Sudgy--yeah that's sort of the point I was trying to make. The notes *don't* wrap around like that. Yet in our musical culture it is deeply ingrained that they should. In band, when we played the circle of fifths, it was a given that you should end up where you started, it didn't occur to me to think why for some time. Sometimes you even see blurbs about the mathematics of music theory, which emphasize the chromatic scale as a cyclic group in order to talk about group theory. My point is this is not the most fundamental way to describe music, it's just a consequence of the neat fact that (3/2)^12 ALMOST equals 2^7.
You can certainly construct keyboards with different tunings than equal temperment. For example the "well-temperment" of Bach's time, which tries to maintain rational intervals when possible but also divides the difference between (3/2)^12 and 2^7 over several intervals so it does wrap around. This means that different keys would have sounded different on such a keyboard (as opposed to equal temperment, where all intervals/keys are identical), which may have inspired some of the differences between the Inventions and his other compositions.
Going back further than that, the most ancient form is "just-temperment", where all intervals are beautiful and rational and Pythagorean. The price of this is that things sound funkier the further you are from the intended key--just temperment is really established around one tonic. So if you have a keyboard in C, don't expect to play songs in F sharp or G flat (which are different notes here!) and expect it to sound nice.
So let's say you were constructing a piano in the key of C, which we'll say has tonic frequency f. A reasonable set of frequencies for defining the rest of the major scale might be:
C = 1f
D = (9/8)f
E = (5/4)f
F = (4/3)f
G = (3/2)f
A = (5/3)f
B = (15/8)f
C = 2f
The chromatic scale is a little trickier and open to interpretation, but you can figure it out. For example, a reasonable value for C# might be (16/15)f = 2*(8/15)f = (4/3)*(4/5)f, but you might also want it as (25/24)f = (5/6)*(5/4)f. Even the simpler ones: 5/3 is a nice simple fraction and definitely a good choice for A, (5/3) = (4/3)*(5/4) = (5/6)*2. But if it were part of a V of V chord, you might prefer it as (9/8)*(3/2) = (27/16).
So yeah this is pretty academic, and of course I understand the practical reasons why equal temperment came into favor. But I think it's pretty cool to think about, and if you're singing or playing violin/trombone/whatever, you can really revel in it and try to tune those rational numbers just the way your brain likes 'em.
Ramble ramble ramble
