Last week I started to tell you about my life epiphany. There is a game I used to play in dance class. Now, you can’t play this game in any tap class. You need to be in a rhythm tap class as apposed to a show tap class. If you have yet to be exposed to the difference: show tap looks like this:

Whereas rhythm tap usually looks little more like this:

Both styles of tap dancing offer challenging steps, interesting rhythms, and personal enjoyment. But rhythm tap offers the regular rhythmic syncopation which is necessary for the game I used to play. Let’s pretend you are repeating a step which takes up three counts and has a downbeat on the 3rd count in a standard 4/4 timed song. So we’ve got: 1 2 **3** 4 5 **6** 7 8… Then weird things start happening because the next downbeat happens on **1**. So the next set of 8 counts sounds like **1** 2 3 **4** 5 6 **7** 8. When will it end!? How many times do I have to repeat the step before I get to satisfactorily end on **8**?

This is an example of modular arithmetic. Modular arithmetic is often called *Clock Math*. Because the most often employed practical use of this type of math is with clocks. When we say 1 o’clock we mean 13 o’clock. This is because 13 (mod 12) = 1. So once we pass the 12 o’clock mark, we circle back around to 1. Okay, back to tap dancing.

My game would not be interesting if I had a step that took 2 counts. Then I would go 1 **2** 3 **4** 5 **6** 7 **8 **and I would be done. The interesting part of the game comes from, 8 and 3 being *relatively prime*. That is to say, there are no proper divisors of 8 or 3 which are also divisors of the other number. So when we look at multiples of 3 (mod 8), we get 3, 6, 1, 4, 7… Mathematically, we can compute something called the *orbit* of 3 of this mapping (multiplying by 3). That’s a lot of math words when I’m supposed to be talking about tap dancing! Maybe now is a good time to insert a drawing:

As it turns out, you need 8 repetitions to get back to **8**. Because the numbers are relatively prime, you need exactly as many repetitions as there are numbers in the original modulus (so, in this case, 8). When I was 16, I certainly didn’t know about numbers being relatively prime and how that interacted with the pattern of the rhythm and how is fits and/or doesn’t fit with the music. Instead what I thought was:

1 2 **3** 4 5 **6** 7 8 **1** 2 3 **4** 5 6** 7** 8 1 **2** 3 4 **5** 6 7 **8**

So elegant! So mathematical! As an audience member or a tap dancer this sequence offers a small measure of excitement. The beat starts off well but then we get confused. For the whole second count of 8 the rhythm feels wrong and terrible. That’s because this syncopation often puts the downbeats of the steps in conflict with the downbeats of the song. But as I slowly made my way around the numbers again, the worlds started to feel better and better. By the time I got to that final **5**, I’m feeling pretty good. And it was the height of pleasure to stamp out that last **8**. In my head, I called this sequence of 8 repetitions of a 3 count step *square*. In my head that meant that the whole sequence would need to end on **8** and would leave the audience feeling warm and fuzzy. In mathematical terms, I wanted the sequence to end on zero (mod 8). But I didn’t know that yet because I was only a 16 year old watching Donald O’Connor do back flips off the walls.

…Tune in next week to learn how one piece of my PhD level mathematics dissertation can boil down to the repeating tap rhythms and modular mathematics!

Pingback: How Tap Dancing is like Climate Change | Social Mathematics

Pingback: Why do we play video games for so long? | Social Mathematics

nothing to do with modular forms

Yes, you are correct! I edited the incorrect reference to modular forms with something that is hopefully more clear and more correct. Thanks for the note!