...But what is natural here?
First, do not read too much into the meaning of "natural" in some sense like "not bizzare", or that a pitch that is not natural (using a sharp or flat) is somehow "unnatural!"
Historically this all stems from the gamut of pitch letters
ABCDEFG. A long time ago, the Dark Ages/Medieval period, musicians used only those letters and there was not yet any symbols for sharps or flats. The flat symbol was introduced first, then the sharp and natural. "Natural" is only a reference to that original gamut of letters
To respond you bullet list of questions about the natural sign and
C major the answer is an emphatic "no" to all the questions. Natural signs are not referring to
C major, they refer to the gamut of pitch letters.
...Given the fact that we are in F major, from my point of view the "natural" B here is the B from the F major scale, which is B flat! That sounds natural!
I think the word you are looking for is diatonic. For this discussion diatonic simply means the tones of the key signature.
In the key of
F major the diatonic
B♭. Playing that
B♭ sounds "right", it sounds "natural" in the sense of it is correctly diatonic. But that description is using the general meaning of "natural" rather than the specific music theory meaning of the word.
In my example of the F major staff, why isn't it more logical to write the "natural" (white) B note as a B sharp, to show that it is outside the F major scale? In other words, have a notation that is relative to the current key signature.
This question is confusing. I assume that you're just getting started with learning key signatures, and things can seem pretty arbitrary in the beginning. A common reaction seems to be a desire to reinvent notation. But, suffice to say, after you learn to read key signatures and accidentals, a
B♮ in a key signature of one flat will definitely be understood as outside of the key signature.
Part of the issue is confusing the concrete sharp/flat/natural signs with the relative pitch changes they effect. Your example of
F major in relative terms is a
B raised a half step from the key signature. There is no specific sign for that. You need to read it in relative terms depending on context. If you had a note with a double flat
𝄫, it would be raised a half step with a single flat
♭. A flat
♭ is raised by a natural
♮, a natural is raised by a sharp
♯. I think part of the confusion in your question is thinking that
♮ means "in the key" in all cases. It doesn't work that way.
Notice earlier I said "outside of the key signature" and not "outside of
F major." Stop thinking of these things as relating to specific keys but rather related to the key signature. They are not the same thing. A key signature of one flat could also be used for
C mixolydian. The use of
B♮ in that context would still work exactly the same way: the natural sign just sets the
B to the plain
B of the gamut. It's just telling you: not
B sharp, not
B flat, but
I don't know if this will help you, but one way to think of key signatures and accidentals is a transposition of the diatonic gamut.
So, you know that the gamut arranged as
CDEFGABC is all natural pitches and gives us a major scale. For the moment try to gloss over the fact that the particular scale is
C major. In fact, to make the point more emphatically, realize that gamut of all natural letters could be either
Whichever tonality you work with the key signature can be thought of not as a collection of accidentals, but that the sharps/flats of the key signature indicate a transposition of some tonality using the gamut of plain (natural) letters. For example, a key signature of one flat indicates transpose the gamut down one perfect fifth, and so
C major becomes
F major, or
D dorian becomes
The point here is subtle. The
B♭ in the key signature is not an accidental. It is a diatonic tone in
G dorian, etc. The
B♭ is not "unnatural." What really happened, when viewed as a transposition, is the tonic of
C dropped to
F, the gamut letter which is a fourth above transposed from
B, but in the case that we are transposing a major scale, that fourth above the tonic must be a perfect fourth,
C up to
F is a perfect fourth, but
F up to
B is not, it is an augmented fourth. We need that
B a half step lower at
B♭. We could make similar points about the transposed intervals in
D dorian, etc. but we can skip over those details.
So, the key signature puts the flat sign on the
B, and that provides the specific spelling of
B♭, but it also works like a sign of transposition. There is no accidental involved. The
B♭ is diatonic, it is not, so to speak, "unnatural." There is no specific key/mode indicated. It's only a key signature. It's just the gamut of letters transposed down one perfect fifth.
We can also think of accidentals in terms of transposition. In the major/minor system of keys, for most of the typical harmony found in common practice era music, accidentals (whether sharps, flats, or naturals) tend to happen on specific scale degrees and for specific functions:
- raising the fourth scale degree a half step to transform the subdominant degree to a leading tone to the fifth scale degree, to tonicize the dominant
- lowering the leading tone a half step to transform the leading tone to the subdominant degree of the subdominant key, to tonicize the subdominant
- to modally "color" the major mediant or submediant scale degrees to their lower minor mode form, probably most commonly on the tonic chord and next perhaps the subdominant chord, to use modal mixture
Technically modal mixture is not a transposition, but it seems worth adding to the list. I'm using the terms transpose/tonicize/key change synonymously.
The important point here is to not think of sharps/flats specifically, but instead to understand that the common accidental use is to raise or lower certain scale degrees in conventional ways. Really, it is another way that staff notation signals a transposition. If, for example, you are in
F major, and you see an accidental on
B, on the fourth scale degree, the subdominant, the most likely thing is to raise it. In
F major the accidental to do that job will be a natural, but in another key, like
D major, a sharp will get the job done. Notice that the accidental differs, but the the scale degree and raising are the same. I don't mean to suggest you can totally disregard reading the accidental. But it is a sort of "cheat" for reading accidentals. In solfege terms the three common things are
FA transforms up to
TI transforms down to
FA, and major
MI transforms down to minor
ME (or MA).
I worry this second part of my answer may just add to your confusion. But for me personally, it was a breakthrough moment when I understood these ideas. It helped me understand key signatures better, helped me with correct chord and enharmonic "spelling", and help with reading scores.