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Comments
I can't get away from this feeling like a D major progression:
Dmaj7 - the I chord
E7 - secondary dominant replacing the A (V chord)
G - the IV chord (wants to resolve to I)
You can feel the pull:
Dmaj7 - xxx222
E7 - xxx434
G - xxx433
Of course, tonal centre can be influenced by the specifics of the progression.
So for me, unless I start to see it differently, the E7 is the interesting chord here...I'm sure I read in my copy of 'Jazz Hacks for Dummies' that you can throw a lydian dominant scale at any surprising 7 chords that pop up...depends how much time you've got: is it a beat or is it four bars? That's not my bag but there are lots of jazz dudes on here that could go to town in this area.
You're already suggesting your G chord is minor, but normally when a 1 chord goes to a bIII chord (in bluesy music) it's a major tonality, in which case you'd also have the option of playing G lydian (which preserves the E7th's 6 note, which I prefer), or G major, which has a standard 4 note, and is a departure from your E7 scale.
You could also play G mixolydian (ie G7), or even G Lydian Dominant, which has the raised 4, AND the flattened 7. That flattened 7th destroys the E's major 2nd.
Of course, if you're Dream Theater you'd be doing a G with minor tonality anyway, so fill your boots!
This is why I think it's essential to know what chords you are solo'ing over and what notes they contain
An A major scale (Ionian) over the Dma7 to E7, whilst giving regard to the chord tones. Unless you want to think on each chord, which would then be D Lydian to E Mixolydian. It's the same notes anyway
For the G chord (assuming it's major) I'd just change one note of the scale and play a G instead of a G#. So if I was thinking in terms of an A scale, I'd think 'A Mixolydian' (or you could think of it as G Lydian, or even D major (Ionian)). It's all the same notes anyway
An A7 might be a good chord to insert after the G, to modulate back to the Dma7, which would be a good fit to the scale I'm already playing over the G chord.
Far better to think of melodies that work in this full context and play those. Then if you really want to you can work out the underlying theory.
If I was composing a solo (which I only seem to do if I'm participating in the forum's SotM), I might go digging but I wouldn't want it to sound too contrived (although it often does
(all my theory nonsense is just for those who like a bit of maths
I like all approaches, including going for it. But since I started learning solos by ear in the early days, then started to improvising, I've always felt the need to categorise things I hear, or hit upon by accident, so that I can create a library of reusable information and extend my vocabulary. But that's just the engineer in me.
I can explore these things if people are interested through future SotM challenges.