Can anyone put a name to this Circle of Fifths observation?

Question

I wasn't able to find anything via search, and don't know enough about the Circle to be able to ask using accepted terminology. As context, I'm a guitar player working out the theory/geometry behind the keys, using the circle of 5ths.

Please see my Circle of 5ths "number line" below. There are three rows. From top to bottom row, each row denotes number of sharps/flats, key name, and note added, respectively. In the middle column, we have 0, key of C, and no added note. To its right, we have 1 sharp, key of G, adding F#. The other way, we have key of F adding Bb, and so on and so forth in both directions, adding one flat or sharp at a time.

I noticed that each flatted note added becomes the name of the next flatted key. I'd known about this for a while, but I was surprised to discover the same pattern continues past C, and into the "sharp keys"! It is clear that 'BEADGCF' is seen in the order of key names, the order of flats added (Bb/Eb/Ab/.../Fb), and the reverse order of the sharps added (F#/C#/G#/.../B#).

My question is, what do repeated interval jumps of perfect fifths have to do with the business of naming keys and the order in which notes are sharped/flatted?

Answer 1

My biggest question is, what do repeated interval jumps of perfect fifths have to do with the business of naming keys and the order in which notes are sharped/flatted?

Recall the pattern of whole steps (W) and half steps (H) that make up a major scale: W-W-H-W-W-W-H.

That creates the following pattern of scale degrees, which of course repeats in each octave:

  W   W  H  W   W   W  H
1   2   3 4   5   6   7 1

Now look what happens when you create a new scale (below) that starts on scale degree 5 of your old scale (above):

  W   W  H  W   W   W  H  W   W  H  W  
1   2   3 4   5   6   7 1   2   3 4   5

                W   W  H  W   W   W  H 
              1   2   3 4   5   6   7 1

Notice how 5 in the old scale becomes 1 in the new scale. Similarly, 6 becomes 2, 7 becomes 3, and so on. However, note that 4 does not quite line up with 7. This means we have to raise scale degree 4 in order to get the new scale's degree 7. By 'raise' I mean that if it is just a natural note, we sharpen it, but if it is already flat, we have to naturalize it (this will be important when I answer 3 in just a moment).

So to summarize, in order to create a new scale on the fifth degree of the current scale (i.e the key to the right of the current key), we have to raise (sharpen or naturalize) the fourth degree of the current scale (i.e. the note to the left of the current key). This explains the pattern on the sharp side. For example, to go from D to A (to the right = up a fifth), you have to raise the fourth of D, which is G (to the left = up a fourth).

Why is it key of F, but not key of F#? Why key of G, not G#? I would expect it to be symmetric to the flats

Why does this pattern span two keys in the sharp side, but only 1 in the flat side?

These questions both deal with the difference between the sharp and the flat side, and I think the answer to them is the same. The reason is that your final row changes definition partway through. On the flat side, you are saying how the key is different from the one to its right. On the sharp side, you are saying how the key is different from the one to its left. At C itself, you aren't comparing to either side -- that's why there's the gap. If you remain consistent with one or the other, the discrepancy goes away, and the pattern become uniform.

For example, to stick with the sharp side definition, we've already seen why G is like C (one note to its left) but with a raised F (two notes to the left). But you could also say that C is like F, but with a raised B♭. In this case, 'raised' means natural. Similarly, you could say that the key of B♭ is like E&Flat; but with a raised (naturalized) A♭.

I'll let you work out going the other direction, but I'll point out that in this case we are lowering notes (either flatting naturals, or naturalizing sharps) instead of raising them. So the key of C could be considered like the key of G, but with a lowered F♯.

What happens to this pattern in enharmonic equivalent keys past 4 #/b? Key of Db/C#, Gb/F#, Cb/B.

You could very well continue this pattern indefinitely, moving to double sharps after you've exhausted all the single sharps, and so on. There is such a concept as the infinite line of fifths. However, at some point, it will probably become convenient to wrap around to the flat side via enharmonic equivalence. When this occurs, each note can be thought of as occupying two spots along the line: one on the right side, and one on the left. They are just two different names for the same relationship. But the same pattern will hold for both (remember, once you rewrite your final line to be consistent as above, it will be the same uniform pattern on both sides).

