Introduction to the special issue on pedagogies of mathematical music theory

A discipline defines itself by its pedagogy. In established fields, standard curricula provide a basis of shared knowledge, organize, classify, and categorize that knowledge, and tell the story of the discipline, its classic results, and its canonical figures. Mathematical music theory, as the present volume attests, is a field of study that is relatively new, even while being at the same time ancient. Many of the contributions testify to how mathematical music theory traces its origins back to the musica speculativa tradition of the middle ages and its roots in Classical thought. After Boethius, however, our shared intellectual forebears seem to skip ahead some generations to the trailblazers of the later twentieth century, with Milton Babbitt, David Lewin, and especially John Clough, standing out as those who shaped an emerging area of study. Institutionally, mathematical music theory is quite young, with the present journal now in its eighth year and the Society for Mathematics and Computation in Music between its fourth and fifth biennial meetings. This youth is revealed in this special issue also, with many of the authors asking the essential questions of an emerging field: Where is our place in the academic institutions of the twenty-first century? How do we teach our subject, and in what contexts? How do we advertise what we do to the general population? What are our canonical results?

This special issue coalesced out of two recent events. The first was the panel discussion “Mathematical Music Theory in Academia: Its Presence, Role, and Objectives in Departments of Mathematics, Music, and Computer Science” organized by Mariana Montiel at the Fourth International Conference for Mathematics and Computation in Music, June 2013, in Montreal. Many of the practical and existential issues confronting this intrinsically multidisciplinary field were addressed in the comments of panelists Guerino Mazzola, David Clampitt, Thomas Noll, Thomas Fiore, Emmanuel Amiot, and Anja Volk, a group representative of the field's diversity of disciplinary backgrounds and professional affiliations.

The second, more direct catalyst was a panel discussion hosted by the Mathematics of Music Analysis interest group at the October 2013 meeting of the Society for Music Theory (SMT). The panelists Jonathan Kochavi, Timothy Johnson, and Mariana Montiel discussed mathematical music theory in the pedagogy of music and mathematics. Jonathan Kochavi's paper here is adapted from his talk at the SMT meeting, while Mariana Montiel's and Francisco Gómez's contribution is partially based on her presentation. The idea for the session originally came from Robert Peck, whose work also appears in this issue. In addition to the invited papers of Kochavi, Montiel–Gómez, and Peck, we also solicited reflections, reports, and pedagogical essays from music theorist Thomas Noll and mathematicians Rachel Hall and James Hughes.

We intentionally construed the scope of this special issue quite broadly. Rather than proposing a specific theme to the invited authors, we merely gave a few examples of potential topics: pedagogical strategies, materials for a past course on math and music, challenges one faces (whether in the classroom or institutional), theoretical reflections, reports on experiences, proposals for courses and homework assignments, inter-departmental teaching collaborations, their successes and challenges, discussions about the role of mathematical music theory in different kinds of mathematics and music programs, etc. Nor did we specify length: short and mid-length papers were equally encouraged. The papers that came in were far richer in their array of topics and techniques than even we had expected.

Many themes emerged in the invited articles and essays that were not specifically anticipated in this very general mandate. One theme is creative and discovery-based learning activities as a chief pedagogical asset of mathematical music theory. James Hughes has made creative interaction with mathematics and music the centerpiece of his course, and Jonathan Kochavi and Rachel Hall use the problem-solving orientation of mathematical music theory to engage students more deeply with music fundamentals and acoustics. Thomas Noll also advocates for this strategy in his essay, and Robert Peck's essay suggests that flexibility of approach and an emphasis on discovery are essential pedagogical strengths of mathematical music theory.

The issue also attests to some of the challenges of mathematical music theory, and one that stands out is accessibility. Mathematics and music theory are both fields that tend to require a great deal of specialized knowledge and skills to access basic research, and these challenges of accessibility are multiplied in mathematical music theory. Mariana Montiel and Francisco Gómez emphasize the importance of popularization in their paper, and propose ways to promote mathematical theory to a wider audience. The museum projects and mobile device applications described in Noll's essay and Gómez's popularization articles are a great start, but much more could be done. Many of the courses described in this issue (Hall, Hughes, Kochavi) are taught to general undergraduate populations under the auspices of different kinds of departments, and show that a range of topics in mathematical music theory can be effectively taught to students who may or may not have a strong prior grounding in music fundamentals. Montiel and Gómez propose that even more sophisticated recent topics in the field can be a valuable component of graduate studies in mathematics or computer science, giving these students entrées into a variety of important areas of mathematics.

