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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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?