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Unit 10

Harmonious Math

10.1 Introduction

"Music is the pleasure the human mind experiences from counting without being aware that it is counting."

-Leibniz

You may have heard the notion that music and mathematics are connected. Perhaps you've heard one of those stories about the violin prodigy who also excels at calculus, or the composer whose works are based on prime numbers. Indeed, musical and mathematical talent seem to go hand in hand at times. Why should this be so?

The answer to this complex question touches on more than just music and mathematics. There are undoubtedly societal, psychological, and perhaps even biological factors involved that can lead to the coincidence of music and mathematical talent in the same individual. Nonetheless, it is safe to say that perhaps talent in both areas has something to do with the fact that the two disciplines are related in many fundamental, even abstract ways. It is the connections between music and mathematics, some of which are surprising indeed, that will be the focus of this unit. Our discussion will be concerned not with explaining the connections between abilities in these two disciplines, but, rather, with how the two disciplines relate to one another on a conceptual level.

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. Tools that have been developed to help us understand the nature of sound, such as Fourier analysis, can be generalized to shed light on many areas of mathematics. In return, the mathematical understanding of sound has helped foster the development of new technologies that extend the possibilities for musical exploration.

To the mathematician, wave structure and theory open the door to the examination of periodic functions, some of the most basic forms of patterns in mathematics. In this unit, we will examine how music and math have influenced each other throughout the ages. In particular, we will view both music and sound as "the math of time, an idea that can be traced back to the Greeks. From there we will look at our current understanding of sound and the mathematical tools that have helped us reach that understanding. We will look at waves and periodic functions in one-dimension, see how Fourier analysis can break these into combinations of simple sine and cosine functions, and then move on to see how these ideas can generalize to more complicated phenomena. Finally, an exploration of the question, "can you hear the shape of a drum? will introduce us to the ways in which the mathematical study of periodicity and patterns can be applied to interesting and challenging problems.

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Next: 10.2 The Math of Time


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