# Drawing a circle of 5ths without memorizing beforehand



## Guest

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. 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. Why don't we use C-sharp instead of D-flat? We can but we want to build a flatted side of the circle because it's customary. 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.


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## Guest

Each of the letters we added to the circle in my previous post represents a major scale. Each major scale has 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. Remember: you're always moving "15 minutes" AHEAD of your current position and never behind. Another way to look at it is to count forward three positions on the circle. So what is the relative minor of D major?










Key signatures determine the scale (and vice-versa). 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:


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## Guest

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 flats 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 whose 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 A. So A 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.


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## Guest

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.

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










Using this chart, suppose you run across a piece that's in a key you can't play or can't sing in. How can you transpose it? Well, if you know what key you want to transpose it to, you would imagine the pointer on this chart highlighting that scale. Say it was in the key of G major. Then imagine sliding the point on C over one notch to G. G becomes I, A becomes ii, B becomes iii, C becomes IV, D becomes V and so on. Transposing a standard I-IV-V blues would be easy because they are lined up in a row on the circle. If we want to transpose a blues to the key of D-flat (although god only knows why we would), then D-flat is I, G-flat is IV and A-flat is V(7). Once you have the circle in your head, you can do it on the fly.


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## Guest

I love to study the circle of fifths because you never what it's going to tell you next. I was thinking about hexachords used in solmization and I realized you can use the circle to plot out hexachords. We know there are three types of hexachords--soft, hard and natural (or hexachordum molle, durum and naturale).









Looking at the chart above, one can see that the hexachord always has a half-step in the middle and this half-step is always designated mi-fa. All the other adjacent notes are whole-steps apart. The natural hexachord starts at C then D E F g a. Notice the mi-fa half-step between E and F.

The soft hexachord goes F g a b-flat c d. Notice the mi-fa now falls on a and b-flat which are a half-step apart. The Bb was seen as a soft or rounded B (B rotunda) and hence its designation at the soft hexachord or hexachordum molle.

The hard hexachord goes G A B C D E and mi-fa falls on B and C which are a half-step apart. Its B is natural and symbolized by a squared off "h" (used for "B" is Germanic languages and so is called B quadrum) and so the hexachord was called the hard hexachord or hexachordum durum.

Now how do we find the hexachords on the circle of fifths? If mi-fa are a and b-flat of the soft hexachord, go to the circle and find a and b-flat. Start at mi and go clockwise to fa and you get the notes b-flat F C G D A. Now, start on the nite following fa on the circle which is F and then skip every other note: F G A now to the beginning so B-flat C D. There you are--the hexachordum molle.

Let's find the hexachordum durum: mi and fa are B and C respectively so we locate B and C on the circle. Start at mi or B and go to fa or C, this time counterclockwise (you always want the section of the circle that encloses six notes) and you get B E A D G C. But to assemble the notes in proper order, you go backwards and start at the second note from the right or G and skip a letter so G A B C D E and there you go.

The natural hexachord has mi and fa at E and F. Again, we go counterclockwise and get E A D G C F. Start and C and go to D then E then F then G then A and there is your hexachordum naturale.

As time went on, more hexachords had to be added to account for chromatic melodies. With the circle of fifths, you can make any hexachord you want. For example, let's designate mi and fa as C and D-flat. Go to the circle and find both C and D-flat and go counterclockwise to obtain C F B-flat, E-flat, A-flat and D-flat. Now assemble the hexachord to get A-flat, B-flat, C, D-flat, E-flat, F. It always works.

Strangely, while the half-step in the hexachord is always in the middle, the half-step defines the ends of the run of notes on the circle of fifths. I don't know what use this is but it's just cool that you can find them on the circle if you know how to use it.


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## Majed Al Shamsi

Thank you for sharing this, Victor. Quite informative.
I've noticed that in the picture showing the key signatures, B-flat major has 3 flats instead of 2. Probably a typo?


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## Guest

Yes. I was wondering when someone would notice that. Good job!


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