Jon,
Christopher Leuba (former Principal Horn of the CSO) suggested that a tuner should be used to identify the intonation tendencies of your instrument. With your best sound (vibrant, colorful, resonant), find where each note on your horn falls with respect to the tuner and make a chart of this information. For a low D you might write “21 cents sharp” and for a 4th space E you might write “14 cents flat”. This gives you an idea where your instrument wants to play (using your best sound) with respect to Equal Temperament and no slide adjustments.
With this information, you are more knowledgeable about how YOU play on this instrument, with this particular mouthpiece. If you have different mouthpieces or different horns, he suggests that you do this exercise for every combination that you play on.
You might think that the 4th space E is a flat note and needs to be compensated for in every situation. This would be an over generalization. In C major, the third of the chord needs to be 14 cents flat to align with Just Intonation. So in that case, you would play the E right down the middle (no need for alternate fingerings with slides pulled out).
I wrote an article called
A440, Savings Bonds, and Symphony Trumpet Players and another called
10 Question Music Quiz! Good Luck. Those will give you a theoretical idea of how to apply the “tendency chart” that you put together with your tuner.
With all that said though, simply using your ears is the best way to
find elegance in your intonation. I like a product called TuneUp for this. You will hear people talk about playing against a drone pitch and finding the place where each interval has “powerful serenity”. This is when your ears are telling you that you have arrived. You need to do this over and over to really allow your ear to learn what to listen for and then guide your body to that sound when playing with other musicians.
Use the tuner occasionally as a visual guide, but train your ears to do this on autopilot. That’s the only true answer to this question!