## A review

Previously we discussed a clock in which the numbers going clockwise from the 12 o’clock position went up by sevens. Of course, we want to keep the numbers between 1 and 12, so when the number gets larger than 12, we subtract the 12 out. We start with the number 7 at the 1 o’clock position. When we add 7 to that, we get 14, but that’s bigger than 12, so we subtract 12 from 14 and the next number becomes 2. Notice that adding 7 and subtracting 12 is the same thing as subtracting 5, so we could also have explained what we’re doing by saying that for each step around the clock face, we either add 7 or subtract 5, whichever keeps us in the 1-to-12 range. When you do this, you get all the way around the clock, using each number exactly once. It sounds more complicated than it is; look at this picture:

We then realized we could do the same thing with musical notes, as there are 12 hours on the clock and also 12 different notes in an octave, and we ended up with this:

We mentioned that this was the equivalent of something called “The Circle of Fifths”. When we discussed intervals and inversions and all that, we determined that a perfect fifth is seven half steps in size, so you’d think we could keep making the notes go up by perfect fifths (or, to stay within the same octave, go down by the inversion of that interval, the five-half-step perfect fourth) and get all the way around our clock face.

## Making a true circle of fifths

But there’s a catch! Recall that the designation “fifth” comes strictly from the note letter names. So C4 to G4 is a perfect fifth, but C4 to Abb4 is a diminished sixth, even though the pitches are the same. With this in mind, something odd happens if we try to go up by twelve perfect fifths.

Let’s try going clockwise. We start with C at the 12 o’clock position. Then, going up by perfect fifths (or, equivalently, down by perfect fourths) we have G, D, A, E, B, F#, C#, G#, D#, A#, E#, and finally B#. We did get to the same pitch, but not to the same note name.

Let’s try going counter-clockwise now, going up by perfect fourths (or, equivalently, down by perfect fifths). Here we go: C, F, Bb, Eb, Ab, Db, Gb, Cb, Fb, Bbb, Ebb, Abb, Dbb. Once again: same pitch, different note name.

So it would seem that we can’t literally create a circle of perfect fifths. Is music theory toying with us? How do musicians resolve this and represent a circle of fifths? Basically, they start from C and start going in both directions, making two tracks around the circle. If you want to travel all the way around the circle back to C, somewhere near the 6 o’clock mark you have to hop onto the other track. You can imagine in our original diagram that the clockwise direction continues on the outside track where the counter-clockwise direction moves on the inside. But in the approved musical version, the track ends after seven steps in either direction from C, so somewhere from 5 to 7 o’clock, you had better jump tracks. A picture should make this clear:

The tracks of the same color move by perfect fifths/fourths. Going clockwise from C, you’d follow the blue track, but to go all the way around, you must eventually jump to the red track near the bottom. Similarly, going all the way counter-clocwise, you’d have to jump from the red track to the blue track to get back around to C.

Why does the crossover occupy three positions? Why does each direction go seven steps? Why didn’t they just keep it simple and make you jump tracks at the 6 o’clock position? There are actually good reasons why the tracks go as far as they do and no farther, which we’ll explore next time.