The actual note doesn't matter at all. Just the ratio. We do have a reference note, but this is only for convenience. (Indeed, the reference note in Europe progressively changed, from 415Hz to 440Hz).
I'm not sure the ratio of two notes played one after the other really counts by itself. However in many instruments, two consecutive notes will tend to overlap (piano, for instance). Also, many instruments (especially those with strings, like claviers and violins), resonate better when the note you play has a "good" ratio with the natural notes of the instruments, even if you don't play them! Some instruments were even designed around this principle. So you have to maintain a good ratio with respect to these "base" note at all times.
In practice, the ratio of two notes played one after another is indeed important.
I'm not sure I understand your last question. Actually, you can change the tune of a piano. So, the optimization you speak of is possible even on a modern piano. Just re-tune it.
A final note about why "simple ratios" sound better: When you make a string resonate, it doesn't do only one note. It does its base frequency (say f), and many others (every multiple of f). So, in a piano, when you play a G at 100Hz, it also plays at 200Hz (a G), 300Hz (a D), 400Hz (a G) etc. Note that there exist actual keys whose main frequencies are 200Hz, 300Hz, 400Hz and so on. They will resonate, which will make the sound richer, louder.
The problem with equal temperament is that the only ratio which is really respected is the octave (2 to 1). In such a case, the D I mentioned above won't be quite at 300Hz, and won't resonate well. So equal temperament became practical only when instruments became loud enough to make up for the loss in resonance.
I'm not sure the ratio of two notes played one after the other really counts by itself. However in many instruments, two consecutive notes will tend to overlap (piano, for instance). Also, many instruments (especially those with strings, like claviers and violins), resonate better when the note you play has a "good" ratio with the natural notes of the instruments, even if you don't play them! Some instruments were even designed around this principle. So you have to maintain a good ratio with respect to these "base" note at all times.
In practice, the ratio of two notes played one after another is indeed important.
I'm not sure I understand your last question. Actually, you can change the tune of a piano. So, the optimization you speak of is possible even on a modern piano. Just re-tune it.
A final note about why "simple ratios" sound better: When you make a string resonate, it doesn't do only one note. It does its base frequency (say f), and many others (every multiple of f). So, in a piano, when you play a G at 100Hz, it also plays at 200Hz (a G), 300Hz (a D), 400Hz (a G) etc. Note that there exist actual keys whose main frequencies are 200Hz, 300Hz, 400Hz and so on. They will resonate, which will make the sound richer, louder.
The problem with equal temperament is that the only ratio which is really respected is the octave (2 to 1). In such a case, the D I mentioned above won't be quite at 300Hz, and won't resonate well. So equal temperament became practical only when instruments became loud enough to make up for the loss in resonance.