Two elephant topics have been in the room of my recent postings: center of gravity (CG) effects in grand action mechanics and part clearance (PC?) constraints in grand action design and setup. They are big topics, and neither easily fits a blog posting. But I've had this notion to consider them together, at least where they apply to hammer travel and strike.
So, first, one at a time.
We feel center of gravity when we stand up. Good balance depends on thoroughly understanding the complex mass of our body and how it relates, in motion or at rest, to the pull of gravity. There might be good reasons to defy it a little, but past a certain point, the cost is likely to outweigh the benefit, and we'll pay a price. We take the subtle play with gravity for granted, and for the most part, we're expert at the application of our CG knowledge, as it relates to body balance, anyway.
A piano hammer is a complex mass, particularly a flared grand hammer. Let's take a tenor hammer as our example. Often, tenor hammers are flared to match the flaring of the strings they address. So the hammer's crown strikes perpendicular to the height of the strings, but it also strikes perpendicular to the alignment of the speaking length of those strings.
A typical calculation would have the molding at right angles to the strings at strike, with the shank horizontal, parallel to the keybed. At rest, the crown, tipped back a little at the bottom of the arc it will follow to the strings, is level with the hammer center. At strike, now over its shank, it is the bore hole and the shank glued within it that is level with the hammer center. Hammers at rest, the action can be removed without the hammers catching on or scraping the pinblock, which is neatly fit with plate webbing, etc between the action and the strings, where they are held at tension to emerge from their agraffe at strike height. (Caveat: strings generally climb a little from agraffe to bridge surface, applying downbearing by being offset from their line to the hitch pin - one small detail in many...).
Looking from the side, if the hammer had access to the full circle around its centerpin, the CG would find balance at the bottom of the circle (hammer hanging with gravity) with the shank tilted toward the tail, i.e. the hammer head has greater mass than its tail and offsets the balance. Standing all the way at the top, again the shank would be off-vertical to balance the extra weight of the crown end over the tail.
At the bottom the hammer would hang with virtually all its weight as tension stress in the length of the shank. At rest position, the CG of the hammerhead would be extended furthest out from the center (effectively heaviest) and stress would be split between tension on top and compression at the bottom of the shank along its length. At the top, stress again would be aligned in the shank as compression over the center. And at the back way down, maximum weight would have the shank above horizontal, with stress as tension in the back side, compression in the front.
Thinking this artificial path around the center, illustrates that both center of gravity and effective weight are changing as the hammer moves. And the stresses in the driving shank are shifting (both with greater complexity than is possible to render here). And that the stresses and relative weights are at their extremes at the four points of a compass, except that the shank as compass needle is offset because the center of gravity of the hammer (and shank) is offset by the head side being heavier and extending farther from the shank than the tail side.
If we now take our consideration to a top view looking down, we see a similar shifting relative to the center. At north and south of our artificial circle around the center (we're looking down the edge of the "compass", now, down the axis of gravity toward its source), a least profile presents, and at east and west the greatest. The greatest reach and, thereby, greatest weight, of the hammer is in rest position, with less reach and effective weight at strike. Again, because the hammerhead above the shank is heavier than the tail below it.
In our tenor example, the hammer is flared, with the near (to the centerpin) shoulder on the treble side of the shank and the far shoulder on the bass side. The bass shoulder, being farther from the centerpin than the molding, which is farther from the centerpin than the treble shoulder, travels in a circle with a wider radius than the molding or the treble shoulder. The far (bass) shoulder travels a longer distance through its arc than the near shoulder, travels the same path from rest to strike at greater speed than the near shoulder, because it's traveling a greater distance, and effectively has more weight than the near shoulder, because it's farther out the lever arm of the shank.
Similarly, but with a 90 degree twist, the crown of the tenor grand hammer travels farther and faster at its treble side, compared with its bass side. This may account for an aural perception that the treble side is harder (or, anyway, louder) than the bass side. It is actually striking with slightly greater speed, slightly farther out from the center pin. Isaac Oleg mentioned this discrepancy in one of his comments.
Blaine Hebert (in one of his comments) pointed to a radical change of spacing in this area of the piano that varied with force of blow. This comes about by CG being to one side of the shank's driving force (worsened by insufficient constraint of bushing tightness).
Balance of the mass of our tenor hammer over its shank would have it slightly tilting toward the lighter side, toward the treble.
One other visualization might be useful before we move on. Holding a tenor hammer at its centerpin in four positions, shank horizontal, feel the difference between the hammer with crown to strings (strike position) and the hammer crown facing the bass, the keybed, and the treble. This exaggerates the stresses and imbalance of the tilted hammer. The closer the mass of the hammer is to being centered over the shank, the less stress and greater stability.
Obviously, part clearance is important and a place in the action where it is trickiest is between flared hammers. Let's do another brief visualization exercise. If we spin the tenor hammerhead around its shank at strike, most of the way around its compass points would be clearly unuseful.The worst case would be with one hammer crown heading toward the treble and its neighbor to the treble heading its crown toward the bass. Spacing for clearance would be a little more than bore distance times two!
If we now imagine the axis of our hammer vertical but the hammer spins on this axis to the points of that compass. Where the bore goes straight through the molding from the front side or the back side, we have easiest clearance (the situation in the top hammer sections in most grand pianos). If the hammer is flared 90 degrees one way or the other (the bore through one side or the other), the spacing again would be way overly compromised. Any flare raises a clearance issue, with the worst possible at 90 degrees. The issue is least pressing when neighboring hammers are in the same spot in their arcs, say rest position, or strike position.
But to the extent that flared hammers are at different places, the problem increases. In the case of two tenor hammers, when the arc of the treble hammer's far shoulder, which flares toward the bass hammer, travels up its arc passing the stationary bass hammer's near shoulder, which flares toward the treble hammer, the hammers are closest to each other, even though their shanks and their hammer parts never veer from their respective arcs. If the arcs of travel between the two hammers are not parallel, these relationships are more complicated. If they move apart, relief would be had between the two hammers, but the next neighbors to the bass and to the treble would have to move their arcs even further apart to sustain the benefit. Not a viable solution with 88 hammers to clear.
If the tenor hammers have parallel arcs of motion but those arcs are tilted slightly toward the bass, some spatial relief happens by splitting the difference of top and tail clearance. There is an optimal amount of tilt per degree of flare.
And here's a point to all this description: the tilt to balance a flared hammer's center of gravity is in the opposite direction from the tilt to maximize clearance. A cost/benefit analysis of pros and cons is needed to deny the benefits of one sensibly for those of the other.
I feel that what happens at strike is most crucial to the piano's TPR (tone, power, and repetition), and that the best result shares compromises proportionately to favor the hammer crown-to-string impact. Mostly, we must accept the general design and layout of action and strings. But tilting from vertical for clearance negatively affects every aspect except clearance. If clearance can be had by hammer and tail shaping, shaving, or sizing, I would go there before tilting. Tilting to center of gravity may improve balance and stability a little but at a cost of complication to the strings above, the filing integrity of the crowns, and/or the relationships to whippens and keys below. A version of shaping, shaving, or sizing could be applied to improve the verticality of CG: a larger near shoulder and smaller rear shoulder, for instance.
In any case, setting travel to vertical and hammers to vertical at strike offers the most sensible balance between considerations and the greatest simplicity of application. Simplicity of application is significant to consistency, which offers maybe greatest benefits to player, regulator, and voicer. And also significantly improves time management, pricing predictability, and, by extension, bottom line!!