How Music and Mathematics Relate

Course No. 1373
Professor David Kung, Ph.D.
St. Mary's College of Maryland
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4.6 out of 5
150 Reviews
89% of reviewers would recommend this product
Course No. 1373
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What Will You Learn?

  • numbers Explore timbre, and learn why each musical instrument sounds different.
  • numbers Learn how Schoenberg and other composers used math to create unforgettable works of concert music.
  • numbers 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 4.5 out of 5 by 150.
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
Rated 5 out of 5 by from ... excellent!!! ... if you love music and love math then you'll love this course ... nothing more needs to be said!!!
Date published: 2016-07-26
Rated 1 out of 5 by from art is emotion not statistics I couldn't get through ten minutes of the first for me is about emotion not mathematical analysis. it might be interesting for some but I found this lecture unable to captivate me...
Date published: 2016-07-18
Rated 5 out of 5 by from Fascinating course Professor Kung does a great job combining his knowledge of music and mathematics in this course. My one disappointment was his explication of rhythm. Nevertheless, this course is certainly worth anyone's while.
Date published: 2016-06-25
Rated 5 out of 5 by from Excellent Introduction I concentrated heavily on the sciences all through high school, college, and grad school and mostly neglected my education in the arts. Lately, I've been looking to increase my understanding and appreciation of music. I thought "How Music and Mathematics Relate" was perfect for bridging this gap. The earlier lectures on timbre and tuning were really mind-blowing, and I also really liked the lecture on the digital delivery of music. Dr. David Kung covers a broad variety of topics in just 12 lectures. I bought the video download on sale and throught it was an excellent value.
Date published: 2016-06-19
Rated 5 out of 5 by from Truly a great course A very interesting presentation for those that are both mathematically and musically inclined. As a retired teacher of advanced math, I wish I seen this thirty years ago.
Date published: 2016-06-18
Rated 5 out of 5 by from Calculus instead of fractions I was hoping to learn something about fractions, so I can use music to teach fractions to my K-1 and middle school students. So far, the math used is trig and calculus. I'm familiar with both so, I'm learning quite a bit.
Date published: 2016-06-05
Rated 5 out of 5 by from Excellent Course! As an engineer and a musician (who has built a few synthesizers and tuned some pianos over the years) I came into this course with a pretty decent background in the math and physics behind music. This course went well beyond what I thought I knew. I learned a great deal and, as the professor noted a number of times, I find myself listening to music from a different perspective now. I do a little composing and the segment that covered transformations was especially enlightening. With that in mind I plan to try a few things on the keyboard in the upcoming months. Probably the biggest overall gain I got was the clarification of some different mathematical concepts that I had learned in other subject areas. Very well done indeed!
Date published: 2016-05-14
Rated 5 out of 5 by from Enjoyed the audio version I see other reviews talking about the need for the visuals from the video. i only listen to Teaching Company courses while commuting and hope they continue to offer audio only versions. I found this course thoroughly enjoyable and engaging in the audio format, but do have to admit that if you don't have some background in music and math it might be harder to follow in audio format. Wonderful course. I highly recommend it.
Date published: 2016-03-06
Rated 5 out of 5 by from Lecture 3 incorrectly uses Harmonics & Overtones I am enjoying the Doctor's presentation, and will soon purchase a Keyboard to experience the fun of hearing the interactions of Notes. I notice that the verbiage in his Blue Book (page 27-28) (How Music & Math Relate) concerning overtones and octaves is wrong. Within the 2nd, 4th, & 7th Bullets; the terms Harmonic & Overtone are repeatedly reversed. Also, the formula for Harmonics is printed wrong on page 27. It should be (2 raised to the n power) times (x). NOT (2nx). Dr. Dave also mis-speaks frequently in the video by referring to Overtones as Harmonics. "The 7th Harmonic of F2 ... " should obviously be spoken as the 7th Overtone.
Date published: 2016-01-30
Rated 4 out of 5 by from Gift for my grandson who is a musician My grandson (18) tells me the course was "cool" so I assume it was good value and well presented.
Date published: 2016-01-22
Rated 4 out of 5 by from math and music The course was way above my head. You should be very well versed in mathematics and physics to understand this course. It seemed like a very good course and well presented but i only listened to a couple of lessons and gave up. I do recommend this course but only if you have a solid background in math and physics but not to my friends.
Date published: 2016-01-20
Rated 5 out of 5 by from Very interesting to an engineer and music lover I am only at Lecture 6 but I really enjoy the presentation. For me math is a tool in my work, music is what I like for relaxation. It has always been clear that music and mathematics are related, Professor Kung explains it very well.
Date published: 2016-01-20
Rated 5 out of 5 by from An enjoyable course Let me begin with a caveat. I am a retired high school mathematics teacher, who played clarinet and tenor sax as a pastime, so I was predisposed to like this course. That said, this course will appeal to anyone who has an interest in either one of the disciplines, and at least a vague interest in the other. The material is presented from a musical point of view, beginning with pitch and timbre, then going to rhythm and phrasing and composition. The examples are given, live, on violin (Professor Kung is concertmaster of a community orchestra), piano, timpani, sitar, gamilan and laboratory equipment. The math behind the acoustics is explained. The presentation is sometimes very clever. I heartily recommend this course.
