According to popular wisdom - which may be incorrect of course, feel free to correct me - Gustavus Adolphus revolutionized battle tactics by introducing light weight field cannons and by assigning one or two of those to each of his infantry regiments/brigades. Those cannons were of around 3-pdr caliber if I understand correctly.

Now my questions is: why exactly was the size of 3-pdr chosen? There were plenty of cannons of both higher and lower caliber existing at the time. 6-pdr and 12-pdr were seemingly the typical cannons of the artillery train, but I can guess they were considered to large and heavy for the intended use. But lists of existing cannon types of the time (e.g. this) show a lot of smaller caliber cannons, to give some english examples: falcons, falconets, robinets, bases.

Why hasn't any of those types been chosen as the archetype of infantry support gun? They appear even lighter than the 3-pdr? Is there some reason behind the 3-pdr to give it a clear advantage? Do we have some insight into thought process of Gustavus Adolphus or whoever was the first to propose those?