Jason,
Sorry to jump in before Ed, but I’m afraid I’ll forget to do this if I don’t answer now.
When you mention changing the characteristic of sound when you are playing, you seem to be limiting yourself to only focus on the tone quality. While manipulating the tone is one way to change your overall sound, there are so many other ways (vibrato, articulation, decay, etc.) that you may want to consider these other possibilities (in addition to modifying your tone).
Please go to the Yamaha web site and watch the video with James Thompson called
How To Manipulate Sound Quality. You will need to click on “Video” in the bottom left corner and then scroll down to find Jim’s masterclass topics.
When you’ve watched his presentation, consider some standard literature that you have played on different instruments. Now, get your C trumpet and try to sound like you are playing the opening to the Hummel on your Eb. Choose a good cornet solo from Arban and explore getting that specific characteristic cornet sound on your C trumpet.
If this sound picture in your mind is strong enough, you will be able to make these differences. I didn’t think that I could do this. I talked to my instructor about it and he had me go through this little drill. I was amazed that I was actually able to change the “tone” of my horn to pretend I had a smaller horn in my hand.
Exploring vibrato, articulation style, etc. will modify your sound much more dramatically than just changing the tone alone.
I hope these ideas help!