The Head of the Order of the TriMagus is the title given to the ruling Master of the TriMagus, who ostensibly still chairs his own Order as well. His vote is the tie-breaker in any contest, and his particular Order tends to echo the state of magical affairs.
This position is voted on every seven years, and all mages who have formally passed their Tests can vote. Over the years the number of Towers, and the number of Orders, has dwindled. There are only three Orders currently, but each mage can vote and hold positions within their Order. This determines the head of each Order [ie Master of the White, Master of the Black] as well as the 'Head of the Order' every seven years.