What is the difference between
"as a side note" and "on a side note"?
Are they interchangeable? Which one is preferred over the other?
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I think "as a side note" makes more sense, given that a side note is literally "a notation made in the margins or sides of a page; also called a marginal note." So when you're adding a bit of additional detail, you can preface it with "As a side note to this information,...."
The NGram for as versus on a side note shows that as a side note was more widely used in English books from 1800 to 2008. (Neither show up much in just the British English corpus.)
With the similar idioms "on another note" or "on a different note," I would use the preposition on.
"As a side note" is a metaphorical phrase, and will need to be used as a predicate, or subordinately to a predicate. e.g. "This acts as a side note..." it makes a statement about something. This form is to be equated with something IN the sentence itself.
"On a side note" is a locative, and is to be used more logically in a sentence. As an interjection, it introduces the following sentence which is itself to be considered as secondary/complementary. This form is to be equated with the statement as a whole.
The latter, "On .." should be used to begin a sentence, as an interjection. I imagine this would be your preferred option, as the former really ought to be part of a sentence, and if it is used at the beginning will be a subordinate clause, i.e. "As a side note, the ... is ...".