sign in sign up
<<
Home , Augmented sixth, Diminished seventh, Enharmonic, Major seventh, Major sixth, Minor seventh

Mathematics and Music

http://dx.doi.org/10.1090/mawrld/028

MATHEMATICAL WORLD VOLUME 28

Mathematics and Music

David Wright

Providence, Rhode Island 2000 Mathematics Subject Classification. Primary 00–01, 00A06.

For additional information and updates on this book, visit www.ams.org/bookpages/mawrld-28

Library of Congress Cataloging-in-Publication Data Wright, David, 1949– Mathematics and music / David Wright. p. cm. — (Mathematical world ; v. 28) Includes bibliographical references and index. ISBN 978-0-8218-4873-9 (alk. paper) 1. Musical intervals and scales. 2. —Mathematics. I. Title.

ML3809.W85 2009 781.01′51—dc22 2009014813

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by -mail to [email protected].

⃝c 2009 by the author. Printed in the United States of America. ⃝∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 21 14 13 1211 10 09 About the Author

David Wright is professor of mathematics at Washington University in St. Louis, where he currently serves as Chair of the Mathematics Depart- ment. He received his Ph.D. in Mathematics from Columbia University, New York City. A leading researcher in the fields of affine algebraic geometry and polynomial automorphisms, he has produced landmark publications in these areas and has been an invited speaker at numerous international mathemat- ics conferences. He designed and teaches a university course in Mathematics and Music, the notes from which formed the beginnings of this book. As a musician, David is an arranger and composer of vocal music, where his work often integrates the close style called barbershop harmony with jazz, , gospel, country, doo-wop, and contemporary a cappella. He is Associate Director of the St. Charles Ambassadors of Harmony, an award winning male chorus of 160 singers. He also serves as a musical consultant and arranger for numerous other vocal ensembles. He is active in the Barbershop Harmony Society and was inducted into its Hall of Fame in 2008. As arranger and music historian David has been featured in national radio and TV broadcasts at home and abroad and has authored several articles on vocal harmony.

Contents

Introduction ix

1 Basic Mathematical and Musical Concepts 1

2 Horizontal Structure 17

3 Harmony and Related Numerology 31

4 Ratios and Musical Intervals 45

5 Logarithms and Musical Intervals 53

6 Chromatic Scales 61

7 Identification and Modular Arithmetic 69

8 Algebraic Properties of the Integers 85

9 The Integers as Intervals 95

10 Timbre and Periodic Functions 103

11 The Rational Numbers as Musical Intervals 123

12 Tuning the Scale to Obtain Rational Intervals 137

Bibliography 147

Index 149

vii

Introduction

The author’s perspective. Mathematics and music are both lifelong passions for me. For years they appeared to be independent non-intersecting interests; one did not lead me to the other, and there seemed to be no obvious use of one discipline in the application of the other. Over the years, however, I slowly came to understand that there is, at the very least, a positive, supportive coexistence between mathematics and music in my own thought processes, and that in some subtle way I was appealing to skills and instincts endemic to one subject when actively engaged with the other. In this way the relationship between mathematical reasoning and musical creativity, and the way humans grasp and appreciate both subjects, became a matter of interest that eventually resulted in a college course called Mathematics and Music, first offered in the spring of 2002 at Washington University in St. Louis, the notes of which have evolved into this book. It has been observed that mathematics is the most abstract of the sci- ences, music the most abstract of the arts. Mathematics attempts to under- stand conceptual and logical truth and appreciates the intrinsic beauty of such. Music evokes mood and emotion by the audio medium of tones and rhythms without appealing to circumstantial means of eliciting such innate human reactions. Therefore it is not surprising that the symbiosis of the two disciplines is an age-old story. The Greek mathematician Pythagoras noted the integral relationships between of musical tones in a consonant interval; the 18th century musician J. S. Bach studied the math- ematical problem of finding a practical way to tune keyboard instruments. In today’s world it is not at all unusual to encounter individuals who have at least some interest in both subjects. However, it is sometimes the case that a person with an inclination for one of these disciplines views the other with some apprehension: a mathe- matically inclined person may regard music with admiration but as some- thing beyond his/her reach. Reciprocally, the musically inclined often view mathematics with a combination of fear and disdain, believing it to be un-

ix x INTRODUCTION related to the artistic nature of a musician. Perhaps, then, it is my personal mission to attempt to remove this barrier for others, since it has never ex- isted for me, being one who roams freely and comfortably in both worlds, going back and forth between the right and left sides of the brain with no hesitation. Thus I have come to value the ability to bring to bear the whole capacity of the mind when working in any creative endeavor.

Purpose of this book. This short treatise is intended to serve as a text for a freshman level college course that, among other things, addresses the issues mentioned above. The book investigates interrelationships between mathematics and music. It reviews some background concepts in both sub- jects as they are encountered. Along the way, the reader will hopefully augment his/her knowledge of both mathematics and music. The two will be discussed and developed side by side, their languages intermingled and unified, with the goal of breaking down the dyslexia that inhibits their men- tal amalgamation and encouraging the analytic, quantitative, artistic, and emotional aspects of the mind to work together in the creative process. Mu- sical and mathematical notions are brought together, such as scales/modular arithmetic, octave identification/equivalence relation, intervals/logarithms, /exponents, /integers, tone/trigonometry, tim- bre/ analysis, tuning/rationality. When possible, discussions of musical and mathematical notions are directly interwoven. Occasionally the discourse dwells for a while on one subject and not the other, but eventually the connection is brought to bear. Thus you will find in this treatise an integrative treatment of the two subjects. Music concepts covered include scales (standard and non-standard), intervals, rhythm, meter, form, melody, chords, progres- sions, octave equivalence, overtones, timbre, formants, equal temperament, and alternate methods of tuning. Mathematical concepts covered include integers, rational and real numbers, equivalence relations, geometric trans- formations, groups, rings, modular arithmetic, unique factorization, loga- rithms, exponentials, and periodic functions. Each of these notions enters the scene because it is involved in one way or another with a point where mathematics and music converge. The book does not presume much background in either mathematics or music. It assumes high-school level familiarity with algebra, trigonometry, functions, and graphs. It is hoped the student has had some exposure to musical staffs, standard clefs, and key signatures, though all of these are explained in the text. Some calculus enters the picture in Chapter 10, but it is explained from first principles in an intuitive and non-rigorous way. INTRODUCTION xi

What is not in this book. Lots. It should be stated up front, and empha- sized, that the intent of this book is not to teach how to create music using mathematics, nor vice versa. Also it does not seek out connections which are obscure or esoteric, possibly excepting the cursory excursion into serial mu- sic (the rationale for which, at least in part, is to ponder the arbitrariness of the twelve-tone ). Rather, it explores the foundational commonalities between the two subjects. Connections that seem (to the author) to be distant or arguable, such as the golden ratio, are omitted or given only perfunctory mention. Yet it should be acknowledged that there is quite a bit of material in line with the book’s purpose which is not included. Much more could be said, for example, about polyrhythm, harmony, voicing, form, formants of musical instruments and human vowels, and systems of tuning. And of course there is much more that could be included if calculus were a prerequisite, such as a much deeper discussion of harmonic analysis. Also missing are the many wonderful connections between mathematics and music that could be established, and examples that could be used, involving non-Western music (scales, tuning, form, etc.). This omission owes itself to the author’s inexperience in this most fascinating realm.

