Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

# Unit 10

## Harmonious Math

Waves Moving Through Air

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.

# Video Transcript

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.

# Textbook

One of the most fundamental ways that music and math are connected is in the understanding of sound specifically, and wave phenomena in general. Understanding sound as an instance of wave phenomena provides a nice forum for the interaction of ideas from music, physics, and mathematics.