...NB, it's the notes in the scale I want to memorize, not the keys on the piano keyboard. If I've memorized that C-minor is the naturals except for A flat, B-flat and E-flat I'll have no trouble finding those notes on the keyboard.
When worded that way it sounds to me like you really want to know the keys signatures not really the scales. The two are sort of the same, but suffice to say you just want the names of the tones, the letters and sharps and flats.
I think the way to go about it is think in terms of ascending and descending fifths.
Sharps are added to key signature by ascending fifth starting with
F# and flats are added by descending fifths starting with
C minor example, we start at
A minor with no sharps or flats in the key. We then descend by fifth through the tonics and also add flats by descending fifths.
A down to
D and add a
D down to
G and add an
G down to
C and add an
Ab. We are at
Bb Eb Ab for
The "melodic" and "harmonic" minor a just modifications raising the sixth ans seventh degrees of the minor scale as needed.
A major you ascend by 3 fifths from
C and add three sharps
F# C# G#.
When you get comfortable reciting the letters by fifths you can pick out those sharps and flats quickly.
C G D A E B F# C# G# D# A# E# B#
C F Bb Eb Ab Db Gb Cb Fb
Or in a circle:
After reading Albrecht Hügli's answer about tetrachords I had another idea.
It's a combination of tetrachord thinking and leveraging tonal versus modal scale degrees.
Two perfect fourths joined by a whole step give use the outer parts of two tetrachords (shown in brackets)...
^1 ^4 ^5 ^1
[ C . . F ][ G . . C ]
...those are the tonal scale degrees. The tonic, subdominant and dominant.
Add to that the second scale degree...
^1 ^2 ^4 ^5 ^1
[ C D . F ][ G . . C ] <<< this is the foundation
...the important thing is that in all the major and minor scales these tones do not change.
I showed the tones with a
C tonic, but you can start on any tonic just add the tone a perfect fourth above and below the tonic and it's octave and then the whole step above the lower tonic.
Once you have your tonal foundation you fill in the remaining modal tones.
Start with the third scale degree. For major just add a major third above the tonic (whole step above the second scale degree) or for a minor scale add a minor third (half step above the second degree.) Example in minor...
^1 ^2 ^3 ^4 ^5 ^1
[ C D Eb F ][ G . . C ]
(Note: first tetrachord for ^1 to ^4 is called a minor tetrachord.
The remaining two scale degrees are the sixth and seventh and this is where things get complicated. The upside is if you get a handle on this part you understand are really important part about the major/minor system.
It's probably best to just learn the sixth and seventh from modifications of the major scale pattern which is a whole step above
^5 and a half step below
^1. Showing just the top tetrachord...
^5 ^6 ^7 ^1
[ G A B C ]
...a major tetrachord which is used for major and "melodic minor"...
^5 ^6 ^7 ^1
[ G Ab B C ]
...a harmonic tetrachord used for "harmonic" minor...
^5 ^6 ^7 ^1
[ G Ab Bb C ]
...a Phrygian tetrachord for "natural" minor.
The benefit of this method over learning the key signature stuff by fifths is you can fill in the tonal foundation with simple whole and half steps. As long as you understand there a whole steps between all the letters except the half steps between
EF it should be easy to fill in the full scale.
4 fixed, unchanging tonal degrees filled in with 3 variable modal degrees.