I fun little exercise is to work out what would be different if we found aliens that also used 12 tone equal temperament, and also used a subset of 7 of those 12 notes to make their major scale, and also followed the same convention that we do of naming the notes of the major scale C, D, E, F, G, A, B like we do and naming the other 5 by naming a major scale note and adding modifier to tell how far away the not is from that like our # and b modifiers, and we both have the convention that when naming the notes of a transposed major scale we use each letter exactly once--except they picked a different 7 notes to be their major scale.

Our 12 tone scale with the major scale note names and notes not in the major scale marked with dots looks like this:

  C.D.EF.G.A.B

Suppose the aliens have a very different idea of what makes a good sounding major scale, and their system is:

  CDE.FGAB....

Suppose we transpose ours up 7 tones. We can represent this graphically by drawing out major pattern for two octaves, and below that drawing our major scale pattern with names replaced by X (X.X.XX.X.X.X) shifted over by 7, then look above to see how to name the notes (remembering we can only use each letter once):

  C.D.EF.G.A.BC.D.EF.G.A.B
         X.X.XX.X.X.X

Our major scale transposed up 7 is G A B C D E F#.

Let's do the same 7 tone transposition for the aliens. Their major scale pattern is XXX.XXXX..., so we get

  CDE.FGAB....CDE.FGAB....
         XXX.XXXX...

That's B Cbbbb Dbbbb Ebbb Fbbbb Gbbbb Abbbb.

At first that seems very different from our 7 tone transposition. We only need one sharp and they need 23 flats! But wait...mod 12 we have -23 = 1.

It turns out for all transpositions if you count sharps in your key signature as +1 and flats as -1, the key signature for a transposition by N tones will have 7N sharps or flats mod 12 in both our systems.

In general if you have a T note equal temperament scale with an M note subset major scale, transposing the major scale up N tones gives a key signature with NM mod T sharps/flats.

I'll leave it as an exercise to prove that. Hint: you can think of a transposition as a two step operation: (1) a shift that keeps the same letters and just adds sharps or flats to move the notes, and (2) a renaming that changed the name you use to name the first note of the transposed scale. Think about what each of those operations does to the number of sharps and flats needed.

Music Theory for the 21st-Century Classroom