A good friend of mine is a cellist. We talk from time to time about the nature of her instrument, and as I don’t know much about music and she doesn’t know much physics we have a hard time understanding each other in these conversations.
In order to understand the cello, I started by the usual (second- or third-semester physics) simplest understanding of a vibrating string fixed at both ends. This gives a very simple equation for the frequency of the string’s vibration when it is plucked.
Joan and I both agreed that this was an oversimplification. It did not predict much of the cellist’s experiences, nor, we agreed, was it meant to. Yet we obviously cannot capture everything about even the simplest of systems. Where should we stop? Here are some features of the cello we left out by pretending it was a single plucked string:
- The body of the cello, which seems to amplify some sound waves and deaden others
- The fact that the bow does seem to do more than pluck the string (otherwise, why isn’t it plucked? It sure sounds different!)
- The fact that one can in fact use the bow several different ways to different effect
- The rosin on the string, which seems to work by increasing friction between the bow and the string (this isn’t even in the model the equation proposes)
- The temperature dependence of the length of the strings and (perhaps) the body shape
- The multiple strings influencing each other (“sympathetic vibrations” are, in a way, real, but not mysterious)
There are no doubt more, which is precisely the point. I have also ignored the psychological aspects for the time being (how the sound waves affect a human — or other — ear and nervous system).
So what goes into a good model of a physical system? How do we know when to stop? Does it matter that we are attempting to understand something used for aesthetic rather than (say) commercial purpose? (But of course Joan earns money by playing in concerts, and so on — so maybe that’s a false dichotomy.)
Fields to consider: scientific realism, metaphysics and epistemology of approximations and idealizations, physics, music, differential equations, the relation between an aesthetic understanding and a scientific one.
P.S.: If you have any good references for the physics of stringed instruments that go beyond the infinitely thin monochord attached at two ends and plucked at a single point, etc., let me know!