator Planet, San Francisco, California.

Video performances of solo and ensemble compositions at the Chrysalis New Music Studio. https://www.youtube.com/user/CrisForster/feed

Thomas, R., Videographer and Editor; Forster, H., Writer (2016). Support the Evolution of Acoustic Music. Production by Sponsored Films, San Francisco, California.

Musical Mathematics: On the Art and Science of Acoustic Instruments © 2000–2011 Cristiano M.L. Forster All rights reserved. www.chrysalis-foundation.org February 18, 2011 Index Internet Version: no diacritics

A Latin ratios; superparticular and superpartient, Abul-Salt, 628, 630–632 384–385 Acceleration Ptolemy, 321–322 of bar, 156–158 string length of 120 parts, 316–317 definition, 4 Ramamatya, 568–573 dimensional analysis, 4, 6 Rameau, 431–434, 442–447, 452 English Gravitational System, 9. Ssu-ma Ch’ien, 486–487 consistent system, 9–10, 14–15 vs. frequency ratios, 75–76, 91, 93, 305, 307, 427, experiment, 8–11, 15 431, 487, 576–577, 587 gravitational, 6–7, 10–12, 14 vs. modern length ratio, 305–307, 427, 440–441 of mass vs. “vibration ratios,” 85–86, 93, 435–436 mica, 14–16 vs. “weight ratios,” 83, 85–86, 93 slug, 10–11 Zarlino, 378, 436, 452 Newton, 5–6 Antinodes. See Bars, rods, and tubes; Flexible standard gravity, 11–12 strings; Just intonation, Sauveur; Resonators, of string, 2–3 Adiabatic bulk modulus tube of air, 22 Arabian musical terms. See also Scales, Arabian definition, 21–22 24-TET Admittance. See Flutes 5 nim (Pers. lowered) notes, 756, 759 Ahobala, 93, 587–591 5 tik (Pers. raised) notes, 756, 759 Al-Din, Salah, 761–762 7 fundamental notes; naghamat (notes), Al-Farabi, Abu Nasr, 93, 326, 354–355, 366, 375, asasiyyah (fundamental), 756, 759 378, 401, 610, 622, 625, 628, 632–666, 669, 7 half notes, or “semitones”; arabat (sing. 673–674, 676–677, 679–681, 696, 701–707, arabah, half note), 756, 759 709–710, 712, 715, 717, 720–722, 727, 733–736, 24 quarter tones, Mashaqah, (sing. rub, 749–752, 754–755, 757–758, 771 quarter; pl. arba, quarters), 756, 759 Al-Faruqi, Lois I., 641–642, 648, 760 interval Al-Isfahani, Abul-Faraj, 619–620, 624 Al-Farabi Al-Jurjani, 401–403, 448, 610, 626–631, 707, awdah (whole step), 355, 647 717–722, 726–727, 731–733, 744, 746, 748 fadlah (half step), 355, 647 Al-Khulai, M. Kamil, 749, 755 Al-Jurjani Al-Kindi, Ishaq, 93, 334, 611–617, 619–620, baqiyyah, limma, [L], 721–723 642–643, 646, 709–710, 755 fadlah, comma of Pythagoras, [C], 722–723 Al-Mauaili, Ishaq, 619–620, 622 mutammam, apotome, [L×C], 722–723 Al-Sanusi, Al-Manubi, 402, 720 tanini, double-limma plus comma, Alexander the Great, 610 [L×L×C], 721–723 Ambisonance, 779n.74 tatimmah, double-limma, [L×L], 722–723 Ancient length ratios (a.l.r.) mode definition, 75–77, 287 6 hierarchical functions of tones: qarar, Al-Farabi, 378 quwwah, zahir, ghammaz, markaz, mabda, Needham, 487 738 equations, for stopped (bridged) string, 77 46 Modern Maqamat (sing. maqam, position Euclid, 76, 81–82, 302–307 or place; mode), 738, 761–771, 786n.209 Greek ratios; multiple, epimore, and epimere, Al-Farabi, 8 Ajnas (sing. jins, genus or kind; 286–287 mode), 640, 646–655

895 896 Index

Arabian musical terms (Continued) “,” oldest ud tuning, 3-limit ratio, Al-Munajjim 620–622 8 majari (sing. majra, course or path; Persian middle finger , Al-Farabi’s ud s, mode), 622–625 17-limit ratio, 634–639, 650–651, 653 and Philolaus’ , 624 “neutral second” Ibn Sina Arabian theory, modern 12 shudud (sing. shadd), primary or 46 Maqamat, 763–766 principal mode, 727 of basic 16-tone scale, 770–771 Mustaqim (direct, straight, or regular mode), Safi Al-Din, 84 Melodic Modes, Second Melodic Mode 8, 681–682, 685, 727, Ud Tuning; in six of nine unchanged 754 modes, 767–768 majra (course or path; mode), 617, 622. tik Zirkulah [D], 756 Safi Al-Din origins in near-equal divisions of intervals; 6 awazat (sing. awaz; also shubah), Ptolemy, Al-Farabi, and Ibn Sina, secondary or derived mode, 727, 677–681 730–733, 744 Persian theory, modern 12 shudud (sing. shadd), primary or of basic 17-tone scale, 770 principal mode, 726–731, 737, 740–743, Farhat 767–768 of 2 tars and 3 setars, two different Rast (Pers. direct, straight, or regular kinds, 686–689 mode), Melodic Mode 40, 727–730, 12 Dastgaha, 692–696 733, 740–741, 746–748 as tone on ud Arabian ratios. See also Greek ratios; Latin ratios Al-Farabi apotome Fret 5 on 12-fret ud, 641–643 Al-Farabi, 636, 655 Frets 3 and 4, Mujannab frets, on 10-fret Al-Jurjani, 722–723 ud, 632–635, 639, 673–674 Al-Kindi, 334, 611, 615–616 Ibn Sina apyknon, Al-Farabi, 659–663 Fret 2, Assistant to the middle finger of comma of Pythagoras (ditonic comma) Zalzal fret, 668, 671, 673–674 Arabian theory two different kinds, 672 Al-Farabi, 655, 702–706 Safi Al-Din, Fret 2, Mujannab, Anterior Al-Jurjani, 722–723 fret, Second Ud Tuning, 714–716 Al-Kindi, 611, 615–616, 620 as tone or interval of a genus or mode of tunbur and ud, ancient, 696–697 Al-Farabi, Jins 2/Jins 8, 651–654, 676–677, variants of, notation, 697–701 679–680 Al-Farabi’s tunbur of Khurasan and Ibn Sina original tunbur, 696–706 11 Melodic Modes, 681–685 Safi Al-Din, First Ud Tuning, 712–713 Diatonic Genus 4 and 7, 674–677 Turkish theory, modern Safi Al-Din, 84 Melodic Modes, Second Ud of tunbur, 733–737 Tuning, 724–725, 729 variants of, notation, 734 “neutral seventh” epimere, Al-Farabi, 661–663 Arabian theory, modern epimore Awj [B], 749, 756, 758–759 Al-Farabi, 661–663 of basic 16-tone scale, 770 Ibn Sina, 673–676, 678–679 used to justify 24-TET limma D’Erlanger, 755–758 Al-Farabi, 633, 636–640, 643, 655–656, Marcus, 760–761 702–707 historical context of, Al-Farabi’s and Ibn Al-Jurjani, 722–723 Sina’s ud s, 749–754 Al-Kindi, 334–335, 611, 615–617 “neutral third” double-limma, 697, 699–700, 703–705, 722–723 Arabian theory, modern Ibn Sina, 667–668, 671, 682–685 of basic 16-tone scale, 770 Safi Al-Din, 708–713, 722–723, 741–743 Safi Al-Din, 84 Melodic Modes, Second triple-limma, 722–723, 739–745, 767 Ud Tuning; in six of nine unchanged Turkish theory, modern, 736–737 modes, 767–768 triple-limma, 739–745, 767 Sikah [E], 749, 756, 758–761 Index 897

fasilah (genus), 761–762, 765 schisma maqam, 767–769 definition, 373, 697 used to justify 24-TET variants of, notation, 697–701 D’Erlanger, 755–758 Al-Farabi’s tunbur of Khurasan and original Marcus, 760–761 tunbur, 696–707 historical context of, 749, 754–755, 761 Safi Al-Din Farmer, 716 First Ud Tuning, 373–375, 710–713, 717 origins in near-equal divisions of intervals; Ptolemy, Al-Farabi, and Ibn Sina, Second Ud Tuning, 716 677–681 superparticular (epimore) Persian theory, modern Al-Farabi, 660–661 of basic 17-tone scale, 770 Ibn Sina, 678–679 Farhat superpartient (epimere), Al-Farabi, 661 of 2 tars and 3 setars, two different Arabian scales. See Scales, Arabian kinds, 686–689 Arabian tetrachords 12 Dastgaha, 692–696 Al-Farabi as tone on ud 8 Ajnas middle finger of Zalzal fret description of, 646–648 Al-Farabi, 11-limit ratio fractional and integer parts of tones, Fret 8 on 10-fret ud, 634–640 Fret 9 on 12-fret ud, 641–645 Aristoxenian theory, 648–649 Ibn Sina, 13-limit ratio, Fret 5, 667, interpretation of 671–674, 676–677 as length ratios on ud, 650–655 Safi Al-Din, Fret 5, Persian middle finger modern, in cents, 650 fret, Second Ud Tuning, 714–717 Jins 2, 676–677, 679–680 as tone or interval of a genus or mode Jins 8, 679–680 Al-Farabi, Jins 2, 651–654, 676–677, conjunct, three standard modes, 644–645 679–680 genera, soft genus and strong genus, 659–660 Ibn Sina classification of 15 tetrachords 11 Melodic Modes, 681–686 soft ordered consecutive and non- Diatonic Genus 7, 675–677 consecutive, 660–663 Safi Al-Din, 84 Melodic Modes, Second Ud strong doubling, strong conjunct, and Tuning, 724–725, 729 strong disjunct, 661–663 Ptolemy’s classifications in Greater Perfect System. See also Greek Al-Farabi, 658–659 tetrachords Ibn Sina, 678–679 three conjunct/disjunct systems for the Safi Al-Din, 711–713 distribution of, 662–666 emmelic/melodic intervals inversion of intervalic order B (baqiyyah or fadlah), 721–723, 740 Aristoxenus’ tetrachords, 648 J (tatimmah or mutammam), 721–723, Philolaus’ tetrachord, 648–649 740, 744 T (tanini), 721–723, 727, 740, 744 Al-Kindi’s ud, 612 pyknon, Al-Farabi, 659–663 description of, 613 “quarter-tone” Philolaus’ tetrachord, three harmoniai, 4 modern musical symbols of, 713–714 616–617 half-flat and half-sharp signs, Racy, 760 classification, modern koron and sori signs, Farhat, 640 9 fasail (genera): seven tetrachords, one Arabian theory trichord, and one pentachord, 762 Al-Farabi, origin and function of ratio CC|cx, 46 Maqamat, 763–766 354–355 Safi Al-Din, 84 Melodic Modes, Second Ud Ibn Sina, origin and function of ratio CN|cb, Tuning 679 9 modern maqamat, interval patterns Marcus identical to, 767–768 ambiguity of “quarter-tones” in musical 32 modern tetrachords and modes, practice, 760–761 traceable to, 763–768 description of Mashaqah’s twenty-four fasail (sing. fasilah, family or genus), maqamat “quarter” tones, 759 with similar lower tetrachords, 761–762 898 Index

