An investigation was launched after the body was discovered by a neighbour in the garden of flats in Warrender Park Road at around 8.30am yesterday.
She was named locally as mother-of-two Marilyn McLellan.
Police said there did not appear to be anything sinister about the death, but it was still being treated as “unexplained”.
A post-mortem will be held to determine the cause of death.
Neighbours yesterday were shocked at news of the discovery. One man, who lives in the next-door block of flats, said: “I got in very early last night and went straight to sleep so I didn’t know anything about until this morning.”
And a woman who lives nearby said: “The police asked us to shut our curtains out of respect. I did see the men in white forensic suits out there and there have been police there all day. I think it was an older woman who died but they seem to think it was natural causes.”
A worker at a nearby pizza shop said: “Yes, Marilyn died this morning - the police told me. She was very quiet, but troubled - she never worked, she just tended the garden. She would always say hello to me though. She was very petite in appearance.”
It is understood Ms McLellan had a son James, who lives in London, and a daughter Mhairi, who is based in Barcelona.
A police car was stationed outside the flats, with an officer speaking to locals and passersby.
Two joiners were seen measuring the front windows of number 62 and then boarding them up. Neither joiners or police would explain why.
A police spokesman said: “A woman’s body was discovered in the back garden of a property in Warrender Park Road by a neighbour at around 8.30am yesterday and an ambulance was called,
“Inquiries are ongoing into the circumstances surrounding her death, which is currently being treated as unexplained.”
Anyone with information is asked to contact police.