I just stumbled upon this thread! I knew my ear was itching for a reason.
Follow my line of reasoning:
1. We play with weight.
2. The amount of weight we use varies with the requirements for a musical event, i.e. we will use more weight for louder chords and less weight for softer ones. So, the amount of weight we use for an event will increase the louder we get.
3. Our fingers eventually transfer the weight into the key.
4. Our fingers need a support system in order to receive everything we are throwing at them.
This is where your bridge comes in.
Your bridge is that support system. To me, the bridge is the bones, joints, tendons, and muscles that make up the mass of your entire hand - all the internal, invisible parts inside your hand. Many times the bridge includes the wrist as well.
If the bridge is not "formed", then your hand/wrist/fingers will collapse when you continue to throw weight at them note after note.
And just as our weight we use will increase the louder we play, the structure of our hand must also vary in its strength in order to accommodate this increase.
Here is an analogy:
Jack is on the roof. Jill is on the ground. Jack will drop things down to Jill, and she must catch them.
Jack drops a tennis ball. Jill catches it with no problem.
Jack drops an elephant. Jill catches it, but it took more effort on her part so as not to be crushed by it.
Jack throws an elephant down to her. Jill must use even more strength in order to bear that enormous weight.
Jack is your weight. Jill is the structure of your hand.
A firm bridge simply means that when the weight of our arm drops, and power/energy is directed toward the fingers, that the support structure of the fingers (the bones, joints, tendons, and muscles) is sufficiently prepared to accept and transfer that power into the fingers.
To clarify the fingertip thing:
I believe there are three basic ways to use the tips of our fingers:
1. No bones, using the hand and fingers like a brush. It is as if you are playing with only your skin.
2. Active fingertips (my term), which means that we are attentive, and very subtlety grip the keys as we play them. The word active in this case means not passive.
3. Completely with bones, which means to imagine that your bones are literally in contact with key, with no skin getting in the way to soften the blow.
Obviously, these three ways will create different sounds. My default way is the active idea. If I want more brilliance, I will imagine bones. If I want a less focused sound, I will imagine only skin, with no bones. True mastery comes when you can individually command your fingers to constitute several of these ideas at once, as in a chord, or a fugue.
Incidentally, many people ask why it is important to not bend their fingertips - this is why! How can you vary the intensity of your fingertips if they always collapse?!
If you read my 'Jack and Jill' analogy above, the fingertips would be Jill's feet. And since the feet are the last link in the chain before we actually touch the key, we must tend to them as well.
The final thought with regards to structure is that we must not use more (or less) structure than is necessary to get the job done. The danger in talking about structure over the internet is that some might misinterpret it to mean that we firm our hand up for much of the time, as allchopin innocently did. In fact, the beauty of thinking in terms of structure is that it actually requires less work. The hand should "firm up" for only a millisecond; then we completely relax again.
As I've said before, there are three parts to a note: the preparation, the execution, and the recovery.
Let's say a note takes one second to play - 99% of that second you will be relaxed. The preparation (weight drop) lasts from 1% to 49% of the way through the second. Exactly halfway through the second (50%) you will touch the key and sound the note in one of the many ways we've talked about (the execution). You will then immediately relax and recover for the remainder of the second ((51% to 100%).
Eventually, the preparation phase and the recovery phase will actually merge. In other words, the recovery from an event will be the preparation for the next event, and so on. That is the goal. Much of our practice involves making the execution phase of the note as short as possible. Maybe you are taking too long to recover. I have had students come to me who are working 100% of the time - they never relax.
All of us should be assessing our physical state at all times. That some (in other threads) are talking about reading or watching TV while practicing is beyond my comprehension. I can do most anything while playing too, but I choose to devote 100% of my concentration to my art. Some are proud to be able to split their concentration between several activities. I would encourage those people to reassess their practice habits. There are many threads on this forum which discuss the dangers of practicing and performing on "auto-pilot", and the dangers are many.
I hope this helps.
Robert Henry