Confused about shared notes between keys

Question

It is said that keys that are opposite on the Circle of Fifths only share one note. However, opposite keys on the Circle also share the same tritones, which would imply that they share two notes. For example,
C Major and F Sharp Major are opposite. Their circle positions seem to imply they only share one note. But, they share a tritone. In C Major, the tritone between the fourth and seventh is F and B. In F Sharp Major, the tritone between the fourth and seventh is B and E Sharp, which is the same as F. Therefore, these two Circle opposite keys actually share two notes. So, I am confused about how position on the circle indicates the number of shared notes. Each key contains seven notes. If the six O'clock position on the Circle contains five differing notes from the twelve O'clock position (two being the same), and if there is one less differing note counting back each hour , then that would imply that there are zero differing notes between G Major and C Major. But that is clearly wrong. So I must be missing something in my understanding of the Circle as it relates to shared notes between Keys.

Answer 1

They share a note by name and another note by enharmonic equivalent.

In your example, both C and F# have the note B. C has the note F, and F# has the note E#, which enharmonically equivalent to F. Of course B to F AKA E# is a tritone.

The assertion that they only share one note discounts enharmonic equivalents. That’s all there is to that.

Answer 2

In addition to the musical answers already given, there's a simple arithmetical answer: given 12 different tones (not counting enharmonic equivalents), since any diatonic scale uses 7 of them, any other scale must share at least two tones in common.

Answer 3

They might share only one note, but they share two pitches - hence the tritone shared.

And actually, those shared pitches are only true in 12tet. Played in j.i., for example, I guess even those two pitches aren't exactly the same as each other.

So, it's really, as other answers tell, down to note naming. In both keys (C and F♯), there's B, in key C there's note F, which (sort of) corresponds to key F♯'s E♯, at least enharmonically, if not exactly pitchwise.

All somewhat pedantic - but that's the way some things are...

Answer 4

@Todd_Wilcox is on the money with his explanation (+1). Here is a little theoretical expansion of that idea. Please excuse my lame sloppy drawing of the cycle of 5ths:

The way to visualize and understand this is to think of the cycle all in sharps (although you can also do it in all flats too).

In relation to the key of C:

At B, 5#’s you have 2 common tones, B, E

At F#, 6#’s you have 1 common tone and 1 enharmonic equivalent, B is common, F-E# is enharmonic equivalent

At C#, 7#’s you have 0 common tones and 2 enharmonic equivalents, F-E#, C-B# (which equals Db, 2 common tones)

At G# you still have 0 common tones and add Fx, the next enharmonic equivalent (3).

At D# you add Cx, etc.

As you continue you have more enharmonic equivalents (or common tones in flat keys), until you get to B sharp, which is 7 enharmonic equivalents, same as the key of C.