Arabian tetrachords (Continued) Rameau, 434–436, 441–444 jins al-asl or jins al-jidh, primary lower of “octave,” ratio X|z, 91–93, 437 tetrachord, 761, 767–769 of “triple-octave,” ratio -|z, Rameau, 428–433 jins al-far, secondary upper tetrachord, 761, of harmonic series, 254–255 767–769 length ratios munfasil, disjunct, 767 of “double-octave and a fifth,” ratioN|z mutadakhil, overlapping, 767 Rameau, in context of dual-generator, muttasil, conjunct, 767 444–446 Ibn Sina Salinas, 402–403 11 Melodic Modes, 681–686 Stifel, 386–390 12 Dastgaha, Persian theory, modern, 692–696 Zarlino, 392, 394 16 tetrachords, based on strong and soft of “fifth,” ratioC|x , 325, 396, 437 genera, 676 Rameau, 442–444, 446–448 construction of, with epimore ratios, Safi Al-Din, 381 673–674 Stifel, 401 diatonic genus, preferred over chromatic and Zarlino, 377–378, 382–384, 385–386, genera, 673 399–401 Diatonic Genus 4 and 7, 13-limit ratios, of five tetrachords, Ptolemy, 328 674–677, 679–681 of “fourth,” ratio V|c, 297, 325 Rast or Mustaqim, 727–731 Ptolemy’s Even Diatonic, 329 modal origins of “octave,” ratio X|z, 87–89, 91, 295–296, 437 Arabian, on Al-Farabi’s and Ibn Sina’s ud s, ancient length ratios vs. frequency ratios, 747–754 87–91 Turkish, on Safi Al-Din First Ud, 732–733, Ibn Sina, 379–381 744–748 Ptolemy, 295–297, 324 Safi Al-Din Rameau, 431–432 84 Melodic Modes Zarlino, 382–383 First and Second Ud Tunings, 724–725 of “whole tone,” ratio =|, 35 modes, conjunct tetrachord/ Al-Farabi, 354–355, 636 pentachord combinations of, Cardan, 355 722–726 Galilei, Vincenzo, 355–356 49 modes, conjunct tetrachord Mersenne, 356 combinations of, 722–726 Ptolemy, 354, 636 of seven-by-twelve matrix system, 728–729 Zarlino, compound unities, 390–391 6 Awazat (secondary or derived), Aron, Pietro, 342, 345, 347, 352–353 730–733 Ayyar, C. Subrahmanya, 586 12 Shudud (primary or principle), 726–727, 730–733 B Turkish theory, modern, on First Ud Bach, Johann S., vi, 350 5 of 6 ‘Variant’ Maqamat, 744–745 Barbera, Andre, 82, 308 10 of 13 Basic Maqamat, 740–744 Barnes, John, 349–350 Archimedes, 265, 610 Bars, rods, and tubes Archytas, 86–89, 92, 280, 283, 288–289, 297, 311, antinode (clamped-free bar) 318 bending (BA), 175 Area moment of inertia. See Bars, rods, and tubes location of, first four modes (bar, rod, and Arel, H. Sadettin, 738 tube), 177 Aristoxenus, 280, 289, 308–321, 624–625, 642, displacement (DA), 175 646–649, 651, 655 local reduction at, tuning process, 176–177 Arithmetic divisions/progressions. See also Cents, antinode (free-free bar) harmonic series; Means bending (BA), 153–154 definition, 253–254 bending moment at, 155–157 Archytas, 86–87 local reduction at, tuning process, 160–163 frequency ratios location of, first five modes (bar, rod, and of “double-octave and a fifth,” ratioN|z , tube), 162 426–428 displacement (DA), 153 of “fifth,” ratioC|x , 437 experiment, 153–154 Index 899 area moment of inertia (bar, rod, and tube), 159 shear force, 154–158, 160–161 bending tuning effect on, 160 stiffness, 158 tuning process (clamped-free bar), 176–177 and bending moment, direct mass loading, bars vs. reeds, 178 proportionality, 159–160 tools and techniques, 177 effect on mode frequencies, 159 tuning process (free-free bar), 160–161 as restoring force, 148, 152 bass marimba bar tuning effect on, 160 frequencies, first three modes; before, wave speed, 149–150, 159 during, and after tuning, 165 and bending wavelength, inverse frequency changes proportionality, 149 eighteen analysis/decision steps, 168–170 and dispersion, 148–149 eighteen tuning steps, 165–168 effect on inharmonic frequencies, 148–149 length limitations of, 164–165 triple-arch design, bar profile, 161–163 experiment, 149, 153–154 tuning function of first three antinodes, stiffness effect on, 147–148 161–163 tuning effect on, 160 higher mode frequencies wavelength, 150 effect on pitch perception of fundamental, effect on bending moment, 154 163–164 experiment, 149 tuning possibilities of, 164 dispersion tools and techniques, 164 definition, 149 treble marimba bar variables, indicators of, 152–153 frequencies, first two modes; before and mode after tuning, 171 frequency (clamped-free bar), 174–175 length limitations of, 171 ratios of second, third, and fourth modes, single-arch design, bar profile, 171 174 tuning function of first two antinodes, frequency (free-free bar), 148–149, 151–152 170–171 and bending moment, 154 Bass Canon and height, direct proportionality, 158–160 construction of, 800, 805 ratios of second, third, and fourth modes, dimensions, 805 148 Plate 3, 847 theoretical vs. actual, rosewood test bar, Bass Marimba 163 24 bars of, frequency ratios, 827 tuning effect on, 160–161 dimensions of, longest and shortest bars, 826 shape (clamped-free bar), of first three modes, mode tuning of, 161–165 175 properties of, Honduras rosewood, 172–173, shape (free-free bar), of first three modes, 153 885–886 (clamped-free bar) C2 cavity resonator, Resonator II, 221 bending moment and shear force at, 175 dimensions local reduction at, tuning process, 176–177 inside, 217 locations of, first four modes (bar, rod, and outside, 220 tube), 177 tuning of, 219–222 node (free-free bar) dimensions, 826 bar mounting considerations, 163, 170–171 Plate 8, 852 locations of, first five modes (bar, rod, and Beat rates. See also Inharmonic strings tube), 162 of 12-TET, 136 shear force at, 155–157 “fifth,” 136 plane sections (bar, rod, and tube), 159 “fourth,” 137 radius of gyration (bar, rod, and tube), 159 “,” 136–137, 139–140, 146n.28 restoring forces (clamped-free bar), 154 beating phenomenon, 135–137, 365 bending moment, 175–176 effect on musical quality, 108, 118, 135 shear force, 175–176 limits of human perception, 136 tuning effect on, 176–177 , on the consonance of 7-limit ratios: M|b, restoring forces (free-free bar), 154 Z?|m, and M|v, 364–365 bending moment, 154–161 Ramamatya, description of dissonance on vina, effect on geometry of bar elements, 156 572 900 Index

Beat rates (Continued) Euclid, 76, 81–82 of shimmering “octave,” 509, 515 Plato, 91–92 Balinese gamelan Ptolemy, 78 gender rambat, 529 Canright, David R., v–vi, 101, 156, 877 penjorog (beat rate), 524 Plate 15, 859 Javanese gamelan Cardan, Jerome, 355 gender barung, 516, 519–522 Cents demung, 509–511 calculation of string adding and subtracting, 269 consonance vs. dissonance, 365 conversion flexible canon strings vs. stiff piano strings, cents to decimal ratio, 270 140–142 cents to integer ratio. See Euclidean small integer ratios vs. large integer ratios, algorithm 138–140 frequency ratio to cents, 268–269 harmonics Ellis, 1200th root of 2, 267 coincident, 136. intermediate, 137. increasing or decreasing frequency by cents, Beeckman, Isaac, 406 270–271 Benade, Arthur H., 126, 235, 247 interval comparison, 270 Bending moment. See Bars, rods, and tubes, definition, 268 restoring forces exponent Benedetti, Giovanni Battista, 415–416 definition, 257–258 Bernoulli, Daniel, 426 exponential function, 257 Bharata, 540–550, 552–562, 564, 572, 577, 598–599 as logarithmic function, three aspects, 258 Bhatkhande, V.N., 581, 587, 594–596 first law of, 259 Boehm, Theobald, 235 four laws of, 277n.11 Boethius, 284, 380 Stifel’s description, 276n.6 Break strength harmonic series definition, 35 as arithmetic progression, 254–255 of string human process of adding intervals in high-carbon spring steel music wire, 884 sequences, 260–265 plain, 35, 111, 134 no human perception of interval patterns, wound, 40 254 Brothers of Sincerity (Ikhwan al-Safa), 620–622 as geometric progression, 254–255 Brown, Robert E., 528 human perception of interval patterns, Brun, Viggo, 458 254–255 Buskirk, Cheryl M., viii mathematical process of multiplying frequencies in sequences, 256–262, C 263–265 Canons. See also Bass Canon; Harmonic/Melodic intervals of, verbal terms vs. mathematical Canon; Little Canon terms, 64–65, 256 Arabian , ancient instrument, 628 logarithm Abul-Salt, 628, 630–632 antilogarithm., 259, 262. Al-Farabi, 632 common Al-Jurjani, 629–631 base 1.00057779, 109, 267–268 hamila (moveable bridge), 628, 631–632 base 2, 262–264, 274 Ibn Sina, 670–672 base 10, 258–262 Ibn Zaila, 628 definition, 258 from qanun to ud, evolutionary process, 632, four laws of, 278n.13 669–670 guitar frets, 271–273 construction requirements and human hearing, adding process, 262 bridge, 65 logarithmic function, 258 original design, 792–794 as exponential function, three aspects, 258 string, 65, 118 musical slide rule, 273–275 function of, 77–78, 80, 792 Napier, inventor of, Greek logos + arithmos, Greek kanon 257 definitions, 65 natural, base e, 109, 258 Index 901

“octave” equivalence D definition, 257 D’Alembert, Jean le Rond, 423 as human trait, 257 Dattila, 545, 548–549, 558 ratio, simplification of, 257 de Fontanelle, Bernard Le Bovier, 438 Chalmers, John H., viii, 308 de Villiers, Christophe, 406 Cheve System D’Erlanger, Baron Rodolphe, 625, 666, 675, 755, cipher and cipher-dot notation, 508, 535n.2 758–760 letter-dot notation, 599 Descartes, Rene, 404 Chiao Yen-shou, 502 Diamond Marimba, 13-Limit 49 bars + 5 bars of, frequency ratios, 824–825 Ch’ien Lo-chih, 502 dimensions of, longest and shortest bars, 825 Ch’in (qin) mode tuning of, 171–172 construction of, 488–489 properties of, Honduras rosewood, 172–173, as microtonal instrument, 496 885–886 Chinese scales. See Scales, Chinese construction of, 825–826 Ching Fang, 502 dimensions, 825 Chrysalis expansion of construction of, 788–789 Meyer’s 7-limit Tonality Diamond, 824 dimensions, 790 Partch’s 11-limit Diamond Marimba, 824 Plate 1, 845 Plate 7, 851 Plate 13, 857 resonator score, excerpt from Song of Myself, 800–802 airtight seal, making, 211

stringing and tuning of, for Song of Myself, dimensions of lowest, G3 at 196.0 cps, 198 794–797 Didymus, 280, 319, 329–332, 664 Chrysalis Foundation, 839, 841, 843 Dieterici, Friedrich, 620 New Music Studio, 842–843 Dispersion Workshop, Plate 16, 860 definition, 104, 149 Chu Hsi, 492 variables, indicators of, 152–153 Chu Tsai-yü, 354, 502–504, 542, 569, 749 Downbearing force Chuquet, Nicolas, 339–341 of string on canon bridge, 792 Cleonides, 309–310, 649, 655 string’s angle of deflection, 794 Cohen, H.F., 139 Comma of Didymus. See Greek ratios, comma E Comma of Pythagoras. See Arabian ratios; Greek el-Hefni, Mahmud, 612 Ellis, Alexander J., 265, 267 ratios, comma End corrections. See Flutes, corrections; Compression and rarefaction Resonators, cavity; Resonators, tube alternating regions Equal temperaments of plane wave, in solid, liquid, gas, 186–187 12-TET, 361–362 of sound wave, 183–184 !|1@ ditonic comma of bar-to-cavity resonator coupling, 214, 222 calculation of, 352–353 of bar-to-tube resonator coupling, 196 cumulative reduction of twelve consecutive definition, 130–131, 183, 200 “fifths,” 353–354 dependence on wave speed, 131 cent comparisons in infinite tube, 200–203 to 3-limit, 5-limit, 7-limit, and !|$-Comma in resonator Meantone scales, 376–377 cavity, 213, 219 Ramis vs. Stevin, 375–376 tube, 190–196, 203–208 discovery of of soundboard, inefficient regions, 133 Chu Tsai-yü, 502–504 of string, leading and trailing surfaces, 130–131, Stevin, 358–362 183–184 ditonic comma (comma of Pythagoras), Conductivity. See Flutes, admittance definition, 335, 350 Cremer, L., 123, 131 “fifths” Cristofori, Bartolomeo, 118, 130 circle of 12, That produces, 352–353 Crocker, Richard L., 308 spiral of 12 flat “fifths,” That closely Crusoe, Robinson, 788 approximates, Liu An, 500–501 902 Index

