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Mathematics and Music David Wright April 8, 20092i About the Author DavidWrightisprofessorofmathematicsatWashingtonUniversityinSt. Louis, where he currently serves as Chair of the Mathematics Department. 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 areasandhasbeenaninvitedspeakeratnumerousinternationalmathematics conferences. Hedesignedandteachesauniversity courseinMathematicsand 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 harmony style called barbershop harmony with jazz, blues, 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 Bar- bershop 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.iiContents Introduction v 1 Basic 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 Octave Identification 69 8 Properties of Integers 87 9 The Integers as Intervals 97 10 Timbre and Periodic Functions 105 11 Rational Numbers As Intervals 125 12 Rational Tuning 139 iiiiv CONTENTSIntroduction The author’s perspective. Mathematics and music are both lifelong pas- sions for me. For years they appeared to be independent non-intersecting in- terests; 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, how- ever, 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, andthatinsome subtle way, Iwasappealingtoskills andinstincts endemic toonesubject whenactivelyengagedwiththeother. Inthiswaythe relationship between mathematical reasoning and musical creativity, and the way humans grasp and appreciate both subjects, became a matter ofinterest 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. Ithasbeenobservedthatmathematicsisthemostabstractofthesciences, music the most abstract of the arts. Mathematics attempts to understand 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 tocircumstantial means of eliciting such innate human re- actions. Thereforeitisnotsurprisingthatthesymbiosisofthetwodisciplines is an age old story. The Greek mathematician Pythagoras noted the integral relationships between frequencies of musical tones in a consonant interval; the 18th century musician J. S. Bach studied the mathematical 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, itissometimesthecasethatapersonwithaninclinationforone of these disciplines views the other with some apprehension: a mathemat- ically inclined person may regard music with admiration but as something vvi INTRODUCTION beyond his/her reach. Reciprocally, the musically inclined often views math- ematics with a combination of fear and disdain, believing it to be unrelated 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 existed 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 afreshman-level college course that, amongotherthings, addresses the issues mentioned above. The book investigates interrelationships between mathe- matics and music. It reviews some background concepts in both subjects as they are encountered. Along the way, the reader will hopefully augment his/herknowledge ofbothmathematicsandmusic. Thetwowillbediscussed anddeveloped sidebyside, theirlanguagesintermingled andunified, thegoal being to break down the dyslexia that inhibits their mental amalgamation and to encourage the analytic, quantitative, artistic, and emotional aspects of the mind to work together in the creative process. Musical and mathe- matical notions are brought together, such as scales/modular arithmetic, oc- tave identification/equivalence relation, intervals/logarithms, equal temper- ament/exponents, overtones/integers, tone/trigonometry, timbre/harmonic analysis,tuning/rationality. Whenpossible, discussionsofmusicalandmath- ematical notions are directly interwoven. Occasionally the discourse dwells forawhile on one subject and notthe 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 diatonic and chromatic 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 in- tegers, rational and real numbers, equivalence relations, geometric transfor- mations, groups, rings, modulararithmetic, unique factorization, logarithms, 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,vii 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. 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 oresoteric, possiblyexcepting thecursoryexcursion into serialmusic (the rationale for which, at least in part, is to ponder the arbitrariness of the twelve-tone chromatic scale). Rather, it explores the foundational common- alities 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 estab- lished,andexamplesthatcouldbeused,involvingnon-Westernmusic(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 out the basic mathematical and musical concepts whichwillbeneededthroughoutthecourse: sets, equivalence relations, functions and graphs, integers, rational numbers, real numbers, pitch, clefs, notes, musical intervals, scales, and key signatures. • Chapter 2deals with thehorizontal structure ofmusic: notevaluesand 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.viii INTRODUCTION • Chapter 5 lays out the mathematical underpinnings for additive mea- surement of musical intervals, relating this to logarithms and exponen- tials. • Equaltemperament(standardandnonstandard)isthetopicofChapter 6, which also gives a brief introduction to twelve-tone music. • 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 harmonics 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 dis- cusses some of the basic just intervals, such as the greater whole tone and the just major third. It also explains why all intervals except multi-octaves in any equal tempered scale are irrational. • Finally, Chapter12describes