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In jazz or pop styles, keep in mind that some minor tonalities will be minor key while others will be modal. Patterns like iimin7b5 V7 imin7 indicate a minor key flavored approach where within the pattern we see chromaticism with the dominant and the ^7 scale degree and it roughly conforms to the rule of thumb above. WhereasBut, patterns like i bVII i or ivmin7 imin7 vmin7 indicate a modal approach where the chords within the pattern are diatonic. Purely diatonic minor suggests modal harmony.

In jazz or pop styles, keep in mind that some minor tonalities will be minor key while others will be modal. Patterns like iimin7b5 V7 imin7 indicate a minor key flavored approach where within the pattern we see chromaticism with the dominant and the ^7 scale degree. Whereas patterns like i bVII i or ivmin7 imin7 vmin7 indicate a modal approach where the chords within the pattern are diatonic. Purely diatonic minor suggests modal harmony.

In jazz or pop styles, keep in mind that some minor tonalities will be minor key while others will be modal. Patterns like iimin7b5 V7 imin7 indicate a minor key flavored approach where within the pattern we see chromaticism with the dominant and the ^7 scale degree and it roughly conforms to the rule of thumb above. But, patterns like i bVII i or ivmin7 imin7 vmin7 indicate a modal approach where the chords within the pattern are diatonic. Purely diatonic minor suggests modal harmony.

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The chart gives triad qualities, but the text gives seventh chord qualities. This isn't a big problem, but it makes unclear what style of harmony to are interested in. All seventh chords is indicative of jazz.

Regardless, I think the main point about minor harmony can be made by simply looking at the E chord, the dominant of A minor.

The question asks about minor not Aeolian mode and that is significant. We are dealing with the major/minor system not modal harmony.

Let's some detail of the original post...

If we now apply these [intervals] to the E minor scale (E, F♯, G, A, B, C, D, E) we get the notes back for our chord:

and

If we now apply these [intervals] to the G minor scale (G, A, B♭, C, D, E♭, F) we get the notes back for our chord:

This method of building a natural minor scale over the root and then building the chord from that scale is not how the major/minor system works.

The first problem is setting each scale degree as a new tonic. For example, when you get to scale degree G in A minor you do not make G a new tonic with a G minor scale. You simple move up the A minor scale starting on G. Technically, you can call this the 7th mode of A minor. (The quality of that mode is Mixolydian.)

Second, building chords by stacking up thirds from a fixed scale of tones is a bit misguided and can lead to a misunderstanding of harmony. This is especially the case in minor key harmony.

Stacking up thirds from a fixed scale will produce only diatonic chords. The major/minor system is not a purely diatonic system. It is a combination of diatonic and chromatic harmony. The chromatic aspects are largely a matter of convention rather than a purely logical system. You can't simply use some methodical approach to know what those conventions are. You have to acquire the knowledge from score, a teacher, or good books.

Let's consider the chord on E the dominant scale degree.

If you create a purely diatonic triad on the dominant it will indeed be tones E G B which is a minor triad. This chord may be encountered in a progressions like i v6 iv6 V.

But in the major/minor system an authentic cadence must be from a major dominant chord to the tonic chord: V to i. To create that major dominant triad we use tones E G# B. You can also see the major dominant in jazz progressions like iim7b5 V7b9 imin7.

...have I misunderstood the logic for minor scale chord construction.

Yes. Again the problem is problem is trying to stack thirds from fixed scales and overlooking how to handle the dominant chord.

The best way forward is to read a good harmony textbook and study the section on minor key harmony. I think every harmony text I have seen has a separate section to deal with minor harmony, because of its unique characteristics.

If you want a reasonable rule of thumb for classical, common practice minor key harmony...

  • use the diatonic chords of the natural minor scale
  • in classical, common practice style when a cadence is formed the dominant chord's third should be major
  • if you want to temporarily tonicize one of the non-tonic chords (III iv v V VI VII) precede it with a major triad or dominant seventh chord rooted a perfect fifth above, or a diminished chord root a half step below.

Those three points will not account for all minor key harmony, but it will cover a very large portion of it.

In jazz or pop styles, keep in mind that some minor tonalities will be minor key while others will be modal. Patterns like iimin7b5 V7 imin7 indicate a minor key flavored approach where within the pattern we see chromaticism with the dominant and the ^7 scale degree. Whereas patterns like i bVII i or ivmin7 imin7 vmin7 indicate a modal approach where the chords within the pattern are diatonic. Purely diatonic minor suggests modal harmony.