Minor key harmony, and the field that is not one field
The minor harmonic field comes from three scales, not one, which is why the same degree turns up with different chords; the major fifth degree is the choice that brings the music home.
The major harmonic field is a table: one scale, seven chords, fixed qualities. The minor field does not fit in a table, because the minor key has three scales, and each produces different chords on some degrees. Whoever catalogues "the minor field" without saying which scale each chord comes from is lying a little; whoever understands where each chord comes from plays in minor with nothing to consult.
The natural minor field
Stack thirds on each degree of the natural minor scale, using only its notes, and this comes out:
They are the same seven triads as C major, starting somewhere else, because A natural minor and C major use the same notes. And here is the flaw: the fifth degree, E minor, does not contain the leading tone. G natural is a whole tone from A, and does not pull. An E minor chord before A minor sounds like a lukewarm rest, not a dominant.
The harmonic minor fixes the fifth degree
Raise the seventh degree a semitone, from G to G sharp, and the scale becomes the harmonic minor. The fifth degree now has the leading tone, and turns major; with a seventh, it becomes a real dominant.
It is the chord that brings the music home in a minor key, and it is why the fifth degree usually appears major even under a minor key signature: the sharp on the G is written by hand, every time, because it does not belong to the signature. The same G sharp changes the seventh degree, which stops being G major and turns into G sharp diminished, the dominant's other face.
The melodic minor fixes the rest
The harmonic minor has an odd leap between the sixth and seventh degrees, F and G sharp, an augmented second. The melodic minor raises the sixth as well, and the ascent goes smooth. What changes in the field is the second and fourth degrees: D minor turns into D major, and B diminished into B minor.
The minor II–V–I of bossa nova uses a little of each: the half-diminished second degree comes from the natural, the dominant fifth from the harmonic, and the first is the usual minor.
The three fields coexist
In practice, nobody picks one scale and stays in it. Music in a minor key uses the minor fifth degree and the major one, the major sixth degree and the minor, as the melody asks in each bar. The rule is to look at the melody: if it rises to the tonic, the raised seventh degree and the major fifth; if it falls or rests, the natural material.
The picture, degree by degree, with the commonest chord for each and where it comes from:
- i — A minor, from any of the three.
- ii — B half-diminished, from the natural and the harmonic; B minor, from the melodic.
- III — C major, from the natural; C augmented, from the harmonic, rare.
- iv — D minor, from the natural and the harmonic; D major, from the melodic.
- V — E minor, from the natural; E major, from the harmonic and the melodic, which is what gets used.
- VI — F major, from the natural and the harmonic.
- VII — G major, from the natural; G sharp diminished, from the harmonic, playing the dominant's role.
What to take from here
The minor key has three scales and so three overlapping fields, and what gets played is a choice per bar, guided by the melody. The major fifth degree, from the harmonic minor, is the choice that gives the minor key a dominant, and the half-diminished second degree is what gives the II–V–I its minor version. Knowing which scale each chord comes from is the difference between memorising a table and hearing why it lies.