I’ve undergone some evolution in my thinking about the intended audience for my thesis app. My original idea was to aim it at the general public. But the general public is maybe not quite so obsessed with breakbeats as I am. Then I started working with Alex Ruthmann, and he got me thinking about the education market. There certainly a lot of kids in the schools with iPads, so that’s an attractive idea. But hip-hop and techno are a tough sell for traditionally-minded music teachers. I realized that I’d find a much more receptive audience in math teachers. I’ve been thinking about the relationship between music and math for a long time, and it would be cool to put some of those ideas into practice.
The design I’ve been using for the Drum Loop UI poses some problems for math usage. Since early on, I’ve had it so that the centers of the cells line up with the cardinal angles. However, if you’re going to measure angles and things, the grid lines really need to be on the cardinal angles instead. Here’s the math-friendly design:
There are pros and cons to this look. On the plus side, the beat markers now line up with drum hit onsets, the way they do in real life. On the minus side, it’s now less obvious which beat number refers to which cell, the one to the right or left. Also, it just looks crooked to me this way. Then again, it looks crooked the other way to Alex, so go figure.
Alex gave me the excellent suggestion of skinning the app to create a separate “math mode.” That requires some further interface adjustments. I have the downbeat at twelve o’clock, with playback going clockwise. But angles are usually representing as originating at three o’clock and going counterclockwise. So for math mode I had to mirror-image the circle and rotate it ninety degrees.
I’m imagining how you’d describe beats in math mode. Snare drums usually go on the backbeats, 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. Cool! There are opportunities here to talk about sines and cosines, acute and obtuse angles, complementary and supplementary angles, and all the other fun circle geometry. Even better, by comparing math mode with regular mode, you can satisfy the multiple representations component of the common core.
For the really advanced math kids, how about expressing beats in polar coordinates?
Now the relationship between simple/complex fractions and strong/weak beats is quite a bit more clear. Also, it’s intriguing that the angle π/8 represents an eighth note. You could go even further with polar mode and use it as the unit circle on the complex plane. From there, you could get into powers of e, the relationship between sine and cosine waves, all kinds of heavy concepts. You’d be ready for electrical engineering, signal processing and wave mechanics, all while ostensibly playing around with beats. I’d love to have the Drum Loop do a 3D spiral view like this one:
My ambitions to teach math using music are inspired in part by Jeanne Bamberger and Andrea diSessa’s paper, Music as Embodied Mathematics. They see particular value in music’s ability to help kids understand ratios, proportions, fractions, and common multiples, which are tough concepts to master. More generally, music helps you understand symmetry, transformations and invariants.
The math term for a repetitive beat is periodicity. Music usually has several levels of beats going on at once — quarter notes, eighth notes, sixteenth notes and so on. In math terms, there’s a hierarchy of temporal periodicities. The ratios between different periodic frequencies are intuitive for musicians, but can be tricky and confusing when represented mathematically. For example, a larger frequency or tempo means smaller beat durations, and vice versa. Kids need all the help they can get with this concept. For that matter, so do grad students. Afro-Cuban patterns and other grooves built on hemiola are really helpful if you want to graphically illustrate the concept of least common multiples. If you have a kick drum pattern playing every four units and a cowbell playing every three, you can both see and hear how they’ll line up every twelve units. Bamberger and diSessa describe the aha moment that students have when they grasp this concept in a music context — one kid describes it as if the 12-beat “pulled the other two beats together.”
Time is generally a really weird and difficult thing to represent in any context. Bamberger and diSessa ask what we mean when we say “faster” in a musical context. Trained musicians like me know that it refers to a faster tempo. But novice musicians listen for surface features, so if the feel goes from eighth notes to sixteenth notes, they always hear it as the music being “faster” even if the tempo doesn’t change. Standard music notation is not much help with understanding durations. There’s no relationship between the size of a note and its duration. Actually, a measure crammed with sixteenth notes will usually take up more page space than a measure with a single whole note in it. The MIDI piano roll is a better representation in this case, since it shows longer notes as being physically longer. And I submit that the drum loop is even better for understanding frequency and duration. You can compare the duration of the wedges with the rate at which the playback head sweeps around the circle. You can double or halve the tempo, and compare that to doubling or halving the number of wedges in the pattern. Traditional notation becomes quite easy to learn once you understand the concepts it’s referring to. Same goes with mathematical representations.
Math class is so deadly for a lot of kids because it seems like the study of an arcane symbol manipulation system. It’s awfully difficult to relate any of it to anything you’d want to actually do or know about. If we can get kids to see the patterns and symmetries behind the symbols, we’re going to have a way more math-savvy society. I hope the Drum Loop helps.