How Music and Mathematics Relate

Course No. 1373
Professor David Kung, Ph.D.
St. Mary's College of Maryland
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145 Reviews
89% of reviewers would recommend this product
Course No. 1373
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What Will You Learn?

  • Explore timbre, and learn why each musical instrument sounds different.
  • Learn how Schoenberg and other composers used math to create unforgettable works of concert music.
  • Learn how the digital age has affected the delivery of music and how our brains listen to it.

Course Overview

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
 |  Average 46 minutes each
  • 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

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Your professor

David Kung

About Your Professor

David Kung, Ph.D.
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...
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How Music and Mathematics Relate is rated 4.6 out of 5 by 145.
Rated 4 out of 5 by from Excellent Although I bought this course for my granddaughter, I did view the first part. Anyone interested in both math and music will appreciate it.
Date published: 2018-05-18
Rated 4 out of 5 by from How Music and Mathematics Relate Great presentation, excellent visuals of concepts! I never knew some of these concepts even existed, very enlightening!
Date published: 2018-02-24
Rated 5 out of 5 by from The title describes the content. I was very pleased with the course material and the presentation. Dr. Kung was thorough and detailed in his analysis and explanations. I had several "who knew" moments and the course was very interesting. I think a basic, even if somewhat dated, knowledge of advanced math is helpful.
Date published: 2018-02-13
Rated 5 out of 5 by from eye opening Pleasantly surprised by Dr. Kung's course covering relationships between math and music! The content isn't found elsewhere in such detail. It also is a mind opener with regard to new/contemporary music.
Date published: 2017-12-01
Rated 5 out of 5 by from My whole family loved this series I married into a group of mathematicians AND musicians -- I've needed help! I'm a very practical engineer who failed elementary school piano lessons. Dr. Kung helped me understand this material (amazing!). This course helped me follow the dinner table conversations, and made me a hero with my mother-in-law who finished her long career as a calculus teacher actually teaching at St Mary's College in Md. Dr Kung was a favorite of hers.
Date published: 2017-08-12
Rated 4 out of 5 by from Explanatory, clearly defines subject We had seen a program describing math's relationships to music that was overly simplified. This series is excellent, and though it demands focused attention, it answers questions with layers of detail that satisfyi both musicians and mathematicians. For those of us who can't grasp the math, it does not detract from the stimulating recognition of how the two disciplines are intricately woven together.
Date published: 2017-07-31
Rated 5 out of 5 by from Outstanding Course! This course was excellent! **Dr. Kung's excitement and love of both math and music really help make this course delightful. He is a great lecturer, and obviously knows his stuff. **I loved how Dr. Kung started and ended each lecture with a musical number, and how he wove these two subjects together. **Using the "technical display" from the sound studio really helped re-enforce the concepts. **I have the math background to keep up, but didn't have the music experience--this course has motivated me to finally start learning to play an instrument (something I've been meaning to do for quite some time) This is definitely one of the best Great Courses I have viewed! I absolutely recommend it.
Date published: 2017-07-27
Rated 5 out of 5 by from Very enlighening! Great job! First of all, I previously had almost zero musical background and an extensive applied math background. The capable professor increased my musical knowledge and appreciation many fold. I especially liked the lesson on auditory illusions, that are (mostly) solved by the sum of higher harmonics to produce a representation of the fundamental. Cool. I did an experiment to digitally produce pure sinusoidals at 400,500,600,700, and 800 Hz. I then summed them point by point to create a composite file. Sure enough, the Fourier transform of this file showed some components at 100 and 200 Hz. There are many interesting points made throughout the lectures such as problems with tuning a piano, randomly generated compositions, digitally "adjusting" improper pitches (as in vocal), etc. The lesson on how the ear distinguishes the upper harmonics present in a supposedly "pure" fundamental note was excellent. It is how we distinguish a violin from a clarinet, for example. The numerous demonstrations the prof gave with his violin were enjoyable and instructive. Although no previous math background is required, it is certainly desirable to have at least a working knowledge of trig. Anything beyond that is gravy. There are many graphical displays of the frequency spectrum of instrumentals in real time, i.e., the results of the Fourier transform. This course must be taken on video. It would be useless to listen only.
Date published: 2017-07-25
Rated 5 out of 5 by from We loved this course! My husband is an engineer, I am a musician. This was the perfect course to watch together. It was a little heavy on the math for me at times, but my husband just loved those parts! It was very interesting to learn how music and math relate. The instructor was so interesting and entertaining. He explained things in a manner we both enjoyed. I think this is our favorite course so far, and we've watched many.
Date published: 2017-07-22
Rated 1 out of 5 by from Too much math facility required Despite descriptions to the contrary, a fairly high level of math competence is needed to get much from this course. I am very knowledgeable musically, and somewhat so mathematically, but the math in these lectures was beyond me. Would not have purchased had I known. Unfortunately this was a gift not purchased from Great Courses and thus cannot return it.
Date published: 2017-06-25
