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I am an old fresh water smaller boat sailer and that it a neat rig. I think that is a schooner and I know I have never seen one before. I started sailing when I was 16 59 years ago and from the first time on the water moving under sail was such a great way to travel, owned a handful of boats and sailed and raced 16 foot up to 37 and loved most every minute except when things broke and beat the heck out of me and brought blood and all that.
I suspect there is a different name for this rig. The foremast (front) is shorter than the aft, making this a schooner. The foremast is also gaff-rigged, which you can see by the upper spar along the top edge of the fore-sail; it is hauled up in order to raise sail. But the main mast (aft) is a Marconi-rig - No gaff. I suspect the hybrid configuration would have a specific name, because sailing is traditionally so particular about things when it comes to nomenclature. Just my two-cent guess though. I'm surprised the foremast is gaffed as it is pretty close to the main mast and you wouldn't want the gaff banging into it. Maybe they just did it for the increased sail area.
The gaff rig allows you to get more square feet of sail up without having too high a mast. It also can be faster to lower or reduce for better handling. A topsail can be added above the gaff sail giving even more sail area but more importantly to catch elusive breezes. A two mast schooner can put a sail out on both sides when sailing downwind, called "wing & wing". In a large sailboat this puts a lot of sail out there and feels really fast, but of course downwind sailing isn't really fast. 7 knots feels like you are ripping along with all your sails up and wing & wing.
I sailed on a 133 foot Gaff rigged Schooner and just for fun we put up all the sails. The two big gaff sail; the foresail and mainsail. The fore topsail and main topsail. The jib, the flying jib and the jib topsail. And there was a staysail and a second 'jib' but it was called something else. All put up and taken down by hand. A lot of work.
The rig is called a schooner. It has a Marconi "main" (the larger, aftermost sail) and a gaff "foresail" (the forward smaller sail on a mast on the two-masted the schooner rig), and appears to have a club footed "staysail" (inner jib, on a boom for self-tending). I can't make out any other jib, but it does probably have one.
A pinky schooner would have a distinctive upswept stern (hence "pinky"), but this one does not have that.
A schooner tends to be faster than other rigs on a beam reach, or running off the wind. Does not go as close to windward, due to having more sail toward the stern. The schooner rig was favored by the early Grand Banks fishermen. The triangular mainsail is called Marconi, because the mast due to its height has to be stayed, like the original Marconi radio towers. The use of stays allows for a light hollow mast.
The mast for a gaff rig was solid, thick, and short; having a quadrilateral sail with a spar on top increased sail area, as the peak could be pulled higher than the throat (the part touching the upper mast), as shown in the picture.