In many ways brass playing is similar to singing. Once you have conquered the technical abilities to play concertos, etudes and exerps, what you will be remembered for is your tone and musicality. What matters is not WHAT you play but HOW you play it. We all remember Luciano Pavarotti for his unforgettable voice. Have you ever heard anyone say "Man! That guy Pavarotti can sing so fast!" ? Of course not! It's his tone that everyone remembers.
There are many things you can do to improve your tone. Here are a few ideas:
1) Long tones. Play 8 beats crescendo and 8 beats diminuendo on a chosen note. Try to maintain a nice warm sound through the whole note. Repeat as many times as you want with other notes.
2) Practice easy pieces and make them sound as good as possible. We are so busy practicing high, fast and loud stuff that we forget to practice soft, slow middle range pieces. Most solos in the orchestral repertoire aren't that difficult technically but require great lyrical skills. Also, playing easy pieces is relaxing and makes you embouchure feel good again if your chops are stiff.
3) Hear the tone that you want in your head. Take a few seconds to imagine the greatest sound in your head before you play. Then, do a few long tones while keep producing that tone in your brain. Your brain will send signals to the rest of your body to make it happen. After a few minutes doing this, you should see your tone changing.
Try to play your pieces, etudes and exerps slowly using THAT tone that you just made and speed up to the desired tempo when you feel comfortable.
4) Stay in shape...
5) But don't over practice. Use your judgement to determine how much you really need to practice. It depends on your level, what you need to play, what are your goals etc. It's hard to sound good if you rarely touch your instrument. On the other hand, nothing good comes out of practicing on worn out chops. Practice a lot but also give your chops some rest.
6) Don't over blow. Unless the conductor asks for more, play at a dynamic you can control. Of course, in your practice room, you will work to expand your dynamic range while keeping a nice sound so you'll have it when you need it. Needlessly forcing air into your horn will only give you a harsh tone without that much more power.
7) Don't be too perfectionist. I never play well when I try to be perfect. It makes me tense, it's boring and I still miss notes! Let yourself be human and you'll sound a lot better.
8)Try different mouthpieces. Take all the mouthpieces at a music store or at your school and try them all. See which one feels and sounds the best. A different mouthpiece can change your tone a lot.
9) Think about it when you play. When playing fast passages,we are often so busy playing the notes that we forget about the tone. Sometimes, just reminding yourself to produce a nice sound will solve the problem.
I'm sure there are many other things that can improve your tone. This is what I can think of for now. If you have other ideas, write them in the comment section and I'll add it to the list.
I am associate principal horn of the Singapore Symphony Orchestra and the author of the progressive methods. I'm happy to share my experience as a horn player and teacher with you.