Equal temperaments (Continued) Eratosthenes, 280, 318, 329–332 frequencies of eight octaves, 879 Euclid, 76, 81–82, 86, 92–93, 280, 283, 287–289, frequency ratios, powers of the 12th root of 2, 297–298, 302–308, 311, 318, 337–339, 341, 271 610–611, 619, 624–625, 759 fret equations for, 272–273 Euclidean algorithm on fretted lutes and viols, 340, 348 12-TET, eleven integer ratio approximations, “major third” of, two integer ratio 459 approximations, 140 Brun, 458 “semitone” of and continued , 484n.364 approximate value, rational, ratio Z-|zm for converting irrational decimal ratio to rational Al-Farabi, 354–355 integer ratio approximation, 458–459 Cardan, 355 Exponent. See Cents lute fretting instructions Galilei, 355–356 F Mersenne, 356–359 Farhat, Hormoz, 677, 686–696, 744, 759, 771–772 lutenists’ preference over exact value, Farmer, Henry George, 366, 379, 401, 610, 619–621, 357 623–628, 635, 668, 709, 716, 727 origin of, Ptolemy’s arithmetic division Flageolet tones of “whole tone,” ratio =|, 354 of ch’in, 488–492 exact value, irrational, 353–354 experiments 12th root of 2, 353–354 Roberts, 421 Chu Tsai-yü, 503 Sauveur, 424–426 Stevin, 360–361 of trumpet marine, 408–409 vs. eleven integer ratio approximations, to the Fletcher, Harvey, 99–100, 113, 135 nearest cent, 459 Flexible strings vs. frequencies of the harmonic series, 63–64 antinode 22-TET, Indian theory, ancient definition, 53, 55 Bharata formation of, as regions of maximum motion, 22-sruti scale, 540–543, 544–547 47–48, 51–54 logarithmic pramana sruti, 541–542 locations of, first six modes, 58 no basis in fact, 542–543, 572–573 equations, for stopped (bridged) string 24-TET, Arabian theory, modern, 755–758 frequency, 77 formulaic imposition of, 749, 755–758 length, 77 on fretless ud, as moot issue, 760–761 equilibrium position, dynamic, 48–49 Marcus frequency. See also Flexible strings, mode ambiguity of “quarter tones” in musical definition, 55, 57 practice, 760–761 dimensional analysis, 55 description of Mashaqah’s twenty-four as function of wavelength, 80 “quarter” tones, 759 harmonic series used to justify “incorporation” of “quarter definition tones,” D’Erlanger and Marcus, 755–758, mathematical: length ratios of subdividing 760–671 string, 63–64, 427–428 31-TET verbal: frequency ratios of subdividing Huygens, logarithmic calculations, 362–364 string, 63–64, 427–428 spiral of 31 “fifths,” That closely discovery of approximates, 362–363 infinite series, first thirty-two harmonics, 53-TET Sauveur, 423–424 spiral of 53 “fifths,” That closely limited series, first six harmonics, approximates, Ching Fang, 502 Mersenne, 404–408 Turkish theory, modern, close approximations first sixteen harmonics, two series, 64 of 4-, 5-, 8-, 9-, and 12-comma intervals, intervals of, verbal vs. mathematical, 64–65, 736–737 256 60-TET, spiral of 60 “fifths,” That closely harmonics approximates, Chiao Yen-shou, 502 definition, 63–64 360-TET, spiral of 360 “fifths,” That closely first six on bridged canon string, 66 approximates, Ch’ien Lo-chih, 502 first sixteen of subdividing string, 64 Index 903

frequency ratio, 73 of particle motion, in cord/string, 50–51 ideal characteristics, 44 superposition impedance, mechanical wave, 121 definition, 48 loop interference count (mode number), 59, 105 constructive, 47–48 definition, 59 destructive, 48–49 multiple loop length, 70 , production of, 48, 51–54 ratio, 73 wave pattern, 59 standing single loop length, 59–60 definition, 2, 48, 51–52 ratio, 73 of first six modes, 58 mode motion of, 52–53 definition, 98 and sound production, 53–54 frequency transverse traveling definition, 59, 62 in cord, 50–51 and mode wavelength, inverse definition, 50, 183 proportionality, 61 in string, 2–3, 44–53, 58 shape, first six modes, 58 superposition of, 51–54 node wave speed, transverse traveling, 60–61 definition, 53, 55 wave train (waveform) formation of, as points of minimum motion, definition, 50–51 48–49, 51–54 frequency of, 55, 57 function of, 53, 116n.5 period of, 54, 56 locations of, first six modes, 58 as transverse traveling wave, 50–51 two different kinds, 55–56 wavelength period definition, 54 definition, 54, 56 equations, 60, 71 dimensional analysis, 54 of first six modes, 58 pulse, transverse traveling as function of frequency, 80 crest and trough, 44–46 Flutes collision, 46–49 admittance reflection, 45–46 complex/non-complex, 233 definition, 44 definition, 233 and Pythagoras, 44 terminating, 234 ratios. See also Ancient length ratios and cutoff frequency, 247 frequency of embouchure hole and flute bore, 235 definition, 62, 72 of tone hole, flute bore, and tube-piece, 236 as function of length ratio, 80, 93 conductivity of harmonics, 73 definition, 233 and length ratio, inverse proportionality, of duct, fictitious and actual, 234–236 63, 80 of embouchure hole and flute bore, 234 of non-harmonic tone, 73 of tone hole diameter, limitations, 233–234 interval corrections, 232 definition, 67 at embouchure hole, 228, 231–232 of harmonics, right sides of bridged canon empirical data strings, 68–69 concert flute, 235 of non-harmonic tone, 74 simple flute, 240 length at key pad, 228 complementary, 69, 78 Nederveen and Benade, 237 definition, 59, 69–70 at open tube end (end correction), 228, and frequency ratio, inverse proportionality, 231–233 63, 80 at tone hole, 228, 231–232, 236, 244–245 human comprehension of, 79, 80, 93 Nederveen, 238 of non-harmonic tone, 73 cutoff frequency, Benade, 247 simple harmonic motion (SHM) embouchure hole, typical dimensions definition, 50 concert flute, 235 904 Index

Flutes (Continued) of tension, 18–19 simple flute, 240 unit, 10, 14–15 frequency Forster when known, predicting flute dimension and Cris, v–viii, 839–844 tone hole location, 229–241 Plate 13, 857 when unknown, determining from flute Plate 15, 859 dimension and tone hole location, Heidi, viii, 839–844 242–246 impedance, acoustic, non-complex, 233 Plate 14, 858 intonation, 227 Plate 15, 859 at embouchure hole Forster instruments. See Bass Canon; Bass airstream, 227, 240 Marimba; Chrysalis; Diamond Marimba, blowing technique, 236 13-Limit; Glassdance; Harmonic/Melodic lip coverage, 227, 233, 240 Canon; Just Keys; Little Canon; Simple Flutes flute tube effect on, 235–236 Franklin, Benjamin, 828 making, simple flute, 236, 246–248 Frequency critical variables of bar embouchure hole correction, 240 clamped-free, 174–175 flute wall thickness, effect on timbre, 247 mass loaded, 178 musical interval, Nederveen, 237–238 free-free, 148, 151–152, 158 tone hole diameter, 239, 240–242 speed of sound in, 229 definition, 54–55 subharmonic series, 246 dimensional analysis, 55 standing wave, 231, 234, 245, 248n.10 of longitudinal mode: in plain string, uniform substitution tube, Nederveen, 229–230 bar, and fluid in tube, 187 two different kinds, 230–232 of resonator tone hole cavity, 215–217 dimension and location tube concert flute, 239 closed, 210 simple flutes, 241 closed-closed, 230 from embouchure hole, strategy for length open, 209 calculation, 231–232 of spring-mass system, 185, 219, 223n.9 tube length of string closed-closed (substitution tube), acoustic and flexible, 58–62, 102–103 effective, 230–232 stiff, 99, 105–107, 112 effective and wavelength at embouchure hole, approximate, 234–235 as function of, 80–93 Nederveen, 233 inverse proportionality, 61–62 at tone hole, exact, 236 Frequency ratios. See Flexible strings, ratios Nederveen, 237 Freud, Sigmund, 557 open, measured vs. effective, 229–231 tuning process, simple flute, 236 G variables, twenty-seven symbols, 228–229 Gaffurio, Franchino, 367 Force. See also Restoring force; Weight; Weight Galilei density Galileo, 283, 338, 356 definition, 5–6, 8 Vincenzo, 315, 355–356, 358 English Engineering System, 9, 12, 16 Gamelan. See also Scales, Indonesian pounds-force (1 lbf), 7–9 instruments inconsistent system, 9–10 Bali English Gravitational System, 9 bar percussion instruments, built in pairs: pounds-force (1 lbf), 10. pengumbang (low) and pengisep (high), consistent system, 9–11, 14–15 524, 532–533 experiment, 7–8, 10, 15 penjorog (beat rate) of, 524 of gravity, 6–8, 10–12 shimmering “octaves” of, tuned of muscular effort, 5, 8, 16 sharp, 524 Newton, 2, 5 sharp and flat, 529–530 of spring, 11 (saron), 533 Index 905

gender Euclid’s mean proportional, 297–298 dasa, 523 Zarlino, 338–339 jegogan, 524 of “whole tone,” ratio =|, rambat, 529–530, 532 Ptolemy’s rational approximation, 320 tjalung (jublag), 530–534 Zarlino, 338–339 wayang, 522 Ghosh, M., 549, 556–557, 559 gambuh, 524–528, 530, 532 Glassdance trompong, 523–524, 527–528, 532 48 crystal glasses, frequency ratios, 830–831 Java Ptolemy’s Soft Diatonic, 832 bar percussion instruments, beat rates construction of, 828–830 four different kinds, 522 dimensions, 832 of tumbuk tuning, 519–522 Plates 9, 10, and 14, 853–854, 858 gender tuning process, of brandy snifter glasses, barung, 513–517, 519–521 830–831 panerus, 519 Govinda, 565, 574–576, 578–584 saron Greek ratios demung, 509–511, 517–520 classification ritjik (saron barung), 510–511 epimere, 285, 287 shimmering “octaves,” tuned epimore, 284, 287 sharp, 509–511 Euclid, 287 sharp and flat, 515–516, 519–521 Ptolemy, 288, 321–323, 325–326, 328–329, orchestras, 508 354 Bali Pythagoreans, 288, 323 Court of Tabanan, 526 Zarlino, 397 Kuta Village, 523 equal, 284 Pliatan Village, 524, 530–534 Ptolemy, 324 Puri Agung Gianyar, 529–530 multiple, 284, 286 Tampak Gangsal, 523, 527 Euclid, 287 Java Mersenne, 417 Kangjeng Kyahi Sirat Madu, 520 Ptolemy, 288, 321, 323 Kyahi Kanjutmesem, 509–510, 515–516, Pythagoreans, 288 519–522 multiple-epimere, 285–286 Geometric divisions/progressions. See also Cents, Ptolemy, 288, 319 harmonic series Pythagoreans, 288, 319 definition, 254 multiple-epimore, 284–285 Archytas, 86–87 comma of harmonic series, 254–255 of Didymus (syntonic comma), 315, 350, 363, length ratios 400, 547, 697 of 12-TET Aron, 344–345 Chu Tsai-yü’s solution, 503–504, 542 definition, 344 Stevin’s solution, 358–362, 542 Ramamatya, 572 of “fifth,” ratioC|x , Chuquet, 339–341 Safi Al-Din, 372–374 of “fourth,” ratio V|c of Pythagoras (ditonic comma), 319–320, 487, Aristoxenus, 309–318 542 Al-Farabi’s interpretations of, especially 12-TET, 352–354 Jins 8, 646–655 Al-Farabi, 655 arithmetic-geometric asymptote, tunbur of Khurasan and original tunbur, 313–315, 318 697–706 modern interpretations of, 309–313, Al-Jurjani, 722–723 466n.75, 655 Al-Kindi, 334, 611, 615–616, 620 Ptolemy’s interpretation of, 316–317 Aristoxenus, 314, 318 of “major third,” ratio B|v; meantone ratio, definition, 314–315, 334–335 342–343 Ramamatya, 571–572 of “minor third,” ratio N|b, Chuquet, 339–341 Safi Al-Din, 372–374 of “octave,” ratio X|z, 297–299 Turkish theory, modern, 736–737 Chuquet, 399–341 Werckmeister, 350–351 906 Index

Greek ratios (Continued) Greek scales. See Scales, Greek Euclid, 76, 81–82, 287 Greek tetrachords interval Aristoxenus, six different kinds, 309–310 apotome, 314–315, 319 fractional and integer parts of tones, 309–310 ‘Apotome Scale’, 336, 418–419 interpretation of Ptolemy, 314–315, 319 arithmetic-geometric asymptote, Ramamatya, 571 313–315, 318 apyknon, 326–327, 659–660 modern, in cents, 310 Al-Farabi, 660–663 Ptolemy, as length ratios, 313, 316–317, Ptolemy, 328–329, 334 320–321 , modern, 464n.48 conjunct, Terpander’s heptachord, 289, 291–293 Aristoxenus, 314–315, 318 definition, 289 Philolaus, 299–300 disjunct, Pythagoras’ octachord, 289, 293 diplasios, 81, 86, 286–287 genera of; diatonic, chromatic, and enharmonic, diapason, 81, 407, 416 disdiapason, 407, 410 289–290 ditonon, 384 Archytas, 289 , 314, 416, 633–634 Aristoxenus, 289, 309–310 epi prefix, 81, 384 Ptolemy, 326–332 epitetartos (epitetartic), 286–288, 657 Greater Perfect System (GPS) epitritos (epitritic), 81–82, 86, 286–287 composed of four different kinds, 289–292 diatessaron, 81, 384 diatonic genus, in context of epogdoos (epiogdoos), 81–82, 86, 287 Euclid, 303–304 tonon, 82, 384 Philolaus, 301 hemiolios, 81–82, 86, 287 Ptolemy, 333 diapente, 81, 382, 384, 412–413 Euclid, description of 15-tone “double- limma, 300, 314–315, 319 octave,” 289, 302–303 Aristoxenus, 315–316 harmoniai (modes) of, seven different kinds, ‘Limma Scale’, 336 290–292 Ptolemy, 314–315, 319–320 standard “octave” of Ramamatya, 571 , Greek, 301 pyknon, 326–327, 659–660 Lydian Mode, Western, 301 Al-Farabi, 660–663 three scales, based on three genera, 292 Didymus, 329–330 Lesser Perfect System (LPS), composed of three Eratosthenes, 229–330 different kinds, 289–291 Ptolemy, 326–329 Philolaus schisma, 372–373 Diatonic, ancient and original division, “schismatic fifth,” 372–375 299–301, 462n.17, 464n.47 Nicomachus on Al-Kindi’s ud, 615–617 classification, translated as Latin ratios, 486 in Euclid, 302–307 dimoirou (two-thirds), 84, 96n.30 plagiarized by Plato, 91–92 hemiseias (half), 84, 96n.30 Ptolemy sound (frequency) and string length, inverse Catalog of Scales, 318–319, 330–332 proportionality, 84–85 Even Diatonic, origin of Arabian “neutral “vibration ratio,” 85 second” and “neutral third” tetrachords, “weight ratio,” 82–84, 86 329, 679–681 Plato, 91–92 pyknon and apyknon, three conceptual Ptolemy. See also Arabian ratios principles for the division of tetrachords, classification, 323 326–330 emmelic/melodic (epimores smaller than Soft Diatonic, unique musical quality of a V|c), 288–289, 323–326 minor scale, 334, 832 homophonic (multiples), 295, 323–324 Tense Diatonic, origin of Western major scale, symphonic (first two epimores, C|x and V|c), in Zarlino, 333, 832 322–326 theory of graduated consonance, “quarter-tone,” origin and function of ratios CX|cz mathematical/musical basis for the and CZ|c/, 290–292 division of tetrachords, 321–326 Index 907

