Most of the “work” of breathing is the diaphragm going down. That opens up the lungs, but pushes your abdominal contents down, which pushes your stomach out in turn.
There’s a little bit of “give” in your rib cage that you can exploit to puff up your chest, but a deep breath is going to involve your diaphragm more and push your stomach out.
And yet - if you want to shrink your waist, you will take a deep breath. As an asthmatic, I am aware that if I start breathing with my chest only, it may be time to use an inhaler.
If you really want to shrink your waist you exhale. I have no idea why people have an instinct to inhale for that purpose.
Or sing. It’s not so much the inhaling part that matters but you want that diaphragm down so you can better control the exhaling part with your strong core muscles rather than the chest. In singing, breath control is critical for intonation; for wind instruments the physics of the instrument contributes greatly to intonation, but the voice has no such advantage. For all, breath control is essential to producing good tone.
Diaphragm breathing accounts for well over half of the air taken in. Chest expansion gets you another 20% or so, and clavicle breathing (raising and pulling back the shoulders) gets 10% or so. The actual numbers will vary with the girth and length of your torso. Long-torso folks like swimmer Michael Phelps were issued longer lungs, for example.
I read this stuff a long time ago when I was looking into Zen breathing; my numbers may be off, but the gist of it is solid.
I think that with wind instruments, the goal for deep breathing is not so much the quality of the sound, but to be able to maintain it for longer before needing a breath.
That’s the opposite of what Cooking With Gas pointed out, since what she said was that it was necessary for good tone for singing as opposed to wind instruments, the latter’s tone depending more on the instrument itself.
Now you’ve done it. Ever since I was a kid every time this gets pointed out I think ‘Now I have to remember to breathe for the rest of my life!’. And I do for the next minute or so. Then I always manage to forget…
Breath control is more important in singing vs. wind instruments for intonation, not tone. Intonation is the precision of the pitch. Poor breathers often sing flat, especially in the high register. The pitch of a wind instrument is largely (but not completely) determined by the resonant frequency of the column of air in the instrument, whereas the voice pitch is determined largely by managing the tension of the vocal cords with the air pressure across them. (Wind players, particularly brass, can “lip up” a pitch by fighting the resonant frequency of the instrument, and poor breathing or poor embouchure can throw them off pitch. Reed players can also do this, as heard in the clarinet gliss in the opening of Rhapsody In Blue.)
And to repeat my closing, good breathing is important for good tone no matter what you’re doing.
One physics definition would be the proportions of component frequencies, also called timbre. The type of waveforms come into play too but I don’t know enough about that to elaborate. (I will just say that I heard once on the radio an interview with the man who set the world’s record for the lowest note produced by a human voice and it sounded to me like a square wave. Really awful. Yes, it was a low frequency but did not sound like a “note.”) Improper breath support can result in a “thin” tone, or weakness in the lower overtones. Lack of control of the throat muscles controlling the larynx can push this into the “shrill” category.
For the human voice, tone gets more complicated. Some of the things that come into play is where the singer is placing the resonance; a nasal-sounding tone in undesirable. Singers talk about “chest tone” as rich and desirable; a “head tone” is also well-rounded but usually for higher pitches, not necessarily going to falsetto. (I am not sure how that works mechanically; the vocal cords are in a fixed location but I know from experience that you can adjust things so you can focus resonance in different places.) Here is a pretty good article about vocal tone and technique.
I know less about tone control for reed and brass instruments since I don’t play any, but if you compare old Louis Armstrong recordings to Winton Marsalis playing classical music, you will hear what bad tone vs. good tone sounds like. (Louis was one of the greats in terms of leading the way for jazz but did not have the best technique.)