Overview of the chapters. The book is organized as follows:

∙ Chapter 1 lays out the basic mathematical and musical concepts which will be needed throughout the course: sets, equivalence relations, func- tions and graphs, integers, rational numbers, real numbers, pitch, clefs, notes, musical intervals, scales, and key signatures.

∙ Chapter 2 deals with the horizontal structure of music: note values and time signatures, as well as overall form.

∙ Chapter 3 discusses the vertical structure of music: chords, conven- tional harmony, and the numerology of chord identification.

∙ Musical intervals are explained as mathematical ratios in Chapter 4, and the standard keyboard intervals are introduced in this language.

∙ Chapter 5 lays out the mathematical underpinnings for additive mea- surement of musical intervals, relating this to logarithms and expo- nentials.

∙ Equal temperament (standard and non-standard) is the topic of Chap- ter 6, which also gives a brief introduction to twelve-tone music. xii INTRODUCTION

∙ The mathematical foundations of modular arithmetic and its relevance to music are presented in Chapter 7. This involves some basic abstract algebra, which is developed from first principles without assuming any prior knowledge of the subject.

∙ Chapter 8 delves further into abstract algebra, deriving properties of the integers, such as unique factorization, which are the underpinnings of certain musical phenomena.

∙ Chapter 9 gives a precursor of by interpreting positive inte- gers as musical intervals and finding keyboard approximations of such intervals.

∙ The subject of harmonics is developed further in Chapter 10, which relates timbre to harmonics and introduces some relevant calculus con- cepts, giving a brief, non-rigorous introduction to continuity, periodic functions, and the basic theorem of harmonic analysis.

∙ Chapter 11 covers rational numbers and rational, or “just”, intervals. It presents certain classical “commas”, and how they arise, and it discusses some of the basic just intervals, such as the greater whole tone and the just major . It also explains why all intervals except multi- in any equal tempered scale are irrational.

∙ Finally, Chapter 12 describes various alternative systems of tuning that have been used which are designed to give just renditions of cer- tain intervals. Some benefits and drawbacks of each are discussed.

Suggestions for the course. This book is meant for a one-semester course open to college students at any level. Such a course could be cross-listed as an offering in both the mathematics and music departments so as to satisfy curriculum requirements in either field. It could also be structured to fulfill a quantitative requirement in liberal arts. Since the material interrelates with and complements subjects such as calculus, music theory, and physics of sound, it could be a part of an interdisciplinary “course cluster” offered by some universities. The course will need no formal prerequisites. Beyond the high-school level all mathematical and musical concepts are explained and developed from the ground up. As such the course will be attractive not only to students who have interests in both subjects, but also to those who are fluent with one and desire knowledge of the other, as well as to those who are INTRODUCTION xiii familiar with neither. Thus the course can be expected to attract students at all levels of college (even graduate students), representing a wide range of majors. Accordingly, the course must accommodate the different sets of backgrounds, and the instructor must be particularly sensitive to the fact that certain material is a review to some in the class while being new to others, and that, depending on the topic, those subgroups of students can vary, even interchange. More than the usual amount of care should be taken to include all the students all the time. Of course, the topics in the book can be used selectively, rearranged, and/or augmented at the instructor’s discretion. The instructor who finds it impossible to cover all the topics in a single semester or quarter could possibly omit some of the abstract algebra in Chapters 7 and 8. However it is not advisable to avoid abstract mathematical concepts, as this is an important part of this integrative approach. Viewing, listening to, and discussing musical examples will be an im- portant part of the class, so the classroom should be equipped with a high- quality sound system, computer hookup, and a keyboard. Some goals of the course are as follows:

∙ To explore relationships between mathematics and music.

∙ To develop and enhance the students’ musical knowledge and creativ- ity.

∙ To develop and enhance the students’ skills in some basic mathematical topics and in abstract reasoning.

∙ To integrate the students’ artistic and analytic skills.

∙ (if equipment is available) To introduce the computer and synthesizer as interactive tools for musical and mathematical creativity.

Regarding the last item, my suggestion is that students be given access to some computer stations that have a notation/playback software such as Finale and that the students receive some basic instruction in how to enter notes and produce playback. It is also helpful if the computer is connected via a MIDI (Musical Instrument Digital Interface) device to a tunable keyboard synthesizer, in which case the student also needs to have some instruction in how the software drives the synthesizer. Some of the homework assignments should ask for a short composition which demonstrates a specific property or principle discussed in the course, such as a particular form, melodic , or the twelve-tone technique, xiv INTRODUCTION which might then be turned in as a sound file along with a score and possibly an essay discussing what was done. The course can be enhanced by a few special guest lecturers, such as a physicist who can demonstrate and discuss the acoustics of musical instru- ments, or a medical doctor who can explain the mechanism of the human ear. It can be quite educational and enjoyable if the entire group of students are able to attend one or more musical performances together, e.g., a symphony orchestra, a string quartet, an a cappella vocal ensemble, ragtime, modern jazz. This can be integrated in various ways with a number of topics in the course, such as modes, scales, form, rhythm, harmony, intonation, and timbre. The performance might be ensued in the classroom by a discussion of the role played by these various musical components, or even an analysis of some piece performed. There is only a brief bibliography, consisting of books on my shelf which aided me in writing this book. I recommend all these sources as supplements. A lengthy bibliography on mathematics and music can be found in David J. Benson’s grand treatise Music: A Mathematical Offering [2], which gives far more technical and in-depth coverage of nearly all the topics addressed here, plus more; it could be used as a textbook for a sequel to the course for which the present book is intended.

