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It was thought the idea of a 12-12-18 structure had the support of enough clubs.
Aberdeen's statement said: "Monday's failure to deliver the real benefits to the national game was a huge mistake".
And the Pittodrie club says it will now "work with others who share our belief in delivering real positive change."
As well as changing from the current 12-10-10-10 set-up, the proposed package of change included merging the SPL and Scottish Football League into a single body and a re-distribution of income for the 42 senior clubs.
St Mirren and Ross County voted against the plans and thus scuppered the vote.
A late amendment to the plan, which would have changed the majority needed for future restructuring, was offered in an attempt to achieve the necessary support for approval.
However, St Mirren subsequently explained they wanted a 9-3 majority extended to other areas, such as the distribution of finance, squad sizes, under-21 player rules, season start dates and salary capping.
Aberdeen cite a "unanimous support for the principles" following a general meeting of the SPL in January.
"The time and effort put into this process was huge and the SPL executive, board, restructuring team and legal representatives spent considerable time on this to enable final documents to be sent out to all SPL clubs," the Pittodrie club added.
"It was agreed by all parties that the key to unlocking the deadlock was to deliver far more meaningful games to the supporters that would also be attractive to broadcasters and sponsors alike.
"We found it difficult to understand why the two clubs that rejected the proposal waited until so late in the day to state they would vote against.
"Only by delivering a new organisational structure, as the proposal did, that significantly increases meaningful games enabling financial benefits to be passed down to all clubs will the game move forward.
"Voting reform and an all through financial distribution would also have occurred. It is extremely difficult to see another way of delivering all of these benefits that will receive the backing required."
Hearts made it clear they were also angered by the vote and this prompted beleagured St Mirren chairman Stewart Gilmour to call for a "mediator" to be brought in to help resolve matters.
The SFA has indicated it will look at ways of getting the restructure back on trcak and cited the "acrimony and rancour" following Monday's SPL vote and urged clubs to "uphold the value of mutual respect".
"It should be remembered that 10 out of 12 top-flight clubs voted in favour of change," said the governing body.
"It is with this in mind that the Scottish FA reiterates its commitment to restructuring on the five principles of a single league body, fairer financial distribution, introduction of play-offs, parachute payments and a pyramid below the third division.
"The governing body calls on both the Scottish Premier League and the Scottish Football League to re-affirm their commitment to the five principles.
"We are aware of public comments from club chairmen from both the SPL and SFL inviting the Scottish FA to intervene to find a solution.
"The Scottish FA's rules - and the respective rule books of the league bodies - do not allow for the governing body, or any external body, to 'take control' of a separate company in possession of commercial contracts worth millions of pounds.
"However, if it truly is the will of the clubs, the Scottish FA is prepared to get involved and act in the best interests of the national game.
"This will require a mandate to the Scottish FA from both the Scottish Premier League and the Scottish Football League, so that positive changes to improve Scottish football's future can be proposed without the threat of legal recourse or judicial review."