One good mnemonic
Take On Me. The chorus vaults to its famous high note a major 7th above where it started, and the strain of that reach is the interval's personality in one gesture. Ascending major 7ths are rare in melody precisely because they're hard to sing; composers spend them like money.
Going down there's no famous opening to lean on at all, an honest gap in the songbook. Use the descending button above, or run the near-miss test: sing the bottom note up an octave and feel the target sit one semitone too high. Wanting to sag a half step onto the note you just heard is what a falling major 7th feels like from the inside.
Jazz does have an opening in each direction. Mark Levine's interval table fixes the ascending major 7th to Joe Henderson's "Serenity" and the descending one to Wayne Shorter's "Lady Day."
On the staff
The near miss
Put C and the B above it together and you hear an octave with a splinter in it. The top note is do's leading tone displaced upward, a ti that wants one of two things: to rise the last semitone home or to let the whole interval collapse inward. Theory files it with the sharp dissonances, and bare, it earns the label.
Then jazz found out what happens if you fill in the middle. With a 3rd and 5th stacked inside, the major 7th chord turns the same rub into shimmer, the politest dissonance in popular music. The bare interval and the chord are worth holding side by side in your ear: same outer notes, opposite manners.
What it gets confused with
The octave, one semitone wider, is the big one, and the test is blunt: the octave sounds like the same note twice and the major 7th sounds like a mistake on the way to it. The minor 7th below is the softer seventh, leaning rather than bristling; if the wide dissonance has some warmth in it, it's the minor, and if it's all edge, it's this one. The inversion trick works here too: flip a major 7th and a minor 2nd falls out.
How Aubel handles it
The major 7th unlocks fourth beyond the starting set, earlier than you might expect for a dissonance, because its octave near-miss makes it easy to peg against an anchor the pool already has. The detail screen carries a single mnemonic, Take On Me, which is one more than most people need once the splinter-in-the-octave sound has landed.