Date of Submission

Spring 2016

Academic Programs and Concentrations

Mathematics

Project Advisor 1

Matthew Deady

Abstract/Artist's Statement

Mathematics and music have been lifelong partners since the beginning of time. Rhythm and time are two fundamental aspects of music which rely solely on counting, an under-appreciated skill in mathematics, yet recognized by all mathematically-minded people as the foundation of some of the most important mathematical findings; as John B. Fraleigh would say, “Never underestimate a theorem that counts something!” However, music recordings have evolved through the use of technology further than merely possessing the capabilities to quantify and archive the notes that were played in the recording. In the days before digital recordings, the only way to ensure better sound quality was by modifications to the recording equipment and acoustical adjustments to the setting of the recording. It is an indisputable fact that the overwhelming majority of recordings in our modern day are produced using digital technology. The discovery that sound is quantified by a sum of the disturbances the source of the perturbations created by changes in air pressure, known as sound waves, preceded the use of digital technology, but it was this amazing observation that first shaped how mathematicians and scientists utilized mathematical methods in audio engineering. This group of mathematicians and scientists, known as acousticians, aspire to utilize digital signal processing techniques to reproduce the sound created by a source and enhance the sound quality of the recording. There are many ways this can be accomplished. However, one of the most important methods available to acousticians is filtering. Filtering is the process of accentuating or attenuating certain frequencies in the content of the signal. The focus of this paper is digital filtration techniques and the fundamental workings of the mathematical methods that allow these techniques to be possible, as well as the various kinds of software that utilize these techniques. In addition to detailed explanations of the processes by which these methods were born, there is also a case study comparing the effect of a piece of hardware used for filtration with the effect of a digital filter modeled to exhibit the same behavior and functionality as the analog filter. This intensive and comprehensive study of digital signal processing and the mathematical methods acousticians used to create this field of science has been something I have been interested in for many years since first learning of my passion for mathematics. I am excited to share my research and data with the science and math community.

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