I'd say bell ringing and vibrations of the metal can be both good and bad.  
First of all let me say that horn sound can be a difficult thing to quantify 
objectively because much of the sound is created in the listener's own ear.  
The horn may produce certain overtones that can be measured on a machine, but 
the listener's ear creates it's own overtones from the combination of those 
that are being sounded.  So we're stuck with a little voodoo in trying to 
explain a horn sound.
    The metal of the horn vibrates all over in different ways when it's 
played.  You can feel the location and character of the vibration changing on 
both your left and right hands as you play up a scale.  The bell vibrates a 
certain way, the tubing, the tuning slides, the air column inside the unused 
valve slides.  Changing the location, thickness, shape of a brace can affect 
things.  Think of a violin sound post.  You move it a quarter inch and 
everything changes.  It would be interesting to try to devise a test like the 
violin makers do.  They put powder on the plate of a violin and then bow the 
edge of the plate.  The powder will move into patterns and you can see how the 
plate responds to the different modes of vibration.  Unfortunately the horn is 
too round, all the powder would just fall off.  
    Anyway, to oversimplify, all those various vibrations may be good for what 
you're trying to play, or they might be bad.  Corner one of your trumpet player 
friends and ask them how they like their middle C.  It's kind of a stinky note 
on a lot of even expensive trumpets, usually sharp.  Then take their trumpet 
and without telling them pull out the first slide just a bit and have them try 
the middle C.  Probably much better, but why?  No air is going through that 
slide when you play C so what's going on?  Voodoo of course!

- Steve Mumford
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