Theory Math Question clarification

[jaywhy13]

Just checking to ensure that these are correct...
If "R" represents your root note and m means minor interval and M means major interval then...

R + m + m = Rdim
R + m + M = Rm
R + M + m = R
R + M + M = R aug

I have also heard the term stacking thirds to get the 7th, 9th, 11th and so on... what kind of third is added to get a 7th or 9th and 11th and so on

[Jed]

This kind of thing is more commonly written as:

major triad = 1 3 5
minor triad = 1 b3 5
diminished triad = 1 b3 b5
augmented triad = 1 3 #5

The numbers refer to degrees of the major scale of the root. So 3 = major 3rd (4 semi-tones), b3 = minor 3rd (3 semi-tones), 5 = perfect 5th (7 semi-tones), b5 = diminished 5th (6 semi-tones), #5 = augmented 5th (8 semi-tones)

The intervalic distance between the (major) 3rd and the (perfect) 5th is a minor 3rd. So a major triad can be thought of a maj 3rd with a minor 3rd over it. But it is often more useful to think of the major triad as - 1 3 5 of the root's (major) chord scale.

The intervalic distance between the minor 3rd ( aka b3) and the (perfect) 5th is a major 3rd. So a minor triad can be thought of as a minor 3rd with a major 3rd over it. But it is often more useful to think of the minor triad as - 1 b3 5 of the root's (major) chord scale.

The intervalic distance between the minor 3rd ( aka b3) and the diminished 5th is a minor 3rd. So a diminished triad can be thought of as a minor 3rd with a minor 3rd over it. But it is often more useful to think of the diminished triad as - 1 b3 b5 of the root's (major) chord scale.

The intervalic distance between the (major) 3rd and the augmented 5th (aka #5) is a major 3rd. So an augmented triad can be thought of a maj 3rd with a maj 3rd over it. But it is often more useful to think of the augmented triad as - 1 3 #5 of the root's (major) chord scale.

In order to stack thirds you can rewrite the major scale from the root, skipping every other note. so C major - normally written as C D E F G A B C would be rewritten as C E G B D F A C.

For Cmaj7:
C = 1 (or root)
E = 3 (major)
G = 5 (perfect)
B = 7 (major)
D = 9 (major)
F = 11 (perfect)
A = 13 (major)

C (major triad) = 1 3 5 = C E G
C maj7 = 1 3 5 7 = C E G B
C maj9 = 1 3 5 7 9 = C E G B D
C maj11 = 1 3 5 7 9 11 = C E G B D F
C maj13 = 1 3 5 7 9 11 13 = C E G B D F A

* Note that the 11th is a very dissonant sound over a major chord so in actual practice you won't see chords like C maj11 or C maj13 often. But they are theoretically possible and would be spelled as above *

cheers,

[JonR]

Just checking to ensure that these are correct...
If "R" represents your root note and m means minor interval and M means major interval then...

R + m + m = Rdim
R + m + M = Rm
R + M + m = R
R + M + M = R aug

I have also heard the term stacking thirds to get the 7th, 9th, 11th and so on... what kind of third is added to get a 7th or 9th and 11th and so on

To answer your question directly (following the "stacking 3rds" process, for 7ths etc):

R + M + m + m = R7
R + M + m + M = Rmaj7
R + m + M + m = Rm7
R + m + M + M = Rm(maj7)
R + m + m + M = Rm7b5 (half-diminished)
R + m + m + m = Rdim7 (full diminished)
R + M + M + m = Rmaj7#5

R + M + m + m + M = R9
R + M + m + M + m = Rmaj9
R + m + M + m + M = Rm9
R + m + M + M + m = Rm(maj9)

Going further makes things unnecessarily complicated - if it isn't already!
As Jed says, it's much better NOT to think of stacked 3rds (with the possible exception of aug triads and dim7 chords), and think of intervals measured from the root.
You do need to understand interval language, but this is a more musically useful concept.

Remember that intervals are always counted in note letters (with the root as "1st"), but that one interval type may vary in size (number of half-steps), depending on the scale.
2nds, 3rds, 6ths, 7ths, 9ths and 13ths can be minor or major (half-step difference in each case).
Occasionally 7ths can be diminished (half-step less than minor), and 9ths can be augmented (half-step bigger than major).
4ths, 5ths and 11ths are usually perfect, but can be diminished or augmented (half-step smaller or bigger).
Intervals always include the root, although their name derives from the upper note of the pair.