variousalternative systemsoftuningthat have been used which are designed to give just renditions of certain intervals. Some benefits and drawbacks of each are discussed. Suggestions for the course. This bookis 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 fulfillix 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. Thecoursewillneednoformalprerequisites. Beyondthehighschoollevel 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 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, thecourse mustaccommodate thedifferent setsofbackgrounds, 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. • Todevelopandenhancethestudents’musicalknowledgeandcreativity. • Todevelop andenhance thestudents’ skillsinsomebasicmathematical 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.x INTRODUCTION Regarding the last item, my suggestion is that students be given access to some computer stations that have a notation/playback software such as Fi- naleandthatthestudentsreceivesomebasicinstructioninhowtoenternotes 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 symmetry, or the twelve-tone technique, which might then turned in as a sound file along with a score and possibly an essay describing 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 stu- dents are able to attend one or more musical performances together, e.g, a symphony orchestra, a string quartet, an a capella 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 aidedmeinwritingthisbook. Irecommend allthese sourcesassupplements. 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 projectbyencouragingmetoforgemycoursenotesintoabook,andcarefully readingthefirstdraft. IalsothanktwoofmycolleaguesintheDepartmentof Mathematics at Washington University, Professors Guido Weiss and Victor Wickerhauser, for some assistance with Chapter 10. A This book was typeset by LT X using T XShop. The music examples E Exi were created with MusiXT X and the figures with MetaPost. E David Wright Professor and Chair Department of Mathematics Washington University in St. Louis St. Louis, MO 63130xii INTRODUCTIONChapter 1 Basic Mathematical and Musical Concepts Sets and Numbers. We assume familiarity with the basic notions of set theory, such as the concepts of element of a set, subset of a set, union and intersection of sets, and function from one set to another. We assume famil- iarity with the descriptors one-to-one and onto for a function. Following standard convention, we will denote byR the set of real num- bers,byQthesetofrationalnumbers,andbyZthesetofintegers. Thesesets have an ordering, and we will assume familiaritywith the symbols ,≤,,≥ andbasicpropertiessuch as: Ifa,b,c∈Rwitha bandc 0,thenac bc; + if a,b,c∈R with a b and c 0, then ac bc. We will write R for the + set of positive real numbers,Q for the set of positive rational numbers, and + Z for the set of positive integers: + R =x∈Rx 0 + Q =x∈Qx 0 + Z =x∈Zx 0. + The setZ is sometimes called the set of natural numbers, also denotedN. Some Properties ofIntegers. Given m,n∈Z, we say “m divides n”, and write mn, if there exists q∈Z such that n = qm. Grade school arithmetic teaches that for any positive integers m and n, we can divide n by m to get a remainder r having the property 0≤ r m. For example, in the case m = 9 and n = 123, we have 123 = 13·9+6, so r = 6. This principle generalizes to the case where n is any integer: 12 CHAPTER 1. BASIC CONCEPTS + Division Algorithm. Given m∈Z and n∈Z, there exist q,r∈Z with 0≤ r m such that n = qm+r. We will occasionally appeal to one of the axioms of mathematics called the Well-Ordering Principle, which states: + Well-Ordering Principle. Any non-empty subset of Z has a smallest element. This assertion looks innocent, but cannot be proved without some other similar assumption, so it is taken as an axiom. Intervals of Real Numbers. We will employ the following standard nota- tion for intervals inR: for a,b∈R, (a,b) =x∈Ra x b a,b =x∈Za≤ x≤ b. Similarly, we write (a,b and a,b) for the half-open intervals. Functions and graphs. A function from some subset of R into R has a graph, and we assume familiarity with this notion, as well as the terms domainandrange. We will often use the standard conventions which express a function as y = f(x), where x is the independent variable and y is the dependent variable. When the independent variable parameterizes time, we sometimes denote it by t, so that the function is written y = f(t). A familiar example is y = mx+b, where m,b∈R, whose graph is a straight line having 2 slope m and y-intercept b. Another is the function y = x , whose graph is a parabola with vertex at the origin. 2 y = mx+b y= x y y t ss (0,b) m = t/s x x Two functions which will be especially relevant to our topic are the trigono- metric functions y = sinx and y = cosx.3 y = cosx y = sinx Transformations of Graphs. We will need to understand some proce- dures, called geometric transformations, which move and deform a graph in certain ways. Let c∈R. (1) Vertical shift: The graph of y = f(x) +c is obtained by shifting the graph of y = f(x) upward by a distance of c. (2) Horizontal shift: The graph y = f(x−c) is obtained by shifting the graph of y = f(x) to the right by a distance of c. (3) Vertical stretch: The graph of y = cf(x) is obtained by stretching the graph of y = f(x) vertically by a factor of c. (4) Horizontal stretch: The graph of y = f(x/c) is obtained by stretching the graph of y = f(x) horizontally by a factor of c. (Here we assume c = 6 0.) If c in (1) or (2) is a negative number, we must understand that shifting