The woman, whose identity or age have not been released, is understood to be the victim of an unprovoked attack.
Officers discovered the body after they were called to the flat in Dobbie's Loan Place at 11.30pm on Wednesday.
They were contacted because the house-holder had not been seen for some time and they gained entry to the flat.
A murder inquiry was launched after a post-mortem exam-ination was carried out.
CCTV footage is being gathered and reviewed and officers are continuing with door-to-door inquiries.
The woman was last seen more than three weeks ago, on Wednesday October 9 at Woodside Health Centre in the city.
Detective Chief Inspector Colin Carey said: "We are trying to establish a motive for this attack as it appears to have been unprovoked.
"We know that the woman was seen on Wednesday October 9 and we need to establish her movements from then on - who she saw, where she went.
"If anyone has any information please do get in touch with us."
Additional patrols will be in the area and anyone who has concerns or knowledge of the incident can speak to officers.
Anyone with information can contact police on 101 or via Crimestoppers on 0800 555 111.