teaching

How I Teach Calculus: A Comedy (Fugues and Exponentials?)

April 13, 2011 , 5 Shawn Cornally

I was listening to the radio when the smarmy voice of the public radio host dropped this bomb on me while my mathematical clutch was fully disengaged:

The King of Prussia once sent J.S. Bach a fantastically quirky melody and challenged him to turn it into a fugue. Bach responded by brushing his shoulders off and then entering the party much like this. The King, responding to Bach’s confidence, upped the ante and added the caveat that the fugue must be done with six voices. This is unusual, because most fugues are quite difficult to write for merely three voices. The radio announcer then let the clutch out at 7,500 rpm’s:

Composing a fugue for more voices than three becomes exponentially more difficult.

#MANICMATHGLEE I get to make a lesson out of writing fugues. Yes, Yes, Yusssssss!

So, most calculus students don’t know what a fugue is. Here are a few:

The next thing the students need to do in order to test DJ’s claim is to figure out the rules for writing fugues and musical counterpoint in general. I’m going to lay things out here, but from a pedagogical sense, this is best looked up and toyed with. I’m going to have my students use the sequencer in GarageBand to write simple fugues using easy themes like: Do-Re-So-Fa-Mi-Do

A basic fugue structure might look like this:

Lifted from ComposerFocus.com

The bass plays the basic theme (Subject: Do-Re-So-Fa-Mi-Do), then the alto answers contrapuntally while the bass complements. This is where the fugue form differs from the Canon form. A fugue references it’s basic theme, without being completely tied to it. On the other end of the canon spectrum would be a round, like Frere Jacques.

The third section sees the top voice repeat (verbatim) the basic theme while the alto plays the complement usually raised a fifth or lowered a fourth. Things go bananas from there.

The kicker is that you have to follow the rules of counterpoint the whole time. Turning these into math is where the value of the lesson is, I think, and it turns out the rules for counterpoint are super complicated. So, I’m going to give the kids a simplified set of rules:

1. The Following are Consonant: Octaves, Perfect Fifths, Major & Minor Thirds, and Major & Minor Sixths. All other intervals are dissonant.
2. Only 25% of your chords may be dissonant.
3. Unison is to be avoided at all costs
4. You are writing first-species counterpoint (1:1 notes between voices)

With these very simple rules we can begin to formulate a numerical value for the difficulty of writing a fugue, and, more importantly, we can gauge whether the difficult really does increase exponentially as you add voices.

So, how hard is it to write the first six measures of the fugue in the picture above? (Why six? Because eventually we’ll need six measures in order to add six voices; this is incredibly simplified) This is a tough thing to quantify, so I think I’ll guide them through just figuring out how many new notes Bach has to write. Students are really good at poking holes in something, so hopefully they’ll come up with a more accurate way of measuring “difficulty.”

Three Voices:

6 "measures" for 3 voices

You get the first measure for free; I gave you the basic theme.

In the second measure, you now have to write two new parts each composed of 6 notes (rule 4), so you’re writing 12 new legal notes (Rules 1-3).

In the third measure, the soprano voice plays the subject, so you get those notes for free, and the alto plays the same part that the bass just played (albeit probably transposed up a fifth, but these still don’t count as new notes). So, you’re really only trying to write 6 new notes in measure 3.

In the fourth measure you’re writing three new parts, as the free part from measure 3 can’t just be recycled here, lest we begin sounding repetitive. We can apply this to all free parts, so in measures 3-6 we’ll have amassed 42 new notes.

3-Voice Total: 60 New Notes.

Four Voices:

6 "measure" for 4 voices

Measure 1: freeby.

Measure 2: 12 new notes.

Measure 3: ostensibly only 6 new notes.

Measure 4: 12 new notes.

Measure 5: 24 new notes.

Measure 6: 12 new notes.

4-Voice Total: 66 new notes.

Five Voices:

6 "measures" for 5 voices

Measure 1: freeby.

Measure 2: 12 new notes.

Measure 3: 6 new notes.

Measure 4: 12 new notes.

Measure 5: 18 new notes.

Measure 6: 30 new notes.

5-Voice Total:  78 new notes.

A simple pattern appears to be emerging:

The King’s Challenge: Six Voices:

6 "measures for 6 voices

Things get goofy here. In six measures we’ve had just enough time to introduce all six voices. This limits the amount of free composition time that we have, thus actually lowering the number of new notes we have to write:

6-Voice Total: 72 new notes.

The Seventh Measure:

However, if you include the next measure for the 6-voiced fugue, you’ll have 6 free parts. After all of the voices have been introduced, we see this measure of 100% freedom and it represents a break for the listener from the monotony of the basic theme and it’s closely related answer, which have occurred thus far in every measure. All of the fugues with less than six voices have the opportunity to play this measure, while the 6-voiced fugue has not.

So, if we include a seventh measure for all of the fugues we get:

3-Voices: 66 New Notes.

4-Voices: 90 New Notes.

5-Voices: 96 New Notes.

6-Voices: 108 New Notes.

Not exponential. However, we’ve made some wildly hand-waving assumptions. Very wild. For instance, it is not clear whether adding an additional note within a chord is as equally difficult when it’s the second note of the chord or the sixth. We’ve attempted to wrap that subtlety into the larger concept of just “having to write” more legal notes. The students will hopefully take this rather guided investigation in a much more interesting direction. Perhaps even attempting to compose a fugue to get some experiential data to help inform our assumptions.

This is the kind of mathematical thinking I’d like my student to be comfortable with.

Oh!? Where’s the calculus? I don’t really know, but Calculus is a catch-all course at my high school for students who are math-bound in college. They need to have their algebra skills beefed up (heavily) and they really need to come to understand math’s greater role in actually figuring stuff out. So, fugues.