Tuesday, April 29, 2008

Asymptotic Progress

My favorite class in high school? I’d ask you to guess, but you probably won’t. Algebra II. I ate up Algebra II. On the one hand, it required some crazy abstract thinking. “Ok, today we are going to begin Conic Sections. Now, I want you to imagine two cones, stacked atop each other, connected at their points. Let’s think of how we can slice these cones, and create equations describing those slices…” Wha?

On the other hand, since it was, after all, MATH, that meant the answers were there, awaiting your discovery. Your answers were right, or they were wrong. Unlike English, where you could be simultaneously right AND wrong, depending on how you worded your answer, or built your supporting arguments. On one particular Algebra II test, I remember getting a wrong answer. The fact that I remember this single test question, 20+ years removed from it, should give you some idea as to how annoyed I was to have missed it.

We were learning about asymptotes. Theory being, there is a boundary to a particular function, and as you progress towards infinity, you get progressively, microscopically, asymptotically close to the boundary, but you never… quite… get… there. The question was this:

Plot the volume of a marching band as a function of its distance from the listener.

I gave the question too little thought, and drew a straight line, something like this.

The REAL answer, as you might guess, is that the line is not linear, but rather is initially quite steep, then gradually levels out over time. Like this:

I argued with my teacher on this. "Ok," she said, "Next football game, you measure the sound from the band, and if you can show me that it dies down in a linear fashion, as a function of distance, you can have your points back." The truth was on her side, and I cowered before it.

Ok, how is this piano related? I feel like my progress on the pieces I am working on is asymptotic in nature. I experience some quick, early progression, then some continued steady progress as I get closer to some mythical perfection. "Perfection", in this case, being the ability to play the pieces consistently, at tempo, and without error. Then very little progress as I pile on the time (which is, I admit, not enough these days.) Specifically, I am finding myself putting quite a bit of time into Little Brown Jug and Chiapanecas, and I am making the same mistakes, over, and over, and over. This is one of those times that I think "Man, I wish I had a teacher." But I don't. And I won't for some time yet.

So I am considering, for the first time ever, shelving these unfinished pieces, even though I don't have them completely polished... and moving on.

I mentioned on PianoWorld that when I finish Book One, I figure I will spend some time (weeks? a month?) continuing to improve Book One pieces before I go on. There are a number of pieces (Lone Star Waltz and Lavender's Blue come to mind) that I had down pat at one time, but now struggle with when I pull them out of storage for a go. I don't think I can completely solve this, but I hope to improve the situation a little prior to starting Book Two. Maybe when I return to Little Brown Jug and Chiapanecas later, they'll magically be polished?

On the other hand, if this is the start of a trend, if this is how I do on the rest of this book, it's a very bad trend.

No comments: