I have often tried to visualise the circle of fifths in a way that makes sense both musically and mathematically. Above (click to zoom) is my latest attempt.
There are 12 notes in an octave (7 white piano keys and 5 black piano keys), and the diagram shows these 12 piano keys wrapped into a circle. A fifth is a step of 7 semitones (7 piano keys, counting black ones, e.g. C→D♭→D→E♭→E→F→F♯→G). The coloured spiral in the chart shows the “circle of fifths” resulting from moving up a fifth 12 times (moving left to right, and hence moving anticlockwise).
The reason that this works is that 7 and 12 have no common factor – and therefore the first multiple of 7 that is also a multiple of 12 is 7 × 12. Therefore every time you move up a fifth you get a different note, returning to the starting note only when you have moved up 12 times. In the process, you have hit every other note exactly once.
Scientific Gems 