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An article about Aeolian harmony on Wikipedia begins as follows:

Aeolian harmony is harmony or chord progression created from chords of the Aeolian mode. Commonly known as the "natural minor" scale, it allows for the construction of the following triads (three note chords built from major or minor thirds), in popular music symbols: i, ♭III, iv, v, ♭VI, and ♭VII

Why are the third, the sixths, and the seventh degrees chords written as flattened?

If I understand it correctly, if it's a minor, then these degrees are already flattened. Why are they written with flats?

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This use of flats removes any and all ambiguity regarding the nature of these "three", "six", and "seven" chords. This is especially important when the Aeolian mode is being borrowed from in an otherwise major-key (or Mixolydian-mode, etc.) piece.

(Naturals can be used instead when these chords otherwise have roots with sharps in them. One example is ♮III of D major, which is a F major chord. Note that iii of D major is a F sharp minor chord.)

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It seems as if the Wiki article is describing things from the point of view of the parallel major scale. Thus with C-major as the key of the section of music being discussed, chords taken from the parallel minor (C-minor, note that terms are different in other languages like German) need accidentals.

From a "common practice period, (c.1600 to c.1900)" any chord may occur in any key; the "functional meaning" of these chords is not necessarily the same as diatonic chords with the same root. I like to think of chords a being "tonic-oriented" or "dominant-oriented" or "subdominant-oriented" or "passing" chords. As long as the key is strongly confirmed (over the whole piece), one can use any number of non-diatonic (sort of like non-native) chords without disturbing the sense of key. Think of the II or II7 chord in a major key; it behaves as a V of V or a dominant of the dominant but doesn't seem to disturb any key feeling. Likewise (returning to the original question) the flat subdominant (iv) chord is quite common. In one of the common idioms in American popular music from the 1930s or so, things like I, I7, iv, II7, V7 patterns were commonplace. The II and iv (and I7) all need accidentals as they contain non-diatonic notes. There are lots of other examples (I, III7, VI7, II7, V7) or (I, IV, bVII, V) or (from a couple of country western pieces) (I, III7, IV, V) and (I, IV, II7, V7). Even the popular dominant 9th needs an accidental in C-Major (or C-minor): G-B-D-F-Ab.

The use of accidentals marks non-diatonic notes, but these need not mean a key change.

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