All gems have to be rare, beautiful and durable to be classified as a gem. As old mines become extinct and new mines are found, it is almost impossible to devise a rarity scale. Also, rarity can vary for the same gem, depending on its quality. For example Diamonds are not hugely rare per se - but a 1ct flawless, colourless Diamond is extremely rare. In fact, it is often said that only one white Diamond in a million faceted Diamonds is over 1ct in weight!
Based on the annual amount of carats faceted, Tanzanite is said to be a 1,000 times rarer than Diamond and the likes of Taaffeite and Benitoite are closer to a million times rarer than Diamonds!
Quartz the crystal is certainly not rare, as a large proportion of the Earth’s crust consists of it. However, gem-quality Quartz of a level that Alexandrite is one of the you would be happy to have set in a piece of rarest gemstones on the planet.