Lesson 3

The circle, derived

This is usually the first diagram anyone is shown and the last one they understand, because it is presented as a thing to memorise. It is not. It is what happens when you keep doing the thing lesson 2 finished on, and its shape is decided by a greatest common divisor.

Keep stacking

Lesson 2 ended with the major scale being a chain of seven fifths. The obvious question is what happens if you do not stop at seven.

Start on C and keep adding fifths: C, G, D, A, E, B, F♯, C♯, G♯, D♯, A♯, F, and then back to C. Twelve steps, twelve notes, every one of the chromatic scale visited exactly once, and then it closes.

That is the circle of fifths, and the only remarkable thing about it is that it works. Most intervals cannot do it.

Walking by one interval

Start on C, keep adding the same step, and see where it stops.

Step: 7 (fifth)
gcd with 12: 1
Notes reached: 12 of 12
Closes after: 12 steps

gcd(7, 12) = 1, so stepping by 7 visits every note before returning to C. Only four intervals do this - 1, 5, 7 and 11 - and the fifth is the only one of them that is also consonant, which is the entire reason music is organised around it rather than around semitones.

Which intervals can walk the whole circle

Stepping repeatedly by g around twelve positions visits 12 / gcd(g, 12) of them. That is the entire rule, and everything else in this lesson is a consequence of it.

So an interval reaches all twelve notes exactly when it shares no factor with twelve. There are four such intervals: 1, 5, 7 and 11 semitones. Everything else closes early, and closes into something you have already met.

Stepping by 2 gives six notes: the whole-tone scale.
Stepping by 3 gives four: a diminished seventh chord.
Stepping by 4 gives three: the augmented triad from lesson 1.
Stepping by 6 gives two: a tritone, and then it is back where it started.

Those are exactly the sets lesson 1 and lesson 2 flagged as having no root, and the figure shows why in one picture. A generator that closes early produces a shape too symmetric to distinguish any of its own members, so nothing in it can be a tonic.

Of the four intervals that do reach everything, one is a semitone and one is its inversion, which walk the chromatic scale and organise nothing. That leaves the fourth and the fifth, which are the same walk in opposite directions. So there is essentially one interval that both reaches every note and is consonant enough to build music on, and that is why the entire subject is organised around it rather than around anything else.

A key is a window on the chain

Now put the two halves together. A major scale is seven consecutive fifths. The chain has twelve positions. So a key is a window, seven wide, on a circle of twelve, and there are twelve places to put it.

That single sentence explains the things people normally learn as separate facts. Slide the window one place and it drops a note off one end and picks one up at the other, so adjacent keys differ by exactly one note. That is the "one accidental" rule, and it is lesson 2's finding about the two movable notes arriving from the other direction.

One chain, twelve windows

Slide the window. Every position is a key, and each step changes one note.

The chain of fifths. The highlighted seven are the current key; the gold one is its tonic.

WindowC major

Key: C major
Accidentals: none
Notes: C D E F G A B
New since one flat-ward: B

Every key on that slider is the same seven-fifth window in a different place, so moving one step swaps exactly one note. Walk upward from C and the notes you gain are F♯, C♯, G♯, D♯, A♯, in that order, which is the order of sharps every musician memorises. Nobody sat down and ordered them. It is the chain of fifths, read off the end of the window as it slides.

The order of sharps is not a mnemonic

Walk the window upward from C and watch which note appears each time: F♯, then C♯, then G♯, then D♯, then A♯. That is the order of sharps, taught everywhere as a phrase to memorise.

Nobody had to invent that order. It is the chain of fifths, read off the leading edge of the window as it slides. Going the other way you gain B♭, E♭, A♭, D♭, G♭, which is the same chain read off the trailing edge, which is why the order of flats is the order of sharps backwards.

The figure names every note with sharps, and that is a deliberate gap rather than a bug. Slide the window flat-ward and it will report "A♯ major, 2 flats", which is a sentence no musician would write: that key is B♭ major. Real notation picks the spelling that gives each scale degree its own letter, so a key never uses the same letter twice, and B♭ major spelled with sharps would need both A and A♯. Justifying that rule properly needs scale degrees, which arrive in lesson 4, so rather than half-implement it the figure names pitches honestly and inconsistently with convention, and says so here.

The circle only exists because of tempering

Here is the part worth carrying away, and it is a direct callback.

Foundations lesson 4 stacked twelve pure fifths and found that they do not close. Twelve of them overshoot seven octaves by 23.46 cents, the Pythagorean comma, and that lesson drew the result as a spiral precisely because it never comes back to where it started. Powers of three and powers of two do not coincide, so a chain of pure fifths runs forever without repeating.

The circle in this lesson closes after exactly twelve. It closes because equal temperament made it close, by shrinking every fifth by about two cents until twelve of them fitted into seven octaves exactly.

So the circle of fifths records a compromise rather than an acoustic fact, in the same way that the augmented triad in lesson 1 only exists because three tempered major thirds add up to an octave and three pure ones do not. Two of this course's central objects are artefacts of a tuning decision made in the seventeenth century, and neither of them would exist in a world that had kept its intervals pure.