The 51-year-old was found on Salisbury Road, in Newington, with horrific injuries after falling in the early hours of Tuesday morning.
The IT analyst was rushed to hospital but was pronounced dead shortly afterwards.
Police are treating his death as unexplained but it is not thought to be suspicious.
Mr Barr fell out of the flat window at around 3.15am. Salisbury Road was closed for most of the day yesterday.
Gordon is listed as the sole director of an IT firm and also worked for an Edinburgh company.
He leaves a son, who is understood to work in the US.
A police spokeswoman said: “Police and emergency services attended and the man was taken to Edinburgh Royal Infirmary. He sadly passed away a short time later.
“A report will be sent to the procurator fiscal.”