The main intervals used in building "tertian" ("stacked 3rds") chords are:

major 3rd = 4 half-steps above root
minor 3rd = 3 half-steps above root
perfect 5th = 7 half-steps above root
diminished 5th = 6 half-steps above root
minor 7th = 10 half-steps above root
major 7th = 11 half-steps above root
diminished 7th = 9 half-steps above root (dim7 chords only)
major 9th = 14 half-steps above root
perfect 11th = 17 half-steps above root (on minor chords only)
augmented 11th = 18 half-steps above root (on major chords only)
major 13th = 21 half-steps above root.

In quartal chords (using 4ths mostly instead of 3rds) and in sus chords, you might get these (in addition to some of the above):

major 2nd = 2 half-steps above root
perfect 4th = 5 half-steps above root
major 6th = 9 half-steps above root

On altered jazz dominant 7th chords, you might get these (in addition to some of the above):

augmented 5th = 8 half-steps above root
minor 9th = 13 half-steps above root
augmented 9th = 15 half-steps above root
minor 13th = 20 half-steps above root

(augmented 5ths are not common otherwise)


To put some of Jed's list into full interval terminology:

Major triad (1-3-5) = major 3rd + perfect 5th
Minor triad (1-b3-5) = minor 3rd + perfect 5th
Diminished triad (1-b3-b5) = minor 3rd + diminished 5th
Augmented triad (1-3-#5) = major 3rd + augmented 5th

"7" chord (dominant 7th, 1-3-5-b7) = major 3rd + perfect 5th + minor 7th
maj7 (1-3-5-7) = major 3rd + perfect 5th + major 7th
m7 (1-b3-5-b7) = minor 3rd + perfect 5th + minor 7th
m(maj7) = minor 3rd + perfect 5th + major 7th
m7b5 (half-diminished, 1-b3-b5-b7) = minor 3rd + diminished 5th + minor 7th
dim7 (full diminished, 1-b3-b5-bb7) = minor 3rd + diminished 5th + diminished 7th

Notice the way the names of significant intervals are echoed in the chord names, while other common intervals (eg perfect 5th, minor 7th) are taken for granted. "m" = minor 3rd, "maj" = major 7th, etc.
This is why it's better to think in intervals from the root than in stacked 3rds. Intervals from the root matter, both in sound and in naming. Intervals between other chord tones are irrelevant (unless you're studying counterpoint... )

9 (1-3-5-b7-9) = major 3rd + perfect 5th + minor 7th + major 9th
maj9 (1-3-5-7-9) = major 3rd + perfect 5th + major 7th + major 9th
m9 (1-b3-5-b7-9) = minor 3rd + perfect 5th + minor 7th + major 9th
m(maj9) = minor 3rd + perfect 5th + major 7th + major 9th

Not all chords in practice can take 11ths and 13ths. Here are some of the more common ones:
maj7#11 = major 3rd + perfect 5th + major 7th + augmented 11th
maj9#11 = as above, + major 9th
7#11 = major 3rd + perfect 5th + major 7th + augmented 11th
9#11 = as above, + major 9th
m11 = minor 3rd + perfect 5th + minor 7th (+ major 9th) + perfect 11th
(9th is optional)

13 = major 3rd + perfect 5th + minor 7th (+ major 9th) + major 13th
(9th is optional, 11 is omitted)

Some other chord types:

6, "ADD", SUS, etc

6 = major 3rd + perfect 5th + major 6th
69 = major 3rd + perfect 5th + major 6th + major 9th
add9 = major 3rd + perfect 5th + major 9th
m6 = minor 3rd + perfect 5th + major 6th (NB, not minor 6th)
m69 = minor 3rd + perfect 5th + major 6th + major 9th
m(add9) = minor 3rd + perfect 5th + major 9th

sus4 = perfect 4th + perfect 5th
sus2 = major 2nd + perfect 5th
7sus4 = perfect 4th + perfect 5th + minor 7th
9sus4 = as above, + major 9th
13sus4 = as above, + major 13th (9th optional)
susb9 (1-4-5-b7-b9) = perfect 4th + perfect 5th + minor 7th + minor 9th

ALTERED DOMINANTS

7b9 = major 3rd + perfect 5th + minor 7th + minor 9th
7#9 = major 3rd + perfect 5th + minor 7th + augmented 9th ("Hendrix chord")
7alt = major 3rd + diminished or augmented 5th + minor 7th + minor or augmented 9th (7#5#9, 7b5b9, 7#5b9, 7b5#9)


Lastly, remember that these chord tones and extensions are always measured theoretically upwards from the root.
In practice, chords may be "voiced" in many different ways, including inversions (where the root is not on the bottom). This means that the actual intervals between chord tones will be more varied.
As a simple example, in the common guitar triad chord shapes, there are often perfect 4ths between 5th and root, and "compound intervals" (larger than octave) between other chord tones.