While the Drum Loop is well-suited to individual self-guided study, it is intended for use in a creative classroom environment. Its most obvious application is in general music class, but it can also be a useful basis for lessons and activities in cultural/social studies and mathematics.

## Music

The Drum Loop has always been predicated on a role in the music classroom. There is some flexibility in what specifically that role would be. For teachers of drums and percussion, the answer is clear. I would hope that the Drum Loop could also find a place in the general music classroom, as a way to convey appreciation for the rich history of the rhythms of pop and dance. Furthermore, the Drum Loop offers a new and inviting doorway in for students who find it difficult to engage with music-making through more traditional routes.

The Drum Loop may be a tough sell for more traditionally-minded teachers, administrators and policy makers. The decades-long bitter struggle to include jazz in the classroom is still unfolding; it will be many more decades before hip-hop and techno follow suit. It is possible that the Afro-Cuban content may overcome some doubts. Furthermore, the Drum Loop need not be the focus of the class; if students are already writing and performing songs, the Drum Loop could be easily deployed as a convenient accompaniment tool. This is the intended use case for the O-Generator, which has found widespread classroom adoption in the United Kingdom. I hope for the Drum Loop to find a similar role in the United States.

## Cultural/social studies

It is impossible to separate the study of music from the study of its social, historical and political context. Nowhere is this more true than in the music of the African diaspora in America. The Drum Loop would be valuable as an entry point into a contentious set of social and historical issues. By tracing son clave or the Funky Drummer beat from Africa, through the Caribbean and into the American mass culture, we can tell the stories of the people who played the beats, the people who danced to them, the people who sold and bought them. Specific social studies topics could include:

- The linearity of Eurocentric music versus the circularity of Afrocentric music—does this pattern extend to other art forms beyond music?
- What is the connection between repetition and dance? What makes a beat fill (or empty) the dance floor?
- Are drum machine beats “real” music? Are they authentic? Are electronic musicians “real” musicians? Are they practicing the same art form as violinists and pianists?
- Why is dance music so closely tied up with notions of sin, transgression and excess in America? Do all cultures regard dance music in this way? Why do we expect pop and hip-hop stars to behave in such conspicuously “antisocial” ways?

## Mathematics

The Drum Loop may be an easier sell for math teachers than music teachers. Math teachers do not have a cultural canon to protect, and are eager to find ways to make their subject livelier. The Drum Loop could be used to teach or reinforce the following subjects:

- Fractions
- Ratios and proportional relationships
- Angles
- Polar vs cartesian coordinates
- Symmetry: rotations, reflections
- Frequency vs duration
- Modular arithmetic
- The unit circle in the complex plane

Bamberger and DiSessa (2003) have an epigrammatic credo: “Music is embodied mathematics.” They echo Gottfried Leibniz, who famously said in a letter to Christian Goldbach on April 17, 1712 that “Music is a hidden arithmetic exercise of the soul, which does not know that it is counting.” The mathematical content of music has been appreciated since Pythagoras. Music has rich value for teaching symmetry, transformations and invariants. It is also an effective tool for helping elementary school students understand ratios, proportions, fractions, and common multiples, concepts which they frequently find difficult to master (Bamberger & DiSessa, 2003). It is significantly easier to learn standard notation once you are intuitively familiar with the concepts encoded by the symbols. The same is true of mathematical language.

The mathematical term for a repetitive beat is periodicity. Music usually has several levels of beats operating simultaneously: quarter notes, eighth notes, sixteenth notes and so on. In mathematical language, there is a hierarchy of temporal periodicities. The ratios between different periodic frequencies are intuitive when heard in the context of a beat, but understanding them can be tricky and confusing when they are represented mathematically. Bamberger and diSessa (2003) ask what we mean when we say “faster” in a musical context. Trained musicians know that “faster” music refers to a faster tempo. But novice musicians listen for surface features, so if the feel goes from eighth notes to sixteenth notes, they will hear it as the music being “faster” even if the tempo does not change. Novices further stumble on the idea that a larger frequency or tempo means smaller beat durations, and vice versa.

Bamberger and DiSessa (2003) observe that graphic representations of music should help students come to attend to patterns such as symmetry, balance, grouping structures, orderly transformations, and structural functions. By “structural functions,” the authors refer to medium-level musical entities like phrase boundaries, tension and resolution. Conventional notation does not show structural function, but to novice listeners, these are the most salient features of the music. Here, computer software can be an invaluable aid, with its ability to use dynamically interactive color-coding and spatial organization to convey meaning.

Specific kinds of music can help introduce particular mathematical concepts. For example, Afro-Cuban patterns and other grooves built on hemiola are useful for graphically illustrating the concept of least common multiples. If you have a kick drum pattern playing every four units and a cowbell playing every three units, you can both see and hear how they will line up every twelve units. Bamberger and diSessa (2003) describe the “aha” moment that students have when they grasp this concept in a music context. One student in their study is quoted as describing the twelve-beat cycle pulling the other two beats together. Once students grasp least common multiples in a musical context, they have a valuable new inroad into a variety of scientific and mathematical concepts: harmonics in sound analysis, gears, pendulums, tiling patterns and much else.

The Drum Loop is designed to take the best pedagogical features of the MIDI piano roll and enhance them. The spatial location of events helps reinforce their musical function. Users can see and hear for themselves the difference between a drum hit on a strong beat/cardinal point and weak beat/oblique angle. They can compare the duration of the wedges with the rate at which the playback head sweeps around the circle. They can double or halve the tempo, and compare that to doubling or halving the number of wedges in the pattern.

By rotating beat patterns, students can experience mathematical transformation, and hear its musical effect. This diagram from Toussaint (2005) shows (a) the Bembé rhythm, (b) Bembé rotated clockwise by one unit, and (c) Bembé rotated clockwise by seven units.

Symmetries and hierarchies of beat division are more apparent when reinforced by the rotational and reflectional symmetries of the circle. Furthermore, the Drum Loop can help students distinguish linear from rotational speed, and between linear speed and frequency.

The Drum Loop would be more useful for the purposes of trigonometry and circle geometry if it were presented slightly differently. Presently, the first beat of each pattern is at twelve o’clock, with playback running clockwise. However, angles are usually representing as originating at three o’clock and increasing in a counterclockwise direction. To create “math mode,” the radial grid must be reflected left-to-right and rotated ninety degrees.

In this scheme, math students could describe beats in geometric terms. Snare drums usually fall on the backbeats, at 90 and 270 degrees. In the Funky Drummer beat, there are additional snare hits at 157.5, 202.5, 247.5 and 337.5 degrees. The “round” angles go with the strong beats, and the more “fractional” angles sound more syncopated. More advanced math students could perform a similar exercise using polar coordinates.

Now the relationship between simple/complex ratios and strong/weak beats is quite a bit more clear. Also, it is an intriguing coincidence that the angle π/8 represents an eighth note. One could go even further with polar mode and use it as the unit circle on the complex plane. From there, lessons could move into powers of *e*, the relationship between sine and cosine waves, and other more advanced topics. The Drum Loop could even be used to lay the ground work for concepts in electrical engineering, signal processing and wave mechanics.

The New York State Learning Standards and Core Curriculum for Mathematics state as one of its objective that students “make mathematical connections, and model and represent mathematical ideas in a variety of ways.” In an effort to make the Drum Loop maximally useful to public school teachers and students in meeting these goals, I have examined the state learning standards and identified some pertinent subject areas; these can be found in the Appendix.