The papers in this issue are organized into two groups of three papers each. The first three papers describe specific courses, their goals, and pedagogical philosophies, provide samplings of their contents, and reflect on successes and challenges. The next three papers are essays of a more general nature about the teaching and learning of mathematical music theory, and its place in academic institutions and the public sphere at large.

Kochavi (2014) describes the pedagogical philosophy behind a music and mathematics course originally developed in collaboration with John Clough and subsequently taught by Kochavi at Swarthmore College for a number of years. Kochavi's course teaches music fundamentals, but drawing on scale theory it approaches the topic from a musica speculativa perspective quite unlike the usual regimen of practical rudiments encountered by most students of music.

Hall's (2014) course at Saint Joseph's University, on the other hand, teaches mathematics through music rather than vice versa. The curriculum described in her paper focuses on musical acoustics, and exposes a general population of undergraduate students to important concepts of mathematics and computation through audio processing. Her paper describes nine experiments that she uses as assignments in her course, in which students use audio processing software to discover important principles of acoustics and apply basic mathematics to explain them.

Hughes (2014) reports on his undergraduate honors course at Elizabethtown College, a small liberal arts college in Pennsylvania, USA. The central theme of his course is creativity in music and mathematics. As such, the course is built around six creative learning experiences, one in each of: transcription, acoustics, tuning, modelling, composition, and student teaching at a local school. He consciously aimed for a truly interdisciplinary course, with music in the foreground. In the article, he reflects upon his creativity-based pedagogical strategy and design, curricular requirements, student choices, student ownership of projects, and mathematics as part of creative expression.

Peck's (2014) essay takes up the new “New Math,” an American public-school educational experiment in the 1960s and 70s, as an object lesson for the teaching of mathematical music theory. Peck advocates for the philosophy behind the New Math as a guiding principle for mathematical music theory, even as he takes an unflinching look at its failures in primary education. Those failures, he argues, do not invalidate this pedagogical philosophy, but show us that it should be implemented in a way that emphasizes exploration and conceptualization over rote learning. Peck is deeply involved in music pedagogy as the Music Theory Coordinator at Louisiana State University, where he has taught numerous undergraduate and graduate courses, and directed master's theses and PhD dissertations.

Montiel and Gómez (2014) present two topics: an online math-music popularization effort in Spain and course materials proposed at Georgia State University, USA. The column Música y Matemáticas in the online magazine Divulgamat of the Royal Spanish Mathematical Society has published over 58 installments, many written by Gómez. Translations of excerpts on rhythm and rotations, and distance and musical similarity, are included in the article. Their second topic, didactic materials, is an ambitious program to build courses around research literature in mathematical music theory. They draw from scale theory and combinatorics of words, the music software Rubato Composer^®, maximal evenness, and rhythmic canons. The article is also a call to action, full of proposals for our community to implement.

Noll (2014) incorporates mathematical reasoning into music theory pedagogy via hands-on discovery-based activities. Scale theorists’ insights into diatonic intervals and chords are illustrated with diatonic seventh chords and various diagrammatic materials: the Tonkreisel-Exhibit at the Erlebnisland Mathematik in Dresden, the Garbers–Noll iOS application MIDI SolFa Mode-Go-Round, the Rotating Square of Douthett, and the Diatonic Cord. Key ideas from the research literature, such as generic and specific levels of description, Myhill's property, the Cardinality Equals Variety theorem of Clough and Myerson, parsimonious voice leading, and the decomposition of steps into fifths and fourths, are woven together in thought-provoking, concrete activities. Noll's reflections upon the contribution of mathematics to music-theoretical knowledge, a topic he has investigated over many years, forms an implicit thread throughout the paper. Noll's present paper has been informed by his teaching at the Escola Superior de Música de Catalunya in Barcelona, the Universität Leipzig, and the Hochschule für Musik und Theater Leipzig, and his public engagement in museums and schools.

Lastly, we thank the authors and referees for all their hard work in making this special issue a success.