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How Music and Mathematics Relate

How Music and Mathematics Relate

Professor David Kung Ph.D.
St. Mary's College of Maryland
Course No.  1373
Course No.  1373
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Course Overview

About This Course

12 lectures  |  46 minutes per lecture

Gain new perspective on two of the greatest achievements of human culture—music and math—and the fascinating connections that will help you more fully appreciate the intricacies of both.

Great minds have long sought to understand the relationship between music and mathematics. On the surface, they seem very different. Music delights the senses and can express the most profound emotions, while mathematics appeals to the intellect and is the model of pure reasoning.

Yet music and mathematics are connected in fundamental ways. Both involve patterns, structures, and relationships. Both generate ideas of great beauty and elegance. Music is a fertile testing ground for mathematical principles, while mathematics explains the sounds instruments make and how composers put those sounds together. Moreover, the practitioners of both share many qualities, including abstract thinking, creativity, and intense focus.

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Gain new perspective on two of the greatest achievements of human culture—music and math—and the fascinating connections that will help you more fully appreciate the intricacies of both.

Great minds have long sought to understand the relationship between music and mathematics. On the surface, they seem very different. Music delights the senses and can express the most profound emotions, while mathematics appeals to the intellect and is the model of pure reasoning.

Yet music and mathematics are connected in fundamental ways. Both involve patterns, structures, and relationships. Both generate ideas of great beauty and elegance. Music is a fertile testing ground for mathematical principles, while mathematics explains the sounds instruments make and how composers put those sounds together. Moreover, the practitioners of both share many qualities, including abstract thinking, creativity, and intense focus.

Understanding the connections between music and mathematics helps you appreciate both, even if you have no special ability in either field—from knowing the mathematics behind tuning an instrument to understanding the features that define your favorite pieces. By exploring the mathematics of music, you also learn why non-Western music sounds so different, gain insight into the technology of modern sound reproduction, and start to hear the world around you in exciting new ways.

Among the insights offered by the study of music and mathematics together are these:

  • Harmonic series: The very concept of musical harmony comes from mathematics, dating to antiquity and the discovery that notes sounded together on a stringed instrument are most pleasing when the string lengths are simple ratios of each other. Harmonic series show up in many areas of applied mathematics.
  • "Air on the G String": One of Bach's most-loved pieces was transposed to a single string of the violin—the G string—to give it a more pensive quality. The mathematics of overtones explains why this simple change makes a big difference, even though the intervals between notes remain unchanged.
  • Auditory illusions: All voices on cell phones should sound female because of the frequency limits of the tiny speakers. But the human brain analyzes the overtone patterns to reconstruct missing information, enabling us to hear frequencies that aren't there. Such auditory illusions are exploited by composers and instrument makers.
  • Atonal music: Modern concert music is often atonal, deliberately written without a tonal center or key. The composer Arnold Schoenberg used the mathematics of group theory to set up what he called a "pan-tonal" system. Understanding his compositional rules adds a new dimension to the appreciation of this revolutionary music.

In 12 dazzling lectures, How Music and Mathematics Relate gives you a new perspective on two of the greatest achievements of human culture: music and mathematics. At 45 minutes each, these lectures are packed with information and musical examples from Bach, Mozart, and Tchaikovsky to haunting melodies from China, India, and Indonesia. There are lively and surprising insights for everyone, from music lovers to anyone who has ever been intrigued by mathematics. No expertise in either music or higher-level mathematics is required to appreciate this astonishing alliance between art and science.

A Unique Teacher

It is a rare person who has the background to teach both of these subjects. But How Music and Mathematics Relate presents just such an educator: David Kung, Professor of Mathematics at St. Mary's College of Maryland, one of the nation's most prestigious public liberal arts colleges. An award-winning teacher, mathematician, and musician, Professor Kung has studied the violin since age four, and he followed the rigorous track toward a concert career until he had to choose which love—music or mathematics—would become his profession and which his avocation. At St. Mary's College, he combines both, using his violin as a lecture tool to teach a popular course on the mathematical foundations of music. He even has students invent new musical instruments based on mathematical principles.

In How Music and Mathematics Relate, you see and hear some of these ingenious creations, which shed light on the nature of all sound-producing devices. Across all 12 lectures Professor Kung plays the violin with delightful verve to bring many of his points vividly to life.

