Hi Chris. This is a great observation that I have been struggling with myself in terms of whats the difference between passive and active and how does it all relate to posture. The way I see it:

The diaphragm cannot be controlled or used in order to push air. It simply contracts, which causes air to be drawn into the lungs and then it releases, passivly going upwards while the lungs are being emptied (https://en.wikipedia.org/wiki/Thoracic_diaphragm). There is no notion of this in the balloon mental-model. Nevertheless, the self-shrinking balloon shows what happens when the diaphragm relaxes and no forced exhalation is involved.

If I now want to create more air-support, I use the muscles in the abdominal area, the forced exhalation (active, kick or squeezing of the ballon).

In both scenarios, passive and active, I can use some (actually very, very little) muscular support to keep an upright posture. To me, keeping the chest open and up is more a matter of letting the shoulders float (backwards and down) while keeping my head over the center of gravity, always thinking the body is widening, a feeling that any relaxed motion is possible at any point in time. Alexander Technique gave me some insights into this (https://en.wikipedia.org/wiki/Alexander_Technique).

This support is not to be confused with pushing air. In fact, one can do this while holding ones breath. The benefits however are tremendous. If you keep your chest open this way during (forced) exhalation until little air remains, and you then open the mouth and relax the abdominal muscles, you will find a massive amount of air streaming into the lungs in no time without you doing anything, setting you up for the next phrase. This creates a very relaxed and natural flow of air in-out-in-out.. . Probably the opposite of what people may have in mind when they think “chest up” which can cause the idea of overusing muscles and actively stretching the body. I think this is not covered by Greg – I however experience this quite clearly.

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Related to this: I was taught to belly-breath for playing. This led to a stiff chest, choking, collapsing posture and tension in the body. Also see Greg: breathe to the lungs.

In retrospect I think the belly-breathing exercises that we probably all know just help experiencing the relaxation of the abdominal muscles, which in turn feeds the air-in-out-in cycle mentioned above.
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