This UARTS FEAST team welcomes students from a range of technical and artistic backgrounds—artists, architects, designers, musicians, mathematicians, cognitive scientists, programmers, engineers, data scientists—who are curious about connections between music and mathematics. Ideal team members will have an interest in visual and spatial thinking, a desire to communicate ideas precisely, and an appreciation of the beauty of symmetry and abstract patterns. Some musical experience will be helpful, especially familiarity with the fundamentals of music theory (e.g., reading treble clef, major and minor scales, intervals, triads). No advanced music theory or mathematics background is needed.
In the field of mathematical music theory, music theorists often construct musical spaces: visual arrangements of notes, intervals, and chords that serve as a way for us to conceptualize musical relationships. The subfield of geometric music theory—first launched by a music theory article published in the journal Science in 2008—employs geometry and topology (another branch of mathematics) to construct musical spaces of chords. These spaces can be of interest to both music theorists and music analysts, i.e., they not only provide a way for us to study relationships between chords as abstract musical objects, they can also serve as tools to study (or even compose) pieces of music.
Each point in these musical spaces represents a different musical chord, and paths through the spaces represent chord progressions. Mathematically, these musical spaces begin as ordinary Euclidean spaces: a musical space representing two instruments/voices can be represented as a familiar two-dimensional xy-plane, whereas a space representing three instruments/voices can be represented with a three-dimensional xyz-coordinate system. But music theorists often assume certain kinds of chords to be equivalent to one another (for instance, when the chords contain the same notes, but in different octaves), and when we translate those musical equivalences into mathematical equivalence relations, the geometric spaces get folded up into spaces with more complicated structures and properties. (Think of a Möbius strip!) One of the challenges of this area of scholarship is visualizing these spaces; many of these spaces exist in three (or more!) dimensions and have twists or false boundaries, so they can be challenging to draw and represent.
The main goal of this UARTS FEAST team is to explore new ways of “picturing” and navigating our ways through these spaces. We might try to build physical models of these spaces, or we might explore ways of creating digital models of them—possibly even in such a way that weaves together their sonic and visual elements using sound or animation. One aim of this task is to help make this type of scholarship more accessible, not just for scholarly research, but also for pedagogy and outreach.
In the initial phase of the project, we will study existing literature in the area of geometric music theory and become familiar with ways that musical relationships can be represented using mathematics. Once we’ve learned to navigate the geometric chord spaces, we will begin to brainstorm new ways of “picturing” them using physical materials or digital models.
Wednesday or Thursday – mid to late afternoons
Modality: In-person (interested in the project but unable to be on campus? Contact us to inquire!)
Students apply to a specific role on team as follows:
3D Modeling & Fabrication (2 Students)
Preferred Skills: Team members in this role will be the team leaders for creating physical models of the musical spaces; these members should have experience with tools and methods to construct 3D structures or sculptures.
Likely Majors/Minors: ARCH, ARTDES, BIOCHEM, CHEM, CHEMBIO, COGSCI, ME, NEUROSCI
Data Visualization & Modeling (2 Students)
Preferred Skills: Team members in this role will be the leaders for creating a digital models of the musical spaces; members in this role should have a background in programing languages, software, or animation techniques that allow for multi-dimensional visualization.
Likely Majors/Minors: ARCH, ARTDES, CS, DATA, SI
Musical Structures & Patterns (2 Students)
Preferred Skills: Team members in this role will be the team experts in the music theory concepts (e.g., intervals, chords, voice leading); these members should have taken at least two semesters of music theory at the college level.
Likely Majors/Minors: COMP, CONDUCT, MUSPERF, MUSICOL, PAT, THEORY, VOICE
Mathematical & Cognitive Data Structures (3 Students)
Preferred Skills: Team members in this role will be the team experts in the mathematical concepts and/or knowledge representation and cognitive structures; some knowledge of basic group theory, graph theory, and/or topology would be helpful (though is not required).
Likely Majors/Minors: COGSCI, DATA, MATH, SI, STATS
Faculty Project Lead
Leah Frederick is a music theorist specializing in mathematical techniques for the study of musical structure. She completed her Ph.D. in Music Theory from Indiana University in 2020 and earned a B.S. in Mathematics and a B.M.A. in Viola Performance from Penn State University in 2015.
Frederick’s scholarship examines ways in which musical objects can be represented with mathematical structures. She’s particularly interested in the interpretive nature of this mapping, that is, in the way in which one must make decisions about how to represent musical objects using mathematics in a way that best aligns with musical intuitions.
Her recent research employs geometric and transformational techniques to examine properties of diatonic versus chromatic musical space. Her work on voice leading in mod-7 space appears in the Journal of Music Theory, and has been recognized by Music Theory Midwest’s 2018 Arthur J. Komar Award and the Society for Music Theory’s 2020 SMT-40 Dissertation Fellowship.
Before coming to the University of Michigan, Frederick taught at Oberlin Conservatory, where she was involved in the redesign and launch of Oberlin’s new undergraduate music theory curriculum. While at Indiana University, she served as editor of the Indiana Theory Review and was awarded the Wennerstrom AI Fellowship for outstanding teaching.
Frederick has studied viola with Atar Arad and Tim Deighton, and her writing on the viola repertoire appears in the Journal of the American Viola Society.
Likely Majors/Minors: ARCH, ARTDES, BIOCHEM, CE, CHEM, CHEMBIO, COGSCI, COMP, CONDUCT, CS, DATA, MATH, ME, MUSICOL, MUSPERF, NEUROSCI, PAT, SI, STATS, THEORY, VOICE
Meeting Details: Wednesday or Thursday, mid to late afternoons
Application: Consider including a link to your portfolio or other websites in the personal statement portion of your application to share work you would like considered as part of your submission.
Summer Opportunity: Summer research fellowships may be available for qualifying students.
Citizenship Requirements: This project is open to all students on campus.
IP/NDA: Students who successfully match to this project team will be required to sign an Intellectual Property (IP) Agreement prior to participation.
Course Substitutions: CoE Honors