Acknowledgements. I want to thank Edward Dunne, Senior Editor at the American Mathematical Society, for providing the initial impetus for this project by encouraging me to forge my course notes into a book, and carefully reading the first draft. I also thank two of my colleagues in the Department of Mathematics at Washington University, Professors Guido Weiss and Victor Wickerhauser, for some assistance with Chapter 10. This book was typeset by LATEXusingTEXShop. The music examples were created with MusiXTEX and the figures with MetaPost.

David Wright Professor and Chair Department of Mathematics Washington University in St. Louis St. Louis, MO 63130

Bibliography

[1] G. Assayag, H. G. Feichtinger, and J. . Rodrigues (eds.), Mathematics and Music: A Diderot Mathematical Forum, Springer-Verlag, 1997.

[2] David J. Benson, Music: A Mathematical Offering, Cambridge University Press, 2007.

[3] David Cope, New Music Composition, Schirmer Books, New York, 1977.

[4] Trudi Hammel Garland and Charity Vaughn Kahn, Math and Music: Harmonious Connections, Dale Seymour Publications, 1995.

[5] Leon Harkleroad, The Math Behind the Music, Cambridge University Press, 2007.

[6] Sir James Jeans, Science and Music, Cambridge University Press, 1937, Reprinted by Dover, 1968.

[7] Thomas D. Rossing, The Science of Sound, second ed., Addison Wesley, 1990.

[8] Rex Wexler and Bill Gannon, The Story of Harmony, Justonic Tuning Inc., 1997.

147

Index

a cappella, 132 logarithm, 53, 54, 56–58 vocal ensemble, xiv natural logarithm, 56 abstract algebra, xii–xiii, 73 bass clef, 5, 41, 43, 65 accidental, 7, 37, 38 beam, 18–19 cautionary, 24 , 17–22, 27, 92 rules, 23–24 meter, 22–23 acoustics Beethoven, Ludwig van, 43 musical instruments, xiv Benedetti, G. ., 144 ad lib,17 Berg, Alban, 63 additive measurement, 46–49, 56, 76, 135 bijective, 75 Aeolian, 12, 42 binary form, 27 Ain’t No Sunshine,92 blues, 132 alto, 19 bounded function, 108, 109 amplitude, 111, 120 Brahms, Johannes, 27 loudness, 119 of harmonics, 112 , 11 Ancient Greeks, 11, 50 caesura, 133 Didymus the Musician, 127 calculus, x–xii, 20, 109, 110, 121 Eratosthenes of Cyrene, 88 Fundamental Theorem, 115 Pythagoras, ix, 129, 137 cancellation, 86 arpeggio, 40 cautionary accidental, 24 array (row chart), 64 cent, 47–49, 51, 53, 56–59, 61, 85, 96–99, 101, associativity, 73 112, 125–127, 129, 132, 135, 138, 142, audibility range (human), 4, 5, 109, 112, 116, 144, 145 128 comma of Didymus, 127 augmented comma of Pythagoras, 129 chord, 34–36, 38, 133 detuning, 61–62 fifth, 39, 77 ratio, 48 , 39 chipmunk effect, 117 triad, 37 Chopin, Fr´ed´erik, 27 axiom, 70 chord, x, xi, 31, 41–43, 52, 67, 101, 144 axis, 49 augmented, 34–36, 38, 133 , 49 , 39 horizontal, 17 consonant, 35 logarithmic, 59 diatonic, 140 vertical, 17 diminished, 34, 36 diminished , 35–36, 38, 133 Babbitt, Milton, 63 dissonant, 65 Bach, J. S., ix, 143 , 38 banjo, 52 four-note, 34 bar, 22, 25 half-, 36, 132 line, 39 identification, xi barbershop, 132 incomplete, 43 base jazz, 133, 144 exponential function, 54, 58 just tuning, 131–136

149 150 INDEX

labeling, 31, 36–37, 42, 43 column major, 32–34, 38, 131, 132 row chart, 78, 81 , 35–36, 132 combinatorics minor, 33–34 twelve-tone music, 63 , 34–37, 52, 132, 141 comma, xii, 144 , 133 mean-tone, 142–143 progression, 39–41 of Didymus, 127, 135, 139, 144 ratio, 145 of Pythagoras, 129–130, 139, 145 seventh, 34–36, 39, 42, 132, 136, 141 commutativity, 74–75, 85 spelling, 37–39, 41, 42 ring, 85, 93 suffix, 36–37 compass and rule, 123 tuning, 141, 143 composition type, 36, 42 group operation, 81–82 voicing, 33, 145 interval, 93 chromatic, 10, 61, 100 law of, 73–76, 85 𝑛-, 59, 61, 77, 83, 84, 101, 130 of functions, 73 interval, 25, 26, 57–58, 62, 77, 134 of intervals, 76–77, 83–84, 95, 98 group of, 76 consonance, 35, 37, 61, 95, 123, 132 non-standard, 77, 80, 94 , 131, 139 scale, x, xi, 7, 10, 13, 51, 52, 61–63, 69, 95, , 127 96, 131, 137, 144 twelve-tone music, 63 just intonation, 140–141 consonant, 141 mean-tone, 142 continuity, xii, 53, 108, 109 non-standard, 61–63, 66, 67, 90 definition, 104–105 Pythagorean, 139 convergence, 110, 116 scale tone, 40 cosine, 2, 105, 111, 113, 114, 116, 120–121 transposition, 26, 28 Fourier series, 109, 110 unit, 48, 101 shift and stretch, 106–107 circle, 75, 77 cycle clock, 72 in 𝑚 on 𝑛 pattern, 91–93 comma of Pythagoras, 139 cyclic fifths, 129, 137, 139, 144, 145 group, 82–83 progression, 141 permutation, 10–11, 15, 36 interval, 62, 84 augmented chord, 34 just fifths, 130 diminished , 35 modular equivalence, 71–72 non-trivial, 11, 33 circle of fifths, 36, 39–40 clarinet, 113, 117 decibel, 119 formant, 118 derivative, 109 class, 83 detuning, 61–62, 80, 81, 95 equivalence, 14, 18, 31, 45, 71–72, 76 fifths, 141 interval, 32 diatonic, 10, 26 note, 7, 18, 23, 33, 35–38, 42, 77–80 clock position, 142 row chart, 64–66 major triad, 139 scale tone, 38 note, 13, 26, 38, 137 classical mean-tone scale, 141–143 note class, 42 clef, x, xi, 4, 5, 22, 49 , 143 bass, 41, 43, 65 scale, x, 10, 26, 137 treble, 41 Pythagorean, 138 clock, 137 scale note, 140 𝑚-hour, 72 scale tone, 38 circle of fifths, 40 mean-tone, 142 diatonic position, 142 spelling, 39 mean-tone, 141–142 transposition, 26–28 modular, 80 whole tone, 142 positions, 72 Didymus INDEX 151

