Insomnia

 I was thinking about the circle of fifths as I was falling asleep--or failing to fall asleep--the other night. 

Going around the circle, each new key adds a sharp, on the seventh degree of the scale.  So G has F#, D adds C#, etc... At the bottom of the circle F# adds E#.  At every point of the circle there is one fewer shared notes. So G shares all but one note with C, and F#, at six positions of distance, shares only one note. At first, I was thinking that Gb or F# had five notes in common with C, since there are two white notes in the this scale. But then I realized that the B would have to be spelled as a C flat.  

With the flat keys on the other side of the circle, each new key loses a flat.  Of course, losing a flat is the same as adding a sharp, so the same pattern is observed. With the circle of fourths (reading counterclockwise), each key adds flat, and that flat is always the fourth degree of the scale, so that F adds Bb, Bb adds Eb, Eb adds Ab, Ab adds Db, Db adds Gb, Gb adds Cb.  Usually, I would be asleep by now thinking about this, but this particular night I just kept going.  If I had stayed awake longer I would have figured out why the fourth corresponds to the 7th going the other way.  I do know that the sequence of sharps added follows the same sequence as the circle of fifths itself, starting with F#, C#, G#, D#...  The flats are the same order, in retrograde.  

{The reason why it's flatting the fourth the other way around is very obvious: the fourth of the key becomes the seventh of the next key, in the circle of fifths.} 

[The sharp added will be the 7th, as we said, so that previous sharp added will the 3rd in that new scale. The one added before that will be 6th, then the 2nd, the fifth.  Finally, there will be key that begins on that first sharp added (F#).]