It sounds like the HDD is put into a power saving/sleep mode, spun down, but immediately started up again, and then this repeated 3-4 times, very quickly.
How could I find out what this exactly is, and what causes it?
If the drive has S.M.A.R.T then you could check what it reports with
sudo apt-get install smartmontools sudo smartctl
To narrow the problem down I would recommend to boot up from LIVE CD/USB and see if to happens or not. It it doesn't happen in LIVE CD/USB that means that the issue is related to a software, i.e. your installed OS. Otherwise I tend to think it is may be related to a hardware.