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The properties had been advertised online but an investigation led to them being ordered to stop after furious neighbours complained about noise from guests.
Officials at the Edinburgh City Council ruled there had been a breach of planning laws amid a clampdown on short-term lets across the capital.
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A probe revealed the flats on the capital's Queen Street were being used for 25 days every month by guests who booked in using the popular website.
The owners were given until earlier this month to halt their operations which they also advertise on Booking.com and Expedia.
But they have launched a joint appeal against the decision to the Scottish Government.
The owners, who all use a letting agent, have argued no breaches had occurred and they should be allowed to continue leasing their properties to visitors.
They said they do not allow group bookings and avoid stag and hen parties.
The group also said other nearby properties were leased out in the busy city centre location and had not been told to close.
In a joint appeal document, they said: "Guests are forbidden from allowing any guests or visitors to cause any nuisance, annoyance or disturbance to neighbours.
"Our clients maintain that the use of the property as a short term residential apartment does not constitute development, as it has not resulted in a material change of use.
"Planning permission is only required where development has taken place.
"The council does not therefore have any reason to take formal enforcement action in respect of the existing use.
"There is no 'considerable turnover of people within short periods of time' as the Council has referenced in the enforcement notice."
They added: "The appellants consider that this flat is a well used and relatively unproblematic visitor facility in a very busy location, well located for Edinburgh’s commercial attractions.
"Accordingly, the appellants consider that the use of this apartment contributes positively to the council’s broad policy objectives for the vitality of the city centre."
Issuing their enforcement notice, the council said: "Evidence gathered as part of this investigation has confirmed that the flat is solely used for short stay commercial visitor accommodation and is advertised on several platforms.
"The significant level of occupancy and turnover is significantly different from residential use.
"Irrespective of outside ambient noise, disruptive movements associated with the intensive operation have been reported to the council by neighbouring residents.
"The intensive use of the flats for short-term letting has led to a deterioration in the living conditions of adjacent residents.
"It is recommended that an enforcement notice is served requiring the cessation of the unauthorised change of use."
A government reporter will issue a ruling in due course.
A consultation on a new licensing system for short term let propertieswill run until 13 August. Under the proposed legislation, councils will have until 1 October, 2022, to establish a licensing scheme, with all short-term lets to be licensed by 1 April, 2024.
Existing hosts and operators must apply for a licence by 1 April, 2023.