Ninth, eleventh and thirteenth chords are all extended chords and are named so because they are extended with one or more thirds one octave higher. A seventh dominant chord is extended with a major third to build a ninth chord, a ninth chord is extended with a minor third to build an eleventh chord and, finally, an eleventh chord is extended with a major third to build a thirteenth chord.
See diagrams of extended chords:
As mentioned above, there are several categories of extended chords. The numeric figures indicate the interval (or scale distance) from the root note to the last note.
We can compare the C7 chord with C9, C11 and C13:
C7: C – E – G – Bb
C9: C – E – G – Bb – D
C11: C – E – G – Bb – D – F
C13: C – E – G – Bb – D – F – A
So how do you play a chord with six or more notes? The answer is that you omit one or more notes and in many cases play the chord inverted (i.e. rearrange the order of notes in the chord). You could use both hands and play all notes, but there is a risk for dissonance playing up to seven different notes together. Instead, it's common to use polychord voicings, chord-over-chord structures (see examples below).
To make it easier to memorize these chord types, you can think of the extension notes as follows:
So, the 9th include the same note as an added 2th chord, the 11th include the same note as an suspended 4th chord and the 13th include the same note as an 6th chord.
7th: For the 7th chord it is not necessary to omit any notes, but it is possible to omit the fifth. A C7 with notes that could be omitted in parentheses would therefore look like: C – E – (G) – Bb.
9th: The most standard procedure for the 9th chord is to omit the fifth. A C9 with notes that could be omitted in parentheses would therefore look like: C – E – (G) – Bb – D.
11th: Concerning the 11th chord, the third is normally excluded. A C11 with notes that could be omitted in parentheses would therefore look like: C – (E) – G – Bb – D – F. Notice that without the E, the chord is identical with C9sus4.
13th: For the 13th chord the most common way is to exclude the fifth, ninth, and eleventh notes (especially the 11th is often left out). A C13 with notes that could be omitted in parentheses would therefore look like: C – E – (G) – Bb – (D) – (F) – A.
More specific information and images can be found on the pages with chords for different notes, see links above.
C11: Left hand playing the notes E - G - Bb and right hand playing Bb - D - F.
C13: Left hand playing the notes E - Bb - D and right hand playing A - D - G.
C13: Left hand playing the notes Bb - E - A and right hand playing C - E - A.
The root is sometimes left out, implying that there is a bass player involved taking care of that note.
Besides the most ordinary chord symbols or names, alternatives can occur. C9 could also be written Cdom9. In the same style could C11 and C13 also be written Cdom11 and Cdom13 respectively.
How and when to use extended chords? 9th, 11th and 13th chords are often used as substitutions for regular triads, seventh chords and other extended chords. This is a standard approach in jazz. There are no specific rules in which ways substitutions can be made as long as they keep the same root note: C7 could be substituted for C9, A9 could be substituted for A11 and B11 could be substituted for B13 et cetera.
Major chords Minor chords Seventh chords Extended chords Sus chords Dim chords Aug Chords Add Chords Altered Chords