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It has nothing to do with b specifically as a note, but with the sequence of steps and half steps in the diatonic scale.

The question has no meaningful answer. It's just a mathematical coincidence, coming from arbitrary (frequencies) distances between notes CDEF... and arbitrary distances between major scale degrees.

(Did you read the other replies?)

You're convinced there is a mystery here. Could you clarify what it is you're seeing about this that you find elusive? I personally don't find OP's question that challenging. I think the more interesting question is why we base the whole system off of the Ionian mode in the first place. The harmonic series seems to have an obvious significance to me. Maybe a real expert can elaborate on this.

It is not a coincidence, but it is the result of mathematics.

Maybe I misunderstood the question. Is the question: Why are accidentals introduced in the names of F# and the subsequent sharp keys (likewise with flats)? If that is the question, I still don't see the mystery. F# is the first key (6 movements of a 5th up from C) whose 1 exists as a chromatic alteration of C Ionian.

Yes. IMO it has nothing to do with the leading tone or other functions, it's just a coincidence.

Well, you're wrong.

(That's exactly what it is.)

In a natural diatonic scale (ie without any sharps nor flats) which is simply a sequence of steps and half steps there is only one triton ie an augmented fourth which is worth exactly 3 steps. Because of the distribution of the 2 half step which always fall in a tetracord and the other in the pentacord of the scale, between any given note and it subsequent fifth there is always exactly 3.5 steps except between B anf F which has 3. Thus when transposing (which is replicating the sequence of steps and half steps at a distance of 3.5 steps), you need to sharpen the one note that is 3 steps away to make it to 3.5.

If you take the natural A scale and transpose it by a fifth, you still need to sharpen the F and F is not the leading tone.

If you build a scale that is not a diatonic one like C D E F G Asharp B, this scale has 2 augmented fourth (which transposed by a fifth will give diminished fifth) E to A sharp and F to B. Transposing by a fifth, you will need to sharpen 2 notes: E and F.

This is purely mathematical due to the distribution of steps and half steps between notes.

There is a much simpler answer: The major scale is mirror symmetric (in terms of semitone jumps 2,2,1,2,2,2,1) and therefore its lateral reflection must also be a major scale with the same number of black notes. C and F# reflect into themselves.

Not to offend anyone (including myself, although this probably will) ....but I think all of these various things that we have offered that we think are simple aren't necessarily simple to most people. Like, I don't think any of the things that have been said are simple, except mine.
And I don't expect particularly that many people besides me think mine is that simple.

What would be interesting at this point is to hear back from Pianoloverus I dont know what his motivation was to ask the question in the first place, but since he has now a ton of answers, he can tell us if he got what he was looking for.

Hi. Building on the previous posts the simplest answer to my understanding is that all major scales in the Ionian mode ( the diatonic scale) are built on the pattern of whole tones and semitones ( often referred to as half tones or half steps ) with the pattern being . WWH WWWH and by moving up the keyboard in 5ths or down the keyboard in 4ths keeps this pattern

So for example C to D is a whole tone ( whole step) D to E is a whole tone E- F is a half tone( a semitone or half step ) F to G is a whole tone G-A is a whole tone A-B a whole tone and B to C a half tone. This pattern shows the semitones fall on notes 3 going to 4 and on notes 7 going to 8 or 1 depending on how you count the scale

To keep the pattern going up the scale you start the next scale a fifth above the C which is G . So G is the starting note ( the tonic note) of the next scale the 3-4 note on this scale is the B-C. But to keep the pattern of whole tone and semitones the 6 note E does not go to the F but to the F# as the 6-7 are a whole tone 7-8 is then a semitone bringing the F# down to the G

The next scale is D ( 5 notes up from G) D-E is a whole tone, E-F# is a whole tone abs F# to G is a semitone as is the 3-4 note) G-A a whole tone A-B a whole tone B - C# is a whole tone and C# - D is a semi tone

As you see with the D major scale the pattern means you keep the previous sharp ( the F# in this case) and add another the C# in this case So each scale around the circle going up in sharps adds another sharp each time. Coming down the keyboard in flats you come down in 4ths so from C four notes down is F so this is your starting note for the next scale and following the same pattern of WWH WWWH the 3-4 semitone notes are A-Bb. the 7-8 semitone is E-F

It sounds complicated but it’s not one you understand the patterns.

The key to understanding the circle of fifths is remembering that it goes up in 5ths to keep the WWH WWWH pattern of whole tones and semitones and down in 4ths for the flats.

Yes there is probably some mathematical formula to this but I’m not concerned about this. All I’m concerned about is understanding how the pattern works and understand that going up keyboard in 5ths I’m adding sharps and coming down in 4ths I’m adding flats

Hope this helps explain why.

I get the why after beating my head against the wall and hours of YouTube tutorials and a music theory course at ThinkSpace. I don't really know how important the why is as much as the fact that it is the way it is and the number of sharps or flats are your key signature. Maybe this will help.
Piano Yoga
She runs through the scales in the circle of fifths.

It has nothing to do with the ionian mode in particular.

The transformation from C ionian to G ionian requires the same change in key signature as that from D dorian to A dorian, for example.

I have always found this wildly confusing, this circle of fifths thig does not explain anything.

Here's the deal. If you want to have a proper Ionian scale, classical major mode, you need to keep certain relationship between the notes of your 7 tone scale. If you want the flexibility of starting that scale / mode from any note, you have to add sharps or molls to keep the relationship of whatever interval constant. End of the story.

Now you may ask yourself, in what relationship are the tones once you add one sharp at a time ? as it happens the relationship is a 5th. This is a CONSEQUENCE IMO and not the starting point.

My motivation, like for a lot of threads I start, was just that the question popped into my head. Maybe some thread I was reading was discussing the circle of fifths. Although many of the responses are revealing and maybe close to a clear explanation, I haven't found any of the ones(that I understand!) totally convincing.

To me, this explains what happens but doesn't answer the question of why it happens. For example, when going from C major to G major, why does one only have to change one note(F to F#) to keep the WWHWWWH step relationship? And then going on to D major, why can one keep the notes from G major but change only one?

Again, because you have to raise what is the new leading tone from the previous key to the current one when moving up a 5th to keep it a leading tone. The previous leading tone, which is now the 3rd, was already raised. That is all.

Look at it another way. Think of the tetrachords [first note, tone, tone, semitone] followed by [tone, tone, tone, semitone]

C D E F

G A B C.

If you keep stacking them up [tone, tone, tone, semitone] you get

1/ C D E F

2/ G A B C

3/ D E F#G

4/ A B C# D

5/ E F# G# A

you can go on and on.

For a major scale (for the sake of argument) you take two adjacent tetrachords. The first note of the first one is the same as the last note of the second one.

C D E F

G A B C

You overlap the pairs so the sequence of seven notes repeats indefinitely.

If you take any adjacent pair of tetrachords you get a major scale.

The circle of fifths is just choosing the next pair in a sequence of tetrachords. It works in the other direction as well.

1-2 is C
2-3 is G etc.

The numbers are arbitrary, as are the starting places.