At first I was taking this to be about the inevitability of death. But then I read this interpretation, which takes the story to be about tradition no longer working in a modern environment (more specifically, it looked at Jewish traditions). I quite liked this interpretation when I reconsidered how the clock was behaving. It wasn't losing time, it was gaining it. It wasn't coming up short on its chimes, it was adding them on. I think that's a telling clue as to what Aleichem was getting it. Clever, in any case.
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