If you use Intonia to measure the pitch of an instrument with accompaniment, it would help if the microphone were placed in a way to maximize the contribution of the solo instrument. There are clip-on microphones made for many instruments. I would be interested in hearing how others do with this kind of arrangement.
Intonia was designed to measure the pitch of a violin. If you look at the Spectrum View of a note played on a violin, you'll see many regularly spaced harmonics extending to a high frequency. I haven't tried Intonia with many other instruments: the more an instrument's spectrum resembles the spectrum of a violin, the more likely it is that Intonia will be successful analyzing it. I do know that Intonia does rather poorly analyzing the human voice.
The quality of the microphone being used probably has a great influence on the accuracy of pitch detection. A cheap mike tends to cut off the high freqencies that Intonia depends on.