Magic (splitting hair) lines

[A440A at aol.com]

Ron writes:

<<  it doesn't matter where the capstan/heel location falls 
relative to the wippen body (but it must be on the line of centres at 
half blow, which you have already addressed).  >>

Greetings, 
       I have a different approach to the "magic line at half blow" concept.  
I find better response on actions that see the capstan/heel contact point 
intersects the magic line at let-off.  I think this is why: 

1.  There exists a variable "rate" when two arcs intersect, such as happens 
between the key and whippen. (I believe the technical  reason for this is 
termed "angularity")   In our case, for every unit of key movement, there is a 
corresponding distance the whippen moves. The farther the capstan/heel 
intersection is from the magic line, the less whippen movement per unit of key movement 
there is.  The maximum rate is found when the two arcs intersect on a common 
axis, i.e., the "magic line". 
       Assuming that the arcs' intersection point is below the magic line at 
the beginning of the stroke, the rate is at a minimum in the beginning.  To a 
small degree, this allows more key movement per unit of movement of the 
whippen.  In effect, the key is in a "lower gear" relative to the whippen. In 
overcoming the inertial resistance of the action, the lower rate helps reduce the 
effort to set the parts in motion.  As the intersection point approaches the 
magic line, the rate increases, reaching a maximum when the intersection of both 
arcs is exactly between the centers. It is at this point that the key is 
moving the action at its maximum leverage, (a "higher gear").  This is advantageous 
in transferring power from the finger to the hammer with the least amount of 
effort.  
      
2. Friction is a function of speed.  On the beginning of the stroke, where 
the capstan speed is at its minimum, the effect of friction is lessened.  Even 
though the maximum misalignment is found at this point, the effect of friction 
is minor compared to the inertial effect of the parts at rest, which the lower 
"gearing" helps.  As the contact point approaches the magic line, the 
friction lessens, finally disappearing as the magic line is reached.  Since let-off 
represents the addition of friction from a variety of sources, (the movement of 
the jack against the knuckle, the friction of the tender on the let-off 
button, the contact between drop screw and pad), I think reducing the capstan/heel 
friction to zero at this point helps.  

      With these two items in mind, consider what happens when the magic line 
is reached at half blow.  From that point on the rate is decreasing and the 
friction is increasing.  So, approaching let-off, the "gearing" is beginning to 
decrease as the hammer is supposedly reaching maximum velocity, and friction 
begins to once again increase as escapement approaches.  These two conditions 
are opposed to what I want the action to do.  
    In an attempt to facilitate the extremes, i.e., maximum power when 
needed, and the most delicate pianissimo playing possible, I want greatest 
mechanical efficiency and the least friction at the very point of escapement. I find 
that this is generated by placing the capstan/heel intersection on the magic 
line at the beginning of let-off.  

       My experience says that all this is so, and my logic concurs, but that 
don't make it necessarily true!  I would be interested in what others might 
have observed. 
Regards,    
  

Ed Foote RPT 
http://www.uk-piano.org/edfoote/index.html
www.uk-piano.org/edfoote/well_tempered_piano.html