"Just because the Scottish Parliamentary voice that you hear on the television says that they want to break up the United Kingdom and that Scotland wants to leave, it doesn't mean that the majority of Scottish people believe that".
But even if everyone who voted independent on Thursday would have voted SNP in a general election (a highly improbable supposition in itself), that clearly can not account for all of the difference between the two performances.
SNP MSP Derek Mackay, who has been announced as the party's campaign director for next month's general election, said the council election results would be an "excellent springboard".
The Scottish Liberal Democrats received a 6.8% share of the first preference votes as they won 67 seats, with the Scottish Greens winning 19 seats on 4.1% of the votes.
Mr Robertson, the SNP's Westminster leader, is facing a fight in Moray where the Tories secured 36.1 per cent of first preference votes last week, compared with 31.6 per cent for the SNP.
Ms Sturgeon said: "The issue at the heart of this election, not just in Perth and Perthshire but across Scotland, is a very straightforward one".
A Tory source told a newspaper: "Alex Salmond can't take anything for granted in this election". The party won 431 seats, 155 more than their nearest rivals, the Conservatives.
"Make no mistake - we are the underdogs going into this campaign".
The first minister said it showed the only party that could possibly "stand up" to the Tories were the SNP, and Ms Davidson said it demonstrated the only party that could "stand up" to the SNP were the Scottish Conservatives.
She said: "This week's local government election has shown we are the only party in Scotland with the strength to fight back against the SNP - in every part of Scotland".
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Labour's only Scottish MP in the last Parliament, Ian Murray, also looks poised to lose his Edinburgh South seat to the Tories.
She added that many Scots were angry that their Remain vote had been "assimilated" by the SNP and turned into a "proxy vote" for independence.
The First Minister insisted 8 June is the chance to put "Scotland's in Scotland's hands" and not those of Mrs May.
Meanwhile, leading pollster Professor John Curtice said claims of a Tory resurgence could be overblown - and that the contest to be Scotland's biggest pro-Union party was not yet won.
The analysis also suggests that the party is ahead in Edinburgh South, which is now home to Labour's only MP in Scotland, Ian Murray.
Equally, the official tally of the parties' share of the first preference vote across Scotland as a whole, which was published today, confirms that the SNP was well ahead of the rest of the pack.
"The Tories are the party of the "rape clause" and hard Brexit".
"Voters across Scotland will take a dim view of arrogant Tories taking them for granted". And now the Tories have older people in their sights too - as they gear up to end the pensions triple lock.
The highest turnout was in East Renfrewshire at 57.8% while Glasgow recorded the lowest at 39%.
The Scottish Conservative leader, Ruth Davidson, has told supporters that there are no longer any "no-go areas" for her party, as she tried to establish the Tories as the main opposition to the Scottish National Party.