The third mode of the Melodic Minor scale. Used mostly in jazz and in classical music, this is quite a rarity if one listens to pop music and other such silliness. Outlining a Maj7#5 chord (Augmented Major 7th), the scale is built like so:
Lydian Augmented scale in C
First, we must derive the "i" Melodic Minor scale. So, if our 'III' scale is C, then our 'i' scale must be A:
A B C D E F# G# A

Based on the III, we then outline our scale:
C D E F# G# A B C

Which contains the chord:
C E G# B C
(or, an Augmented Major 7th chord)
Neat! So, how do I apply this wonderful knowledge?
This scale can be played over ANY major augmented chord such as Cmaj7+ or Cmaj7#11#5. Anything that has a raised fifth would sound beautiful with this scale being played over it. Also, any altered scale such as this may be played over unaltered scales as well, to create some interesting harmony and tension.

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