We have all stared at the piano keyboard, admiring its symetrically-placed arrangement of black and white keys. But how exactly are the black keys placed among the white? And why are they placed where they are? In the elementary course in music theory at Mount Holyoke College taught by the first author we answered this question in an original fashion.
The first author gave a homework problem near the beginning of the course which appeared at first to be totally unrelated to music: The students were given a sheet of paper with some circles, each of which had twelve evenly-spaced short cross lines (Fig 1). On the first circle, they were asked to place three dots so that they were "as spread out as much as possible", and continuing with the other circles with four, five, six and seven dots, all as spread out as much as possible.
At the next class session we discussed the students' responses to the exercise. As can be imagined, there was no disagreement on how to place three dots around the circle (Fig. 2). Similarily there were no problems with four or six dots.
Discussions arose, however, as to the best way to place five dots. After comparing various alternatives, we came to the agreement that the best arrangement was Fig. 3. A similar discussion arose about seven dots.
The homework for that evening introduced the musical significance of the first exercise, again through self-discovery. The students were asked to assign note names to the twelve lines around the circle (Fig. 4), and then to determine the musical structures formed by the dots by playing the notes on a piano and trying to recognize the sounds. They discovered that three dots corresponded to an augmented triad (Fig. 5), four dots to a diminished seventh chord (Fig. 6), and six to the whole-tone scale (Fig. 7). Surprisingly enough, they also found out that five dots corresponded to the black notes of the keyboard (Fig. 8). The conclusion, then is that the black notes of the keyboard have a special property, namely that they are "spread out as much as possible". Similarily they found that the seven dot pattern corresponded to the white notes, which were also thus "as spread out as much as possible".
The properties that the students discovered forms part of some recent scholarly work by John Clough and Jack Douthett on diatonic set theory ("Maximally Even Sets," Journal of Music Theory 35 : 93 -173). Clough and Dothett found a mathematically precise definition of "being spread out as much as possible", which they called "maximally even". This formal definition involves counting intervals between all of the notes in the structure. They also proved that many familar musical structures have this property, as the students observed informally in their assignment.
The next class session introduced the concept of maximally even in a formal way for the first time. By this time the students were almost begging for a definition. They had worked with the idea of "most spread out" in an informal way and had seen some musical significance for the notion. But how could they determine whether other musical structures, such as the harmonic and melodic minor scales, fit the pattern? By using the formal definition and carefully checking the distances between every pair of dots in the circle, students could easily determine whether or not the corresponding musical construct is maximally even or not. Their questions could thus be answered.
Through this series of exercises, students learned a definition for maximally even, determined what musical structures are maximally even, and had an opportunity to consider the overall meaning of these ideas. If the definition of maximally even was introduced first, followed by exercises and examples, much of the excitement for students would be lost. The definition likely would be seen as overly cumbersome, and the concept of maximally even might be seen as insignificant. By reversing the process and allowing students to reach conclusions about musical structures on their own, the thrill of discovery and the accompanying wonder about broader issues was encouraged. Our secondary and more traditional goal was to make the students more comfortable with the various types of intervals, triads, scales and so forth in all possible keys, musical material that is customarily introduced in beginning music theory classes.
The other curricular material in music theory that we developed for this project also largely involved exercises designed for self-discovery. By building various patterns and structures according to prescribed stipulations, students were able to observe many of the underlying abstract constructions independently. This material is designed to provide opportunities for "cutting- edge" research in music theory to be presented in a non-threatening and extremely useful way at the introductory level Moreover, the material developed also introduces students to applications of mathematics that appear naturally in introductory music theory courses, and it helps to provide students with a solid abstract foundation for musical thought based on mathematical ideas and reasoning.
The material developed here is part of our Mathematics-Across-the-Curriculum project at Mount Holyoke College. This initiative, supported by a grant from the National Endowment for the Humanities and the National Science Foundation, aims to introduce students to applications of mathematics in a variety of disciplinary settings and to provide students with opportunities to grapple with mathematical ideas and reasoning in contexts drawn from the humanities. This project paired humanists with mathematicians to develop curricular material for introductory courses in the humanities.
Much recent research published in scholarly journals contains concepts and ideas that could be applicable to the material taught in introductory music courses. However, since this scholarship is primarily directed toward other researchers in the field, these articles generally too difficult for students. Furthermore, much mathematically-based material, especially when it appears in the scholarly literature of other disciplines, is often geared toward establishing certain concepts or relationships and then providing mathematical proofs of these ideas. The problem in presenting such new ideas in the classroom is to introduce them in a straightforward and uncomplicated manner while recreating the excitement inherent in the act of discovery. We feel that the approach described above is a solution to this problem.