To gain access you need to buy your ticket across the way from the museum, which is a good idea as people may otherwise miss this amazing collection, complete with carriages and sedan chairs, cabinets full of silverware, papal bulls... you get the idea. There is a room full of coins which you could spend a long time in, and there are several rooms filled with paintings, so it is 5 Euros well spent before you even get to the Cathedral. This is not included in the Heritage pass, as all Churches/the two Cathedrals aren't.
The Cathedral itself is beautiful and if you time it right, very serene. Then the silence is interrupted by noisy tour groups, but they're normally in/out/on to the next thing.
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