Rated 4 out of 5 by from If you're ordering the audio version . . . . . . be aware that it's the soundtrack of a course designed for DVD. Now and then I wished I could see something the lecturer was referring to. But I found the course fascinating anyway. A little (not much, but a little) background in music and in high school trigonometry and algebra allowed me to understand enough, even without the visuals.
Date published: 2017-06-25
Rated 2 out of 5 by from Probably good, but do not buy audio only version I listened to this course, as I did not get the video version. The music part is fine, but without the mathematics visuals there is really no connection.
Date published: 2017-06-16
Rated 5 out of 5 by from Unique, and in a good way I am musically literate, but insufficiently educated, having one but only one course in musical theory under my belt. I grew up with music, was involved in band as a student, play a little guitar, and (above all) sing, to this day. I knew music had mathematic roots, but I didn't know the half of it. Dr. Kung's course was a revelation to me in that respect. I still can't get over learning how no piano is (indeed, CAN BE) completely in tune, that compromises are mandatory due to the nature of the mathematics that govern music. This course helped refine my understanding of many things musical (and the older I get, the more interested I am in understanding underlying principles such as these). Highly recommended.
Date published: 2017-05-09
Rated 5 out of 5 by from How Music and Mathematics Relate I am a retired petroleum engineer (age 78). By the way I am a female. Seeing the fast changing, and increasing complex world we live I wanted to help the local high school students with math skills. The school board principal, and teachers are very supportive. The students are eager to learn. Many of my students are musicians; some at advanced levels in classical piano, etc. Myself, too, love classical music, and learned a bit when I was young, piano and voice. Occasionally we share about our music interest in my tutoring sessions. This course provides me with insight into math and music --> certainly I related what I learn from the course to my students. They will take further joy of learning math and music. Thank you so much Professor Kung.
Date published: 2017-05-04
Rated 5 out of 5 by from Great companion piece to Prof. Greenberg lectures One of my favorite lecturers from the Teaching Company. Very much enjoyed this series and am planning on taking his other available course on mathematics. Professor Kung was also kind enough to promptly answer a few questions I had emailed him. It's always very nice when the professors are more than just talking heads on a computer screen and are happy to engage with the strangers who take their courses--they don't have to do this. So, thanks again.
Date published: 2017-03-23
Rated 5 out of 5 by from A unique and fulfilling experience After years of attempting to find college-level material like this that I could absorb at my own pace, I was pleasantly surprised how this course addressed many questions of mine that had been long awaiting answers. I consider myself an educated consumer of music, but not a musician. Similarly, I am knowledgeable on most of the basics of physics including wave theory but was unaware of the development of music theory from ancient to modern times. Professor Kung is both an accomplished musician and mathematician who explains both fields of knowledge in a graceful, flowing manner. This course has satisfied my long-standing curiosity regarding auditory illusions and the problems that the practical implementation of musical scales and tuning present. Having taken the course, I feel able to appreciate music composition and musical performance to a much deeper level.
Date published: 2017-03-13
Rated 5 out of 5 by from Wonderful Course My husband and I have been watching this video series over the last week and we are just entranced by it! We can only watch one show each evening because it is a workout for the brain. The lecturer clearly loves music but also loves math. He is excellent at making the math as clear as possible for non-mathematicians as well as drawing us into his love of music. Not only have my husband and I learned new things with each lecture, but we have learned to listen to music in new ways. I very highly recommend this course.
Date published: 2017-02-21
Rated 5 out of 5 by from The missing link revealed I studied classical piano while growing up. Most of what I learned was by rote. Now, at age 61, I'm starting over. And Music/Math was a delightful first step. This is the essence of music. Well balanced with theory and examples. Added bonus: wonderful performances by Dr. Kung. Felt a real connection, like what happens in a real classroom. Was actually sad when course ended. Now I move forward to reinvent my musicality with significant gaps filled in.
Date published: 2017-02-05
Rated 5 out of 5 by from One of my all-time favorites. Of the 75 Great Courses I have purchased over the years, this is certainly one of the top five.
Date published: 2017-01-15
Rated 5 out of 5 by from A masterful teaching performance Dr. David Kung delivers insights into the complexities of music (from the perspective of a mathematician who is also a musician) in an accessible, engaging, humorous, and memorable fashion. Dr. Kung takes very complex concepts and makes them relatively clear and much easier to understand, but he also communicates and shares his own affection for both math and music that clearly makes this course an authentic labor of love. I have purchased, so far, over twenty courses from the Teaching Company, and this is my favorite to date.
Date published: 2016-12-26
Rated 5 out of 5 by from Nice mix of music and related mant My title says it all. This is a nicely crafted work.
Date published: 2016-12-19
Rated 2 out of 5 by from Need to be a math major This course was advertised as being useful even if you have limited knowledge of math and music. In fact, it requires a high level of math skill in order to follow the professor's line of logic. I'm a musician and was pretty good at math in school, but I was lost after a few minutes. I'm returning the course for a refund.
Date published: 2016-11-28
