 James Hughes, who's from the undergraduate faculty program. He's a faculty member at Elizabeth Town College in Lancaster County, Pennsylvania. And together with his wife, James has raised three children, two of them are autistic spectrum disorders. I'll mention that the intended audience for this talk is mathematicians with an interest in music. So if you don't have an interest in music, just pretend for five minutes. Okay, okay. Music theorists are using mathematics more and more explicitly in response to the new music of the 20th and 21st centuries. My goal here is to inspire involvement of more mathematicians in this wonderful interdisciplinary endeavor. Reading music is converting notation to performance. Transcription is converting performance to notation. Transcribing multiple simultaneous musical lines is hard and even harder to automate. Notating music is inherently mathematical. For example, this 13th century manuscript represents musical pitch and time by vertical and horizontal location. The articles listed here, fast-forwarding a little bit, show a variety of approaches to automated transcription. Do, re, mi, fa, so, la, ti, do. Do, re, mi, fa, so, la, ti, do. Do, do, do. Every eighth pitch in the scale gets the same name, and every eighth white piano key gets the same name. So we say notes with the same name differ by a whole number of octaves. Notes get the same name because their frequency ratios are powers of two and thus resonant. The resulting equivalence relation leads to the term pitch class in music theory. Associated group actions and covering spaces are also useful. The seven white keys and five black keys in an octave on a piano keyboard occur in a peculiar pattern. Why? More generally, why are certain subsets of the 12 pitch classes preferred among composers as scales or chords? Well, there's a big industry in music called set theory, unfortunately, which is the study of subsets of a 12-element set with lots of extra structure from music. Commentarial approaches have revealed many important properties sets can have two are listed here. A chord is a set of pitch classes sounded together and an interval is the distance between two pitch classes. The interval content of a chord is a big deal musically. A tool used heavily by music theorists is a kind of discrete Fourier transform. The magnitudes of its components measure the strengths of the various intervals. For musicians, it shouldn't come as a surprise that an augmented chord has a stronger major third component than a major chord. The 12 note chromatic scale is standard in Western music. What's so compelling about 12? One answer is symmetry found in all art. Symmetric things are alike enough to seem familiar but different enough to be interesting. And symmetry means group theory. Groups acting on a 12-element set can have a lot of structure for their size and have long been used implicitly by composers. Explicit study of transformation groups is now another big industry in music theory. One, two, three, four, five, six, seven, eight. One, two, three, four, five, six, seven, eight. One, two, three, four, five, six, seven, eight. Okay, pretty compelling, huh? For you, Andrew Lloyd-Reber fans. Yeah, a little more complicated, but still pretty compelling. The cool thing is combinatorial approaches that work for pitch-class sets also apply to rhythmic patterns. A canon is a motif repeated with translates of itself. On the right, a motif and a translate of its retrograde form a tiling canon with every pulse sounded exactly by exactly one part. Composers make heavy use of tiling canons. Classifying rhythmic tiling is another big industry in mathematical music theory. Tools used include combinatorial word theory, Gaoua theory, discrete Fourier transforms, and cyclatomic polynomials. Now my favorite. Modeling chords geometrically is my favorite because I love harmony and chords are the carriers of harmony. With more and more tuning systems in use, modeling chords using continuous spaces makes a lot of sense. We represent the pitches of n voices by Rn, mod by octave equivalents in each voice to get a torus. Chords are unordered sets, so we mod by an action of the symmetric group to get chord space. When n equals two, this is a mobius strip. Voice leading is the assignment of particular pitches to voices in order to form sequences of specific chords. It's an old issue in music theory. Josquin composed the excerpt shown here over 500 years ago, and it shows some interesting voice leading. Can we model voice leading mathematically? Of course. In chord spaces, voice leadings are paths suggesting the use of topological tools. For example, path branching at singularities corresponds neatly to voice leading through chords with doublings. Finally, I would be remiss if I didn't mention an effort led by Gorino Mazola to frame musical problems and solutions in terms of category theory. That's all, thanks. All right. Thank you.