Guitar frets. See Cents, logarithm soundboard Gullette, Will, viii, 845–856, 858–860 critical frequency, 132–133 Gupta, Abhinava, 550 dimensions, typical surface area, 129 H thickness, 125 Harmonic divisions/progressions. See also Means impedance definition, Archytas, 86–87 plate vs. soundboard, 124, 126 frequency ratios radiation, of air at soundboard, 129 of “fifth,” ratioC|x , 437 sound radiation, 133

Rameau, 435 string, D4 (d') of “octave,” ratio X|z, 91–93, 437 dimensions and tension, typical, 125 harmonic series, context of, 63–64, 80, 427–428 Hubbard, 144n.10 length ratios impedance, 125–126 of “double-octave and a fifth,” ratioN|z , Harrison, Lou, v 426–427 Heath, Thomas L., 308 Rameau, in context of dual-generator, Helmholtz, Hermann, 212 444–446 resonator. See Resonators, cavity Salinas, 403 Hitti, Philip K., 611 Stifel, 378, 386–390 Hood, Mantle, 512, 517–519, 521, 524 Zarlino, 378, 392–395 Hsu Li-sun, 486, 492 of “fifth,” ratioC|x , 397, 437 Huang Ti, 485 Rameau, 440–443, 446–448 Hubbard, Frank, 125 Safi Al-Din, 381 Huygens, Christiaan, 289, 362–365 Stifel, 401 Zarlino, 377–378, 382–383, 386, 399–401 I of “octave,” ratio X|z, 89–91, 437 Ibn Al-Munajjim, 619–624, 644–645 ancient length ratios vs. frequency ratios, Ibn Khallikan, 619 87–91 Ibn Misjah, 619 Ibn Sina, 379–381 Ibn Sina, 93, 326, 366, 378–380, 384, 435, 610, 614, Rameau, 431–432 625, 628, 636, 666–689, 692–696, 703, 715–716, Zarlino, 382–383 722, 727, 744, 749–755, 758–759 of “triple-octave,” ratio -|z, Rameau, 430, Ibn Suraij, 619–620 Ibn Zaila, Al-Husain, 628 432–433 Ikhwan al-Safa (Brothers of Sincerity), 620–622 Zarlino, sonorous quantities, 390–391 Impedance Harmonic/Melodic Canon. See also Canons acoustic construction of, 790–792 complex, 197 bridge, original design, 792–794 non-complex, 197 dimensions, 794 of duct, 233 Plate 2, 846 of room, 199 score, excerpt from Song of Myself, 800, 803–804 of tube, 198 stringing and tuning of, for Song of Myself, definition, 120, 127, 129, 197, 233 794–795, 798–799 mechanical Harmonic series. See Cents; Flexible strings; Just complex, 120–121, 124 intonation radiation impedance, 129 Harmonics. See Flexible strings; Inharmonic of soundboard, vs. infinite plate, 122–124 strings, tone quality; Just intonation non-complex Harpsichords. See also Pianos, vs. harpsichords of air at soundboard (radiation impedance), Huygens, keyboard for 31-TET, 363–364 129 impedance ratio of plate, 123 air-to-soundboard, 129 of string, 121 soundboard-to-string, 126 specific acoustic inharmonicity complex, 127 analysis of, 110–111 non-complex, 127–128 coefficients of, plain strings, 111, 126 of air (characteristic impedance), 128–129 908 Index

Indian musical terms. See also Scales, Indian Matanga, 556 ancient 4 suddharagas (pure ragas), 562–563 Bharata, 540 raga (elements of melodic sound), first grama (tone-system, tuning, or parent formal definition, 561–562 scale), 547 Narada, 559 jati (melodic mode), three different types 7 gramaragas, first complete compilation, (1) samsarga (combined), 549–550 559 (2) suddha (pure), 548–550, 552–554 gramaraga (3) vikrta (modified), 548–550, 552, first formal definition, 560 555–556 or gana (song), 559–561, 563–564 4 hierarchical functions of tones: graha, guna (qualities) of, 560 amsa, apanyasa, nyasa, 548, 552, Sarngadeva, 548 gramaraga, 562–564 560, 576, 602n.30, 691 suddha jatis and laksana, definition, 548 laksana (properties) of, 547–548, 552, 554–555, 558–559, 602n.30 suddha, 566 possible influence on, 556–557 vikrta, 566 dastgah, 691 North India maqam, 738 Bhatkhande, 587 patet, 512 10 most popular Thats, 581, 596 rasa (qualities) of, 547, 557–560 Bilaval That, modern suddha scale, 592–593, raga (charm) of a note, 560–561 600 songs, of the theater Narayana and Ahobala, 587 dhruva, 558–560 12-tone vina tuning, 587–592 gana, spurious text, 559–560 samvaditva (consonant intervals), characteristics of, 560 590–592 gitaka, 558 suddha svaras (pure notes), 587 sruti (interval) vikrta svaras (modified notes), 590 of 22-sruti scale, ratio analysis of Sa- raga, 558, 564, 576, 585 grama and Ma-grama, 544–547, Roy, 595 550–551 32 Thats, of eight-by-four matrix system, definition, 540 595–596 of -vina tuning experiment, 541, chromatic Thats, not used, 593–594 543–544 sitar of jatis, 554–555 construction of, 597 pramana (typical), 541–543, 547 melodic, drone, and cikari strings, of Sa-grama and Ma-grama, 540–541, 597–599 544–545 moveable frets, 597–598 svaras (notes) definition, Sachs, 597 7 svaras of Sa-grama and Ma-grama, letter-dot notation, based on Cheve 540–541 System, 598–599 of 22-sruti scale, ratio analysis of Sa- Shankar’s mastery of String II, Junius, 597 grama and Ma-grama, 544–547 tuning of, example, 598 definition, 540 That suddha (pure), 566 definition of svarasadharana (overlapping note), Bhatkhande, 594 ratio analysis of An and Ka, Kaufmann, 594 549–551 vs. modern mela of South India, 594 vina (harp-vina), 541–544, 547, 550–551, South India 553, 564 Govinda, 565 gramaragas mela raga definition, 562 14 most popular mela ragas, 581 and laksana 72 abstract raga-categories, 565 of Kudimiyamalai inscription, 562–564 72 musical heptatonic scales, first formal of Matanga, 562–564 implementation of, 565 of Narada, 562–564 krama (straight) or vakra (zigzag) of Sarngadeva, 562–564 sequences, 565 Index 909

two-directional krama (straight) Venkatamakhi’s avoidance of, Iyer, sampurna (complete) principle, 582–583 579–582 denominative raga, definition, 574 alphanumeric prefixes, 580 example of, Mela Gaula and Raga Gaula, Kanakangi-Ratnangi system, 580–581 574 critique of janya (born, or derived) raga Iyer, 582–584 definition, 574 K.V. Ramachandran, 584–585 N.S. Ramachandran, 580 laksana properties, absence of, 576 Mela musical raga, 564–565 Chakravakam, 580–581 Raga Gaula, 574–575 Kharaharapriya, 574, 581 melakartas Mayamalavagaula, 574, 580–581 14 most popular mela ragas, 581 Natabhairavi, 576, 581 abstract raga-categories, 565 vs. Venkatamakhi’s melakartas, 580 definition, 577–579 raga, 558, 564, 576, 585 Mela Ramamatya, 565 Bhairavi, 576, 579, 581 12-tone practical scale, suddha mela vina Gaula, 579, 581 tuning, 569–573 Malavagaula, 575, 579, 581 Antara Ga and Kakali Ni, elimination of, Sri-raga, 575, 579, 581 572 14-tone theoretical scale, 566–567 Indian scales. See Scales, Indian 7 suddha svaras, 566–568 Indian tetrachords. See also Scales, Indian 7 vikrta svaras, 566–568 North Indian theory denominative raga, definition, 574 Bhatkhande, 587 example of, Mela Sri-raga and Sri-raga, 10 most popular Thats, 581, 596 574–576 Roy, 595 janaka (parent) raga, abstract raga- 32 Thats, of eight-by-four matrix system, category, 574 595–596 janya (born, or derived) raga chromatic Thats, not used, 593–594 definition, 574 South Indian theory laksana properties, absence of, 574, 576 Govinda, 565 musical raga, 565 72 musical heptatonic scales, first formal Raga Mechabauli, 574–575 implementation of, 565 Raga Suddhabairavi, 574–575 Kanakangi-Ratnangi system, 580–581 Sri-raga, 575–576 Venkatamakhi, 565 modern Sadjagrama, Rao, 576 72 Melakartas, of twelve-by-six matrix system, 565, 576–579, 581 anya (foreign note), 575–576 36 vivadi melas, chromatic melas, not vakra (note out of order), 575 used, 579, 582, 593–594, 607n.119 varjya (omitted note), 575 Indonesian scales. See Gamelan; Scales, Indonesian mela (unifier) ragas Ingard, K. Uno, 149 abstract raga-category, 565 Inharmonic strings definition, 574 coefficient of inharmonicity Mela Malavagaula, 574–575 diameter effect on, 110, 118, 134–135, 808–809 Mela Sri-raga, 574–576 frequency effect on, 110, 809 vs. Venkatamakhi’s melakartas, 578 length effect on, 110, 809 Venkatamakhi, 565 steel stringing constants, logarithmic 12-tone scale; new and modern svara plain string, 110 names; ratios by Ramamatya, 576–577 wound string, 115 72 Melakartas, of twelve-by-six matrix of string system, 565, 577–579, 581 harpsichord, average values, 111 19 melakartas, most popular in piano, average values, 110–111 Venkatamakhi’s time, 577–579 piano vs. harpsichord, 126 36 vivadi (dissonant) melas, or chromatic plain, 110–111 melas, not used, 579, 582, 593–594, wound, 114–115 607n.119 tension effect on, 110–111 910 Index

Inharmonic strings (Continued) perception of, dissonance effect, 108, 118, 135, dispersion 138–141 definition, 104 thick string vs. thin string, 104, 118, 135, and wave speed, effective, 104–107 141–142 inharmonicity wave speed, effective, 106 analysis of, stringing and restringing, 110–111, and dispersion, 104–105 118 experiment, 104 cent calculation, strings and modes, 108–112 mode number effect on, 105–106 length, effective stiffness parameter effect on, 106 Inharmonicity. See Inharmonic strings and effective wavelength, 99, 101–102 Interference stiffness parameter effect on, 101–102 definition, 48 vs. measured length, 101–102 in infinite tube, constructive and destructive, mode 200–203 cents equation, 110 in string definition, 98 constructive, 47–48, 51–52 frequency equations, 99 destructive, 48–49, 51–52 shape, first four modes, 101–102 Interval ratios. See Flexible strings, ratios node, two different kinds, 101–102 Isothermal bulk modulus, liquids, 188 piano tuning, 118, 135–137 Iyer, T.L. Venkatarama, 582 stretched octaves in bass, tuned flat, 112–114 J in treble, tuned sharp, 111–112 Jairazbhoy, N.A., 541, 594, 596 restoring force Junius, Manfred M., 597 experiment, 104 Just intonation tension and stiffness, composite force, 104–105 commensurable numbers of, 365–366 stiffness effect Galileo, 283 experiment, 104 definitions, two different kinds, 365 on mode Mersenne cents, 110 harmonic-diatonic system, 404–405, 408, 436 frequencies, 98 harmonic series shape, 101–102 as infinite series, not defined, 404–406, 420 on wave speed, effective, 99, 104–106 as limited series on wavelength, effective, 99, 101–102 discovery of the first six harmonics, stiffness parameter 404–408 diameter effect on, 100 influenced by frequency effect on, 100 personal convictions, religious and length effect on, 100, 105 moral, 407–408 steel stringing constant Ptolemy’s Tense Diatonic, 407 plain string, 100 Zarlino’s Senario, 407 wound string, 114 harmonics of string first 14 trumpet harmonics vs. length ratios plain, 99–100, 111 of “old” trumpet marine, 410–412 wound, 113 “leaps” of natural trumpet vs. harmonics of tension effect on, 99–100, 111 monochord, four identical ratios, 406 tone quality. See also Beat rates “leaps” of trumpet marine vs. harmonics of beat rate, 108, 135–136, 138, 140 monochord, five identical ratios, 406 of harmonics length ratios of “new” trumpet marine vs. coincident, 136. “old” trumpet marine, 410–415 flexible canon strings, 141–142 with prime numbers greater than 5, not stiff piano strings, 140–141 acknowledged, 408, 412, 415 intermediate, 137. problems comprehending cause of; no of keyboard instrument knowledge of traveling waves, 404, Benade, 126 422–423 Fletcher, 135 solved by D’Alembert, 423 mode number effect on, 140–142 three different terms for, 404 Index 911