upward (respectively, to the right) by c actually means shifting downward (respectively, to the left) by a distance of c = −c. If 0 c 1 in (3) or (4) the stretchings are compressions, and if c 0 the stretchings entail a flip about the x-axis in (3), the y-axis in (4). Below are graphs which illustrate some of these transformations for the 2 function y = x : 2 2 2 y =(x+1) y= x y= x +14 CHAPTER 1. BASIC CONCEPTS  2 x 2 y= y =2x 2 Equivalence relations. Let S be a set an let ∼ be a relationship which holds between certain pairs of elements. If the relationship holds between s and t we write s∼ t. For example, S could be a set of solid-colored objects and s ∼ t could be the relationship “s is the same color as t”. We say that∼ is an equivalence relation if the following three properties hold for all s,t,u∈ S: (1) s∼ s (reflexivity) (2) If s∼ t, then t∼ s. (symmetry) (3) If s∼ t and t∼ u, then s∼ u. (transitivity) When these hold, we define the equivalence class of s ∈ S to be the set t∈ S t∼ s. The equivalence classes form a partition of S, meaning that S is the disjoint union of the equivalence classes. Pitch. Amusicaltoneistheresultofaregularvibrationtransmittedthrough the air as a sound wave. The pitch of the tone is the frequency of the vibration. Frequencyisusuallymeasuredincyclespersecond, orhertz,(after the German physicist Heinrich Hertz (1857-1894)) which is abbreviated Hz. Forexample, standard tuning places the note A above middle Cona musical staff at 440 Hz. It is notated on the treble clef as: ¯ G The range of audibility for the human ear is about 20 Hz to 20,000 Hz. We will, however, associate a positive real number x with the frequency x Hz, so + that the set of pitches is in one-to-one correspondence with the setR .5 Notes. In amusical score, specific pitches are called for in amusical score by notes on a staff. We assume familiarity with the usual bass and treble clefs ¯ I G ¯ Middle C as it appears on the treble and bass clefs and the labeling of notes on the lines and spaces of those clefs using the letters A through G. These are notes are arranged as follows on a keyboard instrument. B C D E F G A B C D E F G A B C D Note the presence of “white notes” and “black notes” and the pattern of their juxtapositions. Abstractly, we can envision a keyboard which extends infinitely (and be- yondtherangeofaudibility)inbothdirections,givinganinfinitesetofnotes. This infinite set does not represent all pitches, as there are pitches between adjacentnotes.Werefertothosenotesthatappearontheextended keyboard as keyboard notes. We will be needing a concise way to refer to specific keyboard notes, hence we will employ the following standard convention: The note C which lies four octaves below middle C is denoted C . This note is below the range 0 of the piano keyboard. For any integer n, the C which lies n octaves above C (below C when n is negative) is denoted C . Hence middle C is C , the 0 0 n 4 C below middle C is C , and the lowest C on the piano keyboard is C . The 3 1 other notes will be identified by the integer corresponding to the highest C below that note. The other notes will be identified by the following procedure: First, strip away any sharp of flat alteration, and find the highest C which is lower than or equal to that note. The original note gets the subscript of that C. Hence ♯ ♯ ♯ ♯ ♭ the F below C is F , while the F above C is F . The lowest B on the 4 4 3 46 CHAPTER 1. BASIC CONCEPTS ♭ ♭ ♭ piano keyboard is B , and the B in the middle of the treble clef is B . Note 0 4 ♯ ♭ also that B and C both coincide with C . 4 5 4 Musical Intervals. The interval between two notes can be thought of infor- mally as the “distance” between their two associated pitches. (This is to be distinguished fromthe use ofthe term “interval” in mathematics fora subset of R of the type a,b.) The piano is tuned using equal temperament (to be discussed later in detail), which means that the interval between any two adjacent keys (white or black) is the same. This interval is called a semitone. The interval of two semitones is a step, or major second, hence a semitone is a half-step, sometimes called a minor second. An octave is 12 semitones. Here is a list giving common nomenclature for various intervals: half-step, or minor second (1 semitone) step, major second, or whole tone (2 semitones) minor third (3 semitones) major third (4 semitones) fourth, or perfect fourth (5 semitones) tritone (6 semitones) fifth, or perfect fifth (7 semitones) minor sixth, or augmented fifth (8 semitones) major sixth (9 semitones) minor seventh, or augmented sixth (10 semitones) major seventh (11 semitones) octave (12 semitones) minor ninth (13 semitones) ninth (14 semitones) The meaning of the term “interval” will be made mathematically precise later, but for nowwe will speak in terms of steps, half steps, fourths, octaves, etc.Also, we willlaterdiscuss small modificationsoftheseintervals(e.g.,just and Pythagorean intervals), so to avoid confusion we sometimes refer to the intervals between notes on the abstract infinite keyboard as keyboard inter- vals, or tempered intervals. For example we will introduce the Pythagorean major third, which is greater than the keyboard’s major third. We call intervals positive or negative according to whether they are up- ward or downward, respectively. We sometimes indicate this by using the terms “upward” and “downward” or by using the terms “positive “ and “negative” (or “plus” and “minus”). The interval from C to E could be 4 3

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