Master of Science, The Ohio State University, 2012, Mathematics

The purpose of this study was to show how mathematics can be used to analyze effects on sound. Our hope is that this may inspire student interest in mathematics. We analyzed five common industry standard effects. Research data was gathered using Mathematica and GarageBand software. Three versions of each effect were used to alter pure tone sound waves of ten different frequencies using GarageBand. Then using Mathematica's Fourier command, the frequency spectrum of each altered sound wave was generated. Through observation of each set of 30 frequency spectra, the most prominent and common pure tone components were determined. For each effect, Mathematica's Fit command was used to determine a best fit model of the magnitude of each component as a function of frequency. Our models provide descriptions of the effects that are consistent with the traditional descriptions of the industry standard effects in our study. If similar research is to be conducted, our recommendation is that more versions of each effect, a wider range of input frequencies, and a higher sampling rate would produce function models that are even more consistent with traditionally accepted effect descriptions. Furthermore, an understanding of the hardware and software design used to build effects on sound is highly recommended.

Committee: Bart Snapp (Advisor); Herb Clemens (Committee Chair); James Cogdell (Committee Member) Subjects: Mathematics Education

Honors Theses, Ohio Dominican University, 2021, Honors Theses

The Fibonacci sequence and golden ratio are fascinating ideas to look at on their own. They are analysed a lot in physical art such as paintings and nature, however, not much is researched in their connection to music. You have probably heard a friend or two talk about this connection, but you would not know how factual their statement is because of this lack of prominent research. This led to my attempt to go in depth and present those connections. The golden ratio comes from the Fibonacci sequence, so it is important to incorporate both ideas together. Their connection to music is outlined in detail to help understand what connection lies between the two and music. Then, historical information is provided to show what prominent figures used the golden ratio and Fibonacci sequence in their music. Lastly, I formulated an analysis of the top 25 of the Billboard top 100 decade-end songs of the 2010s to gauge whether those same mathematical and historical ideas are used in popular music today. I wanted to know, is the golden ratio apparent in popular music today? Does this ratio appear at the climax of the song or something different? Beyond that I wanted to see if that relationship had some sort of significance or if it just was an idea with trivial connections.

Committee: James Cottrill (Advisor); Angela Heck Mueller (Other); John Marazita (Committee Chair) Subjects: Mathematics; Music

Master of Sciences, Case Western Reserve University, 2017, Applied Mathematics

The aim of this project is to develop a system of tonal musical generation that can recognize shifts in tonal center via random processes and reorient itself accordingly. One might liken this to a random walk with the added property of boundary crossings. These weights can be conveniently encoded in the discrete probability mass functions corresponding to each scale degree. The result is a self-generating system of music that follow tonal rules in a convincing way.

Committee: Wojbor Woyczynski (Advisor); Wanda Strychalski (Committee Member); David Gurarie (Committee Member) Subjects: Applied Mathematics; Mathematics; Music

Master of Music (MM), Bowling Green State University, 2013, Music Composition

Ouroboros— a single-movement, fourteen-minute work scored for flute, alto flute, B-flat clarinet (doubling bass clarinet), bassoon, horn, B-flat trumpet (doubling B-flat flugelhorn), bass trombone, three percussionists, harp, piano, violin, viola, violoncello, and contrabass— is a work that lacks melodies, motives, clear harmonic shifts, perceivable changes in dynamics and timbre, audible articulations, and a discernible pulse. Every element of this composition was informed by some aspect of the mythical serpent ouroboros. At the broadest level, Ouroboros follows a single, processed-based form. This gesture consists of several subsections that are simultaneously transformed by various processes: registral and dynamic wedges, a timbral rondo, and an exponential accelerando. The algorithms used to develop the material also progressed in a cyclical fashion, terminating in the same way that they began. The harmonic progression, which functions as one giant sequence, is derived from the hexachord 6- 25[013568] and transformations that share at least four common tones. These harmonic materials were arranged across pitch-space in the framework of an ouroboros beginning with a hexachord spanning seven octaves, reducing to a single note, and smoothly spreading back to the fully expanded hexachord. In order to produce many different sonorities, both new and familiar, I developed multi-layered orchestrations that cycle at different rates and are slightly transformed with each reiteration. While inner layers were orchestrated with systematic processes to transform between primary and secondary orchestral choirs, the surface orchestration shifted slowly between “dark” and “light” timbres. In addition, individual pitches were orchestrated by two dissimilar instruments that articulated to and from niente. To further unify these disparate timbres, the majority of the work was written at dynamic levels less than mezzo piano; this also helped facilitate the execution of the unusua (open full item for complete abstract)

Committee: Mikel Kuehn PhD (Advisor); Christopher Dietz PhD (Other) Subjects: Music