comma of, 127, 135, 139, 144 enharmonic, 7, 38, 39 Didymus the Musician, 127 octave, x, 6–7, 10, 18, 31–33, 40, 48–49, diminished 52, 95 chord, 34, 36 ratio, 45 triad, 37, 38, 52 relation, x, xi, 4, 7, 14, 31, 45, 71, 72 diminished seventh scale, 8, 11 chord, 35–36, 38, 133 , 8 discontinuity, 104, 105, 108 Eratosthenes of Cyrene, 88 dissonance, 35, 37, 65, 131 Eratosthenes, sieve of, 88, 93 Pythagorean , 139 Euler phi function, 62, 83, 90 distributivity, 85 exotic divide, 1 interval, 128 Division Algorithm, 2, 13, 31, 32, 82, 84, 87 , 128 generalized, 70–72 , 128 divisor exponent, 53, 82 greatest common, 62 equal temperament, x do,13 in a group, 81 domain, 86–87 properties, 53, 58 function, 2, 53, 54, 105, 113 unique factorization, 124 integral, 94 exponential, x principal ideal, 87 function, 51, 53–54, 83 dominant, 39 interval (musical), xi Dorian, 12, 15, 41 notation in group, 81 dot, 17–22, 27, 29, 110 extending by periodicity, 105 doubleflat,7,9,38 double sharp, 7, 38 fa,13 duration, 24, 27, 28 fifth, 6, 9, 14, 32, 42, 46, 70, 76, 83, 84, 96–99, effected by dots, 19–21 112, 126, 136, 139, 141, 143–145 note, 17 augmented, 39, 77 tied notes, 22 cents, 58 tuplet, 21 circle, 137, 139, 141, 144, 145 durational note, 18, 22, 23, 28 detuning, 141 flatted, 42 ear, xiv generating interval, 83 ear drum, 107 just, 125–127, 129–131, 134, 135, 137–139, ecclesiastical mode, 11 142, 143, 145 Edelweiss,27 keyboard, 121, 125, 129 eighth note, 18, 23, 27, 28, 91, 92 large, 142 element, 10 , 32 of a set, 1 , 35 enharmonic mean-tone, 142, 144 chord, 38 mean-tone scale, 141–143 equivalence, 7, 38, 39 minor chord, 33 spelling, 37 , 35 equal temperament, x, xi, 6, 48, 95, 96, 99, 124, octave equivalence, 32 132–134, 137, 143 parallel, 31 𝑛-scale, 62 perfect, 6 exponent, x progression, 40 fifth, 129 Pythagoras, 129 irrational intervals, 130–131 small, 139 non-standard, xi, 130 spelling, 38 perfect octave, 95–96 tempered, 129, 131, 142 septimal interval, 128 voicing, 33 seven-tone, 80 wolf, 142 equivalence Five Foot Two, Eyes of Blue,27 class, 4, 7, 14, 18, 31, 45, 71–72, 76 flag, 17–19 152 INDEX

flat, 7, 13 inverse, 54, 75 chord labeling, 36 logarithmic, 54–55, 83 double, 7, 9, 38 monoid, 73 fifth, 42 periodic, x, xii, 105, 107–108, 111, 119–120 , 12 Fourier theory, 109–110 twelve-tone music, 64 piecewise definition, 103–105 flip, 3 position, 108 flute, 103 sawtooth wave, 120 folk music, 27 square wave, 113–117, 120 form, x, xi, xiii, xiv, 24, 26–27, 29 vibration, 107 binary, 27 wrapping, 72, 75 ternary, 27 fundamental, 111 formant, xi, 117–119, 122 Fundamental Theorem of Calculus, 115 vowel, 136 Fourier generating interval, 62, 83–84, 94 coefficient, 110, 119 generator, 62 square wave, 113–116 cyclic group, 82–83 coefficients, 110, 120 ideal, 87 series, 110, 113, 120, 121 of ℤ𝑚, 90–91, 94 truncated, 116 geometric transformation, 3, 13, 24 theory, 109–110 Gershwin, George, 25, 91, 133 Fourier, Joseph, 109 Gershwin, Ira, 25 fourth, 6, 9, 25, 46, 70, 76, 77, 99 Glarean, Heinrich, 11 generating interval, 83 golden ratio, xi just, 126, 135, 140, 141 graph, x, xi, 2–4, 13, 24–26, 58, 103–105, 109, keyboard, 63, 126, 128 113 octave equivalence, 32 exponential function, 53 parallel, 31 logarithmic function, 54, 56 perfect, 6 periodic function, 112, 119 progression, 40 sine and cosine, 106 frequency, ix, 4, 48, 51, 55–59, 108, 109, 117, square wave, 113 120, 136 truncated series, 116 axis, 49 greater whole tone, 126–127, 135, 140, 144 formant, 117–119, 122 greatest common divisor, 62, 87–88, 90–91 fundamental, 111 Greeks, Ancient, 11, 50 interval, 45 Didymus the Musician, 127 keyboard notes, 47 Eratosthenes of Cyrene, 88 logarithm, 49, 55 Pythagoras, ix, 129, 137 periodic function, 108 group, x, 74–77, 83–87, 137 ratio, 49, 55, 61, 123 additive notation, 81–82 set, 45 commutative, 75 string, 124 cyclic, 82–83 tempo, 17 examples, 75 vibration, 50, 107 exponential notation, 81 , 50–52, 124 generator, 62, 90 full diminished chord, 35 interval, 93 function, x, xi, 1, 2, 13, 24, 45, 103, 106, 107 interval ratios, 95, 134 bounded, 108, 109 modular, 75 composition, 73 modular chromatic interval, 76 continuous, 53, 108 modular intervals, 76 Euler phi, 62 musical intervals, 76, 123 exponential, 53–54, 83 rotations, 77 group homomorphism, 75–76 identity, 74 half note, 18, 28 increasing, 53, 54 half-diminished seventh, 132 integral, 113 chord, 36 INDEX 153 half-step, 6–8, 13, 138 negative, 70 harmonic, xii, 21, 120–122, 131–133, 136 non-negative, 17 amplitude, 117 , x formant, 117, 118 positive, xii, 14, 21, 22, 59, 61, 62, 70, 71, keyboard approximation, 132 83, 85, 87–90, 93, 101, 119, 123, 131 overtone, 111–113 as intervals, 95–101 series, 133 prime, 93 square wave, 113–117 prime factorization, 89, 93 timbre, xii ratio, 95–101, 111 triangle wave, 121 ring, 86 variety, 143 small, 131 weight, 117, 121 time signature, 22–23 harmonic analysis, xi, 109 integral, 110, 114 basic theorem, xii, 109–110 integration by parts, 121 timbre, x interpreted as area, 113 , 40 sine, 114 harmony, xi, xiv, 12, 31, 35, 43, 67, 81, 92 integral domain, 86–87, 94 implied, 40 integral musical interval, 100, 125 integers, 96 intercept, 2 non-standard scale, 62 intersection, 1 Western, 34 interval hat keyboard, 95 scale number notation, 13 interval (musical), x, xi, 6–7, 14, 35, 45, 49, 51, hemitone, 138, 144 53, 55, 57, 59, 61, 66, 69, 70, 77, 83, hertz, 4, 47, 108, 109, 136 85, 95, 96, 101, 126, 127, 134, 143 Hertz, Heinrich, 4 𝑛-chromatic, 62, 77, 83–84 homomorphism 5-limit, 128 group, 75–76, 83, 84 audible range, 128 horizontal cent, 47–48 axis, 17 chord, 31, 67 reflection, 26 chromatic, 25, 26, 57–58, 62, 77, 134 structure, xi, 17 non-standard, 80, 94 circle, 62 ideal, 87, 93 class, 32 principal, 87 comma of Pythagoras, 129 identity composition, 76, 95 element, 73–76, 81, 85, 87, 95 consonant, ix, 95 function, 74 detuning, 62 implied harmony, 40 exotic, 128 In the Mood,91 exponential, xi increasing function, 53, 54 fifth, 9, 46, 121 inequality, 1 fourth, 9, 25, 46 infinite summation, 20, 109, 111, 116 generating, 62, 83–84, 90, 94 Fourier series, 110 group of, 76, 123 instrument higher primes, 128–129 formant, 117, 118 integral, 95, 100, 111, 125 stringed, 50 inverse, 76 integer, x–xii, 1, 10, 13, 20, 28, 47, 49, 70–71, irrational, xii, 123, 130–131 78, 82, 85, 87, 89, 93, 100, 105, 123, mean-tone scale, 141–143 124, 135, 145 just, xii, 123, 125, 127, 128, 131–134, 137, Euler phi function, 62 144 even, 87 keyboard, xi, 6, 63, 66, 100 harmony, 96 logarithm, x, xi, 55 meter, 22–23 measurement conversion, 47–48, 56–58 modular, 31–32, 72, 77, 86–87, 90 modular, 33–36, 76, 78, 80, 131 modular equivalence, 72–73 group of, 76 154 INDEX

