But there's nothing special about the 1k signal either, because a second is arbitrary. The original signal might have been 878 Hz instead (or equivalently 1000

But there's nothing special about the 1k signal either, because a second is arbitrary. The original signal might have been 878 Hz instead (or equivalently 1000 cycles / 1.139 seconds which is no more or less privileged than 1000 cycles / 1.000 seconds) and then the gating to 439 is easy.