Scientists are a long way from being able to accurately computationally model exactly what distinguishes an excellent instrument from a mediocre one...but they are trying. Complex relationships exist between the physics of instruments - like finite-string stiffness effects in pianos, wall vibrations in brass instruments, and transfer functions for the bridge of a violin - and human elements like haptics (the touch, or "feel," of an instrument). Cross-correlations between different sensory inputs - auditory, visual, and tactile - mean that human enjoyment of music is something that presently eludes non-linear fluid equations. At any rate, one certainly doesn't need to understand differential calculus to be enraptured by a violin sonata - even in the womb, humans can appreciate and benefit from the ineffable poeticism of music: a language as venerable and distinct as mathematics, in its own right.
On another note (pun intended), the explosive sound of a brass instrument being played triumphantly loudly is...well, explosive! The burst of volume is actually generated by sonic shock waves in the long cylindrical tubes of the instrument! Who knew? JJ Johnson and Felix Baumgartner have something in common!
When a trombone is played loudly, the amplitude of the internal pressure wave can exceed 10 kPa. At that level, linear acoustics is inadequate to describe the wave propagation, since the local speed of sound becomes dependent on pressure. The crest of the pressure wave travels faster than the trough, which results in a progressive steepening of the wavefront. At a critical distance along the tube—a distance that depends on the rate of change of the pressure signal in the mouthpiece and the bore profile of the instrument—the almost instantaneous pressure jump characteristic of a shock wave appears. The sound radiated from the trombone’s bell when an internal shock wave is created has a very wide spectrum of harmonics that can extend well into the ultrasonic range.