Date published: 2015-12-12
Rated 5 out of 5 by from I've given this course to two friends The material is outside my comfort zone, but I don't mind repeating these lectures because the presentation is so interesting and engaging. I've now given the course, as a DVD and as a download, to two friends who have a serious interest in music theory. For me, it's a pleasure to be in the "presence" of a teacher who is so knowledgeable and so excited about the subject.
Date published: 2015-12-05
Rated 5 out of 5 by from Perfect Conbination. A Stimulating and Enlightening course expertly taught and presented. Excellent graphics. Highly and well recommended.
Date published: 2015-10-12
Rated 5 out of 5 by from Caution, "Fundamentals" (like Greenberg) required My starting point this spring was an engineering background and life work, including a professorship, in computer science. As a kid, my direction was visual, not auditory, which has served me well in web and app development. However, I've always loved music, so I decided to buy the Greenberg "Fundamentals". Ah, the "Ode to Joy"! I then followed up (as recommended by Greenberg reviewers) with the purchase of a Casio LK-280, and have learned the basics of reading a score and playing simple songs. Had I not done this first, Kung's material would have been way beyond my reach. Similarly, if you had hissy fits in those math courses, expect similar hissys here. So CAUTION, well rounded arts and sciences preliminaries are required! However, the payoff in physical understanding, cross cultural music appreciation, and depth in a direction that Greenberg (for all his great gifts) could not accomplish is worth the effort. I have a grandson, who has already pounded on my Casio LK-280. He's almost three; yes, I mean pounded (as in upbeat, loud, and familiar- as Kung relates)... He is almost at the point where he can play very simple measures from "Up town, funk you up". My greatest hope is that some day, my grandson understands this material. My deepest and sincere congradulations to Prof Kung.
Date published: 2015-08-13
Rated 5 out of 5 by from One of my favorite TTC purchases Professor Kung is the real deal, working both sides of the brain! Thoroughly enjoyed this course.
Date published: 2015-07-01
Rated 5 out of 5 by from Eye-opening and Informative My wife and I just completed the DVD of How Music and Mathematics Relate by David Kung. Being a retired chemist and a violinist, I found the course challenging, fast moving, and it had me asking more questions at the end of each lecture. I can honestly say that I have not been this academically stimulated over music prior to this. Kung touches on many facets of music that I thought I understood or had an appreciation for, only to find out that I didn't know what I didn't know! His lecture style is informative and entertaining, and very down to earth. It has been literally decades since I've worked with partial differential equations and yet he reintroduced them at a high enough level to not cause me serious consternation. Anyone with at least calculus should be able to follow the math. Thank you Great Courses for discovering Dr. Kung for me, and thank you Dr. Kung for doing this course. It is one of the best Great Courses I've watched or listened to.
Date published: 2015-04-08
Rated 5 out of 5 by from Wow I am one of millions of engineers out there who think they can appreciate math and physics, and have their music preferences (including complete silence more then occasionally). I also had my fare share of piano lessons as a child - few happy memories about that part. But this course challenged everything in me. It is difficult, exciting, intriguing, emotional and overall very much unexpected. I do not think I will start torturing my untuned piano more often now, but ... Who knows ;).
Date published: 2015-04-03
Rated 5 out of 5 by from Bravissimo! An engineer of sorts by profession, I have also been educated in music and mathematics. Therefore, I’m always a bit intrigued by what my piano tuner does with the tools he carries. So when this Great Course on the music-mathematics relationship was advertised, I bit. As further background, I should say that I’ve purchased dozens and viewed even more of the Great Courses. However, at least for me, few if any are such a fantastic tour de force as this one. The enthusiasm and exhilaration over explaining esoteric, educational exotica that this instructor exudes are wondrous. Watching his masterful performances in both music and mathematics is a complete joy!
Date published: 2015-03-15
Rated 5 out of 5 by from Excellent Exposition Professor Kung shows how the creative musical artist interfaces with the mathematical aspects of the materials he or she uses.
Date published: 2015-03-13
Rated 5 out of 5 by from Both basic and advanced! Well presented relationships of physics, mathematics, and music. So nice to see it all in graphics and demonstrations versus text alone. And such a versatile and knowledgeable presenter.
Date published: 2015-03-13
Rated 5 out of 5 by from more than I expected this course provided a level instruction that exceeded my expectations; even with my background in math , I was surprised by the amount of detail that the sessions provided to me; it was definitely way more than I expected.
Date published: 2015-03-11
Rated 5 out of 5 by from Excellent lectures on math and music Many of us (most of us?) do know that music and math are somehow related, even if it's just the knowledge that an "octave" is a given note with its frequency multiplied by 2. But Professor Kung -- both a mathematician and an excellent musician - can show any of us MUCH more than we already knew about how math and music relate to each other. There are SO many irresistible Great Courses, but this is one of the very top on my list. Bob Powers Degree in math, and an amateur musician.
Date published: 2015-03-07
Rated 5 out of 5 by from Absolutely brilliant! Coming from a maths background, the course is fascinating and very enjoyable, with many interesting real world examples. Highly recommended.
Date published: 2015-02-26
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