Fun with the Circle of Fifths

[Lord Larehip]

The Circle of 5ths is a remarkable scheme for determining musical scales and is often expressed in a variety of ways. What does the Circle of 5ths tell us? A lot of things. Firstly, it shows us the chromatic scale lined up along the perimeter of the circle. The Circle of 5ths is divided vertically into a sharp side and a flat side. Going clockwise around the circle starting with C at the 12 o’clock position, we descend on the sharp side and encounter G which is a perfect 5th interval (7 half-steps) from C and every note encountered in sequence is a perfect 5th higher than the one before it. The Circle of 5ths can also descend from C at the top and go counter-clockwise down the flat side in perfect 4ths, which is the inverse of the perfect 5th.

While an experienced musician knows his or her scales and chords, the circle of 5ths is still a great mnemonic device. For you novices, though, to really internalize what the circle offers you about music theory, you have to be able to draw one from memory. I'm going to show you a method that requires virtually no memorization. First, let's look at a very basic one:

The pattern we encounter on either half of the Circle is B E A D G C. We encounter it descending on the flat side or ascending on the sharp side. You bass players with 6-string basses recognize this pattern immediately because a 6-string bass is tuned from top to bottom in exactly that pattern.

Use the above image for reference while following these steps:

1. Draw a clock face with 12 marks where the numbers would be. Since there are 12 marks evenly spaced along the perimeter and a circle has 360 degrees of arc, then each mark subtends 30 degrees of arc (12x30=360).

2. You can start at any mark but it is customary to place the key of C at the 12 o'clock position, so draw a C at 12 o'clock.

3. Now draw a line from the 12 o'clock mark across the circle to the 6 o'clock mark. What goes there? We don't know yet. But we can insert the notes flanking C at the 5 o’clock and 7 o’clock positions—B and D-flat respectively. Just go in sequential order--the lower note at lower number position. So place B at 5 and D-flat at 7. We don't we use C-sharp instead of D-flat? We can as you will see but we want to build a flatted side of the circle because it's easier. So buy it for now.

4. So we have three points marked--C at 12, B at 5 and D-flat at 7. Where to next. We can either go straight across the circle from B or from D-flat, makes no difference. So let's draw a line from the B at 5 o'clock to the opposite side of the Circle at 11 o’clock, we fill in the note flanking the lower side of B at the 10 o’clock position which is B-flat. The other note, C at 12 o’clock, is already filled in.

5. We just keep going around the circle this way until we fill in all twelve positions and we are done. How easy is that? Instead of memorizing all these notes in the order they go around the circle, we can just construct it empirically without bothering to memorize any of that. Do this several times until you get the hang of it and do it periodically whenever you're sitting around bored. Do it until it is automatic because the circle can't really help you much until you got it firmly in your head.

In the next post, we'll add in yet more detail to our circle.

Each of the letters we added to the circle in my previous post represents a major scale. Each major scale as an equivalent minor or what is called a relative minor. It is determined by falling back a minor 3rd or 3 half-steps from the major scale designation. So 3 half-steps before C is what? 1. C back to B, 2. B back to B-flat (or we can call it A-sharp), 3. B-flat back to A. So the relative minor of C major is A minor.

But how can we use the circle to find the relative minor?

It bears a 90 degree relationship with the major. Draw a line from C at 12 o'clock to the center where the hands of the clock would both be pinned to the clock face (if this one had hands) and then make a right angle or 90-degree turn to the right towards the 3 o'clock position where A is.

So what is the relative minor of A-flat major? F minor. Another way to look at it is to count forward three positions. So what is the relative minor of D major?

Key signatures determine the scale. In the key of C major, there are no sharps or flats that occur naturally but in other scales they do. Moving clockwise from C to G major, there is one sharp at the note F. Moving counterclockwise from C is F major and it has one flat at the note B. The next scales down on each side have two sharps or flats then three after that and so on.

Rather than putting a sharp by every F in a G major piece, we just put a sharp on the F line at the beginning so that the musician knows to sharp the F throughout. We call that the key signature. The relative minor uses the identical key signature as its relative major. All the available key signatures look thus:

Next, I'll show you how to assemble the correct key signatures using the circle of 5ths.

[Lord Larehip]

The noteworthy thing about the order of the sharps and flats is that they are the reverse of one another. The sharps go in fifths and follow the circle clockwise starting at F and going to B while the flat go in fourths and follow the circle counterclockwise starting at B and going to F.

