Things inside things. A piece of aleph. CC0/Public domain.

Mathematical proofs turned into music? YES!

We didn’t skip the Free Thing of the Week on purpose last week, but we thought if we presented this on April Fool’s Day no one would take it seriously. But serious it is, at least as serious as mathematics ever gets.

Long ago when Bertrand Russell and Alfred North Whitehead dropped their three-volume magnum opus Principia Mathematica on the world, they began a philosophical resolution. The simple version is this: the Principia showed that mathematics could thought of as a steady arrangement of nested ideas (axioms in this case).

When MIT’s Norman Megill began to develop Metamath as a new kind of Principia Mathematica, he discovered that, while running computerized proofs, the nested ideas of mathematics looked an awful lot like the repetition common in musical structures.

Visualization of Adagio from Beethoven Piano Sonata No. 8
Courtesy The Shape of Song.
Visualization of the Schroeder-Bernstein Theorem.
Courtesy The Shape of Song.

So why not turn math into music? Thus began the Metamath Music Page.

There you’ll find a number of mathematical proofs aligned with a tempered Western music notation, creating something that generally sounds like avant-garde classical music, with a notable (sorry) difference:

The music generated from these mathematical proofs stands in sharp contrast to certain other experimental music based on such mathematics as the digits of π (pi). Unlike a proof’s tree structure, which is inherently suggestive of a musical score, the digits of π have no obvious pattern. To make it interesting, a human composer will often add a non-mathematical creative element such as an underlying beat with pre-selected chords and instrumentation. What one hears, then, might be based as much on the originality of the composer as on the essential nature of π: the same algorithm applied to the digits of say e (Euler’s constant), or even a series of random digits, would typically sound about the same after the first few notes. The music here, on the other hand, is a raw and unadorned representation of the mathematics itself, involving few human preconceptions beyond a basic mapping needed to accommodate the Western tonal scale.

The files are all MIDI files and they’ve been speeded up, according to Prof. Megill, so the curious listener can “decide more quickly whether it interests.” If one is so inclined, there are instructions on how to generate the music by oneself, including many of the 7,000 or so mathematical theorems that are not played here. As Prof. Megill says, “Who knows what they sound like?” Give it a try! Or just play the MIDI files as given. (If you’re a QuickTime user, remember to change your settings.)

Check out the Metamath music page here.