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You can tune any instrument to any arbitrary frequency for center-c and as long as all the notes are of the same relative difference between them the melodies will generally work and harmonize. As long as all the instruments are synced it will sound ok in a band too.
In modern day we have a definition of what frequency center C is. But I have to wonder, how do we know the modern day frequency for middle-C is anywhere near the frequency used historically. In fact, when even was the idea solidified at all?
@freemo We can say with certainty that concert pitch (actually set by A4 being standardized at 440hz today) was higher in the past - as high as A = 401hz. Not every instrument is tunable. Bells and some double reed instruments will never change pitch over their life times. By looking at bells from the 15th and 16th c we can tell that music was in general pitched somewhat higher then today.
The big innovation in modern tuning is equal temperament. All modern digital tuning has an precisely equal ratio between each half step. (about 1.059 to 1) This makes music work like you describe. Start on any pitch and the music will sound the same if the relative intervals are followed.
However much more than absolute pitch, relative pitch is hard without modern measurement methods, Prior to the 1700s the most common tuning was to tune in pure 5ths where every 5th was tuned by ear to a 2/3 ratio. It works pretty well until you get to the point where you go from the end of the circle of fifths back to what should be the note you started on. You find you are off by a scotch less than a quarter tone. This terrible sounding interval is a "wolf fifth" because it howls.
In pure fifths each key had its own flavor and so it was important to play a piece in the key it was written in. The major thirds that define the modern major chords are the most effected by these variations which explains why many pieced before JS Back and company are dominated by 4ths and 5ths that are safer in a pure-fifths tuning.
Bach's innovation which caught on across western music was to "fudge the thirds" to make them all sound similar - giving us the last common tuning the so called well-tempered tuning.
"perfect fifths" are not a problem for fretless stringed instuments, winds and vocals. they just keep exact harmonics. only piano and fretted instruments have this problem and they're not really "classical" instruments. they rarely play grand concerts, operas, etc. but yes, the false notes are heard on piano pretty well. this is frustrating so I prefer fretless stringed instruments as more perfect ones.
@antares The bell approach makes a **lot** of sense. that said bells can presumably be tuned by grinding,and i assume that is how they are fine tuned after the initial casting. But yea your point that once tuned they tend to remain fixed and cant be tuned without modifying the instrument makes a lot of sense.
Thanks for a wonderful response, lots of good info here.