Friday, August 22, 2014

Music and Mathematics

I'm reminded of something a classical scholar said to a musicologist many years ago: "do not look into Greek music theory; that way lies madness!" Somehow I have stumbled into some strange mathematical realms while innocently poking around in some Wikipedia articles trying to understand a little bit about the Poincaré conjecture that Grigori Perelman solved. That led me to an article about hearing the shape of a drum:




This little illustration is explained thusly:
Mathematically ideal drums with membranes of these two different shapes (but otherwise identical) would sound the same, because the eigenfrequencies are all equal, so the timbral spectra would contain the same overtones. This example was constructed by Gordon, Webb and Wolpert. Notice that both polygons have the same area and perimeter.
And then I wandered over to an article about the vibrations of a circular membrane (a drum in other words) and saw this very weird thing:


That, by the way, is one way that a circular drumhead can vibrate. Maybe if you hit it really hard in just the right spot? That's known as Mode u_{23} (5d) with \lambda_{23}=11.6198.

And that led me to things that vibrating strings do. Here are the first five overtones shown as standing waves on a string:


And then my head started to hurt and I had to lay down...
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2 comments:

  1. CraigSeptember 2, 2014 at 11:44 AM

    The vibrating drumhead is a classic problem in the theory of partial differential equations: I studied it, if I remember rightly, in a third-year mathematics course. The radial profile of the vibration is described by Bessel functions.

    Typically when you excite the membrane (that is, when you hit it) you don't get just one vibration mode. You get a whole bunch of them superimposed on one another, each with a different amplitude. The one in your animation is one of the more complicated, and is likely to have a low amplitude.

    Music and math is a huge subject, and a very interesting one too.

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  2. Bryan TownsendSeptember 3, 2014 at 8:03 AM

    Despite what they say about music and math being related, like most musicians, my math skills are rudimentary at best! So thanks for dropping by and sharing some insights, Craig.

    But I do know that the same thing obtains with vibrating strings: every time you pluck a string on the guitar, you get a whole bunch of different vibration modes. You can alter their relative intensity by how and where you pluck and this is how you control the timbre of the sound.

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