How flat is that open E on the F side?

Of course, it depends.

But if you make the assumption that the horn is in tune for its C (which is your responsibility to achieve) and that all the other harmonics are perfectly in tune to the harmonic sequence (which isn't true, but you get a decent approximation to it for the better brands of horn, at least if you have a good embouchure) then it is possible to calculate how far out the other harmonics are from equal temperament. So I've done that. Taking the various harmonics upwards from middle C, they are out from equal temperament as follows:

• C = in tune
• E = 13.7 cents flat
• G = 2 cents sharp
• Bb = 31.2 cents flat
• C = in tune
• D = 3.9 cents sharp
• E = 13.7 cents flat
• F = 51.3 cents sharp
• G = 2 cents sharp
• Ab = 40.5 cents sharp
• Bb = 31.2 cents flat
• B = 11.7 cents flat
• C = in tune

Move all those pitches up a perfect 4th to get the tuning of the equivalent harmonics on the Bb side.

From this you can see easily why some harmonics were for the most part avoided by the classical composers. from middle C upwards, the E, G, C, D, E, F (with some bending by use of the hand to flatten it), G and top C harmonics are adequately in tune.

Although the E is 13.7 cents flat (100 cents make a semitone), when playing hand horn, you would generally be crooked in the key of the piece, and so that note would expect to be flat relative to equal temperament - in fact making it so will make it sound in tune for the key you are in.

So if you are playing a transposed part, if you see a written E, you know it is probably the third relative to the key of the piece, and so a bit of flatness on that note is OK.

But if you are playing a romantic or modern piece for valve horns with the part written in F irrespective of the key of the piece, then you can't make that assumption. That means that you have to think about the key and listen out for the rest of the orchestra for your tuning.

Just because you have your horn in tune to the oboe's A doesn't mean that you don't have to adjust note by note during a session. String players of my acquaintance say that when they are playing in a key with sharps, they position their fingers to play slightly sharper for C# and G# than they do when playing Db and Ab, in other words they are using just temperament rather then equal temperament. You will have to match them - whatever key the piece is in.

I also recall a story of a pianist who was the rehearsal pianist for a big choir. When he played the rehearsal piano by itself (tuned of course using equal temperament) it sounded perfectly well in tune, but when he accompanied the choir it always seemed a bit off. The singers would have (without realising it) been singing to just intonation in order to get their chords perfectly in tune, and so the piano would have sounded a bit odd.

Tuning is a complex business. Having a double horn makes it easier to play in tune, but don't assume that simply because you have a good instrument and you have tuned it, that you need do nothing more. When playing in an ensemble, you need to listen to every note, and make instantaneous adjustments where necessary.