## Mathematics Illuminated

# Harmonious Math

## Waves — lightwaves washing against our eyes creating a vision of the world around us, sound waves crashing against our ears — sometimes jarring and other times, beautiful, cosmic waves bathing the Universe. All of it explained, illuminated, and connected via mathematics.

All sound is the product of airwaves crashing against our eardrums. The mathematical technique for understanding this and other wave phenomena is called Fourier analysis, which allows the disentangling of a complex wave into basic waves called sinusoids, or sine waves. In this unit we discover how Fourier analysis is used in creating electronic music and even underpins all digital technology.

### Unit Goals

- The connection between music and math goes back to the ancient Greek notion of music as the math of time.
- Strings of rationally related lengths tend to sound harmonious when played together.
- Sound waves can be expressed mathematically as the sum of periodic functions.
- Trigonometric functions can be used as the building blocks of more complicated periodic functions.
- Frequency and amplitude are two important attributes of waves.
- A mathematical series either converges to a specific value or diverges.
- Any wave can be constructed out of simple sine waves using the techniques of Fourier analysis and synthesis.
- The ability to manipulate directly functions or signals in the frequency domain has been largely responsible for the great advances made in sound engineering and, more generally, in all of digital technology.

### Unit Glossary

### ADDITIONAL UNIT RESOURCES: BIBLIOGRAPHY

# Bibliography

## WEBSITES

http://falstad.com

http://www.ams.org/featurecolumn/archive/199706.html

Boyer, Carl B. (revised by Uta C. Merzbach). *A History of Mathematics*, 2^{nd} ed. New York: John Wiley and Sons, 1991.

Burk, Phil, Larry Polansky, Douglas Repetto, Mary Roberts, and Dan Rockmore. *Music and Computers: A Theoretical and Historical Approach*. Emeryville, CA: Key College Publishing, 2004.

Du Sautoy, Marcus. *The Music of the Primes: Searching To Solve the Greatest Mystery in Mathematics*. New York: Harper Collins, 2003.

Eves, Howard. *An Introduction to the History of Mathematics*, 5^{th} ed. (The Saunders Series) Philadelphia, PA: Saunders College Publishing, 1983.

Harkleroad, Leon. *The Math Behind the Music*. New York: Cambridge University Press and Washington, DC: Mathematical Association of America, 2006.

Kac, Mark. “Can One Hear the Shape of a Drum?” The American Mathematical Monthly, vol. 73, no. 4, part 2:* Papers in Analysis* (April 1966).

Lazzaro, John and John Wawrzynek. “Subtractive Synthesis Without Filters,” *Audio Anecdotes II*. (2004).

Nahin, Paul J. *Dr. Euler’s Fabulous Formula: Cures Many Mathematical Ills*. Princeton, NJ: Princeton University Press, 2006.

Rockmore, Dan. *Stalking the Riemann Hypothesis The Quest To Find the Hidden Law of Prime Numbers*. New York: Vintage Books (division of Randomhouse), 2005.

Rothstein, Edward. *Emblems of Mind: The Inner Life of Music and Mathematics*. USA: Times Books, 1995.

Transnational College of LEX. Translated by Alan Gleason. *Who is Fourier? A Mathematical Adventure*. Belmont, MA: Language Research Foundation, 1995.