Chord Substitution | Lydian Dominant Scale

A very common chord substitution in jazz is the tritone substitution.

Tritone = #4 (or b5) interval

A tritone substitution is the use of a dominant chord that is 6 half steps higher or lower from the original dominant chord. Tritone substitutes are useful for both chordal playing and single note improvisation.

Take for example the G7 chord: G B D F
The 2 most important notes of this chord are the 3 (B) and the b7 (F) (if you didn't know this already I suggest you read the guitar chord tutorial first).
The 3 and the b7 of a dominant chord are a tritone apart. A tritone is a symmetrical interval, it is precisely in the middle of the chromatic scale. This being the case we change the 3 for the b7 and the b7 for the 3 and fill in the 1 and 5 to get a new dominant chord.
This chord would be Db7: Db F Ab B (more precisely: Db F Ab Cb)

So compared to the G7 chord the B and F changed places. Knowing that the F is the 3 and the B is the b7, it's easy to fill in the 1 and 5 (again if you don't agree with me about this being easy, read the guitar chord tutorial first).
The 1 is Db and the 5 is Ab. Relative to the G7 this is a b5 and a b9, so this means that the Db7 can substitute an altered G7 chord.

What scale would you play on the Db7. The first dominant scale that comes to mind is the mixolydian scale (if this scale sounds like Chinese to you, read the tutorial about guitar scales). Problem here is the 4 of the Db mixolydian scale: the Gb or F#. This note is the major 7 of G, while we need a b7 for G dominant.
The solution: raise the 4 (F#) with a half step to the #4 (G).
A mixolydian scale with a #4 is called a 'lydian dominant' scale. It's actually the 4th degree of an Ab melodic scale or the 5th degree of a G altered scale.