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

Agree with Al.

Fundamentally I think it has more to do with mic placement than anything else. Close microphones reproduce high frequencies and low level overtones that we never hear live. Can't think of a single occasion when I attended a live performance and heard screechy violins, even in a chamber. And, FWIW, I have found the balance between the strings and the rest of the orchestra on many recordings to over emphasize the strings, especially the violins.

My method is to very occasionally give my speakers a VERY light dusting with sugar. For a slightly lighter tone substitute Splenda. Seriously, a recording can be messed up at any stage and it is very hard to tell at what point it occurred. I have some CD transfers that sound very little like the original LPs; often different generations of transfers will be markedly different. The original CD issue of "WILL THE CIRCLE BE UNBROKEN' was unlistenable, the remaster sounds very close to the LPs, which were very good. On the other hand the original may be bad, in that case there is little to be done. The recording engineer has a tremendous influence over the sound and some are much better than others.

Als assessment seems accurate (as usual) and I would tend to agree.

It also helps explain why tube gear often produces better results in this department than SS gear, though this is not always necessarily the case.

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.