Is my answer the following question is correct:
Q: What is the augmented seventh from E?
A: D double sharp
Method
Write down the note (of your choosing) on the bottom line of the treble staff which is E.
Note down the same note a generic seventh from above. This will be positioned on the second line of the staff which is D.
On a keyboard, count 12 half steps from E and you end up at E.
Thus an augmented seventh from E is D double sharp.
Correct. A seventh up from E will be 7 letter names up, i.e. some sort of D. Then you can count semitones if you like, though I think most of would relate it to the scale of E major, which has D# as its (major) 7th note. Sharpened again, that's Dx.
(I've got through 50+ years as a professional musician without consciously thinking how many semitones were in a given interval. But I know my scales!)
Not that you often (if ever) FIND an augmented 7th in the wild. It's pretty hard not to hear it as an octave. But you're technically correct.
Your answer is correct. Intervals are counted by letter names, so a seventh starting on E is (disregarding accidentals) E, F, G, A, B, C, D. So you know it will be a D note of some kind.
You then can note that a major seventh is E to D#, because D# is diatonic to the E major scale. To find the augmented seventh, you merely raise the major seventh by a semitone (while keeping the same letter name). Thus, D sharp becomes D double-sharp.
It's worth noting that augmented sevenths aren't common. By ear you would just identify them as octaves.
