We are almost there.
We have just one more step to take before the Circle of Fifths is revealed.
Let us return to the four-note clusters with overlaps shown in full, not morphed together. Let us analyse their overlaps and where that happens. We will do this with three examples. For each, we will be reading notes and clusters from left to right. This is equivalent to reading clockwise around the circle of clustered notes above.
F major, C major and G major scales
F major to C major
Starting at the F major scale, we can see that the C major scale begins at the first note of the second cluster. Another way to describe this is to state that C major starts at the fifth note of F major.
C major to G major
Starting at the C major scale, we can see that the G major scale begins at the first note of the second cluster. Another way to describe this is to state that G major starts at the fifth note of C major.
E major, B major and F# major scales
E major to B major.
Starting at the E major scale, we can see that the B major scale begins at the first note of the second cluster. Another way to describe this is to state that B major starts at the fifth note of E major.
B major to F# major
Starting at the B major scale, we can see that the F# major scale begins at the first note of the second cluster. Another way to describe this is to state that F# major starts at the fifth note of B major.
Ab major, Eb major and Bb major scales
Ab major to Eb major.
Starting at the Ab major scale, we can see that the Eb major scale begins at the first note of the second cluster. Another way to describe this is to state that Eb major starts at the fifth note of Ab major.
Eb major to Bb major
Starting at the Eb major scale, we can see that the Bb major scale begins at the first note of the second cluster. Another way to describe this is to state that Bb major starts at the fifth note of Eb major.
Notice that fifth has been written in italics for all examples!
What we have found in these examples holds true for all adjacent major scales when grouped in continuous manner around a circle as we have done. Reading clockwise, every next four-note cluster brings the first note of a new major scale.
Every new major scale begins on the fifth note of the previous major scale.
We have a system of arranging all twelve major scales around a circle. All twelve scales connect together in a single, unbroken chain. The chain is made of twelve four-note clusters. Each cluster contains half an octave span of a major scale. The first note of each cluster can be taken as the first note (root) of a major scale. Additionally, the first note of each cluster can be viewed as the fifth note of the major scale that precedes it when reading clockwise around the circle.
Put the other way round, the fifth note of any chosen major scale is simultaneously the first note (root) of the subsequent major scale when reading clockwise. Roots connected to fifths which become roots which then connect to fifths etc etc.
We have reached the moment.
We no longer need twelve sets of four-note clusters. We no longer need to try to navigate and read a circle with forty-eight notes around its circumference. We no longer need the clutter of squeezing so much information in to a compact space.
We can pare it down to the bare essentials. All we need now are the first notes of each of the clusters. Everything else is surplus to requirements.
The first notes of all twelve clusters are simultaneously root notes and fifth notes of major scales.
Roots and fifths of major scales in a clockwise circuit that has no set entry or exit point, that carousels around and around no matter where we choose to hop aboard.
Circle of Clusters with each first note shown larger
The next and final step, an obvious one hopefully, is simply to erase all superfluous notes leaving just the prominent first / fifth notes to stand alone. Giving us …
The Circle of Fifths
The result of all this arranging, matching, pattern spotting, morphing and circling is that we have - quite literally - come full circle. We have reached the point where we have a circle of fifths in front of us. And, crucially, we know its origins, its formation, its essence. We know where it came from.
As a small reward for our efforts so far, here is the coloured circle of fifths again.
In hopes of clarity and good understanding, in hopes of removing any remaining confusion, what follows are further depictions of adjacent major scales with their 1st (root) notes and their 5th notes marked above them.
Compare each of these with the full circle of clusters and the twelve notes of the Circle of Fifths shown alongside.
C major → G major
These two major scales overlap and give us the note G on the Circle of Fifths. G is the fifth note in C major (and the root note of G major, our next scale moving clockwiswe).
G major → D major
These two major scales overlap and give us the note D on the Circle of Fifths. D is the fifth note in G major (and the root note of D major, our next scale moving clockwise).
-D major → A major
These two major scales overlap and give us the note A on the Circle of Fifths. A is the fifth note in D major (and the root of A major, our next scale moving clockwise).
-A major → E major
These two major scales overlap and give us the note E on the Circle of Fifths. E is the fifth note in A major (and the root of E major, our next scale moving clockwise).
-E major → B major
These two major scales overlap and give us the note B on the Circle of Fifths. B is the fifth note in E major (and the root of B major, our next scale moving clockwise).
-B major → F# major
These two major scales overlap and give us the note F# on the Circle of Fifths. F# is the fifth note in B major (and the root of F# major, our next scale moving clockwise).
-Note: F# is the enharmonic equivalent of Gb. For the next comparison we convert and use the Gb major scale. This is because we need to recognise that we are moving to the section of the circle containing flat, not sharp, notes.
Gb major → Db major
These two major scales overlap and give us the note Db on the Circle of Fifths. Db is the fifth note in Gb major (and the root of Db major, our next scale moving clockwise).
-Db major → Ab major
These two major scales overlap and give us the note Ab on the Circle of Fifths. Ab is the fifth note in Db major (and the root of Ab major, our next scale moving clockwise).
-Ab major → Eb major
These two major scales overlap and give us the note Eb on the Circle of Fifths. Eb is the fifth note in Ab major (and the root of Eb major, our next scale ).
-Eb major —> Bb major
These two major scales overlap and give us the note Bb on the Circle of Fifths. Bb is the fifth note in Eb major (and the root of Bb major, our next scale moving clockwise).
-Bb major → F major
These two major scales overlap and give us the note F on the Circle of Fifths. F is the fifth note in Bb major (and the root of F major, our next scale moving clockwise).
-F major → C major
These two major scales overlap and give us the note C on the Circle of Fifths. C is the fifth note in F major (and the root of C major, our next scale moving clockwise … and also the scale at which we started meaning we have gone full circle).
-I may be missing the point, but I think the labels “upper group” and “lower group” should be swapped in the diagram.
Excellent work! Must have taken hours to produce. Thanks.
Thanks for reading and commenting József
Do you mean you think upper and lower need to be swapped within the structure of the quiz answer sheet?
Here is question 11, which is completed on both sides:
The major scale in this question is G major.
The lower group of G major (to the left of the grey cell) is an exact match with the upper group of C major (they make a pair).
The upper group of G major (to the right) is a match with the lower group of D major (another pair).
I hope that answers your query and clarifies the presentation.
Cheers, Richard. 
@jacksprat Thanks Chris, I appreciate you reading and commenting, and yes, it took hours and hours (and there is much, much more to come, yet more hours). It’s all a labour of love.
Ah OK, I thought about it the other way around, i.e. the upper part of C major (“upper group”) and the lower part of D major (“lower group”) would yield the G major scale.
Anyway, I’ll spend more time on this as it’s a cool mental exercise.
Will you write about some practical (“real life”) musical applications as well? I know the Co5 comes handy when determining the key signature and it’s common to play a certain riff a perfect fourth higher as a variation, but I can’t recall the same done with a perfect fifth.
And thanks a lot for your efforts! 
Hi Richard,
I just want to say what a fantastic piece/reference book you have put up here 
… you should make your work of it 
Thanks and greetings…
Thanks Rogier, I appreciate it. 
Hang on a minute, there could be a pattern emerging here
All good. It did take me a minute to double check if the question was meaning the Lower / Upper group of the “given " scale or the” solution" scale
