European Sword Weight II - "LIGHT" Rapiers and "HEAVY" Broadswords

I keep reading and hearing that Rapiers were a "huge leap forward" if compared to Medieval swords; they were much lighter, quicker and finally made advanced fencing possible. It was the dawn of true swordsmanship as it is understood by modern sport fencers... That myth utterly shatters on weights of historical swords! Arming swords, Renaissance era riding swords, basket-hilted Broadswords, Waloon swords, Mortuary swords - and even Late Medieval Longswords did weigh EXACTLY THE SAME as 16-17th century