It is not generally accepted that the origin of not has anything to do with note meaning 'piece of paper' or 'bill of money'. But we have to look at the history of money in Scotland. At one time both the pound Scots and pound English were in use in Scotland but the pound Scots was worth about 12 times less. Whilst there was ambiguity in English, the Gaelic terms only ever referred to the Scots money
peighinn = penny Scots
sgillinn = shilling Scots
punnd = pound Scots
The peighinn was so small that it was forgotten about. The sgillinn was worth the same as a penny English, so came to mean that, but they had no word for the pound English. This may have been (I'm not sure) the same time that some banks in Europe (with the Bank of Scotland amongst the first in 1695) began to issue paper money. If, as I assume, this was denominated using pounds English, and as Scottish money was not legal tender but a promissory note, then it is easy to see how 'note' (originally nota in Gaelic, now not) came to mean one of these notes that was denominated in pounds English even if it was worth some number or fraction of pounds). Continuing this logic, this use in Scotland of promissory notes instead of legal tender could have given rise to the use of note to mean paper money in Britain, unlike in some other places where they use bills.