HezhenHarmonics
Math Research Experience
My journey - Harmonies of Mathematics: Unveiling the Power of the Fourier Transform
My journey with Fourier Transform and music started during my multivariable calculus course. I chose the Fourier Series for my final project, which was drawn by its connection to math and music. However, I quickly became overwhelmed by the complexity of Partial Differential Equations (PDEs) that go way beyond the curriculum, essential for understanding the Fourier Series. Frustrated but determined, I broke the material into manageable steps, learning resilience and problem-solving along the way.
This perseverance led me to a summer research program at Yale, where I explored Fourier Transforms alongside a PhD student. Diving into these advanced concepts reignited my curiosity and pushed me beyond my comfort zone. Inspired, I designed a senior course, “Discrete Fourier Transform and its Application in Electronic Music,” integrating PDEs and focusing on analyzing sound through math.
This journey taught me that persistence and creativity are essential for overcoming obstacles, turning challenges into opportunities for growth.
Project Development
Spring 2024
Multivariable Calculus
During my multivariable calculus course, I explored how Fourier Series describe sound waves.
Summer 2024
LearnSTEM
Research Program
I studied Fourier Transforms in solving partial differential equations over the summer, composing a research paper on its applications in electronic music.
Spring 2025
Capstone Course
This upcoming semester, I will take the senior course I designed that bridges these two worlds, exploring how the Discrete Fourier Transform can be used to modify sound quality.
Research Abstract
Harmonies of Mathematics: Unveiling the Power of the Fourier Transform
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The Fourier Transform, a mathematical tool for connecting the time and frequency domains, underpins advancements in fields as diverse as signal processing and music analysis. This research applies its theoretical and practical framework to Chinese folk music, focusing on the bamboo flute and its pentatonic and seven-tone scales. Giving rigorous proofs of Plancherel’s Theorem and the Convolution Theorem, this study examines the Transform's foundational properties, including linearity and convolution, to establish their relevance in both mathematical and cultural contexts.
Extending beyond existing applications, such as Audacity’s spectral analysis tools, this work analyzes the bamboo flute’s tonal qualities through spectral decomposition, highlighting the harmonic simplicity and distinct cultural identity of pentatonic music. Through practical experimentation, it achieves high-fidelity digital synthesis of bamboo flute sounds and conducts a comparative study between synthesized and physical performances, offering quantitative and qualitative insights into the preservation and innovation of musical traditions.
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By bridging mathematical theory with cultural heritage, this interdisciplinary study demonstrates the Fourier Transform’s capacity to preserve tradition while enabling modern innovation. It concludes with a mathematical exposition, an original digitally synthesized bamboo flute composition, and a detailed comparative analysis of tonal structures, reflecting both originality and depth in advancing the interplay between mathematics and the arts.