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If I am asked to write major, minor, diminished and augmented 6/3 chords on the bass note of E. What would be the best way to determine the qualities of each sixth and third?

  1. I could think about first seeing what the chord is in root position and then get all the note names and stack them in the right order with any accidentals and then invert them so the bass is the E note.

  2. I could also learn exactly what intervals go into the inverted 6/3 chords by learning the qualities of the intervals.

The first way is easier since I may already know the interval qualities in root position chords more than those in the the inverted positions.

What is the right thinking when doing an exercise like this?

It would also be helpful to know how to handle 6/4 chords.

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  • Without knowing the context of your assignment, it's difficult to know whether "major" refers to the quality of the related uninverted triad or of the 6/3 chord itself. What is the context of the assignment?
    – phoog
    Jul 15, 2021 at 14:19

3 Answers 3

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6/3 chords

6/3 chords by definition contain a sixth and a third. So every chord will contain notes written on the lines/spaces a sixth and third away from the bass note. Then just add accidentals to adjust the sixths and thirds to be the appropriate size.

The interval qualities are according to the below table:

chord quality third quality sixth quality
Major m3 m6
Minor M3 M6
Augmented M3 m6
Diminished m3 M6

One way to derive this table is to consider the intervals for each chord in root position:

Maj = M3+m3; Min = m3+M3; Aug = M3+M3; Dim = m3+m3.

Then recognize that the upper interval (between the chordal third and fifth) will remain the same, while the lower interval (between root and third) will be inverted (to third and root). The inversion of a third to a sixth flips the quality between minor and major.

Thus:

Maj = m3+m6; Min = M3+M6; Aug = m3+M6; Dim = M3+m6.

6/4 chords

6/4 chords by definition contain a sixth and a fourth. So every chord will contain notes written on the lines/spaces a sixth and fourth away from the bass note. Then just add accidentals to adjust the sixths and thirds to be the appropriate size.

The interval qualities are according to the below table:

chord quality fourth quality sixth quality
Major P4 M6
Minor P4 m6
Augmented d4 m6
Diminished A4 M6

One way to derive this table is to consider the intervals from the root for each chord in root position:

Maj = M3 & P5; Min = m3 & P5; Aug = M3 & A5; Dim = m3 & d5.

Then recognize that the "inner" interval (between the chordal root and third) will remain the same, while the outer interval (between root and fifth) will be inverted (to fifth and root).

Thus:

Maj = P4 & M3; Min = P4 + m3; Aug = d4 + M3 ; Dim = A4 + m3.

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  • so is the right way to think about this to not even consider the chord in root position and just use these formulas to write the chords with the correct accidentals?
    – user35708
    Jul 13, 2021 at 22:13
  • @armani The "right way" is whatever method serves you best. I included one way to derive each of the tables as an alternative way to think about it without just memorizing the tables themselves. You might come up with a way that works even better for you.
    – Aaron
    Jul 13, 2021 at 22:15
  • Do you have any basis for the conclusion that "major 6/3 chord" means "inverted major triad"? In figured bass, it would be the quality of the 6th, not of the 3rd of which it is the inversion. For example, an augmented sixth chord contains an augmented sixth, not a diminished sixth, and it is not an inversion of an augmented triad (which of course contains neither an augmented 3rd nor a diminished 3rd).
    – phoog
    Jul 15, 2021 at 16:06
  • @phoog I'm not sure I understand your comment. My "conclusion" about major 6/3 chords is based on the the OP being about inverted triads.
    – Aaron
    Jul 15, 2021 at 16:19
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First thing is you can't get around memorizing the teritian stacks for the triad types and qualities of inverted intervals.

ROOT POSITION
Major      M3 m3
Minor      m3 M3
Diminished m3 m3
Augmented  M3 M3

For 6/3 chords the third above the bass will just be the second third of the tertian stack, the sixth above the bass will be the inversion of the first third of the stack.

So, root position major triad of M3 m3 becomes 6/3 inversion m3 m6.

Just invert the lower third for all four types. It will end up being...

6/3 inversion
Major      -- m3 m6
Minor      -- M3 M6
Diminished -- m3 M6
Augmented  -- M3 m6

You can do something similar for 6/4 chords, but you invert above the fifth two tones, the root and the third. The intervals to invert can be viewed relative to the fifth. The root to fifth - P5 or d5 inverts to P4 or A4, and the upper third of the root position stack inverts from M3 or m3 to m6 or M6.

So, a root position triad is P5 plus M3 and the 6/4 inversion is the inversion of P5 to P4 plus the m3 inverted to M6. Or, for example, a diminished triad inverts from d5 to A4 plus the upper m3 inverted to M6.

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  • “Tertian” I just found out. And a more minor point is that those intervals need not be memorized. Fifths in pop music are either WWHW or WHWW. Jul 18, 2021 at 21:52
  • @RandyZeitman, are you seriously suggesting to not learn the intervals of the four basic triads?!? The most definitely should be learned along with all intervals names and their inversions. From a harmonic point of view to not learn that stuff is musical illiteracy. Jul 19, 2021 at 13:22
  • No. I mean they can be derived. Same for the circle of fifths. Jul 19, 2021 at 16:10
  • I don't see the distinction you're trying to make. Terms and descriptions like P5, perfect fifth, "seven half steps" are all fundamentals of intervals. They aren't derived, they're fundamental. And describing a perfect fifth in terms like WWHW or WHWW isn't so much about the interval, but about scale, major versus minor. Anyway, you should post an answer. I don't see how you would derive the answer for the OP. Jul 19, 2021 at 17:12
  • I read that they are all derived as harmonic series – Equal or Just temperaments. Ok, WWHW and WHWW are about scale. But aren't intervals the distance between two scale notes? I don't have an answer for the OP. I responded to you as I don't think the M3/m3 needs to be memorized. That is, it need not be taken as-is and it's underpinnings aren't that hard to understand. Jul 19, 2021 at 20:11
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6/3 and 6/4 refer to the inversions of the chords. The inversion concerns itself with what note of the chord is in the bass. If you have a C major chord in root position, you have the notes that for the purpose of this discussion are from the bottom C-E-G.

If you count the intervals between the notes, it is C - E a third and C - G a fifth. Hence the notation of a triad in root position C 5/3. If you take that same chord and have the third in the bass you now have E-G-C. This would be a C Major chord in first inversion. If you count E-G is a third and E-C is a sixth, hence the notation C 6/3.

Lastly, if you have the C Major chord in second inversion you have the notes from the bottom G-C-E. G - C being a fourth and G - E being a sixth, hence notation C 6/4 for that chord in second inversion.

The tetrads have a further set of rules, but this should do as a concise explanation of the inversions of the triads and how they are notated.

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