theory of consonance subharmonic series, Rameau’s recantation based on Benedetti’s observations, of, 446 mechanical motions of strings, 415–416 Generation harmonique (1737) contemplated ratios vs. endorsed ratios, 7-limit ratios, rejection of, 440 416–419 generator (dual-generator), function of, 7-limit ratios 444–446 detailed analysis and conflicted harmonic progression of fractional string rejection of, 417–420 lengths, modern length ratios, 440–443 qualified acceptance of ratios-|m and and Zarlino’s ancient length ratios, same M|n, 420 definition of major tonality, 441 origin of, Ptolemy’s theory of graduated harmonic series consonance, 321–326 consonances produced by nature, Meyer 440–441 7-limit Tonality Diamond, original design, major tonality, directly related to 448–452, 824 fundamental, 441 8 of, 449–451 minor tonality, not directly related to 13-tone scale of, 449 fundamental, 441–446 minor tonality as inversion of major subharmonic series, fallacy of, 441–446 tonality, 452 Nouveau systeme... (1726) moveable boundaries of, 451–452 harmonic generation, theory of, 436 origins of harmonic series in Rameau’s dual-generator, 444–446, hierarchy of consonances, 438 448–449, 452 “major third,” directly related to in triangular tables of fundamental, 438 Al-Jurjani, 401–402, 448 “minor third,” not directly related to Salinas, 402–403, 448, 452 fundamental, 439–440 “table of spans,” original description, 451 of single strings, 436–438 on 11-limit ratios, 448 Mersenne and Sauveur, acknowledged, plagiarized by Partch, 452–453 436 Partch Traite de l’harmonie (1722) 11-limit Diamond Marimba 7-limit ratios, rejection of, 434 36 bars of, frequency ratios, 452–455 arithmetic progression of vibration minor tonality as inversion of major numbers, frequency ratios, 428–434 tonality, 454 vs. Zarlino’s ancient length ratios, Otonality and Utonality, theory of, 453–454 diametrically opposed descriptions 43-tone scale, 454–457 of tonalities, 7 ratio pairs, to fill chromatic gaps of 434–436 11-limit Tonality Diamond, 454–455 harmonic generation, theory of, 428–429, 29 ratios, 67% of 11-limit Tonality 434 Diamond, 454–455 harmonic progression tuning lattice of, Wilson, 455–457 of fractional string lengths (length Meyer’s 7-limit Tonality Diamond, three ratios) changes to, 452–453, 824 knowledge of, 433–434 Monophony, theory of, 453 not included, 431 Zarlino’s philosophy of Unita (Unity), 429, of Stifel’s string length integers (length 484n.359 ratios), 428–430, 432–434 Rameau’s dual-generator, 444–446, 454 least common multiple (LCM), of length Rameau numbers, 432–433 Demonstration du principe... (1750) Zarlino’s minor tonality philosophy of Unita (Unity), 429–431 attempted to rationalized, 446–447 Senario, 434, 440 failed to rationalize, 447 Ramis as inversion of major tonality, Shirlaw, 5-limit ratios, four different kinds, 366–367 448, 452 12-tone scale, 367–373 ‘relative’ minor, origin of, 447–448 with a default schismatic fifth, 372–375 912 Index

Just intonation (Continued) Zarlino vs. Safi Al-Din’s scale, with six schisma major and minor tonalities, origins of, variants, 373–375 377–378, 391–397, 399–401 vs. Stevin’s 12-TET, 375–377 arithmetic and harmonic divisions of Roberts “fifth,” ratioC|x , 399–400 harmonic series Stifel’s arithmetic and harmonic divisions as infinite series of ratio N|z, 392–397 of discrete flageolet tones, 421 numero Senario (Number Series 1–6), of natural trumpet tones, 421 377–378, 381–384, 390–397 as series of simultaneously sounding rationalized consonance of “major sixth” harmonics, not defined, 421–422 and “minor sixth,” 397–399 Western theory of consonance, modern, harmonics, with prime numbers greater That 400–401 5, considered defective, 421 philosophy of Unita (Unity), corporeal or Salinas spiritual manifestations, 429, 484n.359 7-limit ratios, rejection of, 403 Just Keys harmonic division of ratio N|z, 403 Plate 6, 850 triangular table, 402–403 score, “The Letter,” from Ellis Island/Angel arithmetic division of ratio N|z, 402–403 Island origin of, Al-Jurjani, 401–402 description of, 811, 824 Zarlino’s Senario, 401–403 score, 812–817 Sauveur traditional score, 818–823 ‘acoustique’, science of sound, 422 tuning of, 808–811 discoveries on vibrating strings ‘noeuds’ (nodes) and ‘ventres’ (antinodes), K 422 Kanon. See Canons significance to other vibrating systems, Kasyapa, 562 426 Kaufmann, Walter, 490, 578, 594 simultaneous subdivisions/mode Kepler, Johannes, 336, 418 frequencies, 422–423 Khan, Vilayat, 599 ‘sons harmoniques’, consistent with Kitharas intervals of harmonic divisions and description harmonic series, 422, 427–428 Maas and Snyder, 618–619 standing wave of fifth harmonic, 424–426 Sachs, 617–618 harmonic series, as an infinite series tension, only variable after stringing, 323 discovery of, 422–424 tuning limitations historic significance of, 426–427 four significant structural problems, 617–619 no knowledge of traveling waves, 423 GPS not tunable, 619 first thirty-two harmonics of strings and of gut strings, 618 wind instruments, 423–424 Krishan, Gopal, 599 scales Kudimiyamalai inscription, 561–564 3-, 5-, and 7-limit scales vs. !|$-Comma Meantone and 12-TET scales, 376–377 L prime number limit of, 282. See also Limit of Lachmann, Robert, 612 3; Limit of 5; Limit of 7 Lath, Mukund, 545, 550 Stifel Latin ratios division of “double-octave and a fifth,” classification ratio N|z, into four arithmetic and four multiple-superparticular, 284–285 harmonic means, 386–390 multiple-superpartient, 284–286 on the importance of both divisions, 401 superparticular (epimore), 284, 385 plagiarized by Zarlino, 378, 389 Al-Farabi, 661 Wallis Ibn Sina, 678–679 experiments, sympathetic resonance, 420–421 Mersenne, 417–418 nodes Zarlino, 397 discovery of, 420–421 superpartient (epimere), 285, 385 Sauveur’s acknowledgment, 426 Al-Farabi, 661 infinite number of, not defined, 421–422 Zarlino, 398 Index 913

interval Mersenne sesqui prefix, definition, 384 conflicted rejection of, 417–420 sesquialtera, 384–385 qualified acceptance of ratios-|m and M|n, 420 Mersenne, 417 Meyer, of original Tonality Diamond, or “table Zarlino, 385–386, 392 of spans,” 448–451 sesquioctava, 384–385 Ptolemy, of Soft Diatonic Scale, 334, 832 sesquiquarta, 384–385 Rameau, rejection of, 440 Mersenne, 417 Salinas, rejection of, 403 Zarlino, 385–386, 399 Limit of 11 sesquiquinta, 385 Al-Farabi Mersenne, 417 of Jins 2 tetrachord, 653–654, 679–680 Zarlino, 385–386, 398–399 of middle finger of Zalzal fret on ud s, 636–339, sesquitertia, 384–385 643 Mersenne, 417 Javanese and scales, close rational Zarlino, 398 approximations, 510–511 Length ratios. See Flexible strings, ratios Partch, of Diamond Marimba, 452–454 Lieberman, Fredric, 486, 492, 496 Ptolemy Limit of 3 of Even Diatonic Scale, 329, 679–680 Al-Kindi, of original 12-tone “double-octave” ud of Tense Chromatic Scale, 331 tuning, 615 Limit of 13 Chinese 12-tone scale, spiral of eleven ascending Forster “fifths,” 487–488 of Bass Marimba, 827 compared to 12-TET, 377 of Diamond Marimba, 824–825 definition, 282 of Glassdance, 831 Philolaus, of original Diatonic Tetrachord, Ibn Sina 299–301 of Diatonic Genus 4 and 7 tetrachords, 676, Pythagorean theory, 288, 336, 367 679–680 Safi Al-Din, of original 17-tone ud tuning, 712 influence on modern Persian , tunbur of Khurasan, 704 686–689, 692–696 Western and Eastern music, two 12-tone scales, of Zalzal frets on ud, 671, 674 spirals of ascending and descending “fifths,” Limit of 17 335 Al-Farabi, of Persian middle finger fret on ud s, Limit of 5 636–639, 643, 674 Limit of 19 Al-Farabi, on consonance of ratios B|v and N|b, Eratosthenes, of Enharmonic and Chromatic 326, 655–658 Scales, 330–331 Bharata, of Sadjagrama and Madhyamagrama, Forster 546 of Chrysalis, 796–797 compared to 12-TET, 377 of Harmonic/Melodic Canon, 798–799 Gaffurio, rejection of, 367 Ibn Sina, of Chromatic Genus 11 tetrachord, 676 Mersenne, of 12-tone lute scale, 358–359 Ling Lun, 485 Ptolemy Litchfield, Malcolm, 308–309, 311, 317 argumentative acceptance of, 288 Little Canon original emmelic/melodic classification of, construction of, 834–836 325–326 dimensions, 837–838 of Tense Diatonic Scale, in Zarlino, 233 Plate 12, 856 Pythagoreans, rejection of, 288–289 tuning of, 837 Ramis, of original 12-tone monochord scale, Liu An, 500–501 372–373, 376 Locana, 594 Zarlino, of numero Senario, 381–384, 397–399 Logarithm. See Cents Limit of 7 Loops. See Flexible strings compared to 12-TET, 377 Lutes. See also Tunbur; Ud Forster, of Just Keys, 808–811 description of, Sachs, 597 Huygens, consonance of ratios M|b, Z?|m, and M|v, etymology of, Farmer, 610 364–365 experiment with, Wallis, 421 914 Index

Lutes (Continued) Rameau, harmonic progression of modern length inheritance of, from the Arabian Renaissance, ratios, 440–443 610–612 of dual-generator, 444–446 moveable frets of Marcus, Scott Lloyd, 749, 759–762, 767–771 lute and ud, 632, 669–670 Marimba bars. See Bars, rods, and tubes sitar, 586 Marimbas. See Bass Marimba; Diamond Marimba tunbur of Khurasan, 696, 701–704 Martopangrawit, Raden Lurah, 513, 517, 519–521 of North India Mashaqah, Mikhail, 759 sarangi, 600 Mass, 5–6, 12, 17, 23. See also Mass per unit area; sitar, 597–598 Mass per unit length; Mica mass of Persia, tunbur of Khurasan, 696 air mass, of cavity resonator, 212–215, 221 of South India, vina, 543, 586 definition, 1–2, 8 tuning density, 1, 23 !|$-comma meantone vs. approximation of 12- of air, 23, 892 TET, 340, 348–349, 354 of the earth, 6 of “semitone,” ratio Z-|zm of fluid (liquid or gas), 4, 21 Al-Farabi of solid, 4, 16 12 equal “semitones” per “octave,” 355 English Engineering System, 9, 12, 16 on two different ud s, 354–355, 639, 643 pounds-mass, 7. approximation of 12-TET experiment, 7–8 in concert music, Cardan, 355 inconsistent system, 9, 16 fretting instructions lbm-to-mica conversion factor, 14, 880 Galilei, 356 English Gravitational System, 9–10 Mersenne, 356–359 slug, 1, 9. origin of, Ptolemy, 354–355 consistent system, 9–10 Lyres experiment, 10–11 description slug-to-mica conversion factor, 880 Maas and Snyder, 618–619 English mass unit, undefined, 1, 13, 16 Sachs, 617–618 experiment, 7–10, 15 tension, only variable after stringing, 323 and frequency, 1, 4, 16, 18–21 tuning of glass, snifter, 830–831 designs inertial property, 1–2, 4, 8 of ancient Greece, 289 metric system of Arabian simsimiyya, modern, 787n.228 gram, 1 of Pythagoras, 293 gram-to-mica conversion factor, 880 of Terpander, 291–293 kilogram, 1 limitations kg-to-mica conversion factor, 24, 880 four structural problems, 617–619 Newton, 1–2, 5–7 GPS not tunable, 619 of spring-mass system, 183–184, 223n.9 of gut strings, 618 of string plain, 2–3, 18–19, 338, 792 M wound, 806 Maas, Martha, 618 Mass per unit area Major tonality. See also Minor tonality of plate, 123 of 7-limit ratios of soundboard 16-tone scale, 420 harpsichord, 125–126 Meyer’s tonality diamond, 449–452 piano, 123–124 of 11-limit ratios, Partch Mass per unit length 43-tone scale, 454–457 of bar, 158 Diamond Marimba, 453 of cylinder Otonality, 453–454 hollow, 38 Al-Farabi’s ‘Diatonic Mode’, 645 solid, 38 definition, harmonic division of “fifth,” of of stiffness parameter ancient length ratios C|x, 396–397 plain string, 100 Zarlino, 377–378, 399–401 wound string, 113–114 Index 915

