Help me play the Trumpet
Python 3 2, 125 119 81 bytes



lambda e,f,s,t,n=2*'A G# G F# F E Eb D C# C B Bb'.split():n[n.index(e)+2*f+s+3*t]


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Saved a lot of bytes thanks to Jonathan Allan.



My original solution (in Python 3):



n=2*'Bb B C C# D Eb E F F# G G# A'.split()
e,f,s,t=str(input()).split()
print(n[n.index(e,9)-2*int(f)-int(s)-3*int(t)])


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Saved 6 bytes thanks to @HyperNeutrino.



Explanation

First, I make an array of notes, but doubled in length so I don't have to worry about looping around from Bb to A.

Then, I take input in the following format (for example):

Bb 1 0 1


I then find the index of the starting note using n.index(e,9) (the 9 is there to make sure that I start well in the middle of the (doubled) list. I calculate the desired offset with the expression:

2*int(f) - int(s) - 3*int(t)


Where f is the first valve, s is the second valve, and t is the third.

Finally, it simply prints the note found in the list by subtracting the offset from the starting index.