Quibbles and Bits

Like a Circle in a Spiral

On a piano keyboard, most of us know that all the white notes are identified by the letters A–G, while the black notes are the sharps and flats … written A♭, B♭, C♯, D♯, etc.  If I’ve lost you already, then fair warning – this column may not be for you … J.

If we list all the notes in ascending (or descending) order, then we find that we go through each of the 12 white and black notes once, after which the sequence keeps on repeating itself ad infinitum.  It’s as though the notes were all arranged in a circle, and we simply kept on going round and round.

Here we can see the second thing that stands out about musical notation.  All the black notes have two names.  We can choose either C♯ or D♭, and likewise G♯ or A♭, and so forth.  This circular arrangement of musical notes is called the Circle of Semitones, because each adjacent note is separated by an interval of one semitone.

I wrote about some of the theory underlying musical notes back in Copper 31, and in particular I wrote about how the different notes are related to each other by the ratios of their frequencies.  In particular, I noted the important interval of one-fifth, and how we could jump from any one note to the note one-fifth above it by multiplying its frequency by 1.5.  So let’s take our Circle of Semitones, and draw lines connecting every pair of notes which are an interval of one-fifth apart.  Here’s what we get:

It is a nice star shape, and if we trace the star path around the notes we see that it stops off once at each of the 12 notes before eventually returning to where it started from.  We can do something interesting with this information.  We can draw a new circle, formed by notes which are one-fifth apart.  Here are the two circles, side-by side:

I have left the same pattern of connecting stars in each circle.  If you look carefully, you will see that the pattern of stars in each circle traces out the order in which the notes appear around the outside of the other circle.  As it happens, mathematically speaking, both diagrams are actually topologically identical (an observation which, while undoubtedly interesting, yields no useful musical insights that I am aware of).  In any case, the circle on the left is known as the Circle of Semitones, while the one on the right is known as the Circle of Fifths.  The Circle of Fifths is by far the more interesting of the two, so we are going to focus on it.  And I am going to start by observing that all of the white notes and all of the black notes are grouped together, as illustrated in the diagram below:

All of the white notes are in the shaded section, and they happen to be the notes which form the scales C-major and A-minor, which, most of you will know, require none of the black keys.  So now I’m going to rotate the shaded section by one position in an anticlockwise direction, as illustrated below:

The shaded section now includes one of the black notes, the one which is shown as being either A♯ or B♭, and at the opposite end has moved one position away from B, which is no longer included.  So if we choose to call the new note B♭, then the seven notes within the shaded region still contain the letters A – G.  If we had chosen the A♯ representation instead, we would have both A and A♯, and no B, which is more confusing, and harder to work with.  With the shaded region rotated this way, the note which now occupies the position previously occupied by C (the major tonic position) is F.  So the shaded region represents the notes comprising the scale of F-major (and, likewise, the scale of D-minor).  And in both of those scales, every time the note B appears, it is always actually B♭.  This is why the key signature of F-major has one flat, which always appears in the position of B:

Now we can simply repeat the procedure, and rotate the shaded section a further position anti-clockwise:

Notice how the note which is shown as being either D♯ or E♭, has now entered the shaded region, and E has dropped off the other end.  As before, we will call the new note E♭ rather than D♯ so that we still have all of the notes A – G included in the shaded region.  And now the note which occupies the major tonic position is B♭, so the shaded region comprises the 7 notes of the scale of B♭ major (and, likewise, G-minor).  Because all the B notes are now B♭, and all the E notes are now E♭, we can see that the key signature for B♭ major (and G-minor) will have two flats, located in the positions B and E:

As we repeat this procedure ad infinitum, it is easy to see that each time we rotate the shaded section anticlockwise we uncover a new black note, and drop off one of the original white notes.  On each occasion, the black note has the choice of being the ♯ (’sharp’) version of one note, or the ♭ (flat) version of the other, and we always end up choosing the ‘flat’ version because that is the note which has dropped off the scale at the other end.  So, for each position the shaded region rotates counterclockwise, we simply add another flat to the collection on the key signature, until we end up with five of them, signifying the keys of D♭-major and B♭-minor:

Things are very similar when rotating our shaded seven-note segment clockwise instead of counter-clockwise, but not exactly the same, and I’ll give you a couple of examples to illustrate the point.