Rated 5 out of 5 by from Lots of interesting stuff This course was full of new and interesting information. Things that I wondered about and many completely new ideas were explained. Prof. Kung presented the material in a very easy to understand way. I was not able to follow all of the detailed math, but I understood enough of it to follow his concepts and examples.
Date published: 2016-11-15
Rated 5 out of 5 by from Music & Math Interesting explanation & music examples which causes the viewer to wan to know more.
Date published: 2016-10-10
Rated 5 out of 5 by from This would have helped me understand Fourier Serie My background is high school band, electrical engineering, and mathematical simulations. I wish I had this course in the '50s and in EE to help understand waves. Music is the key to making this science interesting. The instructor is brilliant. I have not completed all the sections, but I am sold.
Date published: 2016-09-19
Rated 5 out of 5 by from You'll Never Think of Music in the Same Way Again The librarian at my town library noticed that I had taken out a DVD of Polanski’s “The Pianist” and suggested I watch Professor Kung's lecture series for Great Courses, a recent library acquisition. I’m glad she suggested it: What an engaging and illuminating nine hours! Such visual and auditory invention to present essentially abstract material and make it not only intelligible but striking. I confess that, as I am not a mathematician, much was difficult to absorb — not a shortcoming of the presentation, but a products of my limitations (math studies have long since been in my rear-view mirror). Even so, the experience was eye-opening. (Or should that be ear-opening? Probably both.) For example, and it's but a single example out of many, Professor Kung's treatment and illustrations of the effects of transposition, inversion, retrograde, retrograde inversion had such clarity -- especially of their interplay and how they may be graphed along x or y axes -- that the concepts will stay with you. The use of historical images and information was particularly keen, as was his treatment of the atonal or pantonal revolution. (I liked his brief discussion of Camille Saint-Saëns as a child prodigy: as a small boy, he played the piano for King Louis Philippe and yet lived to write a chorus celebrating French airmen (Aux conquérants de l’air) after World War I — the latter just as Schoenberg in Mödling was turning things upside down.) I especially enjoyed how well-integrated and well-structured the course and its internal references were. No small feat. I’ve been to many concerts of all kinds of music in the years stretching from 78-rpm disc to digital downloads; but I’ve never seen anything to equal your lecture series for giving people a glimpse of the correspondence and difference between math and music. Professor Kung, his production engineers, and Great Courses have produced a masterpiece of pedagogy. It’s a real gift and well worth watching. There will be parts that you may find difficult (not all that many), but work through them and you'll get a rich reward.
Date published: 2016-09-14
Rated 5 out of 5 by from Informative Lecture Series I know a lot about music and play piano with classical lessons. This lecture is heavily about mathematics. The mathematics is very difficult and would require an advanced mathematics degree to fully understand. Fortunately, one does not need to fully understand the mathematics to greatly appreciate the linkage between music and mathematics. The presentation by the Professor is excellent with fascinating anecdotes and musical demonstrations. It would be most helpful for students to have some fundamental knowledge of music theory but not essential. The last 3 lectures are an excellent summation of the previous lectures and place everything into a practical perspective. Highly recommend for all people interested in music.
Date published: 2016-08-21
Rated 2 out of 5 by from Disappointed Perhaps it was my mistake to believe that this course would be of value to me. It involved more math than I could handle. I have written several compositions and was hoping it would increase my understanding of music theory and how to apply it to composing. Unfortunately, the two lessons I viewed, one and four, were more intimidating than enlightening. The professor, however, is quite knowledgeable and, obviously, a skilled violinist. He might consider simplifying the curriculum to make it more comprehensible to individuals with limited math skills. i have returned the course. It would be appreciated if you would consider reimbursing me for at least a portion of the $16.85 postage and handling costs I incurred in both receiving and returning the course. If possible, please advise when you have received it. Thank you.
Date published: 2016-08-18
Rated 2 out of 5 by from Audio version not fit for purpose. Gutted. Had I known that the audio version of this course would be impossible to follow, I would have bought the video version. I had assumed that all course material would be available in the pdf, as it says that there are illustrations etc in there. However, without the examples and visuals at hand in the book, the audio version is a complete waste of money. Don't get me wrong, the lecturer makes it sound fascinating, and I'm desperate to know more, there's so much information being alluded to, it sounds terrific. Such a shame that I can't access any of this knowledge. I really do think that it should be made clearer what is included in the audio version, because had I known that basic material was missing from the audio, I would have obviously gone for the video. Or perhaps the missing visual aids could be added into the accompanying pdf, which would probably also benefit the students who buy the video version. At least that way the audio would begin to make any sense at all. I was so excited to begin this, and so gutted that my enthusiasm was popped within a mere 10 minutes.
Date published: 2016-08-07
Rated 4 out of 5 by from Fascinating Course Relating Music and Math David Kung shows interrelationships between music and math that not be apparent to the casual observer. It was fascinating to to see how mathematical functions could be used to show musical progressions. I will enjoy watching this class a second time.
Date published: 2016-08-02
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