Music Theory · Scale Degrees & Identifying the Key

Why is the mediant (3rd degree) such an important note when classifying a scale?

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More in Scale Degrees & Identifying the Key

Which scale degree is the supertonic, and where does its name come from?
Two scales share the same tonic and dominant but differ in quality (major vs minor). Which degree tells them apart, and why?
Why is the leading tone described as "essential" for identifying a piece's key, especially in minor?
A key signature alone never points to exactly one key. How many keys does it give you, and which two?
Given the two candidate keys from a key signature, what is the simplest melodic clue that usually reveals the tonic?
Both candidate keys share a signature. What accidental, if present, proves the piece is in the minor key rather than the major?