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This question is not about some kind of mathematical permutation of notes, but a perceptual categorization, mostly related to jazz and improvisation.

There's a video of Jacob Collier answering what was his favorite chord at the moment and it was the chord with the following voicing: F,D♭,A,B,E,G♯. It would be an F with ♯11, M7, ♯5, ♯9, but he says "it's basically an F major". Likewise, Rick Beato also categorizes augmented chords in the same group as major chords in some of his videos.

For these people, sometimes an augmented chord is not an important entity that deserves a new category. In some sense, it shares many features with the category "major chord", so it's inside the same group. I can think of some "fundamental" categories: major, minor, dominant, diminished. (could a half-diminished always be seen as an inversion of a minor?)

How do you approach this type of categorization? In what sense an augmented chord IS a major chord? What is it being taken into account?

EDIT

Just some thoughts:

It's possible to see the existence of accessory/embellishment notes in some chords. CM7, CM13, C6add9 or even CM♯11 can be seen as C triads with cherries on top. They are treated in similar ways when improvising. Also, Cm7, Cm9, Cm(M7) are treated in a similar fashion (sometimes musicians even play the 7M over a Cm7 and vice-versa and it can sound good). On the other hand, the addition of a minor 7th on a C makes it fundamentally different from a CM7. You don't land on a 7M over a dominant chord. The C7 behaves and carries different expectations than the CM7.

Some categories might also be hidden inside others in another way. All possible triads can be mathematically defined, but the chord with the notes C-F♯-G for example, (which could be called a "lydian triad") is heard as something major in the lydian world. The ear gravitates towards the perfect fifth (it's not a diminished thing with an added perfect 5th, but a triad with a perfect 5th and an added ♯4), and it fills a major third. It's not exactly a new entity.

In this sense, I would have the categories: minor, major, dominant, diminished. And I'd have some doubts about half-diminished and augmented. A better systematization of my thoughts is needed and it might have limited application, but I'm interested in hearing other definitions and opinions.

I'm not sure exactly what point these Youtubers are trying to make, but sometimes chords can be categorized by their function. This generally breaks down into three functions: Tonic, dominant, and pre-dominant

When categorizing them this way, we have to consider the music around the chord to decide how it works with the other chords that come before and after it.

Tonic chords are ones that offer a sense of resolution and don't want to pull somewhere else, while dominant chords contain tension that leads back to the tonic. Pre-dominant chords have some harmonic tension, but do not pull clearly back to the tonic.

So, I think the idea that Collier and Beato are trying to get across is that augmented chords can act in a tonic function.

The list of triads is easy:

• major


• minor


• diminished


• augmented (some theories believe this isn't a bona fide chords as it isn't from the diatonic triads.)

Seventh chords are a little more complex. To start with just the diatonic seventh chords:

• major seventh


• minor seventh


• dominant seventh


• half diminished

Add to that the fully diminished seventh chord which comes from the minor mode with a raised ^7 leading tone:

• diminished seventh

...(could a half-diminished always be seeing as an inversion of a
minor?)...

I don't see any way it could be seen that way. You cannot invert a minor seventh in any way to get a chord other than a minor seventh. Actually, it works that way for any chord. The quality of the chord does not change when it is inverted.

If you meant half-diminished = inverted minor 6 like Cm6, then "yes." But it needs to be stated clearly. The same kind of thing can be said about Am7b5. It is a half-diminished chord. But they b5 must be stated to make clear the chord quality. Somewhat similar confusion can result when chord symbols are handled indiscriminately. G9 and Gadd9 aren't the same thing. The actual chord qualities help make things clear. G9 is a dominant ninth chord. Gadd9 is a G major triad with an A added.

Without getting into a jazz theory versus classical theory battle, the point here is that the multitude of jazz symbols tends to obscure the shorter list of classical chord qualities. m7b5 and madd6 aren't two different types they are both half-diminished sevenths in two inversion forms. Same goes with chords like C6 and Am7. Classical would call them both minor seventh chords. The add6 type is a bit of a special case. There is some classical history about treating it as a triad with an added note, but let's skip that for the moment. From a classical point of view Fadd6 - F A C D - in the key of C major would very commonly be labelled ii6/5 which is a minor seventh chord.

So, the list of 9 chord types above is a pretty good minimal list of types.

... In what sense an augmented chord IS a major chord...

I suppose if you take a major chord - G major in the key of C major V - and then altered it with a #5 G B D#, it could be categorized as a kind of major chord. But I disagree. If that's the view, then clearly major chords are just altered minor chords. If we go that way, the whole system of chord qualities collapses.

In classical style you could have a G major in the key C minor and then move the 5th of the G chord D up to the escape tone Eb before resolving to the i chord. The sonority that results with the escape tone is G B Eb which would be the same notes as a augmented chord built on the mediant. Basically you could relabel that "chord" as III+6/3. The problem with that labeling is that it treats a non-chord tone embellishment of a clearly functional V chord as equivalent to an ambiguous, non-diatonic, non-functional augmented "chord". That seems wrong. Go with the functional description. The point is to not confuse simultaneous notes with chord. Chords need to be identifiable as significant tonal things.

There is one other special case involving so-called augmented sixth chords. These chords exist in classical style and jazz although the function differs a little and the labeling is of course different.

The short version is a German augmented sixth chord (in Cm:Ab C Eb F#) is like a re-spelling of a dominant seventh chord and is like the jazz tritone substitution bII7. In the sense that the chord is just a respelled dominant seventh chord we can say this is not a new category. In classical theory you could call it an altered iv7 as #iv6/5 but it's common label is Gr+6.

The Italian augmented sixth chord (in Cm:Ab C D F#) is like the jazz tritone substitution bII7b5. The flat 5th makes this a unique chord not on the list of nine above. In classical theory you could call it an altered half-diminished chord iiØ7 as iiØ+6/4 but it's common label is It+6.

In jazz these chords resolve by the bass note moving a half step down to the root of the tonic chord. In classical the resolve with the bass moving down by half step to the root of the dominant (and the note of the augmented sixth goes up by half step to the tonic.)

All that detail is just to give the background. The point is add one more category:

• Italian augmented sixth chord a.k.a. bII7b5 or It+6