Comment by hansvm

It...depends.

If you're working in a continuous environment rather than discrete (choirs and strings can fudge notes up or down a bit, but pianos are stuck with however they're tuned), you'll often find yourself wanting to produce harmonies at perfect whole-number ratios -- e.g., for a perfect fourth (the gap between the first and second notes in "here comes the bride") you want a ratio of 4:3 in the frequencies of the two notes, and for a major third (the gap between the first and second notes in "oh when the saints, go marching in...") you want a ratio of 5:4. Those small, integer ratios sound pleasing to our ears.

Those ratios aren't scale-invariant though when you move up the scale. Here's a truncated table:

Unison (assume to be C as the key we're working in): 1

Major Second (D): 9/8

Major Third (E): 4/3

However, E is also a major second above D, so in the key of D for a "justly tuned" instrument, you would want the ratio D/E to also be 9/8. Let's look at that table though: (4/3)/(9/8) is 32/27 -- 5.3% too big (too "sharp").

When tuning something like a piano then where you can't change the frequency of E based on which key you're playing in, you have to make some sort of compromise. A common compromise is "equal temperament." To achieve scale invariance in any key you need an exponential function describing the frequencies, and the usual one we choose is based on 2^(1/12) since an octave having exactly twice the fundamental frequency is super important and there are 12 gaps in normal western music as you move up the scale from the fundamental frequency to its octave.

Doing so makes some intervals sound "worse" (different anyway, but it makes direct translations hard) than they would in, e.g., a choir. A major third, for example, is 0.8% sharp, and a perfect fourth is 0.1% sharp in that tuning system.

Answering your question, at first glance you would expect the scale invariance to therefore not limit your choices. Every key is identical, by design.

That's not quite right though for a number of reasons:

1. True equal temperament is only sometimes used, even for instruments like pianos. A tuner might choose a "stretched" tuning (slightly sharpening high notes and flattening low notes) or some other compromise to make most music empirically sound better. As soon as you deviate from a strict exponential scale, you actually live in a world where the choice of key matters. It's not a huge effect, but it exists.

2. Even with true equal temperament or in a purely vocal exercise or something, there are other issues. Real-world strings, vocal folds, etc aren't spherical cows in a frictionless vacuum. A baritone voice doesn't sound different just because their voice is lower, but because of a different timbre. When you choose a different key, you'll be moving the pitch of the song up or down a bit, exercising different vocal regions for singers, requiring different vocal types, or otherwise interacting with those real-life deviations from over-simplified physics. Even for something purely mechanical like piano strings, there's a noticeable difference in how notes resonate or what overtones you expect or whatnot. Changing the key changes (a little) which of those you'll hear.

3. Related to (2), our ears also aren't uniform across the frequency spectrum, and even if they were our interpretations of sounds also depends on sounds we've heard before, leading to additional sources of variation in the "experience" of a slightly lower or slightly higher key.