You could visit Edinburgh throughout the year, and enjoy the city's beauty in whatever season you wanted, although the only time to really go is over New year, when the centre of the Edinburgh is home to the biggest party in the world for Hogmanay, and revellers fill Prince's Street throughout the bitterly cold night, packed together tightly with drunken smiles on their red faces in a bid to keep warm.
The most popular birthdays in Scotland, rather unsurprisingly, fall around October 1st, some nine months later.