It would be interesting to know what causes it. All three 6,1s I've had make the sounds a little. My quad-core/D300 (2016 build date) was the most quiet. It would almost never make sounds, but occasionally you could hear it if you put your ear up close. My 6-core/D500 (2014 build date) is the most noisy. Sounds like little birds chirping coming from the power supply area. You can't hear it more than a foot or two away though so it's not that annoying. My 8-core/D700 (2017 build date) makes some noises too. Interestingly, when I temporarily put an SSD into the quad-core from a MacBook Pro it got really noisy. Soon after that the SSD bit the dust. That was after I put it into an OWC Envoy Pro enclosure which corrupted it. So you might want to switch SSDs to see if it helps.
That is definitely some very interesting observations and with three machines I can't imagine someone with more experience
Since you have different configurations with different manufacture dates I think we can conclude that the possibility is present with all Mac Pro 2013.
This is the sound that I'm talking about which is recorded with my iPhone placed down in the top of the computer:
I'm glad that your can hear it too, but it still surpises me that so few seems to "know" about it or can simply hear it.
I think my machine is from early 2015 and I bought it used. It was very minimal used and I don't use it for professionel work. I simply use it a my personal computer.
Something tells me that the power supply maybe "wears" over time but then again I think it would be weird considering the minimal use of mine.
Anyone can write "ac" in terminal and see "powered on time" and mine shows "3854 hours" which is quite low?
I have a difficult time letting this go since I want to keep this computer. In my mind it isn't likely that the noise would come from a SSD or the GPU's(since the sound is present all the time and therefor under very easy load)
Since a new power supply board isn't that expensive I took the chance and ordered from china via eBay. I live in Denmark so it will probably take 2-3 weeks before arrivel and especially in these covid-19 times.
I will of course come back with my results when I have installed the new power supply board!