Posted by: phedrick | November 2, 2008

Math + Music

Hey all – I didn’t have any particularly interesting math problems to talk about, so I thought I’d use my post to muse about the connections between mathematics and music (mainly drawn from my research perusing the internet).

I’ve always loved both math and music, so it’s easy for me to imagine that there is a deep connection between the two, but they’re obviously not completely intertwined (I hope that no one finds a “formula” to play Rachmaninoff’s third piano concerto because I’d like to think that it’s up to the performer to create a unique sound every time). Anyway, Pythagoras was actually one of the first people who was interested in the connection–his vision of the world was that math could be found everywhere. He developed what would later develop into our modern system of scales by working out all the possible ratios for string lengths and their corresponding tones.

Obviously the main reason people like music is because it is pleasing to the ear (or maybe it just “fits” a mood someone may currently be in), but the main reason basic scales are “beautiful,” “tragic,” or bad sounding is almost purely mathematical. Certain tones play better with others–a major scale will sound pleasing to the ear, whereas a minor scale will sound “ominous” or sad. There are more than just those two “harmonic ratios,” which can essentially be written out with a mathematical ratio formula (go here to see more http://www.amarilli.co.uk/piano/theory/mus-sci.asp).

Harmonizing isn’t the only musical concept connected to math–rhythm and pitch are also best described using mathematical language. This is probably more obvious–keeping a beat is a very regular business that doesn’t require much creativity or musical genius, but does this mean that one could combine these things to design a mathematical algorithm that churns out beautiful tunes? Sure, it’s pretty easy to set a computer on a scale and have it create tones that are harmonizing and pleasant to the ear, but I don’t think a computer will ever be able to replicate the works of great composers.

I’ve heard people say that there are only two types of geniuses–mathematical and musical, but I think they are really too closely connected to be separated. It’s been shown in many studies that those who are musically inclined tend to be mathematically inclined as well, though the causal effects are less clear. A computer will never be able to exhibit “genius,” and for now, I’m happy about that.

If anyone knows of any good internet articles on this kind of stuff, I’d love to see them.

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Responses

  1. My dad does some work on aesthetics, asking questions relating to some of what you talk about here, so I’m going to ask him to post a comment. I think he’ll have some references for you.

    There’s a math-of-music book here:

    Music: A Mathematical Offering

    One of the interesting things he says in this book is that while we usually think that certain intervals sound harmonious because the wave forms “line up” and this sounds good to our ear, actually the reason has to do with the instruments and their overtones, which on western instruments tend to ”line up”. See the introduction. For example, he points out that Gamelan (which uses percussion instruments which produce a different set of overtones) uses a different scale for this reason.


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