Uncover Musical Structure Using Math

You will discover how mathematics informs every step of the process of making music, from the frequencies produced by plucking a string or blowing through a tube, to the scales, harmonies, and melodies that are the building blocks of musical compositions. You even learn what goes on in your brain as it interprets the sounds you hear. Among the fascinating connections you'll make between music and mathematics are these:

  • Woodwind mystery: Why can a clarinet produce sounds much lower than a flute? Both are vibrating tubes of similar length. A student-designed instrument called the Wonder Pipe 4000 demonstrates how mathematics predicts this phenomenon.
  • Why is a piano never in tune? Elementary number theory explains the impossibility of having all the intervals on a piano in tune. Study the clever solutions that mathematicians, composers, and piano tuners have devised for getting as close as possible to perfect tuning.
  • Timbre: Nothing is more distinctive than the "twang" of a plucked banjo string. But take off the initial phase of the sound—the "attack"—and a banjo sounds like a piano. Analyze different sound spectra to learn what gives instruments their characteristic sound or timbre.
  • Using fractions to show off: Professor Kung plays a passage from Mendelssohn's Violin Concerto to demonstrate a common trick of showmanship for string players. The technique involves knowing how to get the same note with different fractional lengths of the same string.

And you'll hear how one of the greatest philosophers and mathematicians of all time described the connection between music and mathematics. "Music is a secret exercise in arithmetic of the soul, unaware of its act of counting," wrote Gottfried Wilhelm Leibniz, coinventor of calculus with Isaac Newton. What Leibniz means, says Professor Kung, is that music uses many different mathematical structures, but those structures are hidden. With How Music and Mathematics Relate, you'll see these hidden connections come to light.

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12 Lectures
  • 1
    Overtones—Symphony in a Single Note
    Start the course with a short violin passage from Bach, played by Professor Kung. Then analyze the harmonic series behind a single note, which involves a mixture of different frequencies, called overtones or harmonics. Learn about the physics of stringed and wind instruments, and study the sounds produced by a range of instruments, including the violin, flute, clarinet, timpani, and a fascinating instrument invented by Professor Kung’s students. x
  • 2
    Timbre—Why Each Instrument Sounds Different
    After hearing the opening measures of Bach’s “Air on the G String,” investigate why this piece is conventionally played on a single string of the violin. The reason has to do with timbre, which determines why a flute sounds different from a violin and why a melody played on the G string sounds not just lower, but altered. The study of timbre introduces you to a mathematical idea called the Fourier transform—and how it relates to the anatomy of your inner ear. x
  • 3
    Pitch and Auditory Illusions
    The fundamental frequency of a male voice is too low to be reproduced by the speaker of a cell phone. So why don’t all callers sound like women? Learn that the answer involves the way your brain fills in missing information, convincing you that you hear sounds that aren’t really there. Explore examples of auditory illusions that will leave you wondering if you can ever believe your ears again. x
  • 4
    How Scales Are Constructed
    Professor Kung contrasts a passage from Vivaldi with a Chinese folk tune. Why is one so easily distinguishable from the other? Probe the diverse mathematics of musical scales, which explains the characteristic sound of different musical traditions. Learn how a five-note scale is constructed versus a more complex seven-note scale. What are the relative advantages of each? As a bonus, discover why no piano is ever in tune. x
  • 5
    How Scale Tunings and Composition Coevolved
    Compare passages from Bach’s “Chaconne” and a very modern piece, noting how the compositional styles of Western music have evolved alongside small differences in scale tunings. Then explore the mathematics of tuning, focusing on how the exact pitches in a scale are calculated and why there are 12 notes per octave in Western music. Investigate the alternatives, including a scale with 41 notes per octave. x
  • 6
    Dissonance and Piano Tuning
    Dissonance is a discordant sound produced by two or more notes sounding displeasing or rough. The “roughness” is quantifiable as a series of beats—a “wawawa” noise caused by interfering sound waves. Learn how to predict this phenomenon using basic trigonometry. Consider several examples, then discover how to use beats to tune a piano. End with a mathematical coda, proving the beat equation using basic algebra and trigonometry. x
  • 7
    Rhythm—From Numbers to Patterns
    All compositions depend on rhythm and the way beats are grouped under what are called time signatures. Begin with a duo for clapping hands. Next, probe the effect produced by a distinctive change in the grouping of beats called a hemiola. Also investigate polyrhythms, the simultaneous juxtaposition of different rhythms. Listen to examples from composers including Handel, Tchaikovsky, and Chopin. Close with an unusual exercise in which you use musical notation to prove a conjecture about infinite sums. x
  • 8
    Transformations and Symmetry
    Bach and other composers played with the structure of music in ways similar to what would later be called mathematical group theory. Explore techniques for transforming a melody by inversion, reversal, transposition, augmentation, and diminution. End with a table canon credited to Mozart, in which the sheet music is read by one musician right-side up and by the other upside down. Professor Kung is joined by a special guest for this duet. x
  • 9
    Self-Reference from Bach to Gödel
    Music and mathematics are filled with self-reference, from Bach’s habit of embedding his own name in musical phrases, to Kurt Gödel’s demonstration that mathematics cannot prove its own consistency. Embark on a journey through increasingly complex levels of self-reference, discovering that music and mathematics are like a house of mirrors, reflecting ideas between them. For example, the table canon from Lecture 8 can be displayed on the single face of a Möbius strip. x
  • 10
    Composing with Math—Classical to Avant-Garde
    Sometimes composers create their works using mathematics. Mozart did this with a waltz, whose sequence of measures was determined by the roll of dice—with 759 trillion resulting combinations. Learn how Arnold Schoenberg used mathematics in the 20th century to design an alternative to tonal music—atonal music—and how a Schoenberg-like system of encoding notes has more recently made melodies searchable by computer. x
  • 11
    The Digital Delivery of Music
    What is the technology behind today’s recorded music? Delve into the mathematics of digital sampling, audio compression, and error correction—techniques that allow thousands of hours of music to fit onto a portable media player at a sound quality that is astonishingly good. Investigate the difference between analog and digital sound, and explore the technology that allows Professor Kung’s untrained singing voice to be recorded perfectly in tune. x
  • 12
    Math, Music, and the Mind
    Conclude with an eight-part finale, in which you range widely through the territory that connects mathematics, music, and the mind. Among the questions you address: What happens in the brain of an infant exposed to music? Why do child prodigies often excel in the areas of math, music, or chess? And how do creativity, abstraction, and beauty unite music and mathematics, despite being on opposite ends of the arts and sciences? x