of string Zarlino, 401 plain, 4, 18–19, 27–29 vs. X|m-comma , 343 wound, 36–41 based on irrational length ratios; geometric Mass per unit volume. See Mass, density; Mica division of canon strings, 336–337 mass, density Euclid’s method, mean proportional, 297–298 Matanga, 556, 561–562, 572 interpretation of McPhee, Colin, 522–528, 530–533 Chuquet, 399–341 Means, 86 Zarlino, 338–339 arithmetic, and minor tonality, 90–91 on keyboard instruments, 336, 340 arithmetic-geometric asymptote, 313–314 vs. fretted lutes and viols, 348 meantone, definition, 342 definition; arithmetic, harmonic, and geometric, geometric division of “major third,” ratio B|v, 86 on canon string, 342–343 Archytas, 87–89 origin of, Ramis’ advocacy of 5-limit “major harmonic, and major tonality, 90 third,” ratio B|v, 340–342, 375 interpretations of, ancient length ratio vs. Mersenne, Marin, 338, 349, 356, 358–359, 377, frequency ratio, 87–91, 93 404–408, 410–420, 422–423, 428, 436, 438, 610, Plato, description of, 91–92 642 explanation by Meyer, Max F., 448–453, 824 Nicomachus, 92 Mica mass, 14–16, 18, 23 Plutarch, 93 acronym, 14–15 Stifel, division of “double-octave and a fifth,” consistent system, 14–15 ratio N|z, into four arithmetic and four definition, 14–15 harmonic means, 386–390, 401 user defined units, 23 Zarlino, arithmetic and harmonic divisions of density “fifth,” ratioC|x , 399–400 of bar making materials, 885 incorporation of Stifel’s arithmetic and of bronze, modified, 37 harmonic divisions of ratio N|z, 390–397 of copper, modified, 40 origins of major and minor tonalities, dimensional analysis, 23 377–378, 396–397, 399–401 of gases, 892 Meantone temperaments of liquids, 890 Aron’s !|$-Comma Meantone Temperament, 342, of solids, 888 347 of string making materials, 882 !|$ syntonic comma dimensional analysis, 17–20, 22–23 calculation of, 344 experiment, 15 cumulative reduction of four consecutive mica-to-kg conversion factor, 24, 880 “fifths,” 344–345 mica-to-lbm conversion factor, 14, 880 comparisons to 3-limit, 5-limit, 7-limit, and new mass unit, need for, 1, 13, 25n.19 12-TET scales, 376–377 Minor tonality. See also Major tonality harmonic analysis of, twelve usable keys, of 7-limit ratios 346–347, 349–350 16-tone scale, 420 Huygens Mersenne, conflicted acceptance/rejection of, cent comparisons 418–420 comparable intervals of 31-TET, 363–364 Meyer’s tonality diamond, 448–452 “fifth” of 31-TET, 363 of 11-limit ratios, Partch three septimal ratios; M|n, M|b, and M|v, 364 43-tone scale, 454–457 key color of, unequal temperament with two Diamond Marimba, 453 different “semitones,” 349 Utonality, 453–454 syntonic comma (comma of Didymus), Al-Farabi’s ‘Persian Mode’, 645 definition, 343–344 definition, arithmetic division of “fifth,” of tuning ancient length ratio C|x, 396–397 lattice, 345 Zarlino, 378, 399–401, 452 sequence, three steps, 344–346 as inversion of major tonality vs. Werckmeister’s No. III Well-Temperament, Meyer, 449–452 352–353 Rameau, 444–446, 452 “wolf fifth” and “wolf fourth” of, 346–347 Shirlaw, 448, 452 916 Index

Minor tonality (Continued) gushe, individual mode, 690. Rameau, arithmetic progression of ancient vs. raga, 690 length ratios, 441–444 sign of dual-generator, 444–448 koron, 636–640 Mode shape, 98 sori, 636, 640 Modes. See Scales Persian scales. See Scales, Persian Modulus of elasticity. See Young’s modulus of Persian tetrachords, modern elasticity all 12 Dastgaha on Ibn Sina’s ud, 687–689, Mohammed, 610 692–696 Monsour, Douglas, Dedication Page, 843 Farhat Morse, Philip K., 149 of 12 Dastgaha, 692–693 Musical slide rule. See Cents, logarithm chromaticism, not used, 687 of Mahur, intervalic structure, 771–772 N Philolaus, 91, 280, 288, 299–301, 304, 333, 617, Napier, John, 257, 340 624–625, 648–649, 662, 681, 686, 758–759 Narada, 559–560, 564, 572 Pianos Narayana, 93, 587–592 Cristofori, gravicembalo col piano e forte, 118, Nederveen, Cornelis, J., 227–230, 233, 235–238, 244 130 Needham, Joseph, 487 design change ideas, 142 Newton, Isaac, 1, 428 impedance ratio laws of motion air-to-soundboard, 129 first, 2, 5, 8, 11, 158, 186 soundboard-to-string, 125 second, 5–7, 9, 15 inharmonicity Nicomachus of Gerasa, 82–85, 92–93, 284, 286, analysis of, 110–111 293–294, 299, 319, 380, 610, 624–625 coefficients of, plain strings, 111 Nijenhuis, E. Wiersma-Te, 545, 556, 560 effect on Nodes. See Bars, rods, and tubes; Flexible strings; beat rates, uncontrollable, 105, 108, 135 Inharmonic strings; Just intonation, Sauveur; timbre, 110–111, 118, 135 Just intonation, Wallis; Resonators, tube tuning, 111–112, 118 Numero Senario. See Just intonation, Zarlino tuning possibilities, 105, 108, 118, 135, 808–809 O limitations, structural and musical “Octave” equivalence. See Cents soundboard, thickness, 133–134 Old Arabian School, 619–620, 636 string tension, total force, 134–135 Ornstein, Ruby Sue, 524, 530–531 tuning, inharmonicity, 134–135 soundboard P bending wave speed, 131–132 Palisca, Claude V., 356, 415 components Partch, Harry, v, 452–457, 824, 840 bridges, 124 Period. See Flexible strings liners, 124 Persian musical terms. See also Scales, Persian ribs, 123–124 dastan (fret), 628, 630–632 critical frequency, 130–133 dastgah Mahur, pishdaramad (overture), 771–773 dimensions, typical instruments surface area, 129 and setar (three strings), 597, 687–689 thickness, 123 tunbur of Khurasan, 696–697, 701–704 impedance ud al-farisi, ancient lute, 619 data, Wogram, 122–124, 129–130, 132–133 mode plate vs. soundboard, 123–124 5 hierarchical functions of tones: aqaz, shahed, radiation, of air at soundboard, 129 ist, [finalis], and moteqayyer, 690–694, sound radiation, 132–133 772 spruce vs. laksana, 691 European (Picea abies), 886n.8 12 Modern Dastgaha (sing. dastgah), Farhat, Sitka (Picea sitchensis), 123, 885 692–696 stringing scale dastgah, group of modes/dominant mode, 1.88 : 1, need for, 34 690. 2:1, structural problems of, 33 Index 917

strings Poisson’s ratio coefficients of inharmonicity, 109–111, 126 of hardwood plywood, 222 dimensions and tension, typical of Sitka spruce, 123 Powers, Harold S., 580–582, 586 bass, G2, 40 treble Praetorius, Michael, 336, 349, 610 Pressure D4, 121 adiabatic bulk modulus, 21–22 G3 and C4 ... G7 and C8, 111 energy transfer, to soundboard, 118–119, 125 in cavity resonator, 212–214, 216–219, 226n.50 definition, 22 impedance, D4, 122 length vs. diameter, changes in, 115 dimensional analysis, 22 driving stiffness parameters, 111 of acoustic (wave) impedance, complex, 197 tension equation specific, 127–128 plain, 31 of bar, 196, 214, 217–218 wound, 41 definition, 127 wound, need for, 27, 34 of soundboard, 128 tuning process, general, 302 of tube resonator, 198–199 12-TET, 136 English system of “fifth,” 136 of air of “fourth,” 137 at 1 standard atmosphere, 22 of “major third,” 136–137, 139–140, adiabatic bulk modulus, 22 146n.28 psi, 13, 22 vs. harpsichords at flute embouchure and tone hole, 240–241 coefficient of inharmonicity, average values, of gas, 188 126 wave critical frequency of soundboard, 132–134 in air, 130–131, 182–184 impedance ratio of bar, 214, 217 air-to-soundboard, 129–130 in flute, 230, 234, 239, 247 soundboard-to-string, 125–126 of plane wave, 186–187 sound radiation, 133 in tube resonator, 189–195, 198–199, 200–208 soundboard thickness, string tension, and Young’s modulus of elasticity, 13 inharmonicity, 130 Ptolemy, Claudius, vi, 78, 86, 90, 93, 280, 287–289, Plain strings 295, 312–314, 316–334, 338, 354–355, 358–359, equations 399–401, 403–404, 407–408, 413, 415, 553, 564, diameter, 31 610, 624–625, 628, 636–637, 642, 646–647, 651, frequency, 27, 31 657–659, 661, 664–665, 674, 677, 679–681, 707, length, 31 712, 759, 792, 832 mass per unit length, 28–29 Pulse. See Waves tension, 31 Pythagoras, vi, 44, 82–83, 280, 284, 288–289, four laws of, 31–32 293–295 length Pythagoreans, 288, 308, 318–319, 322–324, 336, 657 and frequency, inverse proportionality, 32–33 piano stringing, structural problems, 33–34 Q piano stringing scale, treble strings, 34 Quadrivium, 630 tension break strength R calculation, 35, 40, 134 Racy, Ali Jihad, 760–761 definition, 35 Radius of gyration. See Bars, rods, and tubes considerations; structural, technical, and Rai, I Wayan, 524–525, 529–530 musical, 35 Ramachandran instrument limitations, 35 K.V., 584–585 Plane sections. See Bars, rods, and tubes N.S., 580 Plato, 91–93, 288, 308, 610–611, 624–625, 759 Ramamatya, 93, 565–580, 582, 584, 586–587, 599 Plutarch, 82, 85, 91–93 Rameau, Jean-Philippe, 90–91, 428–436, 438–449, Poerbapangrawit, Raden Mas Kodrat, 513, 451–452, 454 517–519, 521–522 Ramis, Bartolomeo, 93, 340, 342, 358, 366–377, 401 918 Index

Rao, T.V. Subba, 576 tuning process Rarefaction. See Compression and rarefaction attack tone vs. decay tone, 222 Ratio of specific heats at opening, 215–217, 222 of adiabatic bulk modulus, 22 at sidewalls, with tuning dowels, 219–221 of gases, 892 tube of liquids, 890 antinode (closed tube), 203 Ratios. See also Ancient length ratios; Arabian displacement (DA), 205, 207–208 ratios; Flexible strings, ratios; Greek ratios; pressure (PA), 205 Latin ratios antinode (infinite tube) integer, definition, 71, 281 displacement (DA), 201–203 numbers That compose pressure (PA), 201–203 commensurable vs. incommensurable, 365–366 antinode (open tube), 203 Galileo, 283 displacement (DA), 204 composite pressure (PA), 204 definition, 281 closed, 203 prime factorization, 281–282 airtight seal, making, 211 irrational end correction, 207–209 cube root of 3, Archytas, 283 and diameter, direct proportionality, 208 definition, 283 and frequency, inverse proportionality, square root of real 211 Chuquet, 339–341 frequency Euclid, 283, 297–298 actual, 210 Zarlino, 338–339 theoretical, 207 positive natural; odd, even, and prime, 281 length prime cut, 210 definition, 281 measured vs. effective, 207–208 factorization, 281–282 theoretical, 207 limit, definition, of intervals and scales, closed-closed 282. See also Limit of 3 frequency, theoretical and actual, 230 rational length, measured and effective, 230 definition, 281 impedance, acoustic wave, 197 two types, 282–283 bar-to-resonator relation, 198 Ray, Satyajit, 600 of resonator, 196–197 Resonators of room, 199 cavity of tube, 198 air mass, in neck, 212–213 tube-to-room ratio, 199 actual, 215, 221 node (closed tube), 203 theoretical, 214 air spring displacement (DN), 205 in cavity, 212–213, 215–216 pressure (PN), 205, 207–208 at sidewalls, 217–219, 221 node (infinite tube) air spring-air mass system, two different displacement (DN), 201–203 kinds, 212–213, 217–219 pressure (PN), 201–203 end correction node (open tube), 203 of duct opening, 214 displacement (DN), 204 of flange opening, 214 pressure (PN), 204 and frequency, inverse proportionality, 215 open, 203 frequency, 215 end correction, 209 equation, limitations of, 212 frequency theoretical vs. actual, 217 actual, 209 Helmholtz, 212 theoretical, 206 making, size considerations, 218–221 length neck, effective length of, 212–215 cut, 209 reason for, 212 measured vs. effective, 229–230 sidewall stiffness, 221–222 theoretical, 206 Index 919

