The Circle Of Fifths And The Klein Four-Group

Created: Sun Jan 07, 2024

Recently I started learning the piano, and I had mostly taken for granted what gets taught about the relationship between the circle of fifths and key signatures: as you go around the circle, the key signature has the same accidentals as the previous one, plus one new one.

It took some time, but eventually it really clicked, with a nice mathematical connection.

The octave is made up of 12 semitones, and one nice thing about the number 12—proponents of a base-12 number system point this out a lot—is that it divides very nicely. Halves, thirds, quarters of 12, all of those are whole numbers. In fact, the only numbers less than 12 which do not share any factors with it are 1, 5, 7, and 11.

While that might be very nice for arithmetic, it does also add some limitations elsewhere.

For example, if you put down 12 points along a circle and try to join them up into a star-like shape by skipping the same number of points every time. If you go by twos each time (going from point 1 to point 3 to point 5, and so on) like you would for a five-pointed star, you get back to where you started without connecting every point! You just end up drawing a hexagon; A similar thing happens if you go by threes instead, you end up with a square.

This happens because 2 and 3 share factors with 12, and so the only way to reach every point is to jump by a number that does not. As we've seen, there are only four of these.

If you only jump 1 point at a time, you do end up reaching all the points, but you just get this boring 12-sided polygon. Jumping 11 at a time sounds dramatic and exciting, until you realize that that just lands you at the point directly behind where you just were: you're still jumping 1 at a time, just in the other direction.

If you try this with 5 and 7, you do get a more interesting star-shape, but once again, you get the same star shape, just drawn backwards.

This same idea translates to the octave: if you want to jump the same number of notes each time and still hit every note, your choices are to go up 1 note, up 11 (which gets you the same letter as if you go down 1), up 5, or up 7 (similarly, landing on the same letter as going down 5).

You may be aware that there's a special name for intervals of 5 and 7 semitones: a perfect fourth, and a perfect fifth.

So this is where the circle of fifths comes from and why it's so special: other than reversing the octave, it's the only way to rearrange all of its notes while still having consistent spacing between them. (Well, you can also reverse this arrangement too to get a technically different one, but that's not a very interesting change.)

Now what about key signatures? There's another neat property about the way keys on a keyboard are arranged: they alternate between white and black except for 2 places. E-F and B-C are the pairs of white keys right next to each other.

Those pairs happen to be far enough apart that a prefect fifth, a 7 semitone interval, is just wide enough to always cover one of those pairs, but not quite wide enough to cover the other. And since 7 is an odd number, going up a fifth means going up three white-black key pairs and the lone white key out, and so you land on the same colour you started on.

With two exceptions. The span of keys from C to E is also just narrow enough that a 7-semitone span starting just before it can cover both white-white pairs, leading to a colour change. Sure enough: a fifth up from B is F♯, and a fifth up from B♭ is F.

And what this means is that if you start on a white key and start going up fifths, the only place you can change to a black key is when you reach B. When you're in the black keys, the only place you can change to a white key is from B♭. These are the only changeover points in a sequence that contains all the notes of the octave, and so the loop you make has a contiguous region with all and only white keys, and one with all and only black keys.

And so, this is where the key signature rule comes from. Starting with the C Major scale—made of all and only the white notes—and shifting the whole thing up a fifth, all the white keys land on other white keys—except for B, which lands on the black F♯. Keep going up fifths, and this note will keep crawling along the black keys, with the rest of the scale following its path.

And a symmetric thing happens if you go down fifths instead.

This is pretty helpful for remembering which notes are in each major scale! By counting how many fifths up or down the root note is from C, we know how many accidentals have been picked up, and recalling this rearrangement lets us figure out which ones those are.

(And by the way, if you want to impress your friends, you can phrase this observation as "the automorphism group of the octave is isomorphic to the Klein four-group with the permutation in the circle of fifths as its non-reversing generator.")