Music of the Rectangles, Part One

A lute arrangement of Ravel's Pavane Pour Une Infante Defunte, probably by Yoko Kanno

So, Steve Laniel of SteveReads.com asked a very reasonable question about Fourier Series, Taylor Series and possibly a bunch of other series named after long dead men. I wanted to say about a million things in response, but what I ended up saying was unhelpful and vague, so I am gonna try and make up for it here.

So, consider the Dead Princess with the lute in the above video. She asks Space Dandy if he can hear her music. He can, and that's a very complex fact if you think about what is happening physically. Impulses course through her unfeeling fingers and the strings vibrate, which causes the air to vibrate and those vibrations propagate through the air in roughly spherical waves into your ears, in which a membrane vibrates and sends signals into your brain. What interests us here today is the vibrations in the air, the music in itself as it where. So, let's say that I graphed the pitch of some music over time. What would that look like? Maybe something like this:

No, this isn't from any signal analysis of music or anything. Just a random analytic signal.

Well, what about this? What does it tell us? We need to turn this into something we can handle. I only drew a little piece of a function here, I could have easily drawn one that represented hours of music. There are different ways of thinking of a melody. One is that we could think about what is going on locally, what you hear at each point in time. Now, you can't hear any complexity, any harmony or melody, at just one moment in time. This is the above collapsed to just one point in time:

Helpful. Also, thanks MATLAB for changing the background color for no reason

Now if we're allowed to hear a tiny snatch of music (a little temporal neighborhood) we can figure out a lot more. We can figure out maybe the key it is in right now and what kind of instrument is playing. Some composers, such as Jean Sibelius, pride themselves on the internal logic of their compositions.

Brook Taylor

This is the approach taken in the Taylor Series, invented (in generality, special cases are ancient) in secret by Issac Newton. Some things are well understood like this, we call the analytic. You can tear them into the smallest pieces and learn about the whole. Our intuition is that everything is like this, we assume our local experiences give us knowledge of larger experiences. Of course, that intuition is wrong, the experiences of others might be very different than our own. The same thing is true in physics, we expect that if when a metal is heated it expands then we expect it to continue to expand. This is reasonably close to true of iron, but not of plutonium, which makes it devilishly difficult to machine. Let me show you some approximations to the above musical score made with the Newton/Taylor/MacLaurin idea of listening carefully to a tiny passage at a particular time.

With Approximations.

The red is a low order approximation, the equivalent of assuming that a tone is going to increase because it is at the moment. The other approximations involve paying a little more attention to a very small fragment of music. The important thing is that the changes between pitch is "smooth", we can figure out from where we are the location where we are going. This is not the only or even the most interesting way to write music.

 Rather than infinitesimal steps

Other musicians, like Stravinsky or John Coltrane, are proud of their music's movement. You cannot, by listening to one moment, figure out what the next moment will sound like. This corresponds to a score more like this:

Again, this picture is suggestive, not literally from music.

 Consider the moment when changing from one pitch to the next in the above picture. What note are you playing at that moment? It seems - correctly seems - like a whole range of notes would be equally okay to play. What note are you hearing at that moment then? The answer is not defined. In order to understand a signal like this, we have to go from a local point of view to a global point of view? What does that mean? That's the point of view of Fourier Series...

Infante Defunte

The original version of this post bloated to gargantuan size, so I am going to cut this post off here. Tomorrow, I will post the  part of the explanation that is about Fourier series. I'm going to break things up though, tomorrow will be a post on finitists and why they are wrong.