On many telephone systems, each telephone is equipped with a speakerphone button that, when pressed, immediately answers the incoming call. The call is then on speakerphone. The time it takes to answer is limited only by your reaction time and physical acuity.
If you adjust the speakerphone volume to the lowest setting, (so as not to disturb the nearest worker) on an incoming call, you could punch the speakerphone button immediately to answer the call, then lift the handset and the call will be automatically transferred from speakerphone to the handset without any other action on your part.
I understand that you want to answer it "as quickly as possible" but I hope you realize that referring to "1/4" or "1/2" ring is not a realistic perception of how telephone systems work. It would require many minutes of typing to describe ringing cadences, Central Office vs PBX ringing, wink-start vs ground-start vs loop-start lines, etc. During a detailed and lengthy discussion, starting with your telling us much more information as to the who, where, why and what of your request, we might be able to come up with a perfect plan to address your requirements.
As one example, ground-start lines provide the called party with an electrical indication that an incoming call is present, BEFORE the ringing even starts. That would fall under the description of "0/0ths" of a ring.
There are sophisticated (and expensive) systems called "ACD" (Automatic Call Distributors) that detect and answer an incoming call, then assign the call, immediately, to the next available agent, without any ring at all. The calling party hears ringing, of course, but the agent does not. He simply hears a short duration tone to alert him that his telephone is "live" and he is then connected to the call automatically. These systems assume the use of headsets by the agents.
