A monastery has a curious practice: they call it the music of the heavens.
The monks cycle through a cave with cups, to a stalactite at the center, which drips water. But this is not a steady stream: every few seconds there is a burst of several drops, then a pause, then another burst of drops, then a pause, in a simple repeating pattern of drops per burst.
In times long past, the water would simply have fallen to the floor. But now the monks collect them, listening carefully, then each brings his cup outside the cave, and says a mantra when he pours it into the lotus pond outside, and returns to the back of the line.
When the round of chants is over, a new team of monks takes over, allowing several bursts to again fall on the floor as the last monk hurries up to the pond and the cups change hands. They could go on forever without ceasing, a monk explains to me, as long as the drops follow that same pattern—but for the frailty of human bodies and the time each one would eventually have to wait to fill his cup. So they pray this way in these two-hour shifts, organized around a holy number of drips.
As one chanting ends, each of the monks can wait for over two minutes each, just filling their cups from the bursts and listening on the slow march out. Then comes the exciting bit! After the hand-off of cups, and the starting bursts have fallen, there is a flurry of monks who must run out of the cave, to be back in line within the minute. Then one collects two bursts, another collects three, another collects two, and the pace slows back down to something more meditative.
What are the monks doing as they are listening, and what's the pattern of the bursts of drops?
(And if you have solved that, you might be able to answer: what city did they have a visitor from, that they now associate the dripping pattern with the heavens?)
Edit: the “riddle” tag here may be ill-placed, it’s not that the whole thing is an allegory, just that I have not told you what the monks are doing as they are listening. Rest assured that they are indeed monks!
Hint 1
The monks could do what they are doing if the pattern was 3-periodic as either the steady "drip, drip, drip" or the more interesting "drip, drip-drip, drip-drip". But they could not do it for any other 3-periodic pattern, even if you go to 4 or 5 drops per burst. But then the pattern of bursts collected above would not be correct; you would never see one monk collect fewer bursts than the one ahead of him on that 1-2-2-1-2-2 pattern.
