Now we can explain why the circle of fifths looks the way it does. Recall that we had an ambiguous area from 5 o’clock to 7 o’clock:
We’re going to examine the circle of fifths in terms of major scales. To start out, you need to know that the scale you get by going up the white keys of the piano starting from C is a major scale. Thinking about half steps and whole steps, it’s a mixture of both. You can see from this half-step proportional wacky stretched keyboard that sometimes the white keys have black keys between them that get skipped over in our C-major scale, and sometimes they don’t:
What if we wanted to construct a G-major scale? Way back when we talked about transposing, we first introduced this stretched piano keyboard, and we said that transposing is like taking the pattern of notes and sliding it up or down the keyboard. So let’s take two of these keyboards and slide one over so that C on one lines up with G on the other:
Now we can look at the white keys on the bottom keyboard (the ones in the C-major scale) and just read off the notes in the G-major scale from the top keyboard. Look at the diagram, start at C on the bottom keyboard and follow along on the top, reading off each note at the top above a white key on the bottom: G, A, B, C, D, E, F#, G. Note that it’s proper musical etiquette to keep the letter names in a scale in sequence, which is why we use the name F# rather than Gb here. It would be a serious faux pas to use Gb here.
This rule of etiquette actually has a good reason: when transposing, one strives to preserve all the intervals. In the C-major scale, the C up to the B is a major seventh, and in the G-major scale, the G up to F# is also a major seventh. G up to Gb would be a diminished octave. (Intervals were discussed earlier.)
Notice that the seventh note of the G-major scale is sharped (namely, F#) while the other notes are the same old natural white keys that appeared in the C-major scale. If it seems cumbersome to count up to the seventh note, you can just go down one note from G to get F, and that’s the one that gets sharped. The F was raised by a half step to get F#, and we cleverly verbed the noun “sharp” to mean “raise by a half step.”
Certainly transposing major scales up by a perfect fifth ought to do the same thing every time, namely sharp the seventh note and keep the other notes the same. And it does just that. Now we can go ahead all the way from 12 o’clock to 7 o’clock in the circle of fifths and instantly figure out which notes are sharped in all those major scales:
|C major||C, D, E, F, G, A, B, C|
|G major||G, A, B, C, D, E, F#, G|
|D major||D, E, F#, G, A, B, C#, D|
|A major||A, B, C#, D, E, F#, G#, A|
|E major||E, F#, G#, A, B, C#, D#, E|
|B major||B, C#, D#, E, F#, G#, A#, B|
|F# major||F#, G#, A#, B, C#, D#, E#, F#|
|C# major||C#, D#, E#, F#, G#, A#, B#, C#|
The note in red is the note that’s different from the scale above it.
The same example worked backwards can help us understand how going counterclockwise works. To go from G-major to C-major, one note got un-sharped, namely F. F was the seventh note of the G-major scale, but it’s the fourth note of the C-major scale. Let’s verb the word “flat” to mean “lower by a half step.” It would seem that to go counterclockwise in the circle of fifths, each successive major scale flats its fourth note:
|C major||C, D, E, F, G, A, B, C|
|F major||F, G, A, Bb, C, D, E, F|
|Bb major||Bb, C, D, Eb, F, G, A, Bb|
|Eb major||Eb, F, G, Ab, Bb, C, D, Eb|
|Ab major||Ab, Bb, C, Db, Eb, F, G, Ab|
|Db major||Db, Eb, F, Gb, Ab, Bb, C, Db|
|Gb major||Gb, Ab, Bb, Cb, Db, Eb, F, Gb|
|Cb major||Cb, Db, Eb, Fb, Gb, Ab, Bb, Cb|
Again, the note in red is the note that’s different from the scale above it. (If you are ambitious, you can go back and look at the diagram above with the two keyboards, and look at the notes on the bottom keyboard that line up with the C-major scale on the top keyboard; you should see the F-major scale on the bottom, with Bb as its fourth note.)
In each of the tables we can see that the last scale in the table (C# and Cb) has all its notes sharped or flatted. In order to sharp or flat any notes further, we would have to start using double-sharps or double-flats. In theory, you could indeed construct a scale of Fb major containing a Bbb, but in practice, musicians tend not to do that.
And that is why the circle of fifths has those two “tracks” that go seven steps in either direction. It relates to sharps and flats in the major scale. When we later talk about key signatures, that will help to tie this all together even more.