Lecture Titles

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David Kung
Ph.D. David Kung
St. Mary's College of Maryland

Dr. David Kung is Professor of Mathematics at St. Mary's College of Maryland. He earned his B.A. in Mathematics and Physics and his Ph.D. in Mathematics from the University of Wisconsin, Madison. Professor Kung's musical education began at an early age with violin lessons. As he progressed, he studied with one of the pioneers of the Suzuki method and attended the prestigious Interlochen music camp. While completing his undergraduate and graduate degrees in mathematics, he performed with the Madison Symphony Orchestra. Professor Kung's academic work focuses on mathematics education. Deeply concerned with providing equal opportunities for all math students, he has led efforts to establish Emerging Scholars Programs at institutions across the country. His numerous teaching awards include the Homer L. Dodge Award for Excellence in Teaching by Junior Faculty, given by St. Mary's College, and the John M. Smith Teaching Award, given by the Maryland-District of Columbia-Virginia Section of the Mathematical Association of America. Professor Kung's innovative classes, including Mathematics for Social Justice and Math, Music, and the Mind, have helped establish St. Mary's as one of the preeminent liberal arts programs in mathematics. In addition to his academic pursuits, Professor Kung continues to be an active musician, playing chamber music with students and serving as the concertmaster of his community orchestra.

