The sound generators of string and non-electronic keyboard instruments are nearly always coupled via the soundboard or organ case. Sympathetic resonance between two notes is strongest for pure intervals. With tempered intervals, this effect should be weaker, but in addition the coupling should "pull" the two frequencies away from the frequencies of the notes played separately.
People such as Werckmeister who developed tempering systems defined the intervals in terms of length ratios on the monochord. Nowadays, published data are given in cents, presumably on the assumption that the length ratios are an accurate indication of the frequency ratios. I'd like to know whether this assumption ought to be questioned.
A supplementary question would concern possible differences between tuning a harspichord or piano note-by-note with the aid of an electronic device, and doing it the difficult traditional way by listening to pairs of notes played together.
One interesting point is that Bach, himself, tempered his intervals by actually counting the beats. Therefore each interval (be it a 3ed , 4th 5th etc) had a characteristic number of beats. The intervals being determined independent of string length. It was empirical. This approach was a bit different for concert and chamber music, the beats per intervals being some what different. It was said that the cover page to the Well tempered Clavier has a code with this information.
If the cents approach should be questioned depends on your perspective. The difference can be heard. But, Musial instruments are different from Pitch references. But because this effect can vary between instruments it is hard to generalize. My impression with tuning is, that with harpsichords ( I made several Z-boxes), that the resonant characteristics of different sound board may vary, so the same note on different instruments have a different quality in tone and resonance. On key bound instruments it is exactly this quality and the "pull" that gives the instrument its characteristic "color". Bach was able to exploit it for his uses, as were the Impressionists. Though in different ways. The piano being one of the Impressionists most effective instrument because of this effect. Did they know what the frequency was, unlikely.
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