The sharps and flats ALWAYS follow this order. You would never see just C sharped by itself in a key signature. It must appear with F. Only F can appear in a key signature by itself. Likewise B cannot be sharped in the key signature without sharping from F to E clockwise along the circle. The same goes the flat side of the circle.

So we have a key signature with 5 flats in it? How could the circle of 5ths tell us what key that is? We would bisect the circle such that one of the half-circles has five of the flatted scales isolated in it--only five. Which five?

Well, you know that if there are five flats in the signature they must be in a certain order, in this case, BEADG. So draw a line from G-flat to C. Now you have the five major scale designations with flats in the title isolated in one of the half-circles. Now turn this line one notch clockwise. You end up on D-flat and that is the key with five flatted notes (and difficult to play bass in because there are no open notes).

But look at the other half-circle! It contains only one sharped scale designation--F-sharp. When you turned the line one notch clockwise, the other end of the line (which is in the sharp side of the circle) ends up on G. And what is the major scale who only sharp note is F? Yes, that would G major.

Let's try another one. Suppose we have a key signature with three sharps in it. Which is it? Well, it if it has three sharps then what is the sequence? F-C-G. So we bisect the circle of fifths so that one of the half-circles contains F, C and G all sharped. Look carefully. If you picked the FCG at the top of the circle, you would be wrong! They do not have sharps in their titles. Instead, look at the bottom of the circle. At 6 o'clock, there is an F-sharp. Going clockwise, there is a D-flat but D-flat is also C-sharp so there is our F- and C-sharp. We need G to be sharp too so we also include the A-flat since A-flat is also G-sharp. So we draw our bisecting line from A-flat straight across to D. Now we rotate the line one notch clockwise and the end of the line in the sharped side of the circle moves to D. So D major is the key with three sharps--FCG.

What about the other half circle? That side has three flats in it--BEA. That is the correct sequence for flats. Now when we rotated the line clockwise one notch, the end of the line in the flatted side of the circle comes to rest on E-flat. So E-flat major is the scale with three flats--BEA.

Okay, okay. So the circle can show you how to determine a key based on the number of sharps of flats. But can it tell us which notes are sharped or flatted in each key signature? Yes, it can.

Use the above figure for reference. To determine the flats, drop back two notes on the circle. G major gets one sharp so drop back two spaces on the circle to F major and that is the note that gets sharped in G major. Trigonometrically, draw a line from G to the center of the circle and go 60 degrees counterclockwise (remember that each note represents 30 degrees of arc). Moving to D major, which has two sharps, drop back 60 degrees to C and that is the next sharp so D major has two sharps—F and C. And so on.

For flats, we look at the note on the opposite side of the circle. So the note opposite F is B so that is the note that gets flatted in F major. Trigonometrically, we drop back (or move forward) 180 degrees to find the flats. The next note of the circle is Bb and will get two flats. We already know that one will be B and the other is 180 from Bb which is E so Bb major is flatted at B and E. And so on.

The special case is C which has no key signature (no sharps or flats). If we drop back 60 degrees as in the manner of finding sharps, we find ourselves at Bb which, if sharped, simply becomes B. Or if we cross the circle 180 degrees to find the flats, we end up at Gb, which, if flatted, simply becomes F. So, as the circle shows, C has no naturally occurring sharps or flats.

Next we'll investigate how chords figure into the Circle.

[Lord Larehip]

When we write out the C major scale on a staff, we start off this way (above). This figure shows which notes are used to make chords. The point is, to make chords, we only use the notes available to us in the scale. In a B-flat scale, for example, where the B and the E are flatted, then all the available chords that use a B or E use them flatted. Otherwise, the chords would be out of tune. To make the chords, we just pile the notes up thusly:

Using only the notes available in the scale, they form chords as shown on the figure. The first chord is a C major 7th. We call it I--the Roman numeral one in upper case because it is major. The second chord is a D minor 7th and we label it ii--or the Roman numeral two in lower case because it is minor. Note the sequence--I, ii, iii, IV, V, vi, vii. No matter which major scale you plot out, the chords will ALWAYS follow that sequence. What is notable is that the vii chord always has a diminished 5th (it's 6 half-steps above the root or lowest note in the chord instead of the usual 7) and the V chord has a major triad (root-major third-fifth) but always uses a minor 7th instead of a major 7th as a true major chord would. Notice the true major chords are notated as the letter followed by a triangle and then the numeral 7. That's the symbol for a major 7th chord. The V chord is just a letter followed by a 7 or simply a 7th chord. A 7th chord is major-minor, i.e. a major triad topped by a minor 7th (it's 10 half-steps above the root instead of 11 as with a true major 7th chord). The V chord is extremely important as it usually resolves back to I in most songs and also allows us to change keys in a song.

