There are definitely no hard and fast rule, and I bet we can find outliers to almost anything. However, a good amount of research has come to the conclusion that in the significant majority of cases, tinnitus volume is within a few decibels of hearing threshold at any given frequency -- going all the way back to Fowler's work in the 1940s. Reed duplicated that result in the 60s, and I believe was the first person to find that trying to connect objective volume/pitch measurements to distress, doesn't tend to show the 1-to-1 correlation you might expect (that is, some people with relatively "loud" or "high pitched" tinnitus based on sound matching, report low distress, and vice-versa). Much more recent research has shown tissue density deficits in two different parts of the brain which appear to correlate separately with tinnitus volume and distress; it's tempting at this time to see that as a smoking gun, but probably that needs another 10-20 years to bake, since we're really in the infancy of a neurological model of how tinnitus operates.
Perceptual experiences are pretty weird and hard to pin down in the lab, in general. So much of what we "experience" is related to the way we construct our internal world. It remains to be seen, but my strong belief is that you can't really separate physiology from psychology, they are just two different models assessing the same things.