They have all been off the site for a while, but in the early years of Horn Matters there were a series of articles by a certain “Professor Corno” which addressed some issues of horn playing and teaching in humorous and pointed ways.** One article in particular was on the topic of the dreaded C on the third space.
The gist of the article was this: C on the third space, fingered T-0, should be a note you have absolute faith in that it will be in tune.
Certainly, you can find very reputable players that firmly believe that 3rd space C should be looked at as being the most in-tune note on the horn. Their idea would be in fact to tune the horn to that note; start the process of adjustment of all the other slides on your horn with the idea that note is correct, adjusting everything to agree with that note. Embrace it!
On the other hand, you can also certainly find very reputable players who note that the harmonic involved is naturally sharp. The idea being that C is the sixth partial of the overtone series when played on the Bb horn, and therefore tends to be slightly sharp. If the note truly is a sharp overtone for you depends on the horn you have (how the maker handled the tapers) and your mouthpiece. Theoretically the C might be slightly sharp, but in reality it might actually be completely in tune due to the skill of the maker.
From the above I think you can see I am in the camp of people who think that note should be thought of as your best note, one you are never afraid of. You really should not be afraid of any note, but if you are afraid of one, be sure it is not third space C! If you are in fear it will be sharp, give the system I describe a try, tune the horn to that note. You will likely need to do less adjusting around your horn if all the notes you normally finger T-0 are ones you don’t fear due to an imagined intonation tendency.
But what if you really can’t lower that note? Well, actually you can, even if your horn does not have a separate Bb horn tuning slide. See this article for more.
**At least that was the idea. The series was cut in 2014 as part of a content review, the Professor was ready to move on. But there were some good topics cut, which will be returned to in a few upcoming articles.