modular arithmetic, 31–32 chord, 133, 144 modular chromatic Joplin, Scott, 27, 43 group of, 76 JoyTo The World ,24 non-negative, 32 just octave, 45 fifth, 125–127, 129–131, 134, 135, 137–139, octave equivalence, 7 142, 143, 145 opposite, 46, 76, 93 fourth, 126, 135, 140, 141 orientation, 46 interval, xii, 123, 125, 127, 128, 131, 137, positive integers, xii 144 prime, 93, 96 intonation, 134, 139 Pythagorean, 6 chromatic scale, 145 Pythagorean scale, 137–139 scale, 139–141, 143, 145 ratio, 45–50, 52, 59, 61, 66, 76, 93, 101, , 127, 135 134, 135, 138 major third, xii, 125–127, 131, 134–136, mean-tone comma, 142 139–142, 144, 145 rational, xii, 93, 125, 131, 134, 135, 137 minor seventh, 133–135 mean-tone scale, 141 , 127, 129, 132, 134, 135, 138, row chart, 64 140, 141, 144, 145 scale, 7–8 semitone, 127–129, 135, 140 semitone measurement, 31 seventh chord, 132 septimal, 128, 141 tuning, 131–136 sequence, 11, 42 , 131 key, 9, 12–13, 36, 38, 41–42, 66, 144 interval (numerical), 2, 93, 108, 113, 115 major, 143 continuity, 105 minor, 143 equally divide, 123 problems in unequal temperament, 143 half-open, 105 tonic, 31 time, 17 white, 80 intonation, xiv key (keyboard), 11 𝑝-limit, 134 detuning, 62 just, 131–134, 139 key signature, x, xi, 8–10, 12, 15, 41 just chromatic scale, 145 mode, 12 just scale, 139–141, 145 keyboard, 5–7, 11, 62, 80, 85 inverse approximation, xii, 96–101, 125, 129, 132 additive, 78 of harmonics, 111 element, 74–75 fifth, 125 uniqueness, 75 fourth, 126, 128 function, 75 instrument, 143 generating interval, 84 interval, xi, 63, 66, 95, 96, 100, 125 interval, 76 major sixth, 127 multiplicative, 86 major third, 52, 125 inversion, 78, 81 minor sixth, 129 row chart, 64, 65, 67 minor third, 62, 127, 129 Ionian, 12, 24, 37 note, 37, 47, 128 irrational piano, 14 interval, 123, 135 scale, 95 equal temperament, 130–131 semitone, 127, 129 mean-tone scale, 141 step, 126 interval (musical), xii synthesizer, xiii number, 130, 134, 135 temperament, 134 tritone, 128 tuning, 132 tuning, ix, 132, 143 isomorphism, 83 keynote, 9, 12 group, 75–76, 123 iteration, 47, 48, 62, 94, 138, 141 la,13 labeling jazz, xiv, 23, 35, 132, 133 alternate, 37 INDEX 155

