Circular Reasoning

(entry for 1/3/2025)


If you’ve been around music much, particularly if it’s music theory rather than mere performance, I’m sure you have by now run into the idea of the Circle of 5ths. If you’re a typical musician, you’ve also been bewildered, confused, and perhaps even angered, by the concept. The reason for this is that a lot of theory teachers think that the Circle is the answer to everything, and it simply isn’t. But their promotion of that notion is what sets off the negative reactions.

The Circle is important, but not for the reason the typical theory teacher thinks it is. The mistaken idea is that the Circle somehow drives harmony. It doesn’t. It’s the other way around.  Harmony drives the circle. By that I mean that the way harmony works is most easily explained by using concepts from the circle. But the way harmony works would be true, whether there was such a thing as the circle or not. The circle just helps explain and understand it.

But before we can see why that is true, we need to go back to the idea of Series, which we’ve been covering in the last three posts, and particularly the ranking of intervals that we covered in the last post.

If you’ll recall, as we go up a fifth to reach a new ‘key,’ we add a sharp to the key signature of that key. And if we go down a fifth to reach a new key, we subtract a sharp or add a flat to the signature. (Going up a fifth can also subtract a flat if there are any flats to subtract from.) We can go either direction in this process until we reach a maximum of seven sharps or seven flats (because there are only seven different letters to sharp or flat). 

This sounds like a long continuous line stretching off to the left in the flat direction, and off to the right in the sharp direction. And if we’re talking pure pitches, not tempered ones, that’s exactly what it is. Just one long continuous line. Here is the middle of the line, showing the keys with the fewest sharps or flats in their key signature. (You can imagine the line stretching off in both directions.) We've put the Major Keys above the line as capital letters, and the related Minor Keys below the line as lower case letters.



However! If we’re talking tempered pitches, where A-sharp is the same as B-flat, etc., then it isn’t a continuous straight line. It’s a circle! Because you can wrap the line around on itself and meet where the sharps and the flats are equivalent to each other.

Look at the design at the head of this post. Note how it looks somewhat like a clock face, with 12 at the top and 6 at the bottom. Also note, please, that the bottom three clock positions are divided. That is, the perpendicular lines crossing the circle at the five, six, and seven o’clock positions are ‘forked’ at both ends. We’ll come back to those ‘forks’ a little later.

For now, let’s apply our supposedly straight line ‘map’ of keys and put the empty signature at the top of the circle, at the 12 o’clock position. Put the major keys as Capital letters on the outside of the circle and the minor keys as lower-case letters on the inside of the circle. Since the empty signature represents 'C' major and 'a' minor, as we’ve discussed in previous posts, we put them in the 12 o’clock position. We put 'F' major and 'd' minor (one flat) at the 11 o’clock position, and we put 'G' major and 'e' minor (one sharp) at the 1 o’clock position. The top three positions will then look like this:



It's probably a good idea to show what the key signatures indications are for each clock position, so we'll add those, and now it looks like this:



As we go counter-clock-wise around the circle to the left, we add one more flat at each position, and put the new key names (major on the outside, minor on the inside) at each position, going down a fifth for each new key name.

The next three positions on the flat side of the circle will then look like this:

       


We can do the same on the right side of the circle by doing the same thing going up a fifth each time, and adding one sharp to the signature. The right side of the circle looks like this:

             


Now comes the tricky part. The bottom of the circle has two names for each position. I’ll first show you what that looks like, and then explain. Here’s the bottom of the circle (but don't freak out just yet!):




Let’s take the 7 o’clock position first. Its two major names are ‘D-flat’ and ‘C-sharp’ (which, in a tempered scale, are the same note). The two relative minor names are ‘b-flat’ and ‘a-sharp.’

The key signature for ‘D-flat’ major is five flats. (If you count down by fifths from C major at the top of the circle, going counter-clockwise, ‘F’ major is one flat, ‘B-flat’ major is two flats, E-flat is three, A-flat is four, and D-flat is five.) But the same key, if you spell it with a C-sharp instead of a D-flat is seven sharps! (Going clockwise from C, one sharp is the key of G major, two sharps is D, three is A, four is E, five is B, six is F-sharp, and seven is C-sharp.)

Likewise, the 6 o’clock position is either G-flat major or F-sharp major (and their relative minors), which is either six flats or six sharps, depending on which way you spelled it.

And now it gets weird, because we have a ‘white-note’ flat key. The 5 o’clock position is either ‘B’ major (five sharps), if we’re coming clockwise down the sharp side of the circle, or it’s ‘C-flat’ major (seven flats) if we’re coming around counter-clockwise from the flat side. Most musicians, when exposed to this position on the circle, become baffled, because they're not used to the idea of a white-note flat, but the fact is, you can create a white sharp or flat note just as easily as you can create a black on. Look at a piano keyboard and you'll see why. There are no black notes between E and F or between B and C.  The pitch 'E' can also be called 'F-flat,' and the note 'F' can also be called 'E-sharp.' Likewise, the note 'B' can also be called 'C-flat' and the note 'C' can also be called 'B-sharp.' Why would you want to do this? All will be explained when we get to the post on 'Scales.' Meanwhile, just trust me that there are good reasons for doing so. I promise not to let you down.

You see now why we had to have the ‘forks’ on those bottom three positions: it was so we could accommodate all seven letter names in the series of sharps and flats.

Also, please notice that, if you look at each of the three positions at the bottom of the circle, for each position, the number of sharps plus the number of flats equals twelve. That's because there are twelve different notes available within each octave, before you start all the letter names all over again.

That’s it. That’s the whole circle of 5ths. I suggest you draw a fairly large circle yourself, and fill in all the spots using the above information.  Or you can click on the blank circle at the head of this post and print copies of the circle that comes up as a picture file. It won't be quite as large as one you draw yourself, but it's easy and it will do. 

You can now see at a glance how all the keys, both major and minor, are related.  This is important, because knowing how the relationships work is the core concept of understanding harmony. Next time we’ll talk about how to use the circle to make it even more valuable!



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