Everyone knows the major scale goes Tone Tone Semitone, Tone Tone Tone Semitone. But to see the reason why this particular scale is so common you need to examine the circle of fifths.
The most important interval in music is the octave (12 semitones.) The second most important is the perfect fifth (7 semitones.) The diatonic scale is built from a series of six stacked fifths, which is why it has a total of six chords with perfect fifths available in it (three major and three minor.)
Below is the circle of fifths opened up and arranged in a zigzag
Gb Ab Bb C D E F# G# A#
Cb Db Eb F G A B C# D# E#
A couple of examples:
The notes of the key of C major are
F C G D A E B or alternatively
C D E on the top line and
F G A B on the bottom line.
The notes of Db major are
Gb Db Ab Eb Bb F C alternatively
Db Eb F on the bottom line and
Gb Ab Bb C on the top line.
Note: as you are looking for minimal memorization its worth noting that the above diagram can be made by writing out two whole-tone scales one above the other with an appropriate shift. This is a good way of constructing the diagram, but does not explain what is going on.
Steelpan and the circle of fifths
In order to avoid dissonant notes being close together, the notes on the Steelpan are arranged in a circle of fifths. This means all the notes of any given key are grouped together. To my knowledge it is the only instrument that uses this layout and the diagrams linked below are simpler and perhaps clearer than the wikipedia article linked above.
For example C is at the bottom (6 O'clock position) and all the notes of the key of C major are grouped from 1 O'clock to 7 O'clock. Bb is at the 8 O'clock position and the notes of Bb Major are grouped from 3 O'clock to 9 O'clock.