Here is how to figure out the circle of fifths.
Start with the scale of C...which has NO sharps or flats.
If you have forgotten the pattern for a Major scale, here it is:
Whole step,Whole step, Half Step, Whole step, Whole step, Whole step, Half Step.
Got C!?
Now got to the fifth step of the C scale
C-D-E-F-G-A-B-C
Start you next scale from that fifth note G:
Whole step,Whole step, Half Step, Whole step, Whole step, Whole step, Half Step.
You get:
G-A-B-C-D-E-F#-G
There you are: the key signature for G scale is F# and G is the first scale in your cicle of fifths.
Let's try one more;
Find the fifth step of the scale of G:
G-A-B-C-D-E-F#-G
Start a scale from the note D:
Whole step, Whole step, Half Step, Whole step, Whole step, Whole step, Half Step.
D-E-F#-G-A-B-C#-D
So, C# and F# are in the scale of D and D is the next scale in the circle of fifths.
Continue and you will see that each time you start a scale on the fifth note of the previous scale and add a sharp that is five steps up from the last sharp you added, voila, you have the next scale in the circle of fifths.
Once you hit F#, you may want to change then to the circle of fourths and re-name the sharp scales as flat scales.
The circle of fourths is done like the circle of fifths, only you find the next starting scale tone on the fourth step of the scale instead of the fifth.
Attached is a visual of this circle for those that would like to "see" this in action.
Practice these scales until you can go from one to the other without problems.
Chords coming up......