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Reviews

Rated 4.7 out of 5 by 52 reviewers.
Rated 5 out of 5 by UNIQUE PERSPECTIVE Although I have a undergraduate degree in instrumental music, I was a little skeptical about purchasing this course, since, Math was not my best school subject. However, after completing this course, I discovered that my fears were base-less. Dr. Kung gets right to the material in the first lecture, and he had my un-divided attention from then-on-out . For example, He drops dark seeds onto a vibrating tympani head, and it is interesting to see the seed form into a mathematical curve, that can be graphed. His illustrations of dividing a vibrating string into different lengths, and showing how the ancient Greeks used this to relate to musical intervals, were very clear. There are several key points that really make this course "tick" (many of these have been mentioned by other reviewers) 1) Dr. Kungs' knowledge and enthusiasm for his material. He also teaches a course in the same vein, at his home university. 2) The professor's use of the violin to illustrate key ideas, although, in some cases, he also uses other instruments, when appropriate. For example, in one lecture, he explains basic procedures for piano tuning, demonstrates how, and why, a piano will always be slightly out-of-tune. 3) The excellent graphics and illustrations. I found these very useful, when he would discuss a mathematical equation, and highlight the part that he was discussing. I don't have any negative feedback, but I would caution not watching these lectures when ones' mind is fatigued. It does take considerable mental concentration at times, even sometimes stopping the player, in order to digest the wealth of material being presented. There were a few times, when it seemed the math was going "over the top of my head," but once I took time to pay closer attention, I readily understood. I especially appreciated some of the material and resources covered in the final two lectures.. In one of these, of special interest to musicians, Dr. Kung gives information about a website, in which one can "Plug in" musical notes, or motives, into a data base, and one will be given compositions in which these musical sequences or motives appear. Some other reviewers have mentioned the relation of this course to those of Dr. Robert Greenberg, especially Dr. Greenberg's course on, "Understanding the Fundamentals of Music." The style of Professor Kung is very different, but it evident, I think, that he has knowledge of Dr. Greenberg's material. For example, those of you that have taken Dr. Greenberg's course on Understanding Great Music, know that one of his key points is that the term, "Classical Music," should more correctly be called, "Concert Music," unless one is talking about music from the Classical Era. Dr. Kung mentions this same point in an early lecture. Some of the basic material in this series, and the Fundamental of Music series, complement each other quite well. In closing, this is one of the best courses the Teaching Company has ever produced, and I can't recommend it enough, to anyone that has any interest, in either Mathematics, or Music. April 12, 2014
Rated 5 out of 5 by A Masterful Course Well Designed and Presented I have two Great Courses somewhat related. This one which focuses on mathematics and music with the presenter using his violin to demonstrate many of his points and one by Professor Greenberg on "Understanding the Fundamentals of Music" in which he focuses more on the piano to demonstrate. Both courses are excellent and they make a fine pair. I had wondered if there would be too much overlap between these courses but found instead that they compliment each other. Where one (Greenberg) discusses a bit of mathematics (differently from Professor Kung) the other from Professor Kung approaches music mostly from a mathematical standpoint. As a result putting both together I have a much better understanding of music than when I started. One word of warning and that is that both courses get into a great deal of technical material that for those of us who are not serious students of math or music can make for a challenge to stay with them. Especially as each gets deeper into their course. That said one does not have to follow the math or the details of thirds, fourths and augmented sevenths to come away with a much better understanding of why music "works" or does not "work" in terms of our appreciation of music. Having played piano and clarinet in my younger days did help with this material but even without that background I would have gotten a great deal from these courses. I especially found the graphics in this course very useful, especially the showing of wavelengths and frequencies and I came away with a far better understanding of overtones from this course and why they contribute to timbre. His discussion of tuning was excellent and I now understand the difference in various methods of tuning from Just Tuning to equal tempered tuning and why pianos are never fully in tune as well as how they are tuned. His explanation of various scales and how they are useful as well as why there are 12 keys in an octave was enlightening. In this regard he and Professor Greenberg cover the same areas but very differently and the result, at least for me, was a much better understanding having watched both courses. I wondered which of the two courses to watch first and frankly I came away thinking it did not matter much as the two courses will require watching more than once to absorb well the materials they are covering. These are not courses to sit back in your recliner and relax watching as they will require mental engagement to get the most out of them. But I certainly know that the next time I listen to a symphony or a string quartet or a piano solo I will have much more understanding of what I am hearing and think it will give me more enjoyment from the experience. If you are interested in the technical aspects of music this course and the one on "Understanding the Fundamentals of Music" make a fine set and I can recommend either or both. November 25, 2014
Rated 5 out of 5 by Fusion of music and math worked I loved the course. Professor is super talented at both math and music. My math is better than music, but professor pulled the two together to make the course very valuable. If you know either topic, you will learn a lot about the other. Never knew there was so much math in music. Highly recommend if you are interested in these disciplines. November 23, 2014
Rated 2 out of 5 by great subject, disappointing presentation On its face it looked like we found the best fit for a perfect presentation: a mathematician who is also a musician. Unfortunately presentation is not his strong suite, and even worst this course seems like a presentation from the 70's judging by the very limited aids and demo tools being used. It might as well be a radio/ audio course. Moreover, whereas the course description specifically claims you need neither math nor music education to adequately follow the lectures, we found it not to be the case - perhaps it is possible, but certainly not in the way it was presented in this course - naming complicated mathematical concepts (as described by the presenter himself) without clear demonstration of how they apply to their musical does not provide much added value when presented as statements . We did like his character and enthusiasm for the subject, but found him very limited with presentation skills - constantly turning from one camera to another with no natural need for it, emphasizing sometimes trivial issues, but speeding through complicated ones, and certainly not using sophisticated demonstrations of complicated issues. Again on the musical side he could have actually played many of the references (short parts of course, not complete) instead of just listing them. We tried to hang on for about three quarters of the course, It is relevant to mention that my prior education in both fields is much more than the "none required" , and yet I eventually gave it up due to all of the above shortcomings. A real disappointment especially in light of our real interest in the subject and the raving reviews for this course. November 23, 2014
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