What are the diatonic extended chords of C major?

Question

I saw this question about 9ths, but I'd like to know what the diatonic chords of 11ths and 13ths as well. This isn't strictly for jazz, I just want to know what the basic diatonic chords are.

I'm confused at the notation, For example the iii and vii chords of the 9ths chords look like Em7(♭9) and Bm7(♭5 ♭9) why do they flat certain notes and call them seventh chords? It makes me unsure how to notate 11th and 13th chords.

7th chords of C major:

• Cmaj7
• Dm7
• Em7
• Fmaj7
• G7
• Am7
• Bm7(♭5)

9th chords of C major:

• Cmaj9
• Dm9
• Em7(♭9)
• Fmaj9
• G9
• Am9
• Bm7(♭5 ♭9)

So what are the diatonic 11th and 13th chords in C major?

Answer 1

The diatonic extended chords of C major appear below.

Perhaps your confusion stems from the fact that the chord symbols contain sharps and flats when, in the key of C major, the constituent notes of diatonic chords never do. These alterations are necessary because chord symbols consistently assume a particular quality for every interval that participates in the chord.

The alterations you see in chord symbols (♭9, for example) are exceptions to the assumptions about those intervals. To keep the symbols as compact as possible, each interval within the chord symbol has a default or assumed quality.

• The third of a chord is assumed to be major unless the symbol says it's minor, which is indicated with an m, min, or -. (Diminished and half diminished symbols also include a minor third in their meaning.)
• A fifth is assumed to be a perfect fifth from the root. A ♭5 indicates that it should be a diminished fifth. (The diminished and half diminished symbols also include a diminished fifth in their meaning.)
• A seventh is assumed to be a minor seventh from the root, so major 7ths need to be indicated with M, maj, or △. (A diminished symbol—but not a half diminished symbol—indicates that any seventh is diminished as well.)
• A ninth is assumed to be a major ninth from the root, so minor 9th intervals are indicated with a ♭9. (You'll also see ♯9 for augmented ninth intervals.)
• An eleventh is assumed to be a perfect eleventh from the root, so augmented eleventh intervals are indicated with a ♯11. (They are probably more common than regular eleventh chords.)
• A thirteenth is assumed to be a major thirteenth from the root, so minor 13th intervals are indicated with a ♭13.

Since you asked about the chords that are diatonic to C major, the alterations are necessary to keep the constituent notes in that key. That's exactly the case with the ninth in the E and B chords in the key of C major. You would see the same alterations on the iii and vii chords in every major key.

If you were to use the "default" major ninth interval with an Em7 chord, you would be playing an F♯. Nothing prevents you from doing so; Em9 is a perfectly good chord. However, F♯ is not diatonic to C major. To get an F which is diatonic, we need to apply a flat symbol to the ninth, thus taking the F♯ down to F. The same applies to the eleventh on the F chords. A perfect eleventh up from F is B♭. To get some sort of B that's in the C major scale, we raise it. Thus, a ♯11 in the symbol takes us up the semitone we need to get from B♭ to B♮.

You may also wonder why the extension numbers are written the way they are. For example, why, among the thirteenth chords, is it Am11(♭13) and not Am♭13? By convention, you write the highest unaltered extension (11 in this case) and then list the alterations.

Answer 2

This isn't a real answer, but an addition to @trw's chord chart.

...I'm confused at the notation, For example the iii and vii chords of the 9ths chords look like Em7(♭9) and Bm7(♭5 ♭9) why do they flat certain notes

The jazz chord symbols - in my assessment - are based upon an assumed diatonic dominant chord that extends up to the 13th. Any chord that isn't a diatonic dominant 13th chord uses various modifiers symbols altering a dominant 13th chord.

For example G dominant seven G7, to make a G minor seven from a G dominant seven we must change the third to min indicated by the min symbol Gmin7.

G dominant nine G9, to make a G dominant seven minor nine from a G dominant nine we must lower the 9th indicated by the b9 symbol G7b9.

We can compare diatonic iii and V chords extending all the way up to the 13th and note which intervals above the roots need modification in the iii chord relative to the intervals above the root of the V chord...

        Em11b9b13   G13 

root    E           G   root
m13*    C           E   M13
P11     A           C   P11
m9*     F           A   M9
m7      D           F   m7
P5      B           D   P5
m3*     G           B   M3
root    E           G   root

        iii13       V13

(*) These intervals differ from a diatonic dominant 13th chord and so the jazz chord symbol requires modifiers that detail the changed intervals.

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I hope that isn't too confusing changing roots and comparing intervals.

Look back to @trw's chart and notice all the G chords are simply the root letter G plus the highest extension number. All the other chords have one or more modifiers for each interval above their roots that is not the same interval above the root of the dominant chords.

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Sanity check:

The complete list of all diatonic chord with extension up to the 13th, written in jazz chord symbols, is a nightmare! But maybe we can take a step back and look at vi ii V I which is the practical basis for a lot of jazz harmony.

All four of those chord types start with the root letter and extension figures without any accidentals (except the crazy Am11b13.) Then only min is needed for the vi and ii, and maj for the I. This covers over half of the chart! That isn't too hard to manage.

Less than 1/4 of the chart uses true awful symbols like Bm11b9b5b13 and they will surely be much less frequently encountered. I don't think you need to worry about instantly recognizing their diatonic identities.