I hope others can give a more detailed answer, but my advice would be to start by studying those cultures and genres that use microtonality. Middle Eastern music springs to my mind; especially music for the oud; this might be the closest to the Western concept of "scale," in which we conceive of a given scale degree as being a certain frequency, "always" or in the abstract.
The violin is also used in Carnatic music, with non-Western techniques and tunings. The concept of scale is more complicated in this and other genres, because it's not just a matter of static pitch, but of rules about how you raise, lower, and bend pitches in different contexts. For that matter, there are plenty of Western practices that encourage us to "bend" or alter pitches for contextual reasons, from the "blue note"(s) in blues to the way classical players narrow the gap between the leading tone and the tonic. In the history of classical music, the notion that all half-steps are equal distances ("equal temperament") is relatively new; for centuries, the frequencies would be adjusted or "tempered," and even instruments with fixed pitch like lute or harpsichord would be able to adjust the temperament (in the case of lute, with moveable frets). For a deep dive into how this system still affects piano tuners (at least, smart ones) today, see The Seventh Dragon: The Riddle of Equal Temperament by Anita Sullivan.
Scandinavian folk music also exhibits "altered" tunings; I think of the willow flute which plays the overtone series with its "out of tune" notes, and both Norwegian and Swedish fiddle traditions adjust certain pitches noticeably out of the standard scale (see this article).
It's debatable whether "microtonal" is the right word for these practices of temperament or altered intonation, though. In the case of pre-Romantic Western music's "temperaments," this was still a system built on the tetrachord and the familiar 12 semitones that we derive from it; it's not a matter of dividing the octave into more or fewer parts than 12 and identifying "other notes."