Armed with this knowledge, here is the most useful circle of fifths chart I have seen:

[Lord Larehip]

The Circle of Fifths also unites music and geometry in a number of ways and we'll investigate one of them here.

How can you construct the proper diatonic scales from each note on the circle?

There are a number of ways. Chords are said to consist of root, third and fifth. But that scheme, although widely used, is not particularly useful. I was classically trained on the double bass and we construct chords in thirds only: root, major third and then another minor third on top of that for a major chord or root, minor third and then a major third on top for a minor chord. Using this method, let’s construct our diatonic scales on the Circle of Fifths using the pattern shown above: root, major 3rd, minor 3rd, major 3rd, minor 3rd, minor 3rd, major 3rd, minor 3rd.

For example, let us use C. Starting at C we go up a major 3rd (or 4 half-steps) to E. Then go up a minor 3rd (or 3 half-steps) to G. Then go up another major 3rd to B. Then another minor 3rd to D. Then another minor 3rd to F. Then a major 3rd to A. Then a minor 3rd to C'. So the order is:

C E G B D F A C'

Now arrange these notes starting at C in a circle in alphabetical order: C (C') D E F G A B and draw lines starting from C to each note in the order obtained as shown above. You get a 7-pointed star! And this works with any key and whether major or minor. Try it!

Pick another key—say Ab. The order we obtain is:

Ab C Eb G Bb Db F Ab'

Now lay them in a circle in alphabetical order and then draw a line from one note to the next in the sequence above—a 7-pointed star!

[Lord Larehip]

Or we can make a circle of the notes in the original sequence Ab C Eb G Bb Db F and then connect the notes alphabetically with lines to obtain a completely different type of 7-pointed star.

Pythagoras was right!

What is also noteworthy is that the sequence of the C scale by thirds--C E G B D F A--contains how the notes are laid out on manuscript paper. FACE represents the spaces between the lines on the treble staff and EGBDF represents the staff lines themselves. Needless to say, it also describes the layout of the bass staff: ACEG GBDFA respectively.

But what's even more noteworthy is that no matter what scale on the Circle we plot out, the order of the notes follows that layout if we ignore the accidentals for the moment. For example, the Ab flat scale:

Ab C Eb G Bb Db F

Ignore the accidentals and there it is again ACE GBDF

Pick another scale—say F:

F A C E G Bb D F'

And there it is again: FACE EGBDF

It will always work out that way. It will always show the layout on staff paper and form a 7-pointed star. Doesn’t matter if you use the equivalent minors and it doesn’t matter if you use enharmonic equivalents. It always produces the same pattern.

[Lord Larehip]

Is the star a coincidence? Let’s see if it works with the Chinese pentatonic scale. This scale is generated by starting with the note called gong which drops a 4[SUP]th[/SUP] to zhi which rises a fifth to shang then drops a fourth to yu then rises a fifth to jue. From lowest to highest, the notes are zhi, yu, gong, shang and jue.

Now, make a circle and space these notes out equally along the perimeter (72 degrees apart) in the order of their generation--gong, zhi, shang, yu and jue--and connect them with lines in order going from lowest note to highest--zhi, yu, gong, shang and jue. What geometrical figure do you get? This one:

Or we can reverse the layout and mark the perimeter of the circle with the notes in order from lowest to highest and connect them with lines in the order of their generation and we will get the identical pentagram again.

[Lord Larehip]

Now, if you go to the Circle of Fifths and connect every 5th or 7th mark with a line you will get the identical geometrical figure:

We could also make a 12-sided polygon if we connect every mark in sequence or by counting by 11s. If we count by any other intervals, however, we will not hit all 12 marks on the circle. They will form only squares, triangles or hexagons. Why is that? Because a 4[SUP]th[/SUP] is 5 half-steps, a 5[SUP]th[/SUP] is 7 half-steps, a minor second is 1 half-step and a major 7[SUP]th[/SUP] is 11 half-steps, and the octave is 12 half-steps. 1, 5, 7 and 11 are what we say in mathematical jargon co-prime in modulo 12. That is, they are not divisible with any number other than 1 and themselves and also not factors of 12. All the other intervals are not prime or are factors of 12 or both and so end up creating circles that are factors of 12 but never all 12. So here are your 12 chromatic scales united with their related diatonic scales.