Sometimes people ask me when they will use mathematics in everyday life. There are some examples—grocery shopping, calculating a tip, re-carpeting a room—in which math is clearly applicable. But in my life, there are many more examples of tasks where multiple approaches work, some of which are made possible by my background in mathematics. In March, I wrote about one example: making bias tape when I’m sewing. There are many ways to do it, some of which are more clearly mathematical than others. Another example comes from my days in orchestra.
Anyone who has played in a band or an orchestra has probably had to count some rests. Sometimes this stretch can go on for dozens of measures (for timpanists, hundreds). How do you make sure you come back in on time?
One approach is to count the measures of rests in your head like you’d count anything else: 1, 2, 3,…37, 38, play! But sometimes all those rests become unwieldy, and if you forget whether you’re on 27 or 28, you’re toast. If you’re familiar with viola jokes, you might not be surprised to hear that I tend to count rests on my fingers. But you might be surprised that I use modular arithmetic as well.
Modular arithmetic is a fancy way of talking about clocks. 15 hours after 10:00 isn’t 25:00. We call it 1:00, though we may specify further with am or pm. We use the same label for two times if they differ by 12 hours (24 for military time). In other words, we only care about the time up to its remainder when divided by 12 (or 24), also known as the remainder mod 12. Modular arithmetic is the same system, but it can apply to any base.
Arithmetic mod 2 is simply sorting numbers into even or odd: do they have a remainder of 0 or 1 when divided by 2? Mod 10 is keeping track of last digits. And at the music stand, base 4 modular arithmetic comes in handy.
Most western orchestral music has phrases that are composed of 4 measures, or 8, 12, or other multiples of 4. In music with 4-measure phrases, measures 1 and 5 serve similar functions: each is the beginning of a phrase. Likewise, measures 4 and 8 are the ends. If you count rests on your fingers, you will naturally tend to use the 5 fingers of one of your hands, so you will use the same finger for measures 1 and 6, 2 and 7, and so on. This system fights against the natural base 4 system of the musical phrases.
My heroic orchestral innovation was to count rests using only my index finger through pinkie so each finger would correspond to the same measure in the musical phrase every time I used it. For me, apt as I was to zone out momentarily during a long string of rests, counting rests mod 4 gave me a built-in way to check for errors. If I was using my first finger to count a rest that wasn’t at the beginning of a phrase, I’d know that something had gone wrong. Generally I didn’t zone out for a full four measures, so I could get back on track by using the position of the measure in the phrase.
Do I need to know what modular arithmetic is to play in an orchestra? Definitely not. But knowing a little modular arithmetic not only helped me count the rests, but it gave me something to think about while I was doing it.