The Lydian augmented mode comes from the 3rd note of the melodic minor scale and is good for improvising over chords with a flat or sharp 5th or a sharp 11 in it.
This is for altered 5th or 11th notes. A flat 5 and an augmented 11th are the same note.
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This is played against an E♭ma7♯5, this chord is mostly used as a passing chord. A chord leading you into a main chord or another passing chord.
The E-flat Lydian augmented played over an E-flat major 7th sharp 5 chord
This is a little rough, this isn’t a very melodic chord to play against.
Lydian Augmented List
The Lydian Augmented Formula
This one goes 1-2-3-♯4-♯5-6-7
This ones not too bad to remember.
I hope you found this page useful.