Question on the formation of the Harmonic field in general

Given a key note, you can build up any sort of scale, for scales are just patterns of whole and half steps. The minor and major are examples.

Now, if the scale is the set of notes, the harmonic field is the set of chords.

So, suppose we want to write a music in the C key. The next natural step is to search chords. Well, here I became confused; I will explain:

It seems that the "harmonic field construction" is formed, every time, just with the major scale! Example:

1. key: c
2. major scale: c d e f g a b
3. Harmonic field: C Dm Em F G Am Bdim

My question is:

How to build the harmonic field in general?

Note that, I know how to pick a key note, and I know how to establish a scale pattern and write the scale for a given key note. But, the harmonic field in 3) is very specific, the II, III and VI degrees are minor chords, the I, IV and V are major chords and the last degree is diminished. So, I suspect that if I write the minor scale in 2) the harmonic field will not follow the rules of 3).

Here’s how you build a harmonic field from a heptatonic or octatonic scale:

1. Start with the first note of the scale and use it as the root of the first chord in the harmonic field.
2. To determine the second note of a chord in the harmonic field, go up the scale two notes from the root note of the chord. For the first chord, this will be the third note of the scale.
3. To find the third note of a chord in the harmonic field, go up the scale two notes beyond the second note of the chord, or four notes from the root of the chord. For the first chord, this will be the fifth note of the scale.
4. You have now formed the first chord (triad) in the harmonic field. Determine its quality (e.g., major, minor, augmented, diminished, etc.) and make a note of it.
5. Repeat steps 2-4 based on each note of the scale to determine the triads that make up the harmonic field for the scale.

For example, the harmonic field of the harmonic minor scale, is as follows (using A harmonic minor as the example scale):

• A minor (i)
• B diminished (ii°)
• C augmented (III+)
• D minor (iv)
• E major (V)
• F major (VI)
• G# diminished (vii°)

Note: A harmonic minor is A - B - C - D - E - F - G#

• "which was often used in the common practice period" the harmonic minor scale is more of a theoretical abstraction than something that composers actually used. Commented Dec 22, 2022 at 1:30
• @phoog I’ve seen a lot of sharpened seventh scale degrees in common practice music, but I think you’re right that it’s not exactly that that was done because the composer was thinking about the scale. I definitely oversimplified that bit - I thought it would be a more interesting example and wanted to motivate it in some way, so I sort of fudged the justification. Commented Dec 22, 2022 at 4:16
• "it’s not exactly that that was done because the composer was thinking about the scale": for most of the common practice period the harmonic minor scale hadn't yet been invented as a theoretical concept. People had been raising seventh scale degrees for centuries before the common practice period even began, but the harmonic minor scale was invented in the 19th century to account for this practice. Anyone "generating the harmonic field" in the 18th century would more likely have started with the proposition that the seventh scale degree may be raised or not (and the sixth, for that matter). Commented Dec 22, 2022 at 7:56
• @phoog Um. Ok. What is the goal of your comments? Commented Dec 22, 2022 at 9:36

The question supposes that all music is diatonic - using only the notes from the particular scale involved. As in key C, only notes CDEFGAB are available for playing. That seems to be a common misconception reading quite a few questions posed on this site alone. It's nowhere near reality, but let's go with it for now.

True, only using diatonic notes, and tertian harmony (stacks of thirds), we do indeed end up with M m m M M m d as the 7 available chords. Quartal harmony gives different chords.

But, you ask about minor. By stacking thirds, you will get a set of chords. More than you do with major, due to there being harmonic and melodic minr scales, giving more triad and four note chords.