Det Ch Insp Graham McMillan, who is leading the investigation, said that finding the woman's identity was a main priority.
"The family are very traumatised by the event," he said.
"That's a key line of investigation for us now," he said.
"The children who made the discovery are very frightened by it.
"Whilst we are continuing our inquiries here we need the assistance of everyone to help us identify this woman, which will hopefully lead us to the cause of death."
"I've spoken with the mother who is also very upset and her first concern is for her children.
"Yet she still expressed concern and regret for the murdered woman and her family."
Specialist officers are planning to speak to the two sisters again.
Forensic examinations are continuing on the body parts in an attempt to identify the dead woman.
However, police said it could be several days before they had the test results which could shed further light on the inquiry.
The woman does not match the records of any local missing person and Tayside Police are liaising with every other UK force to try to identify her.
Det Ch Insp Graham McMillan said: "We've had some positive lines of inquiry coming into our incident room which we are grateful for.
"But it is still early days and we're keeping an open mind on a number of lines of inquiry."
He appealed for anyone who saw anything suspicious in the area, or anyone with concerns about a friend or relative they have not seen for a while, to get in touch.