pulse reflection tetrachords; Turkish musical terms; Turkish at closed end tetrachords; Well-temperaments compression, 190, 192 Arabian rarefaction, 190–191, 193 8 tetrachords, origins in near-equal divisions at open end of intervals; Ptolemy, Al-Farabi, and Ibn compression, 191, 194 Sina, 677–681 rarefaction, 191, 195 24-TET, modern scale, 755–761 reason for, 196 46 Modern Maqamat, 761–771, 786n.209 standing waves, pressure/displacement 9 fasail, for the construction of, 762 closed, 203, 205, 208 16-tone scale, for the playing of, 770–771 closed-closed, 230 of Al-Farabi infinite, 201 3 standard modes, 7-tone scales, 644–646 open, 203–204, 230 3 tetrachordal conjunct/disjunct systems; Restoring force in the context of GPS, 662–666 of bar 8 Ajnas, tetrachords, 640–641, 646–655 clamped-free, 174–177 10-fret ud tuning, theoretical, 632–639 free-free, 147–148, 154–162 15 tetrachords, 2 genera, 658–663 bending stiffness 17-tone scale of bar, 152, 154–155, 158–160 original tunbur, 703–707, 735 of plate, 123 tunbur of Khurasan, 696–697, 701–704 of rod and tube, 158 22-tone scale, ud tuning, 640–643 of soundboard of Al-Kindi, original 12-tone scale, ud tuning, harpsichord, 125 611–617 piano, 123, 130–132 of Al-Munajjim, 8 majari, 7-tone scales, of cavity resonator, 226n.50 621–625 definition, 2 of Brothers of Sincerity, 9-tone scale, ud elastic property, 2–4, 21 tuning, 620–622 of fluid, 4 of Ibn Sina of glass, snifter, 830–831 11 Melodic Modes, 681–686 of plane wave, 186–187 16 tetrachords, 3 genera, 673–676 of solid, 4 17-tone scale, ud tuning, 666–673 of spring-mass system, 226n.50 of Safi Al-Din of string. See also Downbearing force 17-tone scale flexible, 1–4, 18–19 ascending spiral of “fourths,” original stiff, 104–105, 107–108 construction, 731–733 Roberts, Francis, 421–422 First Ud Tuning, 707–713 Robinson, Kenneth, 503 monochord/ud tuning, 717–719 Rods. See Bars, rods, and tubes tunbur/ud tuning, 720–721 Roy, Hemendra Lal, 595–597 Second Ud Tuning, 714–717 19-tone scale, theoretical; ascending and S descending spirals of “fifths,” 710–713 Sachs, Curt, 489, 556, 597, 617–618 84 Melodic Modes, 721–726, 728–729, 768 Safi Al-Din, 93, 326, 366, 373–375, 379, 381, 384, 6 Awazat, 730–731 401–402, 610, 622, 626–627, 673, 696, 705, 12 genera, for the construction of, 726 707–733, 735, 737, 740–745, 747–749, 754–755, 12 Shudud, 727–730, 737, 740–744, 758–759, 763–768, 771 767–768 Salinas, Francisco, 401–403, 448, 452 Chinese Sarngadeva, 548, 562, 566–567, 572, 576 1-lu interval Saunders, Lawrence, viii two different kinds, 497–499 Sauveur, Joseph, 390, 422–428, 436, 438 of two tiao (mode tuning) cycles, 496–499 Scales, 86. See also Arabian musical terms; 5-tone (pentatonic) scale Arabian ratios; Arabian tetrachords; on ch’in Equal temperaments; Greek ratios; Greek 12 tiao (mode tunings) tetrachords; Indian musical terms; Indian ascending cycle of six; descending tetrachords; Just intonation; Meantone cycle of six, 496–499 temperaments; Persian musical terms; Persian interval sequences of, 497, 500 920 Index

Scales (Continued) Bhatkhande, 10 most popular Thats, 581, open strings 596 post-Ming, new scale, 490–493 letter-dot notation, after cipher-dot pre-Ming, old scale, 490–493 notation of the Cheve System, 508, String III as chiao, dual identity of, 535n.6, 598–599 492 Narayana and Ahobala, 12-tone vina tuning sequence of, 492 tuning, 591 stopped strings (hui fractions), pre-Ming, Roy, 32 Thats, of eight-by-four matrix old scale, 492–496 system, 581, 595–596 cipher notation of, Cheve System, 490, 493, sitar tuning and fret locations, 598 496, 508 South India original, 487–488, 490–493 14 most popular mela ragas, 581 12-TET, discovery of, Chu Tsai-yu, 502–504 Govinda 12-tone scale 72 musical heptatonic scales, first formal formula of, Ssu-ma Ch’ien, 486 implementation of, 565 generation of, 486–487 lu, definition, 487 Kanakangi-Ratnangi system, 580–581 spirals of “fifths,” That closely approximate Ramamatya equal temperaments, 500–502 12-tone practical scale, suddha mela vina Greek tuning, 573 Archytas, three means for the construction of 14-tone theoretical scale, 567 tetrachords, 86–87 Venkatamakhi of Aristoxenus, 309–311 12-tone scale, 577 Catalog of Scales, seven theorists, 330–332 72 Melakartas, of twelve-by-six matrix Dorian and Lydian modes, 292, 301, 333 system, 579, 581 of Euclid, 304 Indonesian. See also Gamelan Greater Perfect System (GPS), 289–292, Bali 301–304, 333 gamelan Semar Pegulingan, 525–530 Lesser Perfect System (LPS), 289–291 patutan, definition, 523, 537n.43 of Philolaus, 300–301, 304 pelog of Ptolemy, 328–329 7-tone scale (saih pitu), 523–530 of Pythagoras, 289, 293 5-tone scale (saih lima), derived, seven harmoniai (modes), 290, 292 524–530 ecclesiastical names of, 290, 292 pemero (two auxiliary tones), of Terpander, 289, 291–293 penyorog and pemanis, 525 three genera, Archytas and Aristoxenus, cipher notation of, 523 289–292 intervals of, 523 Indian patutan Selisir, 525–530 ancient six-tone, Pliatan Village, 530–534 Bharata cipher notation of, 532 Madhyamagrama (Ma-grama), 540–541, penyorog, 525, 531–534 546 Patutan Tembung, Selisir, Baro, Lebeng, and svarasadharana, 549–551 525–529 Sadjagrama (Sa-grama), 540–541, 546 and Sunaren, 527–529 Suddha Jatis, 554 slendro, 5-tone scale Arsabhi and Naisadi vs. Ptolemy’s cipher notation of, 522–523 Tense Diatonic, 553 intervals of, 523 Vikrta Jatis, 555 saih gender wayang; shadow puppet Gandhari vs. original Chinese theater only, 522–523 pentatonic scale, 556 , slendro and pelog, 525 gramaragas and laksana Java of Kudimiyamalai inscription, 563 patet, definition, 512–513 of Matanga, 563 pelog, 7-tone scale of Narada, 563 5-, 7-, and 11-limit ratio analyses of, 511 of Sarngadeva, 563 cipher notation of, 510–511 North India definition, 510 12-tone scale, modern svaras of, 592 intervals of, 510–512, 517–521 Index 921

Patet Lima, Nem, and Barang, 517–521 of Mersenne’s gong tones and , 517, harmonic-diatonic system, 403–405 519–521, 537n.29 lute, 356–359 slendro, 5-tone scale “old” and “new ” trumpet marines, 5-, 7-, and 11-limit ratio analyses of, 510 410–415 cipher and cipher-dot notation of, Cheve of Meyer, 449 System, 508–511, 535n.6 minor scale, Ptolemy’s Soft Diatonic, 334, 832 definition, 509 of Partch, 454–457 intervals of, 509–517 Pythagorean ‘Apotome Scale’ and ‘Limma Patet Nem, Sanga, and Manyura, Scale’, 336 513–514, 516 of Ramis, 366–377 gong tones and cadences, 513–516 of Spechtshart, first twelve-tone 3-limit scale, solmization, slendro and pelog, 525 334–335 tumbuk (common tone) between slendro spirals of twelve “fifths,” ascending and and pelog, 519–522 descending, for the construction of, Persian 335–336 12 Modern Dastgaha, Farhat, 690–696 of Stevin, 358–362, 375–376 9 intervals of, 686–688 of Werckmeister, 349–353 Schisma. See Arabian ratios; Greek ratios, interval; basic 17-tone scale of, 770–771 Just intonation, Ramis; Tunbur dastgah, definition, 690 Schlesinger, Kathleen, 227 on Ibn Sina’s ud, 692–696 Seeff, Norman, 857 17-tone scale Senario. See Just intonation, Zarlino modern cent averages of two tar and three Shankar, Ravi, 597, 600 setar tunings, 687–689 Sharma, P.L., 561 tunbur of Khurasan, 696–697, 701–704 Shear force. See Bars, rods, and tubes, restoring Turkish forces 24-tone modern scale Shirlaw, Matthew, 428, 434, 448, 452 6 modern musical symbols of, 735–736 Signell, Karl L., 736, 738–739, 741–745 53-TET, theoretical model, 736–737 Simple Flutes, 833 comma-limma equivalents of, 736–736 Acrylic Flute vs. Al-Farabi’s original tunbur tuning, Flute 1 733–736 construction of, 246 vs. Safi Al-Din’s First Ud Tuning, 735 dimensions, 241 basic 18-tone scale, 770–771 tuning of, 239–241, 833 Signell Flute 3, tuning of, 833 6 ‘Variant’ Maqamat, 744–745 Amaranth Flute, Flute 2 5 of 6 on Safi Al-Din’s First Ud, 744–745 dimensions, 243 13 Basic Maqamat, 740–743 tuning of, 245–246 10 of 13 on Safi Al-Din’s First Ud, Plate 11, 855 740–743 Simple harmonic motion (SHM) Western definition, 50, 182 of particle motion 7-tone modes, ecclesiastical names of, 290, 292 in air, 182–184 12-tone scale in cord/string, 50–51 five examples of, 376–377 in solid, liquid, gas, 186–187 origins of, Chinese and Arabian sources, of spring-mass system, 183–186 334 Sindoesawarno, Ki, 512 transformation from 3-limit to 5-limit scale, Sitar. See Indian musical terms, North India 418–419 Smith, Page, Dedication Page, 839 of Anonymous, in Pro clavichordiis faciendis, Snyder, Jane McIntosh, 618 first twelve-tone 5-limit scale, 473n.175 Sound waves. See Waves, sound of Aron, 342–347, 353, 364 Soundboards of Halberstadt organ, Praetorius, 336 instrument of Huygens, 362–365 Bass Canon, 800, 805 of Kepler, 336 ch’in, top piece, 489 major scale, Zarlino’s advocacy of Ptolemy’s Chrysalis, 788–789 Tense Diatonic, 319, 333, 832 Harmonic/Melodic Canon, 790–793 922 Index

Soundboards (Continued) Spechtshart, Hugo, 334 harp-vina, 543 Speed of sound. See also Wave speed, longitudinal harpsichord, 123, 125–126, 129, 132–133 in air, 23, 126–127, 131–132 Just Keys, 808–809 at 68°F, 189 kithara and lyre, 617–618 at 86°F, 189 Little Canon, 834–836 per degree rise of, 189 piano, 123, 125–126, 129, 132–133 as constant, 188, 208 qanun, ancient, 628–631 inside played flute, 229 sitar, 597 in rosewood, 21 trumpet marine, 408 in spruce vs. steel, 886n.8 ud, top plate, 619–620, 626 temperature effect on, gas, 188–189 vina, 586 Spring constant violin, top plate, 626 acoustical, of cavity resonator, two different piano. See also Pianos, soundboard kinds, 212–219, 221 bending wave, 122, 128 definition, 183–185 speed, 119, 131 mechanical, 183–185, 216, 218, 221–222 and bending wavelength, inverse Spring-mass system proportionality, 131–132 acoustical, of cavity resonator and critical frequency, 132, 134 definition, two different kinds, 212–213, and dispersion, 130–132 217–219, 221 effect on acoustic radiation, 123–124, elastic and inertial components, 212–213, 131–133 217–219, 221 experiment, 131 excess pressure of, 226n.50 of plate, 131–132 frequency of, 215–217 dispersion restoring force of, 226n.50 definition, 131 mechanical effect on radiation, 130, 132 definition, 183–185 energy transfer (coupling) elastic and inertial components, 183–185 soundboard-to-air, 127–128 frequency of, 185, 223n.9 string-to-soundboard, 119, 121, 127–128 two springs in series and one mass, 218–219 impedance, mechanical wave, 120–121, 124 restoring force of, 226n.50 data, Wogram, 122–124, 129–130, 132–133 Ssu-ma Ch’ien, 93, 334, 486 experiment, thick soundboard Stapulensis, Jacobus Faber (Jacques Le Febvre), and thick piano string, 133–134 396 Stevin, Simon, 354, 358–362, 375–376, 502–503, and thin canon string, 142 542, 569, 749 fluctuations, 122 Stifel, Michael, 276n.6, 378, 386–390, 392, 396–397, of plate, 123 401, 403, 427, 429, 432–433, 435 radiation, of air at soundboard, 124, String Winder. See also Wound strings 129–130 construction of, basic components, 805–807 ratio dimensions, 807 air-to-soundboard, 129–130, 134 Plates 4 and 5, 848–849 soundboard-to-string, 125–126, 134–135, Strings. See Flexible strings; Inharmonic strings; 142 Plain strings; Wound strings reactance and resistance, 122–123 Subharmonic series. See Flutes; Just Intonation, resonance and resonant frequency, 122–124, Rameau 132–133 Sumarsam, 512 radiation, 119 Superposition. See Flexible strings bending wave speed effect on, 131–132 Surjodiningrat, Wasisto, 509–511, 516, 521–522 critical frequency effect on, 132–133 data, Wogram, 124, 129–130 T stringing effect on, 125, 130 Temperaments. See Equal temperaments; thickness effect on, 130, 134 Meantone temperaments; Well-temperaments wood Temperature European spruce (Picea abies), 886n.8 conversion factors, 189, 881 Sitka spruce (Picea sitchensis), 885 effect on T’ung wood (Paulownia imperialis), 489 density vs. pressure, 188 Index 923