Answer 2

The major scale of C is built of 2 equal tetrachords (= four step ladders) of 2 whole steps and a half step: W W H => C D EF and G A BC - the half steps are made visible by the closer distance.

notice that between the 2 tetrachords is another whole step (W). Thus the major scale has the pattern: W W H (W) W W H whereby (W) is referring to the whole step between F and G, the whole step between the two tetrachords.

numbering the degrees of this ladder we get the analog scale 1 2 34 and 5 6 78

The relative names of a major scale are Do Re MiFa So La TiDo (MiFa and TiDo are the halftone steps, Ti is called leading tone to Do, Fa leading tone to Mi)

You can now develop all the scales of the sharps by using first the second tetrachord of the C scale:so la tido becomes the do re mifa of a new scale with the root note G. We get G A BC and D EF G.

The first tetrachord G A BC is built like the first tetrachord of the C scale and can be replaced as 1 2 34 ... but the half step of the second tetrachord EF is between the 6th and 7th degree 5 67 8 and we have no leading tone from 7 to 8. By augmenting the 7 degree with a sharp # we construct the leading tone and solve the problem of the half step between 6 and 7. (5 6 78) We have a new upper tetrachord with a halftone between 78 with 1 # (F#). Continuing ths operation the upper tetrachord will become the lower one of the next key in the direction of the circle of 5ths clockwise.

To develop and understand the circle counterclockwise with the sharps we build notate the scales down step:

CB A G ... FE D C (DoTi La So FaMi Re Do)

Now we take the lower tetrachord (FaMi Re Do) of the scale of C major:

FE D C and build a new scale with the root tone F. The upper tetrachord of F major is useful but the lower has a problem: CB A G F is not congruent with So FaMi Re Do. We need to flatten the B to Bb (a half tone lower) and get a whole tone between the upper and the lower tetrachord: FE A G (W) BbA G F and we have also construct the new leading tone Fa to get a half tone to Mi (or Me) BbA. That's the explaining why the scale of F major has 1 b-flat. Following this principal the lower tetrachord of F (BbA G F) will become the upper of the new scale Bb-major with two flats (Bb and Eb) etc.

Now we understand the developing of the scales and sharps and flats:

the new scales in the circle of 5ths have clockwise a (#) more which is the 7th degree building a new leading tone.

the scales in the circle of 5ths counter-clockwise (we also call this the circle of 4ths) have always one flat (b) more which is the new leading tone Fa. As the "old" Fa becomes the Do of the "new" scale you'll understand the name of the scales referring to the root tone is identical with the 2nd last flat:

e.g. Eb was the last flat of Bb and will be the second last flat of Eb (BbEbAb)

It also explains why the row of the keys in the circle of fourths is identical with the row of the developed flats: Bb-Eb-Ab-Db-Gb

If this was too abstract I propose you write the scales in sheet music and make also a pattern of the keyboard and on the guitar frets to fully understand the principals of the scales and circle of fifths.

(we used in the 8th class in secondary school in singing lessons to develope up to over 40 sharps counting in a septimal number system: the key of C = 0 sharp, C# = 7 sharps but continuing this idea might be only confusing you...)

Why does this pattern span two keys in the sharp side, but only 1 in the flat side? For ex: D# is the 4th sharp note added, but key of D has 2 sharps. The difference is 4-2=2 sharps. In contrast, take Ab -- it is the third flat note added, but it is the name of the key of 4 flats. 4-3=1.

The fundamental difference between the right side and the left side of the circle of 5ths is that the additional sharp is the new 7th degree and the additional flat is the new 4th degree (so there is no congruence between the Fa and the Ti, but you will find it if you compare the old Ti with the new Ti (right side comparing the sharps and left side the naturals (♮) ... the difference will be analog in both sides.