Does an A13 chord include the 11th?
The simple answer is that the 11th can be used in a 13 chord,
but it is usually left out because it clashes with the third.
When it is used, you will usually see it in a 'sus' chord.
It is important to remember that the 9, 11, and 13 are simply 2, 4, and 6 but up an octave.
So, the chord tone in question, the 11, has a big problem.
If we were to play each chord tone as stacked thirds ...
1 3 5 b7 9 11 13 ... and we transposed the top three chord tones down an octave, we would get:
1 2 3 4 5 6 b7
That would be the equivalent of mashing all of the white keys on your piano.
Even with that octave cushion, we still get some cringy clashing in the harmony.
So, where do we go from here? Let us diagnose our problem as 'too many notes'.
Then, let us build our chord from the ground up with as few notes as possible in order to see what we can get away with. From there, we can look at some other voicings that might be interesting.
1. We begin with the fact that the two most important chord tones are the 3 and the b7.
These are the main flavor tones. They establish minor and major flavors.
2. Playing the 5 is redundant. It is just more of the same flavor as the root.
Most piano and guitar players will play 1, 3, b7, 13 when playing solo,
but if there is a bass player of some sort in your band,
you can even get away with leaving out the root!
So, if you are in a band with a bass player...
3. You can get way with playing three notes and you will have a 13 chord: 3, b7, 13
Start getting a feel for the chord in root position.
If and when you feel ready to experiment,
you can try adding in the b9, 9, and #9,
and see what floats your boat.
Avoid the 11 unless you substitute it for the 3.
Then, you cross over into 'sus' territory.
If you do elect to have the 3 and the 11 in your voicing,
make sure it is as wide as you can make it.
You will need those octave cushions to make the chord sound good.
Thank you for your question.
Robinson Tyndale G.