Greybeard's Circle of Fifths Table
This is the method that I used to remember the circle (or cycle) of fifths:
1: Remember the sequence F, C, G, D, A, E, B - if can do that, you are about 1/4 of the way there!
2: "C" is always neutral!
3: Make a circle and divide it into sixteen segments.
4: Take the top two adjacent segments and write "C" in them (just once to cover both segments).
5: The right side is sharp and the left flat.
6: Fill out the first sequence. You already have a "C", so, in the segment to the left of it put an "F", you can then see the rest of the sequence, running right round to the bottom right of the circle. You should have 2 segments left in the right half of the circle.
7: Remembering Point 5, we start again after the "B" with "F" again, but this time sharp (F#). Then comes a "C", again sharp. I said earlier that "C" is always neutral,, so how can I have "C#"?. Easy put a "Cb" in the next segment to equal it out. Then fill the rest with the remaining notes in the sequence (G, D, A, E, B) reemembering that this is the flat side, so they are all flat!
8: Number the segments from 0 to 7, starting with "C" as 0.
9: So taking our sequence in hand, we start at the right hand side, in the segment marked "G" with the 1 next to it. We need to put one sharp here, which comes from the start of the sequence F, C, etc, which is F#. In the segment marked "D" / 2, we need 2 sharps, so here we enter F#, C#. We do this for each segment.
10: The left hand side is similar, except that we start at the back end of the sequence. One flat is Bb, two flats is Eb, Bb, three is Ab, Eb, Bb.
That is the cycle of fifths
Once you do this for yourself, you will see how little you actually have to remember to have the cycle of fifths at your fingertips. the most important thing is the sequence F, C, G, D, A, E, B, the rest all follows.
Rule 1: F, C, G, D, A, E, B
Rule 2: "C" must always neutral
Rule 3: Right side is sharp, left is flat
Rule 4: Sharps start from the front, flats from the back
The notes, shown in dark blue, are the tetrachords for each note. If you follow the circle around, you will see that the key of C consists of the tetrachords for C & G (the next note, clockwise, in the circle) - G being the 5th degree of C (which is why it's called the circle of fifths). The same applies to the flat scales - Cb plus Gb gives the key of Cb. Gb plus Db gives the scale of Gb.