chord, 31, 36–37, 42 mean-tone lesser whole tone, 126–127, 135, 140 fifth, 144 LetItBe,29 minor third, 145 Let Me Call You Sweetheart,29 scale, 141–144 letter measure, 22, 23, 26, 28, 40, 43, 91, 93 note class, 6–7, 95 accidental, 23 note identification, 5, 14 beats, 22–23 section, 27 transposition, 25 subscripted, 5, 95, 100, 101 melodic translation, 25 Liebestraum, 143 melodic transposition, 27 limit, 20 melody, x, 12, 15, 24, 26, 40, 43, 81, 94, 133 Fourier series, 110 𝑚 on 𝑛 pattern, 92 Listz, Franz, 143 figure, 24 Locrian, 12, 15 non-standard scale, 62 logarithm, x, 53, 56, 58, 63, 83, 96, 97, 125–127, note, 15, 40 129, 138, 142 pattern, 24 base, 53, 56 symmetry, xiii interval, x, xi, 49 transposition, 25–26 natural, 56–57 meter, x, 22–23, 92 pitch, 55 mi,13 properties, 55, 58 microtuning, 47–48 logarithmic middle , 4, 51 axis, 56, 59 MIDI, xiii function, 54–55 minor, 15 scale, 55 chord, 33–35 loudness, 118 key, 143 lower bound, 70 mode, 12, 36, 38, 41, 42 Lydian, 12, 15, 42 relative, 12 third, 33, 35, 37 major triad, 34–37, 52, 132 chord, 32–34, 38 justly tuned, 140 just, 131–132 minor ninth, 6 key, 143 minor second, 6 mode, 12, 24, 36, 38, 40–43 minor seventh, 6 scale, 13 chord, 34–37, 52, 132, 141 third, 35, 37 just, 133–135, 140 triad, 34–36, 52, 132, 139, 143–145 septimal, 128, 133 justly tuned, 140 minor sixth, 6, 42, 47, 99 Pythagorean, 139 exotic, 128 major ninth, 14 keyboard, 129 , 6 octave equivalence, 32 major seventh, 6 minor third, 6, 14, 28, 33, 42, 46, 51, 77, 84, 144 chord, 35–36, 132 just, 127, 129, 132, 134, 135, 138, 140, 141, generating interval, 83 144, 145 major sixth, 6, 42, 47, 66 keyboard, 62, 127, 129 just, 127, 135 mean-tone, 145 keyboard, 127 Pythagorean, 144 major third, 14, 32, 42, 46, 51, 52, 62, 66, 83, ratio, 47 96 septimal, 128, 134 just, xii, 125–127, 131, 134–136, 139–142, tempered, 144 144, 145 minuet, 27 keyboard, 6 misspelled chord, 38–39 Pythagorean, 6, 139, 144 modality, 11–12 ratio, 47 mode, xiv, 11–12, 15, 42 tempered, 131, 138, 139 Aeolian, 12, 42 Maple Leaf Rag, 27, 43 chord labeling, 36 156 INDEX

Dorian, 12, 41 dotted, 19–21, 110 ecclesiastical, 11 duration, 17–18 Ionian, 12, 24, 37 durational, 17–18, 22, 23, 28 Locrian, 12 eighth, 18, 23, 27, 28 Lydian, 12, 42 half, 18, 28 major, 12, 24, 37, 40–43 head, 17 minor, 12, 41, 42 keyboard, 5, 37, 47, 128 Myxolydian, 12 melody, 15, 40 Phrygian, 12 microtonal, 48 modular, 64, 77 minor chord, 33 clock, 77, 80 non-diatonic, 38 equivalence octave equivalence, 6 integers, 72–73 piano, 5 real number, 71 quarter, 18, 21–23, 27, 28 real numbers, 72 root, 41, 67, 144 group, 75 scale, 7 integer, 72, 77, 86–87, 90, 131 sequence, 11, 40, 133 interval, 33–36, 52, 76, 78, 80, 83 sixteenth, 18–19, 21, 22, 27, 28 𝑛-chromatic, 62 slur, 22 scale, 94 spelled, 37 modular arithmetic, x, xii, 62, 75 subscript notation, 5 𝑛-tone row chart, 79–80 thirty-second, 27 generating intervals, 62 tonic, 15 interval, 31–33 triad, 33 scale, x tuplet, 21 twelve-tone chart, 77–79 value, xi monoid, 73–75, 85, 86, 93, 123 voicing, 33 commutative, 74 white and black, 5, 6, 8, 11 examples, 73–74 whole, 17–18, 21, 22, 28, 29 integral intervals, 95 note class, 7, 23 Moonlight Sonata,43 notehead, 18–19 multi-octave, xii, 94, 131, 135 multiplicative measurement, 46–47, 56, 76 O Christmas Tree,26 multiplicativity O Tannenbaum, 26, 27 interval ratio, 46 octave, 6–7, 14, 32, 42, 45, 47–49, 52, 56, 59, 77, music theory, xii 85, 96–100, 125, 126, 129–131, 135, MyBonnieLiesOverTheOcean,29 136, 138–142, 145 Myxolydian, 12 equal subdivision, 61 equivalence, x, 6–7, 10, 18, 31–33, 40, 48– natural, 7, 23 49, 52, 64, 76, 84, 95, 142, 144 natural logarithm, 56–57 identification, x, 69–70 natural numbers, 1 keyboard, 97 ninth, 6, 7, 76, 98 multi, xii, 94, 131, 135 chord, 133 parallel, 31 major, 14 ratio two, 95–96 minor, 6 twelve-tone music, 63 octave equivalence, 32 one-to-one, 1, 53, 54, 75–77, 82 note, xi, 4–5, 9, 14, 19, 32, 51, 55, 59, 100, 101 onto, 1, 54, 75, 76 accidental, 7, 23 opposite interval, 46, 93 altered, 23 order chord, 31, 37 of group element, 82, 94 chromatic, 10 orientation class, 18, 33, 35–38, 42, 77–80 interval, 46 row chart, 64–66 original row, 64–67, 77, 78, 81, 84 detuning, 61–62 overtone, x, 111–113 diatonic, 10, 26, 137, 140 audible, 112, 116 INDEX 157

