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Invited Paper Session Abstracts - Applications of Mathematics to Music

MAA Invited Paper Session and Jam Session (aligned with an MAA Invited Address)

Applications of Mathematics to Music

Please note: all sessions are listed in Eastern Daylight Time (EDT = UTC-4:00)

Thursday, August 3, 2:00 p.m. - 5:20 p.m. and 5:30 p.m. - 7:00 p.m., Ballroom B/C

Mathematics and music have a long-standing affinity for each other. In this session, our speakers will talk about many topics, including vowel production and a question at the intersection of mathematics, music, physics, communication, and perception; the application of natural Pythagorean intervals to the closure for scales generated by three or more intervals; the circle of fifths and the twelve-tone scale on a torus; the Piano theorem; graphs for music events, including modulation; and a musical pythagorean theorem that Pythagoras missed.

Jason Brown, Dalhousie University
Ezra (Bud) Brown, Virginia Tech

How Do Vowels Work?

2:00 p.m. - 2:20 p.m.
Brian Katz, California State University - Long Beach

In this session, we'll think about a question at the intersection of mathematics, music, physics, communication, and perception. We'll do a quick version of an activity I do in many Calculus courses that uses my training in music theory and vocal performance, and then we'll talk a little about why I value making these aspects of my identity explicit in class.

Closure and Symmetry in Generalized Tonal Systems of More than Two Dimensions

2:30 p.m. - 2:50 p.m.
Brett Stevens, Carleton University

Carey and Clampitt proved that the well-formedness of a musical scale created from iterated series of fixed intervals can be characterized by two equivalent conditions: "symmetry" and "closure". Zabka generalized each of these conditions to Generalized Tonal Systems generated from two or more intervals. He proved that the conditions are still equivalent for two dimensional Generalized Tonal Systems and identified scales from human muscal traditions that can be formulated in two dimensions that cannot be generated from just one interval. Zabka asked if the two conditions remain equivalent for higher dimensions. We show that the symmetry condition implies the closure for scales generated by three or more intervals and provide counterexamples in three dimensions to the equivalence. These counterexamples are generated by natural pythagorean intervals.

Using Mathematics to Compose Popular Music

3:00 p.m. - 3:20 p.m.
Jason Brown, Dalhousie University

An investigation of the best compositional techniques in pop music highlights a role that mathematics plays surreptitiously. In this talk we’ll highlight a number of ways that seeing the math in the background can allow us to write musical “hooks”. Yeah, yeah, yeah!

Comparing Songs without Listening: From Music to TDA and Back Again

3:30 p.m. - 3:50 p.m.
Katherine M. Kinnaird, Smith College

The multidisciplinary field of Music Information Retrieval (MIR) is motivated by the comparisons that we, as humans, make about music and the various contexts of these comparisons. By defining tasks such as building better song recommendation systems or finding structural information in a given recording, MIR seeks to algorithmically make these musical comparisons in the same manner that a human being would, but on a much larger scale. In this talk, we will introduce the field of MIR, including popular tasks and cutting edge techniques. Then we will present aligned hierarchies, a structure-based representation that can be used for comparing songs, and new extensions of aligned hierarchies that leverage ideas from topological data analysis (TDA).

Symmetry and Group Theory in Bach’s Canons

4:00 p.m. - 4:20 p.m.
Brianna Donaldson, American Institute of Mathematics
David Kung, Charles A. Dana Center, The University of Texas at Austin

Mathematics and music come from different spheres (arts and sciences), yet they share an amazing array of commonalities. Group theory gives modern language to the symmetrical structures beneath the surface of Bach's magnificent canons and fugues. These structures will be described mathematically and demonstrated on violin and piano.

Musical Interlude

4:30 p.m. - 4:50 p.m.

Music Is Mathematical, Mathematics Is Musical

5:00 p.m. - 5:20 p.m.
Ezra Brown, Virginia Tech

Music has scales, intervals, chords, melodies, harmonies, and rhythms -- which are of mathematical origin. Mathematics has patterns, themes, variations, and converses -- which are of musical origin. Topics in this talk (time permitting) include the Pythagorean scale, good and bad vibrations, the Piano Theorem, odd and even harmonics, the circle of fifths and the twelve-tone scale on a doughnut; musical theater, science fiction, and one of those bad vibrations; a musical Pythagorean theorem that Pythagoras missed; and what you get when you play a certain famous theme backwards.

Jam Session

5:30 p.m. - 7:00 p.m.