frequency, 189 schisma and comma variants of, 696–707 speed of sound in gas/air, 188–189, 225n.26 of Turkey, modern tuning and musical symbols per degree rise of, in air, 189 of, 733–736 inside played flute, 229 Turkish musical terms Tensile strength, 32 6 hierarchical functions of tones: karar, tiz durak, and break strength calculation, 35 yeden, guclu, asma karar, giris, 738 definition, 35 Turkish scales. See Scales, Turkish of spring steel, 35 Turkish tetrachords, modern of string materials, 882 of Rast, 727, 746–747, 749 gut strings, thick vs. thin, 883n.1 modal origins on Safi Al-Din First Ud, high-carbon spring steel music wire, 884 726–728, 730–733, 744–748 temper classifications; copper, brass, bronze, and Signell steel, 883 6 basic tetrachords/pentachords, 738–739 Tension. See Plain strings Tenzer, Michael, 533 6 ‘variant’ (non-“basic”) tetrachords/ Terpander, 289, 291–293 pentachords, 738–739 Tetrachords. See Arabian tetrachords; Greek Tyagaraja, 578, 583–584, 604n.73, 606n.97 tetrachords; Indian tetrachords; Persian tetrachords; Turkish tetrachords U Theon of Smyrna, 284, 286, 318, 380 Ud. See also Lutes Tonality Diamond. See Diamond Marimba; Just bridge intonation, Meyer; Just intonation, Partch dama, 628 Tone notation, ix faras, 626–628 Toth, Andrew F., 524 hamila, 626–628, 631–632 Touma, Habib H., 760 hamila of canon functions as dastan of ud; Trumpet marines vice versa, 628, 632, 670–672 description, 408 musht, 626–628 flageolet tones of, 408–409 construction and tuning of, earliest, 620–622, Mersenne 775n.27 harmonics of, 406 fret (Pers. dastan), 628, 630–632 illustration of two, 411 Al-Farabi, frets of ud function as bridges of as monochord, 410 canon, 632 length ratios of “old” and “new” trumpet binsir (ring finger), 615, 617. marines, 410–415 khinsir (little finger), 615, 620–622. Roberts, flageolet tones of, 421–422 middle finger of Zalzal Sauveur, rejection of, 422 Al-Farabi’s ud s, 634–640, 643–645, 650–653 Trumpets Ibn Sina’s ud, 667–673, 681–683 harmonics of, first fourteen, 412 Safi Al-Din’s First and Second Ud Tunings, Mersenne, 404–405, 414–416 709–712, 714–716 harmonics of, 406, 413 mujannab (neighbor; also anterior or “leaps” of natural trumpet vs. harmonics assistant), 615, 620, 634. of monochord, five identical ratios, Persian (ancient) middle finger 406–407 Ibn Sina’s ud, 667–672 with prime numbers greater than 5, not Safi Al-Din’s First Ud Tuning, 709–713 acknowledged, 408, 412, 415 Persian middle finger Roberts, harmonics of, infinite series, 421 Al-Farabi’s ud s, 634–639, 642–645 Sauveur, harmonics of, first thirty-two, 424 Safi Al-Din’s Second Ud Tuning, 714–717 Tubes sabbaba (index finger), 615, 617. instrument. See Bars, rods, and tubes wusta (middle finger), 615, 617. resonator. See Resonators, tube zaid (surplus), 709. Tunbur. See also Lutes of Persia, ud al-farisi, precursor of classic bridge and string holder (hitch pin), two Arabian al-ud, 619, 775n.27 separate components, 626–628, 696 Pythagorean diatonic scale, playable on all of Khurasan, Al-Farabi ancient ud s, 755 frets of, fixed and moveable, 701–704 string strings and frets of, 696 Bamm, String I, 612. 924 Index

Ud (Continued) W Hadd, String V Wallis, John, 420–421, 426 Al-Farabi’s ud, 641. Wave speed Ibn Sina’s ud, 669. bending Safi Al-Din ud s, 708. in bar, 149–150, 159–160 Mathlath, String II, 612. and bending wavelength, inversely Mathna, String III, 612. proportionality, 149  mutlaq (free; open string), 615, 617 . and dispersion, 147–149 tuning of, 612–613, 621–622, 633–634, effect on inharmonic mode frequencies, 641–642, 644–647, 667–668, 681–683, 148–151 702, 708–709 experiment, 149 Zir tuning effect on, 160 1st string, only on Ikhwan al-Safa’s ud, 621 in soundboard (infinite plate), 131–132 String IV, 633. and bending wavelength, inverse ‘Zir 2’, String V, only on Al-Kindi’s ud, 612. proportionality, 131 V and critical frequency, 131–132, 134 Vaziri, A.N., 640, 691, 772 and dispersion, 131–132 Velocity driving pressure of, 128 definition, 4 effect on acoustic radiation, 124, 131–133 dimensional analysis, 4, 20, 22 experiment, 131 equations, 5, 7 phase velocity of, 119 particle, 2–3 stiffness effect on, 131–132 of bar, 156–158 longitudinal of complex impedance in fluid (liquid or gas), 3–4, 21, 188 mechanical (wave), 120–121 in gases, 892 specific acoustic (wave), 127 in liquids, 890 of cord, 2–3, 50–51 and mica mass unit; to simplify all of plane wave, 186–187 calculations of, 21 of solid and fluid, 119–120 in solid, 3–4, 21, 188, 888 of sound wave, in air, 183–184 transverse phase in solid, 3–4 of solid and fluid, 119–120 in string of string, 131 flexible, 3–4, 60–61, 102–104 volume, complex acoustic (wave) impedance, 197 as constant, 60–61, 103–104 Venkatamakhi, 565–566, 574–584, 586, 594–596 stiff, 99, 104–107 Muddu, 577 and dispersion, 104 Vina as variable, 105–107 ancient Wave train. See Flexible strings alapini vina, 543, 568–569 Wavelength. See also Dispersion Bharata, 540 bending vina (harp-vina), 543–544 of bar, 149–150 tuning experiment on, 541, 544 experiment, 149 kinnari vina, 569, 586 of soundboard, experiment, 131 zither-vina, development of, 568–569 definition, 54, 60 North India dimensional analysis, 54–55 bin, modern stick-zither, 543 of flute tube, half-wavelength Narayana and Ahobala, 587 approximate, 229, 231. 12-tone vina tuning, 591 exact, substitution tube, 229–232 tuning instructions, 587, 590 and frequency vicitra vina, 585, 599 as function of, 80 South India inverse proportionality, 61–62 Ramamatya, 565 of longitudinal traveling wave, in air, 183–184 suddha mela vina, 569 of string 12-tone tuning, 570, 573 flexible, 60–62, 71 tuning instructions, 569–570 as constant, 61–62 vina, modern lute, 543, 586 of transverse standing wave, 51–53, 58 Index 925

stiff, 99, 101–102 pressure, in tube resonator of transverse standing wave, 102 closed, 203, 205–208 as variable, 101–102, 116n.6 infinite, 200–203 of transverse traveling wave, wave train, 50–51, open, 203–204, 206, 230 54–57 transverse of tube resonator in bar (clamped-free), 175 closed-closed, theoretical and actual, 230 in bar (free-free), 153 closed (quarter-wavelength) definition, 48, 51–53 actual, 210 in soundboard, 122 measured vs. effective, 207–209 in string theoretical, 207 discovery of, Sauveur, 424–426 open (half-wavelength) and flageolet tones, 408–409 actual, 209 flexible, 51–55, 58–59, 182–183 measured vs. effective, 229–230 mathematical model of, D’Alembert, 423 theoretical, 206 stiff, 101–102 Waves traveling bending frequency of, 55 in bar, 147–151 longitudinal in soundboard, 119, 122, 124, 131. in air, 182–184 definition, 44 definition, 183 pulse in fluid, 119 longitudinal, in tube resonator in solid, liquid, or gas, 186–188 at closed end, compression and rarefaction, in tube resonator, 189–191, 196–197 192–193 closed, 203–207 at open end, compression and rarefaction, infinite, 200–203 194–195 open, 203–206 transverse, in string period of, 54–56 crest and trough, 44–45 transverse collision of, 46–49 definition, 50–51, 182–183 incident and reflected, 45–46 in solid, 119 sound in string, 2, 44–53, 58, 80 beating phenomenon, 135–137 Weight. See also Weight density as longitudinal traveling wave, 21, 130, definition, 6 182–184, 186–188 equations, 6, 8, 16, 18 speed of of mica, 14 in air, 23, 128, 131, 188, 203 of object, standard weight, 11–12 and critical frequency, 131–133 of rosewood test bar, 173 in solid, liquid, gas, 119, 188 of slug, 11 temperature effect on, in gases, 188–189 as string tension, experiment standing flexible, 18–19, 336–337 displacement, in tube resonator stiff, 108 closed, 203, 205, 207–208 as “weight ratio” closed-closed, 230, 248n.10 Nicomachus, description of, 82–86, 93, 294 infinite, 200–203 Ptolemy, rejection of, 319 open, 203–204 Weight density frequency of, 55, 58–59 of bar making materials, 885 longitudinal definition, 16 definition, pressure/displacement, 200 English Engineering System, inconsistent in flute, 231, 234, 245 system, 16 in tube resonator of gases, 892 close, 196, 205, 208 of liquids, 890 closed-closed, 230 of rosewood test bar, 173 infinite, 200–203 of solids, 888 open, 204, 230 of string making materials, 882 period of, 54–56 Weight per unit volume. See Weight density 926 Index

Well-temperaments of custom string, three different materials, based on irrational length ratios; geometric 39–40 divisions of canon strings, 336–337 of piano string, two different materials, 40–41 Euclid’s method, mean proportional, 297–298 tuning process, piano, 113 interpretation of wrap wire, modified density Chuquet, 339–341 of bronze, 37 Zarlino, 338–339 of copper, 40 Werckmeister’s No. III Well-Temperament, Wright, Owen, 622–625 349–350, 352 !|$ ditonic comma Y calculation of, 350 Young Robert W., 99 cumulative reduction of four non- Thomas, 181n.38(B) consecutive “fifths,” 350–351 Young’s modulus of elasticity Bach’s Well-Tempered Clavier tuning, Barnes, of bar making materials, 885 349 definition, 21 ditonic comma (comma of Pythagoras), elastic property, 4 definition, 335, 349–350 of hardwood plywood, 221–222 harmonic analysis of, twenty-four usable keys, heat effect on, 164, 177 351–353 of solids, 888 “key color” of, unequal temperament with of spruce, 123, 885 four different “semitones,” 352 grain direction factor, 123 tuning of steel, Thomas Young, 181n.38(B) lattice, 350–351 of string making materials, 882 sequence, 471n.140 vibration test for, 180n.38(A) vs. Aron’s !|$-Comma Meantone rosewood bar, 172–173 Temperament, 352–353 water effect on, 172 Werckmeister, Andreas, 350–353 Western scales. See Scales, Western Z Whitman, Walt, v, 788, 794–799, 802–804, 840 Zalzal, Mansur, 636, 716, 755 Widdess, Richard, 558–559, 560–562 Al-Farabi’s ‘Mode of Zalzal’, 645 Wienpahl, Robert W., 397 Zarlino, Gioseffo, 93, 319, 333, 338–340, 342–343, Wilson, Erv, 455–457, 887n.12 349, 377–379, 381–382, 384–387, 389–401, 403, Winnington-Ingram, R.P., 308 407–408, 428–429, 431–432, 434–436, 440–441, Wogram, Klaus, 122–124, 129, 132 446, 452, 553, 832 Wound strings Ziryab, 614 Bass Canon, 39–40, 800, 805 break strength of, 40 coefficient of inharmonicity, 114–115 experiment, 61–62 inharmonicity, difficulties analyzing, 113 length, 36 making with String Winder, 805–807 mass per unit length, 36–37 composite, 36–38 of custom string, three different materials, 39–40 of piano string, two different materials, 40–41 of cylinder hollow, 38 solid, 38 of stiffness parameter, 113–114 mode frequency, 27–28 stiffness parameter, 34, 113–114 tension, 36, 41