Key Leads and Inertia

[Phillip Ford]

>Stephen, Bill, John, Phil, and the rest of y'all:
>
>Seems from reading the essay posted (below) that there is a direct
>relationship between how far from the balance rail pin any eventual
>leads are (perhaps the center of all key mass is better ??) and the
>occurance of the breakpoint seperating soft/hard zones. At one point it
>is said:
>
>         "moving the location of the lead moves the breakpoint (along the 
red
>line)"
>
>thoough there is no relationship (formula) given. I am wondering just
>what this might be.

I imagine that it wouldn't be too difficult to come up with a formula to relate break point to lead location, but it probably would be for a specific key configuration (a specific location of finger force and a specific location of action mass).  My mind is feeling too feeble at the moment to take that on.  But a couple of observations I can manage:

As the lead is moved closer to the balance point the slope of the balanced line becomes closer to the unbalanced line, so that the response of the key must become more like an unbalanced key.

As the lead is moved closer to the balance point the negative consequences of having balancing lead, in terms of force required at high levels of acceleration, becomes smaller.  Or in other words, the level of acceleration that it takes so that the balanced key feels heavier than an unbalanced key is higher.

>Its interesting that the use (or not) of leads changes absolutely
>nothing relative to the division between hard/soft play.

I don't understand what you mean here.  The charts show that the location of the leads affects the break point, or the level of force required to go from soft play to hard play.  At a given lead location, the break point is independent of the amount of lead - perhaps that's what you mean.  One thing that I question about the charts is the meaning of the break point between soft and hard play.  Stephen's conjecture is that in the soft play area the action is 'harder to control'.  You'll notice that the break point for the unbalanced key moves with the lead location under discussion, and yet the unbalanced configuration hasn't physically changed at all.  Why should the location of the breakpoint, or transition from the hard to control zone to the easier to control zone be changing for the unbalanced key?


>Another point I am a bit confused on. In all cases, the balanced line
>has least slope, then the half balanced a bit more, and the non balanced
>has the most slope. And clearly until the breakpoint is reached the non
>balanced line gives the least acceleration for force. Yet at the same
>time the degree of increase in acceleration for same amounts of increase
>in force is greater for the non balanced line. This seems at odds with
>the idea that the key would be more difficult to control in the soft
>zones for the balanced key then the non balanced key.
>
>It would seem (intuitively) to me that one would have better control
>when the slope of the acceleration is slight. And this regardless of
>which zone we are in.
>--
>Richard Brekne

This is an interesting point.  I have to agree with you that intuitively it would seem to me that the action setup with the most shallow slope would be easiest to control.  Less change in acceleration per change in force.  The action would be less 'touchy', so to speak.  I believe that this also relates back to the phenomenon of jerk that Sarah was referring to before.  She seemed to think that it was a desirable thing to minimize jerk.  Since jerk is the derivative of acceleration, the action having the acceleration curve with the smallest slope would have the least amount of jerk.  From my own experience, when I've played on actions that have no lead, they never seem to feel quite right.  Perhaps this has something to do with it.  Admittedly, I haven't played many examples, and they've been poor quality actions in general, so it's probably not safe to draw conclusions.

The slope of the curves is determined by the inertia of the key and action masses.  So the unbalanced key has the highest slope, since its inertia is lowest.  The absolute level of acceleration is determined by the torque applied to the key.  For the unbalanced key F and W1 are applying opposing torques.  For the balanced key, F, W1 and W2 are applying torque.  The torque from W2 is adding to the torque from F.  So at the same level of F, the balanced key is applying more torque since W2 is 'helping'.  So, at a give level of F the balanced key will have a greater absolute acceleration than the unbalanced key.

Another thing to note (and which isn't really made clear from these charts) is that the slope of the curves is proportional to the distance of the finger force F from the fulcrum and inversely proportional to the action mass M1.  In other words, as you decrease the key ratio (move the finger further out from the fulcrum) the slope of the lines will increase.  As you decrease the mass M1 the slope of the lines will increase.  So as the action mass decreases (as you move up the scale, say) then the rate of change of acceleration for an increase in force will be greater.  Also, a longer key will give a greater increase in acceleration per change in force.  This seems intuitive to me.  As you make the key longer you get more leverage but you sacrifice some control.  As you move up the scale it takes less effort to accelerate the hammer but you have a bit less control than you do lower in the scale.  Perhaps there's some ideal slope for this curve (and it probably varies with the individual).  I think of it a bit like setting your mouse tracking speed.  Set it too high and you can really zip it around but you can't control it.  Set it too low and you've got lots of control but you go crazy because it's so unresponsive.  Perhaps Stephen's investigations will shed some light on this.

Phil F




Phillip Ford
Piano Service & Restoration
1777 Yosemite Ave - 130
San Francisco, CA  94124