Thinking about instruments like saxophones or violins I find it really hard to imagine how people figured out those shapes, let alone without modern computers.
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Most of the work was done before the science necessary to understand what was going on was developed, so mostly it was done through simple trial and error. If you observe that certain forms produce better sound than others, than you don’t bother to keep producing the inferior forms.
Like everything else, trial and error. Hominids have been making music for tens, if not hundreds, of thousands of years. You start with “banging things together makes sound”, move to “banging on hollow things makes different sound”, and after a few different iterations, you have a good-sounding drum. Same with blowing into a shell leading to wind instruments, rubbing or plucking strings for stringed instruments, etc. Anything that sounded good was iterated on, and anything that sounded bad was abandoned.
They did it through trial and error. The original horns were actual rams’ horns that had been hollowed out. Pipes were hollow reeds or bones or sticks. People would make something, and if it worked they’d make another copy. Eventually you’d get professionals who knew they could make good money by coming up with something new.
They didn’t – because the exact shape doesn’t matter that much.
The sound from a trumpet, saxophone, etc. is largely dependent on the mouthpiece and the length of the air column in the instrument. The mouthpiece – and the mouth of the performer themselves – create the initial vibration. The length of the air column determines the standing waves that can form.
The valves on a trumpet or saxophone change the length of the air column. Most of the complexity of a saxophone’s shape is all the extra tubes to get the valves to work.
The bent shape of the “main tube” of a saxophone is not really relevant to the sound. A straight tube and a curved tube of the same length sound almost identical. The bends in a saxophone are for convenience; it’s “folding up” a longer tube into a more compact space, to get a long-tube sound without holding a very long and unwieldy instrument.
String instruments work similarly. The main thing is simply the length and tension of the strings. You can take a stick of the right length, pull the same strings across it, and get *most* of the sound of a violin, guitar, etc. In fact, there are plenty of instruments like that – harps, lutes, etc. are just “strings on a frame”.
It sounds a bit different with a resonance cavity – the body of a typical string instrument – but even then, it’s mostly just the size of the cavity that matters. Exact details of the shape are about ergonomics and aesthetics as much as they are about acoustics.
Particulars of the shape, material, and so on can influence the exact note you get – but the note is a “nice sound” regardless, because that mostly depends just on the frequency. A 329-Hz E note on a guitar, a lute, and a balalaika will sound different, but will all be pleasing notes.
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