Image via WikipediaWhen you are sitting at your keyboard or piano, do you see patterns on those keys? Here's what I mean:
1. Chromatically the notes from C to E are the mirror image of the notes from F# to B flat with F and B natural left out. If you number those notes from 1 to 5 in each pattern, the same number in each pattern is the Tri-tone. C = 1, F# = 1....tri tone. Use this visual to find tritone substitutions.
2. Take the first five notes of the major scale. These keys are mirror images of each other:
D flat and D
E flat and E
F and F#
A flat and A
B flat and B
3. Remember the tri-tone(s) B-F and F-B because it is the only tri-tone(s) with two white notes. All the others have one black and one white note as in your first pattern.
4. When you look at the circle of fifths, you notice that all the black notes are consecutive: Bb-Eb-Ab-Db-Gb, which you can use to help remember the circle. Try to think about why all the black notes would be consecutive in this circle. What you see is that the circle of fifths consists of the two whole tone scales alternated:
C - Bb - Ab - Gb - E - D
F - Eb - Db - B - A - G
It seems to be easier for me to always think of the circle of fifths in the counter-clockwise direction. Normally I am thinking of the dominant to tonic direction so that is why it seems more natural in this direction. If you learn the circle in this direction, it also gives you the ii-V-I sequence starting from any position as the ii chord. Or you can start two positions earlier and it gives you a iii-vi-ii-V-I turnaround sequence.
Within the circle of fifths the pattern B-E-A-D-G is repeated, first with all flats, then with all naturals, then you just have to add C & F and you have the whole circle (that's how I remember it). It also helps that the first four letters of that sequence spells a word. Once you know the circle and can picture it like the numbers on a clock as it is most often represented, you can look at the position directly opposite to get the tritone interval e.g. if C is at 12 O'clock, look at 6 O'clock to get its tritone interval.
5. Also, let us take the pentatonic scale 1, b3, 4, 5, b7. These keys are the mirror image as far as black key white key notes.
C and Db
D and Eb (E and A are the same as D)
F and F#
G and Ab
Bb and B natural
All the best,
Charlie Parker once said "Learn the changes and then forget them."