Here I have taken the original C-major alignment and rotated it one position clockwise.  Now the shaded segment includes the note which could be either F♯ or G♭, but has lost its F which has fallen off the other end.  This time, in order to have each letter from A – G appear once in the scale, we need to choose F♯ rather than G♭.  The major tonic position is now occupied by G, and so this must be the G-major (or E-minor) scale.  In both of those scales, every time the note F appears, it is always actually F♯.  This is why the key signature of G-major has one sharp, which appears in the position of F:

This time, as we repeat this procedure ad infinitum, it is easy to see that each time we rotate the shaded section one position clockwise, uncovering a new black note and dropping off one of the original white notes, the choice of black note is now always the ♯ (’sharp’) version of that note, and not the ♭ (flat) one.  So, for each position rotated, we simply add another sharp to the collection on the key signature, the first being F♯, then, in sequence, C♯, G♯, D♯, and finally A♯, until we end up with five of them, signifying the keys of B-major and G♯-minor:

At this point, some of the more observant among you will be wondering why the keys of D♭-major (with five flats) and B-major (with five sharps) are not the same thing since both of them use all five black keys, and that would indeed be a very good question.  So here are the shaded rotations for both D♭-major (on the left) and B-major (on the right):

We can clearly see the difference between the two, which is that D♭-major (on the left) includes all the black notes plus F and C, whereas B-major (on the right) includes all the black notes plus B and E.

You might also like to observe that as we go round the circle counter-clockwise, the black notes (which we designate with the ‘flat’ notations) follow the same sequence as the white notes.  In other words, if we start with B and go round the circle counter-clockwise, we get B, E, A, D, G, C, F, B♭, E♭, A♭, D♭, and G♭.  The same thing happens if we go round the circle clockwise, except that we designate the black notes with their ‘sharp’ notations) … F, C, G, D, A, E, B, F♯, C♯, G♯, D♯, A♯.  So you could argue that the Circle of Fifths is actually a Spiral of Fifths!

Already this column is way too long, so I don’t have time to go much further into this, but you might ask yourself what would happen if we kept rotating our shaded section beyond the points where we reached five flats and five sharps.  And indeed, there is nothing to stop us from doing that.  It’s just that the results have increasingly limited practicality.  Which isn’t to say that they don’t have any practicality … in fact we can’t realistically do without them.  But if you’ve got this far, and are still following me, I reckon you have a good chance of being able to figure it all out for yourselves.  There are just two additional concepts that you’ll need to arm yourselves with.

First, there is the notion that any given note can have multiple designations, which we have already encountered, since all the black notes have a choice of either a ‘flat’ or a ‘sharp’ designation.  Well, we need to extend that to the white notes as well.  For example, there is not a black note between B and C.  So it should not surprise you to learn that {B and C♭} are different designations of the exact same note.  As are {B♯ and C}.  Likewise, there is not a black note between E and F, so {E and F♭} are the same note, as are {E♯ and F}.

The second concept is that there are things called ‘double flats’ and ‘double sharps’.  Whereas a ‘flat’ lowers a note by a semitone, a ‘double flat’ lowers a note by a whole tone.  Double flat uses the symbol , so that B is the same note as A.  Likewise, double sharp uses the symbol , so that C is the same note as D.  There are even ‘triple flats’ and ‘triple sharps’, but those tend to go beyond the simply esoteric and into regions verging on the pointless.

Thus, in closing, if you took the original C-major ‘shaded section’ of my Circle of Fifths, and rotated it eight positions counter-clockwise, you would reach the key of F♭-major (and D♭-minor) whose key signatures would comprise six flats (including both C♭ and F♭) and a double-flat B♭♭: