What makes strings sound "sweet"?

I believe that what it basically comes down to is harmonic balance, which in turn can be influenced by many things.

As you probably realize, a note consists of a "fundamental" frequency, together with harmonics (integral multiples) of that frequency, in particular proportions. Variations of those proportions (in other words, variations of the amplitudes of those harmonics relative to each other and to the fundamental) will produce the variations in sound quality you are describing.

Those proportions can be affected by mic placement and distance, the amount and character of reflected sound in both the recording space and the listening space, frequency response unevenness, distortion introduced by electronics and speakers, and many other factors.

Over-emphasis of odd harmonics (3rd, 5th, 7th, 9th etc.), and particularly higher-order odd harmonics such as the 7th and 9th, is generally considered to be the most objectionable form of harmonic imbalance.

Regards,
-- Al

Screw high order harmonic distortion. Harmonic distortion is related to the signal itself. IM distortion is, by definition, enharmonic, and much worse. Many factors contribute to sweet sound, frequency response, harmonic distortion spectra, amp behavior at clipping, room treatment, etc., but a lack of IM distortion is certainly one of them.

Viridian, I agree with your comments about the importance of having minimal IM distortion, and I would additionally mention TIM distortion.

But note that I referred to harmonic imbalance, not harmonic distortion. The distinction being that I was using the term "imbalance" to encompass a much broader set of ways in which the harmonic structure of a musical note can get messed up (just one of which is harmonic distortion), including frequency response unevenness and the other variables I mentioned in my earlier post.

Regards,
-- Al