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Starting at C and going up we get G, D, A, E, B, F#, etc. Why does it turn out that those keys have 1 sharp, 2 sharps, 3 sharps, etc. respectively? And same thing for flats moving downward from C.

Because the added accidentals are necessary to maintain the relative Ionian mode.

I am not sure that i fully understand what is the exact meaning of the question. Going up the circle of fifth is transposing the tonic by a fifth, ie taking G as a start note. Every note of the c scale is distant by an exact fifth of another note of the scale, except b which is at a distance of a diminished fifth. Going up you always sharpen the 4th note of the scale in the mode of C.

This by the way works if you take any mode. For example if you are in the mode of D and you want to transpose by a fifth to A, you would still need to sharpen F for the exact same reason. This is a characteristic of the diatonic scale, which contains only one triton. Going up you sharpen the first note of the triton. Going down you flat the second note of the triton.

Another way to look at it is the exact opposite, ie assuming you want to transpose from C by a triton thus to F sharp. Then you would need to sharpen 6 notes out of seven because one and only one of them is already at the distance of a triton, in that case B. All other notes would be sharpened thus giving a 6 sharps signature. Again this would be true whatever is the mode you start with.

It's an interesting question. At an 'inner' level this has always seemed 'obvious' to me, by which I guess I mean I never thought about it and I have no idea ....but that from some harmony or music theory consideration it seems to make instant sense.
But I wouldn't have any idea how to explain it without thinking about it, and maybe not even then.

Sidokar's posts look like as good an explanation as I could come up with, but I think everything in those answers still leaves the question of WHY?


Any answer I could come up with probably would essentially come down to "Because."

I think it remains an elusive question.
I'm sure there's some fundamental answer, in terms of the nature of our traditional scales and what happens when you go to the dominant key, but I can't grab hold of anything that would really do it.

OK -- I think I have it.
Or at least that I'm getting close.

When we go from a given key to its dominant -- which is exactly just what we're doing in the circle of 5th's -- there's one and only one thing that we need to do to the basic scale:
We need to make the note just below the new key into a leading tone.
And that note that we need to make into a leading tone is the 4th step of the old key.

And, the way we do that is by sharpening it.

Hey, I think that's really it....

This was pretty interesting!!

Yes! I love this question and totally get the reason for it. It’s because the leading tone in each key is raised from the key signature in the previous key.

In C, no sharps, leading tone is B. In G, add a sharp to raise the leading tone, it’s F#, and so on.

Flats going down can can just be the incidental opposite of going up, or think of it as lowering the new 4th each time. Not as clear as raising the leading tones, though.

Going one way, the 4 becomes a 7.
The other way, the 7 becomes a 4.

You need to sharpen the 4 to make it a 7. Then it follows that
you need to flatten the 7 to make it a 4.

Many have already given the answer, so i will say the same, from my own approach.

In a major scale there's a whole tone beetwen 4 and 5 grade. And there's a semi tone beetwen 7 and 8.

Now, if you go up the circle of 5s, you just pick the 5 of the previous scale as the 1 of the new one. 4 of the old one becames 7 in the new one. So in order to keep a semi tone beetwen 7 and 8, you put a sharp on the new 7 (old 4).

This is the only change you need to do. You don't need any extra flat or sharp because the structure of the major scale is identical from 1 to 4 as from 5 to 8, regarding distance between each grade. So 5 to 8 works perfectly as new 1 to 4. No need to change anything there. Also 2 to 3 works as new 5 to 6. The only one that needs a change is old 4 in order to become new 7.

That's why one, and only one, new sharp each time you go up a 5.

When you go up a 4 is similar. Old 7 becomes new 4. So you need a flat on it in order to have a semitone beetwen 3 and 4. The rest of the scale fits perfectly so you don't need any other change.

Essentially the reason is unrelated to the role of the dominant or other functional reasons. It is purely mathematical due to the structure of the diatonic scale. It can be explained by translation formulas. If you take the chromatic scale based on C and translate by a fifth, you dont need to sharpen anything. The chromatic scale is invariant when you apply any multiple of half steps translation.

Life is full of mystery.

Aren't all the answers after this one just saying the same thing, but more verbosely?

No.

This is all because our scale originated from the pythagorean scale which is fundamentally built by stacking up perfect fifths. https://en.wikipedia.org/wiki/Pythagorean_tuning

Take the C scale. It contains the notes C,D,E,F,G,A,B

Transpose them all up a perfect fifth one by one. You get the notes G,A,B,C,D,E,F#

The only question remaining I think is why they call the perfect fifth above B F#. That's a much harder question.

I guess the explanation for that question is as follows. suppose you start at D. Then it sounds really good together with a A (perfect fifth), but is getting pretty sharp with a E (major second). I think they decided that more than 3 perfect 5ths out it became too dissonant. So they used only 7 (3 perfect fifths up, 3 down from the D) put these on the white keys, and then put the remainder on the black keys.

Yes, actually.

No, not just because the other thing doesn't flesh it out at all but because one can't tell for sure that what he had in mind was those particular details.

Maybe you can.
I couldn't. I wasn't sure at all what exactly he was thinking.

I don't understand your post.

No, these explanations are different.
Sidokar sees the b as a special note

Mark sees the leading tone as something special.

Ubu sees adding a sharp as a re-arrangement 'trick' to get to a different scale

I say that the b is not different from the other notes, that we're just transposing ALL the notes in the scale a 5th up. f# is just a naming convention.

These are the same opinion.


It's not a trick. It's what's actually happening to the relationships. The point being that Ionian is foundational to how we approach western musical theory, and any chromatic alteration in the circle is only done so to maintain Ionian in any given location. When Mark says that we use a sharp in G because F must become a leading tone, he's saying the same thing I'm saying in a different way. G Ionian requires a new pitch previously nonexistent in C Ionian, which is a raised F.