Very fascinating! I understand 1/10th of it, but I appreciate your effort!

I don't blame you XD I've been studying Japanese swords for about ten years, and half of this stuff was relatively recent knowledge for me, which kind of explains my recent overwhelming obsession. This is pretty advanced stuff that the average person wouldn't know anything about

Very fascinating makes me want to do a little research of my own question the Katana sword that Samuri used what school would have made those? just curios asome work on the swords

These are references of typical characteristics of the Gokaden, the five major swordsmithing schools in the Koto Era (Heian to Azuchi-Momoyama; 794 to 1596 AD). In this time period, tachi were the swords of the time. The differences between katana and tachi are subtle, but noticable. The most noted is that katana are worn in the belt edge-up, while tachi are worn edge down. Other differences include the fact that tachi tended to be longer and have steeper curvatures than katana. All of these aid in the context in which each was used.

Tachi were used from the Heian to Muromachi periods (796-1573), although in the Muromachi period, they fell out of favor to the katana (then known as the uchigatana). The difference, both physiological and contextual, are similar to the differences as cavalry and infantry sabres in Europe. Tachi, like cavalry swords, were mainly used on horseback, hence their larger curvatures and longer, heavier blades. Uchigatana, like infantry swords, were used on foot, and thus the more shallow curvatures and shorter, lighter blades were more practical in that context.

Starting in the Shinto Era, which took place in the early Edo period (starting around 1600 AD), shogun Tokugawa Ieyasu unified Japan after many years of abundant civil wars, ushering a time of peace. As a result, swordmaking has drastically changed. Since there are no wars, horseback combat, and thus tachi, have become completely obsolete, making the katana the primary weapon of the samurai. New schools popped up to overtake the old ones, and new techniques focusing on aesthetics rather than functionality, such as new hamon patterns, and have prospered until the Meiji Era (starting 1868), where in an effort to modernize and "Westernize" Japan, the samurai class has been dissolved, and Nihontou were banned, focusing more on swords closer to the Western sabre.

So to answer your question, all of these schools focused on making tachi, since that was what was needed at the time. In the last parts of the Muromachi period, they made katana, but fell out of favor in the Edo period, mainly because all the wars in the Muromachi period has put strain on availability of materials and training, in favor of newer schools that focused more on aesthetics. Hope that answers your question :3

thank you for answering my question