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Where would you like to live in London and I or someone else could translate it to Edinburgh? I also guess that you could take a place in London and it would be half the price for the same thing in Edinburgh
I think EH1 would be the worst of living in Edinburgh, great for a weekend airbnb but it would feel touristy living there.
Standard word of warning as well, house pricing and sales work differently to England. It's "offers over" in Scotland so that price you see on rightmove, add 25% if you really want it or 20% if you are happy to keep bidding and losing out.
To try to lesson the culture shock of leaving London, mainly- on the assumption that the closer to the centre you are the more amenities there will be ranging from coffee shops to museums.
I know very little about Edinburgh and would be the first to admit it though.