integer, x interval, 93 reinforced, 112–113 eleven, 99 series, 112, 132, 133, 136 five, 96 singing, 112 seven, 97 thirteen, 99 parabola, 2 three, 96 parallel two, 95 interval (musical), 31 number, 21, 88–89, 93, 94, 101, 127, 137, pattern of 𝑚 on 𝑛, 90–93, 133 144 perfect fifth, 6 five, 128 , 6 higher, 128–129 period, 105, 108–111, 114–115, 119–120 infinitude, 94, 128 sine and cosine, 105 seven, 128 vibration, 107 three, 128 periodic function, x, xii, 105, 107, 119–120 two, 128 definition, 105 ratio, 95 Fourier theory, 109–110 relatively, 62, 90 sawtooth wave, 120 row, 64 sine and cosine, 105 three, 137 square wave, 113–117 two, 137 tone, 108 principal ideal, 87 periodicity principal ideal domain, 87 extending by, 105, 113, 120 progression, x, 39–41 permutation fifth, 40 cyclic, 10–11, 15, 34–36 fourth, 40 phase shift, 112, 120 time, 17 Phrygian, 12, 15 pure tone, 109 physics of sound, xii Pythagoras, ix, 129, 137, 138 PID, 87 comma of, 129–130, 139, 145 pitch, xi, 4, 6, 18, 21, 24–25, 49–52, 95, 101, 103, Pythagorean, 6 108, 109, 111, 112, 119–121, 123 chromatic scale, 145 𝑚 on 𝑛 pattern, 90–92 major third, 144 altered by accidentals, 7 minor third, 144 continuum, 132 scale, 137–139, 141–144 formant, 117–118 tuning, 129, 138, 139 harmonic, 111 whole tone, 126, 138, 139 harmony, 31 interval, 45, 46 quarter note, 18, 21–23, 27, 28, 92 logarithm, 49, 55 note, 18 radical, 51 overtone, 112 ragtime, xiv, 26 periodic function, 108 Raindrops Keep Falling On MyHead ,26 ratio, 131 range, 2, 53, 54 reinforced overtone, 112–113 ratio, xi, 53, 125, 135 set, 45 𝑛-chromatic unit, 63 slur, 22 cent, 48 spelling, 37 chord, 145 tone, 108 equivalence, 45 vertical axis, 17 fifth, 145 vibration, 107 frequency, 55, 61, 123 polyrhythm, xi, 21, 93 integer, 95–101 popular music, 27 harmonics, 111 prime interval, 45–52, 56, 57, 59, 61, 66, 76, 93, factorization, 89, 93, 95, 97, 100, 124–125, 95, 101, 126–128, 130–135, 137, 138 131, 134–135, 137, 141 audible range, 128 higher, 136 just intonation scale tones, 140–141 158 INDEX

just major chord, 139 circle of fifths, 40 mean-tone comma, 142 diatonic, 143 mean-tone scale, 141–143 , 35 prime, 95 doubled, 132 Pythagorean minor third, 144 half-diminished chord, 36 Pythagorean whole tone, 139 major chord, 32 semitone, 124 major seventh chord, 35 rational minor chord, 33 interval, xii, 93, 123–125, 129, 131, 134, minor seventh chord, 35 135, 137 movement, 141 mean-tone scale, 141 non-discernable, 34–36, 38, 42, 133 rational number, x–xii, 1, 53, 93, 94, 130, 134, note, 41, 67, 144 135, 137, 141 note class, 37, 42, 43 interval, 123–124 numeral, 37 ring, 86 position, 139, 140, 145 rationality progression, 40 tuning, x scale tone, 41–43 re,13 seventh chord, 34 real number, x, xi, 1, 4, 20, 28, 55, 70, 85, 105, spelling, 38 123 voicing, 33 base of exponential function, 54 row, 77–81 dividend, 70 chart, 64–67, 77–81, 84 Fourier coefficient, 109, 110 𝑛-tone, 79–80 interval ratio, 45, 46 original, 64–67, 77–79, 81, 84 modular equivalence, 71–72 prime, 64 musical intervals, 76 rubato,17 positive, 46, 50, 51, 53 ring, 86 sawtooth wave, 120 semitone, 47 scale, xi, xiv, 7–8, 14, 62, 143 real numbers 𝑛-chromatic, 52, 59, 61–63, 66, 67, 84, 90, group, 83 94, 101, 130 reflection 𝑝-limit, 137 horizontal, 26 chromatic, x, xi, 10, 40, 48, 51, 52, 61–63, reflexivity, 4, 7 69, 95, 96, 131, 137, 144 reinforced overtone, 112–113 just intonation, 140–141, 145 relation, 14, 31 mean-tone, 142 equivalence, 14, 31, 71, 72 non-standard, 61, 62 relative minor, 12 Pythagorean, 145 relatively prime, 62, 90 diatonic, x, 10, 26, 137 remainder, 1 mean-tone, 142 repeating patterns, 24 equally tempered, xii, 95, 97, 99, 143 rest, 19 equivalence, 8, 11 retrograde, 65 just intonation, 139–141, 143, 145 retrogression, 26 major, 13 Rhapsodyin Blue , 91, 133 mean-tone, 141–144 rhythm, x, xiv, 23–26, 28 mode, 11 figure, 91, 92 modular arithmetic, x pattern, 25, 91, 92 non-Western, xi swing, 23 numbers, 13 rhythmic translation, 25, 27 Pythagorean, 137–139, 141–144 ring, x, 85–87, 93, 94 chromatic, 139 trivial, 86 standard, x, 7–12, 33 Roman numeral, 36, 42, 43, 145 tone, 13, 15, 28, 36–38, 40–43, 137, 138, root, 36, 37, 52, 112, 131, 136, 139, 143, 145 140 augmented chord, 34 unequal temperament, 131 chord labeling, 36 scherzo, 27 INDEX 159

