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I'm curious if various instruments require that the player compensate in any way (in terms of technique, e.g., force used to bow, force used to blow, or embouchure alterations) when playing low vs. high notes.

On a violin or cello, say, if you use the identical bowing for a low-pitched note vs. a high-pitched note, will one note tend to sound louder than the other or will they both be the same volume? Are string players taught to, say, dig in a little more to make a low-pitched or high-pitched note project relative to the other?

On a clarinet or saxophone, if the exact same embouchure and breath pressure is used on a low-pitched note and a high-pitched note, will they sound at the same volume? (Also, will the upper note tend to be a bit flat or sharp?)

My experience, on the oboe, is that if the exact same breath pressure and embouchure are used on a low note versus a high note, the higher note will tend to be flat and to be the quieter of the two notes. I'm guessing that double reeds have special characteristics that contribute to this phenomena.

As I try and understand the situation better, I thought it might be useful to know if other instruments also require techniques to compensate for volume characteristics that are dependent upon the pitch of the note being played.

EDIT: I realized after posting that this question is more like a survey than a specific problem to be solved and thus it will be difficult to select a "best answer." I do appreciate the answers for specific instruments that arise from direct experience.

As a brass instrumentalist, I can only speak for that family of instruments.

But the difference required to play in extreme registers is often explained with a simple sentence:

Low notes require more air, high notes require faster air.

And a simple experiment proves this: have a tuba player play, say, a middle C at a forte dynamic level and see how long they can hold it. You'd be amazed at how quickly they need to take a breath when they play that C three (or even just two) octaves lower.

Physics tells us that higher pitches have more energy. So if we apply the same amount of energy--bow pressure and speed for strings, air for winds--then low notes will naturally be louder.

However, this is pure physics. It's not a huge effect, and the technique involved in playing the actual instrument completely overshadows it. It's most visible on piano, harp, and pitched percussion instruments, since exactly the same technique is used to play across the whole range of the instrument.

With wind instruments, and especially brass instruments, the need to use more air to support higher notes generally means that it's easier to play high notes louder. Players work to be able to play any note at any dynamic, but there's always a point above which you can only play notes loudly.

On a violin or cello, say, if you use the identical bowing for a low-pitched note vs a high-pitched note, will one note tend to sound louder than the other or will they both be the same volume?

The low note will be louder. But there's so much that goes into bowing that I doubt experienced string players think about it at all.

Are string players taught to, say, dig in a little more to make a low-pitched or high-pitched note project relative to the other?

I don't think this is something that has to be explicitly taught. It's just something that will be accounted for in the listening/playing feedback loop. If anything, I suspect that you think about it the other way--not letting low notes "honk" compared to higher notes.

On a clarinet or saxophone, if the exact same embouchure and breath pressure is used on a low-pitched note and a high-pitched note, will they sound at the same volume? (Also, will the upper note tend to be a bit flat or sharp?)

The embouchure has to change a bit to support the pitch. If you try to play a high note with the embouchure for a low note, it will come out flat, and may squeak. So when you do a leap from a low note to a high note, you make a bunch of small adjustments, and one of those is applying a bit more air to support, and that negates the physics effect.

Been playing saxophone for 10+ years

The lower notes natrually get louder, and the higher notes tend to get quieter. For lower notes we relax the jaw a bit. Higher notes I'll just be mindful of the tendency and play out a bit more.

As a former clarinet/sax player and current cellist, I have dealt with these issues most of my life. The other answers cover wind instruments pretty well.

For string instruments, There are two basic ways to get louder: either press harder (so the string is extended farther before skipping loose from the bow hairs) or play faster (more grab/skip per second hence more energy transfer). But that's just the start.

WIthin limits, it's easier to generate lound tones by bowing closer to the bridge and soft tones by bowing near the fingerboard. If you look at the absolute amplitude of the vibrating string as a function of position, it's easy to see why.

Now, for various ugly physics and materials science reasons, the pitch produced varies with pressure and with overall amplitude. We have to make tiny fingering adjustments to stay in tune as the dynamic level changes -- this becomes automatic as our skill progresses.

And it should be pretty obvious that, due to the large difference in string weight and thickness, which lead to different elastic response, that each string has to be bowed at different speeds and pressures to optimize the output tone and timbre.