Say I recorded an audio of C major: do re mi fa so la si do.
Now imagine I used some software to change the pitch higher (I think Audacity has a feature like this). I was wondering: if I change the pitch of this to be higher, is that like cycling through each key? (C major, C♯ major, D major, E♭ major, etc).
(I'm sure it's micro-tonal but I mean eventually you get to each key right?)
Yes; if you move each pitch by the right interval, you can move from C major to C♯ major, then D major, etc.
But there is one main caveat to this: assuming you move each pitch by the same interval, this only works in an equal-tempered tuning system (see edit below). Within equal temperament, each half step is precisely 100 cents and each whole step is precisely 200 cents. With those equal and equivalent intervals between each pitch of the initial C-major scale, moving every pitch up 100 cents moves the entire scale up a half step to C♯.
But if you were using another tuning system, you would not have a series of 100 and 200 cents between the pitches of the C-major scale. If you were to move each pitch of, say, a meantone-tempered C-major scale up 100 cents, you would have a mean-tone tempered C♯ major scale that is suddenly in a different meantone system than the original scale.
Doing so would basically destroy the whole point of a meantone-tempered scale. So not only does the action you describe only work in an equal-tempered environment, the very concept is really best fit for an equal-tempered environment.
Edit: I was unclear with the end of my fourth paragraph. The subsequent scale would still be in a meantone temperament, but it would be a different C♯ scale than if you took the C♯ scale within the meantone environment where you took the original C-major scale. As @topo morto succinctly put it, these would be "pitches that weren't in the temperament being originally played, but the notes will still be pitched the same relative to each other." This was my intent.
Furthermore, I didn't realize programs could do work within different tempered systems. As such, this works in all temperaments as long as your program supports it. That's good to know!
Like Richard, I'm going to say yes, but with a slightly different caveat - it will work as long as your re-pitching algorithm works well! You'll generally be fine if you simply resample the audio such that the 'speed' change is what causes the pitch shift (the only cleverness needed there is some filtering to avoid aliasing), but if you use any pitch-shift algorithm that as able to work independently of audio speed, that's more complicated and will introduce distortions that may alter the perceived pitch of some of your notes. Such algorithms are typically very good these days, but it is something to watch out for.
I will assume that you are only considering equal temperament so that the frequency ratio between of any semitone interval is the same.
The answer depends on your point of view. Suppose that you were able to play a certain piece on a keyboard but only in the key of C. Your friend could play it on the guitar but only in the key of E. Played normally, these would not sound good together. However, if you recorded one, moved its pitch as you suggest, and played the other instrument against the recording then it may sound good. Some electronic keyboards can do this in real time. You play as if in C but the keyboard outputs a set interval higher or lower. You would think that you are playing in C but a listener would hear it as if you were playing in E.
I used to have that problem with my piano. It was relatively in tune but slightly flat. So, I could not practise together with a recording. I had considered processing some recordings so that they were equally flat. I no longer need to as I have just had the piano restored to standard pitch.
Another example is historically informed performances. These might not be at the modern standard of A4 = 440Hz. A historical performance in C might sound as if it were in Bb and maybe even A to modern ears. It is a matter of opinion which key it is in.
Finally, you could consider a capo on a guitar which has a similar effect. If you put a capo on the first fret then you could play as if you were in E but give the effect of playing in F.
Old music module players like SoundTracker, on legacy computers where memory was low, and where you had 4 channels to play with, the trick of using the same chord played at different replay sample rates was widely used specially for:
With 1 sample (looped for strings) recorded in, say F (Fa/La/Do), you could play with slower sample rate or higher rate to cover all major chords, and using only 1 channel out of 4.
Natural piano/string play would use reversed chords to stay in the same range (ex: to play A# you'd play Fa/Sib/Re not Sib/Re/Fa), whereas in that case the chord notes are is always in the same order. So those chords sound very "techno".
