This is a strange animal. It has both diatonic and non-diatonic elements present and takes a bit of time to get used to, both in hearing and playing over this chord. Here's a good fingering for an A13(#11):
Note how difficult it is to grab the root. I would recommend playing a voicing without the root if you're playing anything but the A13(#11).Otherwise you'd have to leave out another note, and frankly, the root isn't the most important note in this chord (in my opinion).
There are several scale choices you have when dealing with this chord. The first one I would use is the A Lydian Dominant mode, a sort of cross between Lydian and Mixolydian:
This mode fairly well outlines the chord. There's another more familiar way of thinking of this mode and that is the E Melodic Minor:
Finally, here's an interesting lick that outlines the harmony in theA13(#11)chord. It has an arpeggiated quality and with a little work lays well on the fretboard:
NEXT LESSON: We'll further explore this chord and look at some polytonal scale ideas for improv. See you then!