Why Math and Music are Friends

When I started this blog, I figured I’d mostly write about math topics, but lots of other things have crept in. A big one is music.

There’s general agreement that math skills and music skills overlap in some deep way, but for a long time I had a hard time putting my finger on exactly where (perhaps because I have pretty good math skills, but am very deficient in music skills). The applications of math in music theory (arithmetic mod 12; going up an octave = doubling the frequency of a sound wave = cutting the length of a vibrating guitar string in half on a fretboard) are very nice, but I never felt like they were getting to the heart of the matter.

Here’s what finally got me there. It’s hard to define exactly what math is, but here’s one definition I’ve gotten to like over the years (I believe it’s due to Andy Gleason):

Mathematics is the science of order and mathematicians seek to identify instances of order and to formulate and understand concepts that enable us to perceive order in complicated situations.

I.e., it’s not just about numbers! There’s also geometry (finding order in space), functions and correspondences (finding order by finding rules that relate one kind of thing to another), and a lot more. It’s an inclusive, generous definition, by which I mean that it lets many notions in, and helps explain how, and to what extent, those notions are actually mathematical.

Not accidentally, it echoes Varese’s equally generous definition of music:

Music is organized sound.

Finding order, organizing things, organizing sound. When you think about it that way, of course music is mathematical.


Two Kinds of Model Error

One winter night every year, New York City tries to count how many homeless people are out in its streets. (This doesn’t include people in shelters, because shelters already keep records.) It’s done in a pretty low-tech way: the Department of Homeless Services hires a bunch of volunteers, trains them, and sends them out to find and count people.

How do you account for the fact that you probably won’t find everyone? Plant decoys! The city sends out another set of volunteers to pretend to be homeless, to see if they actually get counted. (My social worker wife gets glamorous opportunities like this sent to her on a regular basis.) Once all the numbers are in, you can estimate the total number of homeless as follows:

  1. Actual homeless counted = Total people counted — Decoys counted.
  2. Percent counted estimate = Decoys counted / Total decoys
  3. Homeless estimate = Actual homeless counted / Percent counted estimate

For example, say you counted 4080 people total out in the streets. And say you sent out 100 decoys and 80 of them got counted. Then the number of true homeless you counted is 4000 (= 4080 — 80), your count seems to capture 80% of the people out there, so your estimate for the true number of homeless is 4000 / 80% = 5000 (in other words, 5000 is the number that 4000 is 80% of).

But it’s probably not exactly 5000, for two reasons:

  1. Random error. You happened to count 80% of your decoys, but on another day, you might have counted 78% of your decoys, or 82%, or some other number. In other words, there’s natural randomness in your model which leads to indeterminacy in your answer.
  2. Systematic error. When you count the homeless, you have some idea of where they’re likely to be. But you don’t really know. And your decoys are likely going to plant themselves in the same general places where you think the homeless are. Put another way, if there are a bunch of homeless in an old abandoned subway station that you have no idea exists, you’re not going to count them. And your decoys won’t know to plant themselves there, so you won’t have any idea that you’re not counting them.

The first kind of error is error inside your model. You can analyze it, and treat it statistically by estimating a confidence interval, e.g., I’m estimating that there are 5000 homeless out there, and there’s a 95% chance that the true number is somewhere between 4500 and 5500, say. The second kind of error is external; it comes from stuff that your model doesn’t capture. It’s more worrying because you don’t — and can’t — know how much of it you have. But at least be aware that almost any model has it, and that even confidence intervals don’t incorporate it.

Litmus Test

The Eric Garner video is bad enough, but then there is the Tamir Rice video. You can find it online easily enough. The gist of it is that a 12-year old black kid is playing with a toy gun by a playground, and a cop car rolls up, and the cops burst out of the car, and in two seconds the cops shoot the kid dead.

There were, as always, attenuating circumstances. The boy was large for his age, and someone had phoned in a tip about a gun, and said the gun probably wasn’t real, but the last part didn’t get relayed to the cops, and the tipster also said the kid might be 20, which did get relayed to the cops… but still. Tamir Rice is a 12-year old kid with a toy gun, the cops see Tamir Rice, they shoot Tamir Rice, and Tamir Rice dies.

As a parent, with boys, who are large for their age, I can see only two possible ways you can react to this:

1.  You can be completely, utterly, genuinely terrified that this might happen to your kid. Terrified enough that you go out and protest, or the equivalent, because if you don’t, you know down deep that you haven’t done all you can to keep your kid safe.

2.  You can be horrified, and even scared up to a point, but down deep you figure that this probably won’t happen to your kid, even if your kid does something goofy or weird at the playground once in a while, because, you know, you live in a different kind of town, and your kid isn’t… isn’t… bl

And then you have to go out and protest too. Don’t you?

Tom Lehrer and Moral Education

Tom Lehrer has done many great things, but one of my all time favorites is a sly dig at McCarthyism that he managed to sneak in toward the end of an Electric Company song. Pay attention around the 1:50 mark of this video:

When I came to the US as a kid (age 7), I watched a lot of PBS kids’ shows to learn English. The Electric Company was my favorite, and for some reason that one song really imprinted itself on my subconscious brain (though I figured out that it was by Tom Lehrer only a few years ago). I don’t know for sure how much it contributed to my moral education, but I like to think that maybe it had some impact.

As a grown up, I love the premise that if you think things through, you really can frame current events, and the issues of right and wrong embedded therein, in terms that kids can understand. That’s helped a lot with my own kids over the last few days as we try to process the events in Ferguson and now New York City.

Rock and Roll Saxophone

Bobby Keys died this week. I only learned his full name a couple years ago when I read Keith Richards’s autobiography, in which he features prominently as Keith’s drinking buddy and just about the only guy who can keep up with him. But I’ve known his sax playing ever since I wore out two copies of the Stones’ Sticky Fingers (on which he is credited as “B. Keyes”) as a teenager. Here he is on the record’s centerpiece, Can’t You Hear Me Knocking:

The first thing you notice about the long instrumental section is how the sax blends in just right, sliding in perfectly between the two guitars. But as you keep listening, you realize that the instruments play against type as well as blending: guitars take on what might have been a sax part, and vice versa. The guitars sound rough and bluesy, but also jazzier, more like the sax. As they play together, the instruments find new voices, new possibilities in each other, and the texture of the sound that results is unforgettable. It’s pure rock and roll, but also something more than that.

After hearing Keys on Sticky Fingers and the Stones’ other early 70’s records, I always kept an ear out for how a sax could add depth and texture to a rock and roll song. Here are two of my favorite examples, from the punk era. I’m not sure either of these songs could be as rich, could grab you and demand your attention the way they do, without Keys’s work as a precedent.