The complicated mechanism that is the grand piano key action has been

mathematically analyzed in many different ways here at KeyForce 1. The

##### goal in this has long been to not only develop an accurate way of compre-

##### hensively balancing the 88 notes of a piano action, but to actually do so in

##### a

*cost-effective manner*##### . Many dozens of pages of calculations, scattered

##### across both space and time, were created along the way. Many of the results

were inevitably discarded, but certain guiding physical principles eventually

##### emerged. An offshoot of all this was a rigorous verification of the age-old Down

##### Weight, UpWeight, Friction equation, and its extension into the world of con-

##### tinuous forces and displacements. That was discussed at length in U.S. patent

##### 8,049,090, along with physical means of making these improved methods a

##### reality. While these methods certainly improved the accuracy of the longstand-

##### ing Down Weight, Up Weight and Balance Weight numbers, they had little to

##### say about the world of

*comprehensive*#####

**key action balancing**##### . In the context

##### of key action balancing,

*comprehensive*##### is herein taken to mean satisfying

##### both dynamic (inertial) and "static" equations.

Solving the equations is not just something that should be done because

it can be, or because it sounds cool or...gasp...geeky. It should be done because

the result is an incredibly powerful mathematical model for *each note* across

##### the keyboard. Each model has unprecedented predictive capabilities, ultimately

##### resulting in a nonobvious note-specific prescription for mass changes required

##### to achieve true key action balancing, both statically and dynamically!

##### Looking at the equations above, one sees that the only "reaction force" terms

##### are BF in eq. P and F

_{tot}##### in eq. Q. These are both measured noninvasively with

the Key Force 1 machine. There are no component-level reaction force terms

##### such as Strike Weight or Front Weight. Rather, BF and F

_{tot}##### are ensconced

##### solidly in equations chock full of mass-based terms. Equation XY has no

##### reaction force terms, and simply expresses the true, effective moment of

##### inertia in terms of the various masses and locations. Having these important

properties allows the three individual equations to "talk amongst themselves"

as it were, and thus be solved simultaneously. Note that Eq. P is the purely

##### static equation, while the other two equations are the dynamic equations. The

##### Dyna-Stat process not only brings much-needed dynamic/inertial equations to

##### the key balancing fight, it brings them in a way that allows the entire shabang to

##### be solved, and solved in a note-specific manner!

The goal was not simply to form equations that describe each key mechanism,

##### but for these equations to have certain properties that would ultimately allow

##### them to be solved together. This simultaneous solving was really the holy grail

##### that would allow tremendous predictive power to be brought to bear on real,

##### measured piano actions. The important two properties the equations would

##### have to exhibit for this "holy grail" to be obtained are:

As I detailed in my 4-Part piano action inertia series in early 2014 issues of

the PTG Journal, this business of accelerating the key and measuring the

resulting total (and inertial) force is fraught with difficulty. As I indicated

##### there, the equation-solving process could also be done using the KRT equation,

##### rather than the inertial force/acceleration equation. The reason is because both

##### the inertial force equation and the KRT equation contain the all-important

##### Inertia at the Key (IK) term. Thus, while there are four potential equations

##### available for comprehensive action balancing, two of these are not independent,

##### and can be switched. In essence, our development of the Key Return Time pa-

##### rameter - and its analytical formulation - opened up a more easily accessible

"back door" into solving real-world

##### key balancing problems.