I want to take a different view of your question.
Sound quality of what? The microphone, amplifier, and speaker system or of the pleasing sound of a particular instrument?
A few examples: An accoustical guitar and an electric guitar with no amplifiers sound almost the same. What makes the electric guitar's distinctive sound is intentional distortions intruduced by an highly non-linear amplifier or a fuzz box. Without distortion you would probably not like the electric guitar.
Consider a horn, a french horn and a trumpet can produce the same note (same fundamental frequency) but they sound very different because of harmonic content. What is distorted?
A long time ago I attended a lecture by the head of the Stanford University Music Department. The topic was electronic music. One point he made is that a pure note (an undistorted sine wave) is easy to produce electronically and sounds about the same as a tuning fork for a simple plucked string. But(!) that is not music (he said). Music also includes the distortions, variations in tempo, attack, sustain, and decay of notes, squeeks of chairs, clatter of feet, fingers sliding on strings, fingernails on the keys, and all that stuff beyond pure tones. Leave that out an it sounds dead.
So, what do you want to measure?
A bass viola makes a sound, do you like it? Record, save, playback, amplify it, run the electrical signal into a speaker or earphones, and so on. Do you still like it? Is it better or worse? Speakers and earphones sound very different.
Another aspect you can measure is the effect of the environment. An open-air bandstand in the park sounds a lot different than a hard-surfaced room. The addition of echos adds another layer of distortion. Do you still like it?
Sound quality is what you call it. Once you define it, then you can measure it.