Not too quickly ( Caller is shocked) ...not too delayed (Caller is annoyed) ... Just right ...ring ~~ ring ~~ ring ~~ Hellooooow
You have heard of the ready >> steady >> go, the red >> amber >> green, the on your mark >> get set >> Go
It is just a convenient and smooth transition time.
Just wanted to add "three on a match..." I know the cliche is that it's bad luck, but the "spot the target, aim at the target... fire," concept to which it refers is analogous to "one ring to identify a call is incoming, one ring to stop what you're doing and one ring to pick up and answer..."
Consider a typical human at home. In the days before wireless phones, 3 to 4 rings were normal if you called someone. It may be a little faster today (cell phone proximity), but only because we often have our phone in hand when a call comes in (texting, using aps etc). If we consider that this is the norm for an individual, then it is what we most readily identify with. Anything deviating too far from that with which we most readily identify, tends to arouse suspicion, from a psychology POV.
Of course there are those who will applaud the companies who deviate on either side of three rings, but the more customers that can be pleased by policy the better. I remember in the 80's, Disney World/Land would let the phones ring for an hour if necessary. They didn't have a toll free number to take reservations (presumably to keep ticket prices low...lol) and to keep their customers from having to pay for long distance fees. Still, I remember my mom and her sister being quite annoyed by it listening to endless ringing before speaking to a human.
Personally, if I owned my own call center, I'd want the highest productivity from my employees (answer on the first ring, if available) combined with making as few waves as possible. While some might appreciate an answer on ring one, very few will think three rings unreasonable. If they did, I probably wouldn't want them as a customer anyway.
I assume the OP is referring to automated incoming call monitors that force 3 rings before the call is assigned to an operator? Seems like the perfect marriage of pleasing the most customers and attaining the highest productivity.