Is there an easier way to remember the notes in 11th and 13th chords, like maybe an alteration of a different chord or a combination of two triads? For example, if I see a 13th chord, I have to think "ok the 9th is this, the 10th is this note", etc. That's slow (for me anyway)!
Subtract 7 from the extension and the note is the same, but an octave up:
9th - 7 = 2nd (an octave up) 11th - 7 = 4th (an octave up) 13th - 7 = 6th (an octave up)
So instead of thinking of it like this:
1 3 5 7 9 11 13
It might be easier to think of it like this:
1 3 5 7 2^ 4^ 6^
(here I used ^ to indicate an octave higher)
While you could think of a Cm13 (for example) as a combination of the two chords Cm7 + Dm...I don't think this is really all that helpful and it seems like this would only be more confusing than anything.
Think of it as two chords. Let's say C13. C E G Bb D F A. The first triad is C major, leaving the 4 note chord of Bbmaj7 (Bb D F A). OR - C dominant 7 - C E G Bb, with D minor over it. So the first way is root triad, bVII major 7 chord, the second - main 7th chord, with ii over it. That way, you have all the main notes, which may or may not be altered anyway, but that would still need calculating in any case.