A Letter About Metronomes
Dear Reuben,
You wrote that you’d been thinking about time again. How a bar of music is only the whole numbers wearing a costume, how a rhythm is a promise to come back. Good. I have a problem for you, then, and it is the one that has lived on my desk for years.
Lay the integers out like beats, going on forever in both directions. Now hand out a few metronomes. Each strikes on a fixed period, every second beat or every third or every fourth, and each begins on a beat you choose for it. Ask a small question. Can a handful of them, ticking together, leave no beat unstruck?
They can, and the first way is dull. Put one metronome on the even beats and one on the odd, and every beat is spoken for. But that is a cheat, because the two share a period; they are the same machine started half a step apart. So make it strict. Insist that no two of your metronomes run at the same speed. Now it feels like it should break. Difference is supposed to be the thing that leaves gaps.
It doesn’t break. Erdős saw this in 1950. Take periods two, three, four, six, and twelve, start them at the right offsets, and the five of them close over the line with nothing missing. You can make the slowest one slower still. This summer two mathematicians in Shanghai pinned down the tightest system whose fastest metronome only strikes every seventh beat: everything in it divides ten thousand and eighty, and not one beat is left out. The whole contraption of odd periods and careful offsets simply shuts, like a lid.
Here is why the problem never left my desk.
You can cover everything. You cannot cover it cleanly.
Try to make the covering perfect, every beat struck once and once only, a true partition of time with no beat doubled. You will fail, always, and not for want of cleverness. Someone proved it can’t be done. Add up the reciprocals of your periods, one over two plus one over three and so on, and for any covering by distinct periods that sum comes out strictly greater than one. The excess is not slack in the accounting. It is beats that get hit twice. A little of your music always lands on top of itself.
The reason is the loveliest thing I know in this corner of mathematics, so let me spoil it for you. Write each metronome as a term in one long sum, the kind that converges inside the unit circle, and the whole covering has to equal a single tidy fraction. Walk that fraction out to the point on the circle that belongs to your slowest metronome. The tidy side stays finite there. Your side does not, because the slowest term has a pole at that exact point, a note that rings to infinity, and nothing in the sum is shaped to answer it. Unless. Unless a second metronome runs at that same slowest speed, tuned to cancel the ring. Which is the one thing you swore off when you made them all different. The slowest beat, if it is alone at its speed, gives the whole thing away.
So it is settled, and it is strange. Distinct rhythms can fill all of time. They can never divide it evenly. To leave no silence, something must be spent twice.
I don’t have a moral for you. I only wanted you to hear it, since you are the one who thinks in beats now, and I am still the one who thinks in numbers, and it turns out we were circling the same thing from opposite sides. Play me the twelve-beat one sometime. I want to know which of the doubled beats you’d lean on.
Yours, M.
