You know your shit Elite, good stuff. I know the basics but really would like to get deep into that side of it much more.
You guys amaze me. Seriously.
I'm a music and guitar teacher and I've studied musicology and music theory. Most of the time my skills or knowledge have limited use
Phrygian Dominant with a raised 7th is called Flamenco Scale or Phrygian Major
Great, I didn't know that. But I can see why it would exist. The maj7 chord on the third degree makes sense to establish it as a 'home' chord, rather than the dominant 7 in its original state.
Elite?
(not laughing at you, but at the notion of me giving the right answer here).
Honestly, I've never seen scales written out like that before, because that's not exactly how scales work. Any seven note scale should have a note on every letter, so the correct way to write down E major, both ascending and descending would be to use sharps on the F, G, C and D. This is to ensure that every letter gets used exactly once.
However, for example when you're writing piece and you have notes out of key, leading up to a certain note (let's say we have a piece in C major and the melody goes D-D#-E), you would use a sharp, when it would go down, you'd use a flat (E-Eb-D). This is to ensure good readability along the way.
Basically, what you want to do is make sure your music is readable when transcribing it. Mixing up various accidentals will only make stuff confusing for the performer. There's a lot of rules about how to properly write down music and since it doesn't exist in a vacuum (a melody is never there solo, there's bound to be some backing as well), there's no way to go over it all in a short paragraph here.
But really, E major is E-F#-G#-A-B-C#-D#-E. Both up and down
Just don't ask me to explain the difference between G# and Ab
Well, this goes back to the 'every letter had to be used once' thing in scales/key signatures. F minor would be F-G-Ab-Bb-C-Db-Eb-F, with an Ab rather than a G#, while the E major scale has G# (See above). F to Ab is a minor third, while F to G# is an augmented second.
Actually the key of G# Major is funny as it goes
G# A# B# C# D# E# F## G#
And that's why you'll never see that signature on a score, but instead see Ab-Bb-C-Db-Eb-F-G, which is the exact same notes. Doesn't mean F## doesn't exist, it does, but there's a good reason for rarely seeing double sharps and flats in pieces.