Celtic proposed a change in the voting structure for future league reconstruction to a 9-3 majority instead of the current 11-1, but both Saints and County rejected the compromise at the Scottish Premier League meeting.
A club statement read: "Clearly we are disappointed that today's proposals have not been voted through. We believe this is a huge missed opportunity for the whole of Scottish football."
The statement continued: "We were pleased a number of months ago to have achieved unanimity across all clubs on the proposals for the way forward.
"We do not understand the rationale now offered for rejecting these proposals and the unwillingness to take the game forward positively.
"It is unfortunate that the vision shown by the vast majority of clubs is now not shared by all. We believe these positive proposals would have fundamentally enhanced Scottish football for the future."
Celtic had agreed to give up a significant proportion of their potential prize money, given they are expected to win the league for the next few years, to secure the 12-12-18 structure.
They believed the game needed freshening up with chief executive Peter Lawwell admitting concern at the number of Parkhead season-ticket holders who have not been turning up this season.
The statement praised SPL chief executive Neil Doncaster for his efforts and continued: "We understand today's events are unfortunate and the implications for the game will generate some uncertainty. However, Celtic has strong foundations and will look forward.
"We have enjoyed a tremendous season, we are on the verge of securing back to back SPL titles and yesterday we ensured our place in this season's Scottish Cup final. This, at the end of a season where we joined Europe's elite as one of the top 16 clubs across the continent."
- Sports & Recreation
- Scottish Premier League