Engines of Our Ingenuity

No. 2803: FOURIER MUSIC

by Andrew Boyd

Today, music in translation. The University of Houstonís College of Engineering presents this series about the machines that make our civilization run, and the people whose ingenuity created them.

Most people who download songs have never heard of eighteenth century mathematician Joseph Fourier. Yet his pioneering work now pervades how we listen to music.

Sound travels in waves. When a stereo sends music to a speaker, what itís actually sending is an electronic representation of a wave. The wave in turn causes the speaker to vibrate.

sound wave image

If we look at a picture of a sound wave itís a line that moves up and down over time. Different shapes correspond to different sounds. One way of storing a sound wave on a computer is to record how high or low the wave is at any given instant. For example, when I speak into a microphone, recording equipment takes periodic snapshots of the height of the sound wave I create. It does this a little over forty-four thousand times a second. Thatís a common rate for good quality recording, and itís the rate used for CDs.

cd picture

The only problem is that recordings made this way create big files. Try sending one over the internet and youíll have to wait a while. Smaller files are needed to speed things up.

Enter Fourier. Fourier discovered a wonderful fact about our universe. Every wave, no matter how complicated, is made up of very simple sine waves. Hereís what they sound like.

[sine wave audio]

Fourier showed how we can decompose any sound wave into a collection of sine waves of different pitch and loudness. We call this process a Fourier Transform. And hereís something to think about: we can reconstruct the original sound simply by playing all the component sine waves at the same time.

wave image

Iíll admit, itís hard to imagine the Brandenburg Concertos as a bunch of overlapping sine waves. But if youíve downloaded Bachís work over the internet, thatís not far from the truth. The MP3 audio file format, which dominates music on the internet, is based on decomposing a sound wave using a Fourier Transform, discarding what our ears wonít miss, then reconstructing a wave that sounds a lot like the original.

MP3 is a lossy format, meaning some information is lost in translation. But the reduction in file size more than makes up for it. MP3 files are about one-tenth the size of the originals. And the sound quality is excellent. Even with high-end playback equipment, most people canít hear any difference. Itís one of those delightful engineering solutions that traces its roots to another time and place — a time and place long before we even imagined recorded music.

[MP3 of Brandenburg Concerto Number 3]

Iím Andy Boyd at the University of Houston, where weíre interested in the way inventive minds work.

(Theme music)


Notes and references:

Special thanks to listener Phil Englander for a communication which led to this episode.

The Fourier Transform finds frequent use in all areas of study involving waves. Readers should be aware that the process decomposes waves into an infinite number of sine waves. From an engineering perspective, all but a handful make infinitesimally small contributions to the original wave.

The creation of MP3 files can involve a number of steps, and the exact means is dependent upon who has written the MP3 encoding algorithm.

For more on Fourier, see the related episode JEAN BAPTISTE JOSEPH FOURIER.

K. Devlin. The Maths Behind MP3. From the website of The Guardian: http://www.guardian.co.uk/technology/2002/apr/04/internetnews.maths. Accessed June 5, 2012.

J. Guckert. The Use of Fast Fourier Transforms and MDCT in MP3 Audio Compression. Class notes taken from the University of Utah website: http://www.math.utah.edu/~gustafso/s2012/2270/web-projects/Guckert-audio-compression-svd-mdct-MP3.pdf. Accessed June 5, 2012.

Introduction to the Fourier Transform. From the website: http://www.thefouriertransform.com/. Accessed June 5, 2012. Contains some excellent descriptive pictures.

Sine waves for this episode were generated from the website: http://www.audiocheck.net/audiofrequencysignalgenerator_sinetone.php.

The picture of the CD is from the Public Domain Images website: http://www.public-domain-image.com/. The picture of a recorded wave and the picture of the wave constructed by adding together three sine waves are by E. A. Boyd.

This episode was first aired on June 14, 2012



The Engines of Our Ingenuity is Copyright © 1988-2012 by John H. Lienhard.