At one point, we began thinking about how you would prove the hexachordal theorem, which maintains that complementary hexachords have the same multiplicity of intervals. So we sat and we worked for many a week together. We were typically equipped amateurs in mathematics (though David, God knows, had technique that I didn’t have), and we found a solution. We used topological methods. We hit this little problem with all kinds of heavy hammers, and we solved it. Anyway, one day, I was speaking to a man named Ralph Fox, who was one of the great knot theorists of the world. (Knot theory is a very complicated field of topology.) At a summer mathematics workshop here in Madison, he’d heard about the property from a colleague of his who’d been told about it by George Perle, then at the University of Louisville. He said, “This is a very interesting problem. Have you solved it?” I showed him our proof and he said, “My God, I don’t even understand your proof.” What he was saying was that amateurs use such heavy-handed methods. He also realized very quickly, as we had by then, that the hexachordal theorem was a generalization of the complementation theorem, which asserts that the weighting of the intervals will be the same for any complementary sets, regardless of the partitioning. No matter how you partition, no matter how many notes you take out, you’re going to have that intervallic weighting remaining the same between and among the parts.

About two weeks later, Ralph called me up and he said, “Milton, I think I’ve got a general solution for this. And not only is it a terrific solution but it’s going to help me crack Waring’s problem.” Waring’s problem is one of the old standing problems in number theory. And by finding a very elegant proof, using group theory, to solve our little empirical musical problem very simply, he had solved Waring’s problem. Strangely enough, David and I certainly knew enough group theory to do it ourselves, but we didn’t know how to use it. Well, Ralph invited us to hear him present the proof to the math club. He began his presentation by saying that these musicians had suggested a problem in partitions and interval weighting, and that’s why we were there. This issue is very esoteric for mathematicians because it would never occur to them to subtract numbers. That’s why the all-interval set makes no sense to them at all, of course. We subtract numbers and call the results intervals, but there is no particular reason why this should ever arise in mathematics. Anyway, he showed the proof and it was wonderful: it was very elegant. It was published as a new way of solving Waring’s problem. In order to solve our problem, he had solved one of the really classical mathematical problems.

Milton Babbitt