Schoenberg, Arnold, 63 equal temperament, 143 section, 26–27 half-diminished, 132 semitone, 6–7, 9, 32, 35, 42, 46, 48, 49, 53, 56, major, 6, 83, 132 57, 59, 63, 70, 77, 83, 96, 97, 99, 100, major seventh chord, 35 135, 142 minor, 6, 132, 134, 135, 140, 141 cent, 48 just, 133 comma of Pythagoras, 130 septimal, 128, 133 exotic, 129 minor seventh chord, 35 fifth, 58 septimal, 136 generating interval, 83 spelling, 38 just, 127–129, 135, 140 sharp, 7, 13 keyboard, 127, 129 chord labeling, 36 modular arithmetic, 31–33 double, 7, 38 ratio, 47, 124 key signature, 12 tempered, 138 twelve-tone music, 64 septimal shift, 3, 13, 25, 120 interval, 128, 141 periodicity, 105, 108 minor seventh, 128, 133, 136 sine and cosine, 106–107, 111 minor third, 128, 134 sieve of Eratosthenes, 88, 93 seventh chord, 141 signal analysis, 117 whole tone, 128 signal processing, 113 septuplet, 28 sine, 2, 105, 111, 114–116, 119–121 sequence, 21 Fourier series, 109, 110 chromatic scale, 10 pure tone, 109 cyclic permutation, 10–11, 33, 34 shift and stretch, 106–107 finite, 10 sixteenth note, 18, 19, 21, 22, 27, 28, 92, 93 Fourier coefficient, 109 tuplet, 21 interval, 11, 42, 78 sixth, 6 key, 9–10 major, 6, 42, 47, 66, 135 melodic, 25 just, 127 modular interval, 33–36 keyboard, 127 note, 7–8, 11, 24, 40, 101, 133 minor, 6, 42, 47, 99, 129 note class, 64, 77–78 exotic, 128 pitch, 7–8, 25, 111 slope, 2 retrograde, 65 slur, 22 row chart, 65–67, 81, 84 sol,13 semitone, 42 solmization, 13 serial music, xi, 63 soprano, 19 series sound pressure, 119 Fourier, 110, 116, 120 sound wave, 4 harmonic, 133 spelling, 37–39, 41, 42 overtone, 112, 132, 133, 136 fifth, 38 truncated, 116 note, 15 set, xi, 1, 4, 10, 14, 53, 73, 76, 91 root, 38 group, 75 seventh, 38 interval, 76 third, 38 monoid, 73, 74 twelve-tone music, 64, 66 non-empty, 74 square wave, 113–117, 120, 121 one-to-one correspondence, 53 staff, x, 4, 17, 19, 42, 49, 59, 95 pitch, 45 stem, 18–19 ring, 85 flag, 17 seven-tone, 80–81 step, 6–8, 42, 56, 62, 66, 70, 84, 98, 138 seventh, 6, 136 keyboard, 126 chord, 34–36, 39, 42, 132, 136 octave equivalence, 32 just, 132 quarter, 99 septimal, 141 whole, 62 160 INDEX straight line, 2 just, 125–127 stretch, 3, 13, 114, 117 major chord, 32 logarithmic function, 56 minor, 14, 28, 35, 37, 42, 46, 47, 51, 62, periodicity, 105, 108–109 77, 84, 129, 132, 134, 135, 138, 144, sine and cosine, 106–107, 111 145 Strike Up The Band,25 just, 127 string, 52 keyboard, 127 fret, 124 septimal, 128 vibration, 50, 123–124 minor chord, 33 string quartet, xiv minor seventh chord, 35 structure Pythagorean, 143 horizontal, 17 seventh chord, 34 subgroup, 137 spelling, 38 𝑝-limit, 134 voicing, 33 subsequence, 10 thirty-second note, 27 subset, 1, 10, 72 ti,13 suffix tie, 17, 22–23, 29, 92 chord, 36–37, 41, 42, 67 timbre, x, xiv, 103, 108, 111, 112 summation formant, 118 formula for sine and cosine, 106 harmonic, xii infinite, 109, 111, 116 harmonic analysis, x swing rhythm, 23 time swing time, 91 horizontal axis, 17 symmetry time signature, xi, 17, 21 cyclic, 33–35 Tin Pan Alley, 27 equivalence relation, 4, 7 tonal center, 12, 66 form, 27 tone, ix, 4, 33, 111, 112, 117, 119, 120, 131 internal, 27 periodic function, 108–109 melodic, xiii, 26 pure, 109 symphony, 24 scale, 13, 15, 36–38, 40–42, 137, 138, 140 symphony orchestra, xiv timbre, 103, 112 synthesizer, 80, 113 trigonometry, x detuning, 61 vibration, 107 tunable, xiii tonic, 12, 31 mode, 12 temperament note, 15 𝑛-scale, 62 triad, 42, 140 equal, x, 124, 128, 129, 132–134, 137, 143 transcendental number, 46 irrational, 130–131, 134 𝑒,56 mean-tone, 145 transformation, 27 unequal, 131, 143–144 geometric, x, 3, 13, 24 tempo, 17, 23 retrogression, 26 tension, 50 transformation of graphs, 3, 107 tenth, 132 flip, 3 voicing, 132 shift, 3 ternary form, 27 sine, 111 tetrachord, 8 stretch, 3, 56 The Rose,29 transitivity, 4, 7 third, 76, 98, 112, 136, 139, 141, 144, 145 translation, 24–25 minor, 140, 141 horizontal, 25 equal temperament, 143 melodic, 25 just major, xii rhythmic, 25, 27 labeling, 37 transposition, 25–26 major, 14, 32, 35, 37, 42, 46, 47, 51, 52, chromatic, 26, 28 62, 66, 83, 96, 131, 134–136, 138–142, diatonic, 26–28 144, 145 melodic, 27 INDEX 161

row chart, 64–65 tone, 108 treble clef, 4, 5, 41 violin, 22, 24, 50, 103, 107, 132 triad, 33–34 vocal cords, 107 augmented, 34, 37 voice leading, 39 diminished, 34, 37, 38, 52 voicing, xi, 33, 36, 136 just, 140 ratio, 131 just intonation scale, 139 tenth, 132 major, 34–36, 52, 132, 141, 143–145 volume, 111 justly tuned, 140 vowel, xi, 103, 136 Pythagorean, 139 formant, 117–118, 136 minor, 34–37, 52, 132, 141 justly tuned, 140 waveform, 113, 117, 121 tonic, 42, 140 Webern, Anton, 63 triangle wave, 121 Well-Ordering Principle, 2, 70–71, 82, 87, 89 trigonometry, x, 2, 106 Well-Tempered Clavier, 143 tone, x Western harmony, 34 triplet, 21, 23, 28, 133 Western music, 31, 33, 34, 61, 141 tritone, 6, 14, 42, 66, 76, 99 When The Saints Go Marching In,25 exotic, 128 whole note, 17–18, 21, 22, 28, 29, 43 keyboard, 128 whole tone, 6 , 103 diatonic, 142 trumpet, 107 greater, xii, 126–127, 135, 138, 140, 144 formant, 118 lesser, 126–127, 135, 140 tuning, x, xi, 4, 51, 109, 130, 141, 143 Pythagorean, 126, 138, 139 𝑝-limit, 128, 134, 137 septimal, 128 3-limit, 137, 141 wolf fifth, 142 5-limit, 139, 141 wrapping function, 72, 75 alternative, xii chord, 143 detuning, 61–62 fork, 109 jazz chords, 144 just, 131–136, 139 just intonation scale, 140 keyboard, 47, 132 micro, 47–48 Pythagorean, 129, 138, 139 rationality, x seventh chord, 34 tuplet, 17, 21, 28 twelve-tone chromatic scale, xi music, xi, xiii, 63–67 row chart, 77–79

union, 1 unique factorization, x, xii, 89, 100, 131 rational numbers, 124–125, 135 unison interval, 46, 70, 76, 84 unit, 86 in ℤ𝑚,90 upper bound, 70, 71 vertical axis, 17 vibration, 4, 21, 107–108 string, 50, 123–124