I have been reading, with interest, some of the PTG email regarding deriving the action ratios.
Here is an opinion which tries to consider the non-ideal behaviour of the action parts.
Getting back to basics, the human finger has a thickness that leads to giving the piano key a certain dip; let's say nominally 10mm, for the purpose of the discussion. This dip then allows for the movement of the finger across the keyboard in the most comfortable fashion. Given that pianists may play the key at any position, not just at the front of the key, the dip is a nominal value (that is, a set-up specification) upon which the action will deliver its function; to strike the string with the hammer, in a controlled manner, to produce the sound.
The mechanics of the action allow for control of this striking action, so as to give the pianist the choice of dynamics and speed, including touch and repetition. By touch, we mean the ability to play smoothly, as legato, or detached, as staccato. And any of the degrees in between. The touch of the piano, to the pianist, is the single property which determines the quality of the instrument; if he can control the sound then he can produce the music he is seeking.
In order to set up the piano action correctly, for touch, the hammer travel distance, as hammer blow, is regulated to fit the given dip at the front of the key. For any given piano, there is little choice about how much blow to set, once the dip is determined.
Firstly, the dip must include a small amount of aftertouch, so as the pianist can feel the bottom of the key stroke, and come to rest in a comfortable manner, rather than a hard landing. There is some choice with aftertouch, and the technician looks for this in ensuring the jack has sufficient clearance from the knuckle. Pianists may also ask for a “softer” and “closer” aftertouch.
Secondly, the hammer letoff must allow for escapement whilst at the same time give control at all dynamics; for the pianist, a close letoff will give nice control for pianissimo, but for the technician, it may lead to double-striking or bobbing of the hammer. Again, there is some small choice with let-off.
Once these two variables have been established, the blow can be set. This assumes here that the jack and repetition lever have been set to nominal position first. Regardless of how any of the individual action components move, the key travel will raise the hammer by a given amount, which will satisfy the requirement for aftertouch and letoff.
The response of the hammer to the key travel is non-linear; that is to say, the ratio of displacements (also called the velocity ratio) varies as the key is depressed. This is because the internal ratios of the action components vary as they move.
The key pivot centre is not precisely at the balance pin centre; rather it pivots about a centre which moves towards the front edge of the key balance felt punching as the key is depressed. This causes a change on the ratio of the key-capstan pair.
The capstan, in turn, slides across the whippen heel, so that the velocity ratio varies for this capstan-jack pair. We can regard the jack to whippen pair as non-variable, and does not need to be considered as an additional pair here.
The jack-to-knuckle contact point varies as the jack slides (rolls) across the knuckle as the hammer lifts, and changes the velocity ratio across this pair.
With these three pairs of varying component ratios within the action, a simple multiplication of the three ratios to give a number consistent with the overall velocity ratio is not so easy; this is due to the difficulty of determining exactly which ratio we should use for each pair.
One obvious answer is to use the final ratio, at the instant (more or less) of the hammer travel at letoff. This can be quite easily measured in the piano, as the distance travelled by the hammer to letoff, divided by the distance travelled by the key to aftertouch (corresponding to the same instant at letoff). Arc distances are theoretically more correct, although linear measurements (point-to-point) using vernier calipers give a very accurate result, the aim being to reconcile this overall measurement with the product of the component ratios.
If the component ratios are measured in a similar fashion, at the end of the stroke, then their product will agree with the overall action ratio. Any error can be put down to measurement accuracy, and can be progressively eliminated as accuracy improves. Using dial gauges is useful, but becomes tedious when working in the real world.
The point here is that to try to derive this result theoretically, by assuming certain positions for lever centres and points of contact, is fraught with frustration, unless you recognise that the ideal calculations must be modified by the non-ideal behaviour of components turning on wool and leather about movable rotation centres. And theoretical ratios can only be taken at one position, being at letoff.
All of this recognises that the action component ratios, within each of the three pairs, are themselves able to be modified to give different responses within the same overall action ratio. But that's another story.
Cheers and thanks for the chance to express my ideas,
Peter Sharp
"Nothing is ideal, expect possibly theory. Even that can be wrong. Which is not ideal."
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