The man, who is believed to be in his 40s, was discovered in Spey Terrace on Sunday afternoon.
Police are treating his death as “unexplained” although it is understood that his death is unlikely to have been suspicious.
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Hide AdOfficers have been carrying out inquiries in the common stair. A report is set to be sent to the procurator fiscal.
A police spokeswoman said: “Lothian and Borders Police are investigating after the body of a man, believed to be in his 40s was discovered within a common stair in Spey Terrace at around 4.05pm on Sunday.
“Inquiries are ongoing into the full circumstances surrounding his death, which is currently being treated as unexplained.”