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I'm attempting to fully understand the circle of fifths, and I noticed that in the clockwise direction, the sharps are added in their own circle of fifths starting at F. I'm wondering why this pattern appears; I've tried looking at a piano and seeing if there is any special change in the pattern of black keys that happens when you go up a fifth, but I haven't found anything. To me it just seems that each key just happens to have those sharps due to the pattern of the major scale, but I'm wondering what the underlying reason is that the sharps are added around the circle of fifths in their own little circle of fifths.

As the first answer indicates there is a specific pattern to the major scale. It is built from whole and half step intervals, specifically

(W - W - H) - W - (W - W - H)

The parentheses are placed around a smaller grouping of intervals called a tetrachord (a 4 note sequence). Notice that the same exact tetra chord appears twice in the construction of the major scale. This means that scales that differ by a 5th have a lot of common tones and are sometimes referred to in music theory as being compatible keys.

I disagree with some of the statements made in the previous answer. While it is somewhat true that the structure of the major scale is a coincidence the relation between different keys is not. You can't just stack them in any order are see a simple pattern. Only the stacking in 5ths (or 4ths) will reveal this pattern and it is related to the structure of the scale using tetrachords.

As for changing key, this construction makes it very easy. To move up a 5th from where you are all you have to do is augment the 4th of the key you are in. Example: Modulate from C to G, raise the F. To modulate down a 5th all you need to do is drop the 7th of the key you are in. Example: from C to F, drop the B to Bb. On an instrument like guitar this makes reading key changes very easy without moving position by several frets. Now you may wonder how this helps if a key modulates to the relative minor third above the current key. Perhaps it doesn't, but a lot classical music (and this is classical music theory) makes use of compatible keys in modulation (even a lot of modern music does too). So this is a common type of modulation and it helps to memorize it. In Jazz there are other types of modulations that make use of modern harmony ideas. A few that come to mind are the Coltrane and Parker cycles, that modulate by thirds instead of 5ths and 4ths. Another is Pat Martino's way of seeing multiple keys as connected by the diminished form.

If you abandon the major scale as the building block of melody you will lose the circle of 5ths connection, it may be replaced by another circle.

To me it just seems that each key just happens to have those sharps due to the pattern of the major scale,

That is exactly the reason. When the diatonic scale is transposed to start on tonics other than C (or A for relative minor) you must make adjustments with sharps/flats to maintain the diatonic pattern.

...but I'm wondering what the underlying reason is that the sharps are added around the circle of fifths in their own little circle of fifths.

I think you may be misunderstanding the key signature circle of fifths. The tonics are arranged in fifths to show the alteration of sharps/flats one by one. It's a circle of key signatures arranged in fifths.

You may be mistaking the key signature circle of fifths for a chord progression.

There is a circle of fifths chord progression but it moves by diatonic fifths through the seven diatonic chords.

They are similar, but not the same thing.

I'm not sure this is what your asking about but...

The diatonic major scale is two tetrachords (series of four tones), those two tetrachords are major consisting of two whole steps followed by a half step. The starting notes for the two tetrachords are a perfect fifth apart. So, when we start on C...

[C D E F ] ...is a major tetra chord and gives us the first half of the scale, then go up a perfect fifth for the second major tetrachord and the second half of the scale...

[G A B C ] ...the second tetrachord is major, so we do not need to add sharps.

The full scale is [C D E F ][G A B C ].

Now we go up a perfect fifth to start the next scale, and so start on G...

[G A B C ] ...for the first half of the scale, go up a perfect fifth for the second half...

[D E F A ] ...that tetrachord is minor! we need to make it major with F#. Notice that it is the third we changed.

The full scale is [G A B C ][D E F# G ].

Now we go up a perfect fifth to start the next scale - we will will keep the previous sharps added - and start on D...

[D E F# G ] ...is our first half major tetrachord and we go up a perfect fifth for the second half...

[A B C D ] ...this tetrachord is also minor, so we put a sharp on the third and use C#...

The full scale is [D E F# G ][A B C# D ].

If you keep following this pattern of moving up to the next tetrachord by a perfect fifth, and maintain the sharps added in the previous stages, you only need to add one sharp to the next new tetrachord - to alter the third of the tetrachord - to change it from minor to major.

I hope this helps illustrate why moving up by perfect fifths results in sharps added one by one.

It might help to see these as overlapped lines...

C major
[C D E F][G A B C]
--------------------------------------------------------------------------------
 G major
 [G A B C][D E F! G] ...second tetra chord from C major scale is minor
 [G A B C][D E F# G] ...so raise the third to make G major scale
--------------------------------------------------------------------------------
 D major
 [D E F# G][A B C! D] ...second tetra chord from G major scale is minor
 [D E F# G][A B C# D] ...so raise the third to make D major scale
--------------------------------------------------------------------------------
 A major
 [A B C# D][E F# G! A ] ...second tetra chord from D major scale is minor
 [A B C# D][E F# G# A ] ...so raise the third to make A major scale

I put an exclamation point after the third of each minor tetrachord to show that it is the tone that need to be raised.

You can see that every time we move up a fifth diatonically to the second tetra chord it is minor and the third tone needs to be raised.

One small addition. The diatonic scale in relation to the chromatic scale can be a little confusing. Arranging the letters in fifths can help see the connection. Below are the musical letter arranged in perfect fifth, numbered 1-12 to show the connection to a chromatic scale, and with enharmonics in parenthesis. In this arrangement each unique pitch class is assigned a unique number. Notice how a letter's pitch identity changes when sharps or flats are added, but enharmonic re-spellings involving same letters do not change number: F=1 but F#=8, while F=1 and E#(F)=1. The oddness of the system is because it is based on a series of both 7 and 12.

...Bbb|Fb Cb Gb Db Ab Eb Bb|F C G D A E B|F# C# G# D# A# E# B#|Fx...
 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3
 (A) (E)(B) (F)(C)(G)

In practical terms the major key signatures are Gb (six flats) to F# (six sharps.)

I started my numbering series with F so the natural letters would begin the series.

To me it just seems that each key just happens to have those sharps due to the pattern of the major scale.

This is exactly correct, and there's not much more to it. The major scale requires the seventh scale degree to be a half-step below the tonic, as with C major. When you build a major scale on G, you have to raise the seventh scale degree, which is F#. When you go up a fifth to build a major scale on D, you keep that F# as the third, and you have to raise the seventh scale degree from C to C#. When you go up a fifth to A major, you again have to raise the seventh scale degree from G to G#.

but I'm wondering what the underlying reason is that the sharps are added around the circle of fifths in their own little circle of fifths.

It's because it's always the seventh scale degree of the corresponding major scale. Just like the tonic of each scale, the seventh degree is a fifth higher than the seventh degree of the preceding scale in the circle.