Practical applications of the Fourier analysis to identify pitches and synthesize sounds in music

The fundamental principles of music theory are demonstrably intertwined with mathematical concepts. Mathematical principles have long served as a cornerstone for understanding and manipulating musical elements. Musical notation, the written language of music, also uses concepts such as set theory and ratios to represent notes and their duration. This paper delves into this intricate relationship, specifically focusing on the application of Fourier analysis in music. Currently, while the Fourier analysis is used in audio and signal processing, its direct application in theoretical music analysis and music composition and creation, is less explored. The primary purpose of our research was to explore the potential of the Discrete Fourier Transform (DFT) as a powerful tool for music theory and composition, which is a specific algorithmic implementation of the Fourier Analysis designed for discrete data such as sampled audio. We employed the DFT to effectively analyze and synthesize musical elements with a high degree of accuracy. We hypothesized that pitches of the chords, scales, and intervals can be accurately identified by applying the DFT. Utilizing the DFT command in Wolfram Mathematica, we precisely identified the fundamental pitches in both digitally generated and recorded audio samples. Our results demonstrate the potential for the DFT to serve as a powerful tool for music analysis, enabling precise harmonic transcription and spectral differentiation of audio sources. Moreover, we investigated the potential of DFT for sound synthesis. This opens doors for novel approaches to music composition and sound design. It presents the DFT as a powerful tool for musicologists, composers, and sound designers.