The Application of Fourier Transforms in Music
Coming into this final project, it was somewhat hard to land on just one topic to dive into. Calculus has had so many applications in various fields, and it was honestly mindblowing to see the impact that this branch of mathematics has had on the real world. However, the applications of calculus in music intrigued me the most. Specifically, the role that Fourier analysis plays in representing periodic functions of sound waves and musical notes excited me. In essence, this project is an exploration of Fourier series and Fourier transforms, with an extension on isolating frequencies to determine the notes playing in a chord.
Research:
Fourier analysis is the study of how general functions could be approximated or represented by a summation of trigonometric functions. These components are oscillatory, and therefore can apply to numerous fields that consist of wave analysis. A Fourier series is an infinite sum of sines and cosines that adheres to either periodic functions or non-periodic functions in a given domain. The general form of a Fourier series is given by the equation below:
In this equation, the left hand side will equal the right hand side as N approaches infinity. Otherwise, the right hand side of the equation would be an approximation of f(x), increasing in accuracy as N tends to infinity. The five factors that we need to modify based on the curve that we want the Fourier series approximation to output are a0, an, bn…
