Weight, Friction, Inertia, and Distance:

The mechanics of a grand action are brilliant, a series of levers that reaches inside the piano while converting finger force into a hammer force that can play strings with musical expression. Such a feat has many complicated details, but observed individually, they are each logical, pragmatic – mechanical. Geometry is the design engine for this collaboration. And its leverages both transfer these forces and transform the distances they travel. Each finger’s mechanical advantage (up to 2 kilograms to play a 12-or-less gram hammer) is divided by 5 in a tradeoff multiplying finger travel distance by 5 into a hammer travel distance that accomplishes fff. Or ppp.

For these mechanics to work well, friction is needed. Too much necessitates a heavy downweight and challenges repetition. Too little produces instabilities that can cause tonal and mechanical problems. When we weigh off, friction clouds our ability to observe the place of best balance, but it will be found halfway between the least weight that will depress the key (downweight) and the most weight the key will lift on its own (upweight) by matching their speeds. If speeds for a particular note are slower than for the neighbors, it has more friction somewhere, and if faster, less friction. The key weights (leads) used to fine-tune this weight balancing add inertia to the system, and where some lead is expected, too much is a problem, particularly at fff.

Let’s review our model for assessing what happens with weight and friction. Then we’ll add inertia and distance to be able to anticipate more fully what happens when an element is changed.

Downweight (DW), balanceweight (BW), and friction (F) from hammer weight (HW):
      DW = WR • KL  (whippen resistance • key leverage)
      WR = SR • WL + WW   (shank resistance • whippen leverage + whippen weight)
      SR = HW • SL + SW   (hammer weight • shank leverage + shank weight)
      DW = ((HW • SL + SW) • WL + WW) • KL  
      UW (upweight) = 20 minimum  
      BW = (DW + UW) ÷ 2       
      F = (DW – UW) ÷ 2

Let's choose low bass hammers for the natural and sharp sampling. With their fixed minimum/maximum upweight of 20g (minimum to assure good repetition, maximum to not make the heaviest hammers feel heavier), we only need to solve downweight to know balanceweight and friction. By positioning key weights on top of the key until the speed of downweight depressing key matches the speed of 20 grams upweight being lifted by key, we can by trial and error optimize materials, geometry, and regulation specs against the weighoff they produce. And we can use the model to guide us in our choices of what to try.

You may notice that our model leaves out the key’s backweight, which includes wood, backcheck, capstan, and key lead (sometimes) and the key’s frontweight, which includes wood, key covering, and key lead. Frontweight and backweight get zeroed out in weighoff so they don't appear in the formula. But they are significant to our assessment of inertia and may influence what we choose in our preparation to regulate and weigh off.

Action inertia is the attribute of parts to remain at rest when at rest and in motion when in motion. More hammer weight, coupled with more lead in the keys, increases inertia in the system: the mass that a finger must set in motion (and that friction and soft materials must overcome in slowing and stopping that motion). We adjust key weights in keys as a compromise between the benefits of lower touch-weight in soft playing and better responsiveness and finger comfort in loud playing. We seek a sweet spot: least lead (or other key weight material) with appropriate friction for normal specs, fast repetition, consistent stability, and good power. Repetition can benefit by a shorter blow distance, for instance, but power will be sacrificed and some amount of control lost.

Returning to our weight formula:
      DW = ((HW • SL + SW) • WL + WW) • KL       
      UW = 20 minimum   
      BW = (DW + UW) ÷ 2       

      F = (DW – UW) ÷ 2

This presents a list of what in keys and topstack can be changed to modify playability and tone: downweight, hammer weight, shank leverage, shank weight, whippen leverage, whippen weight, key leverage, balance weight, and friction. Weighoff and inertia should also be on the list to be adjusted and considered, respectively. And the weights of backchecks and capstans should be added. But to try these possibilities, we need a distance version of the formula to anticipate how proposed changes will affect distances.

First, let’s see how changing knuckle-to-center distance changes shank leverage.

Shank leverage changing with knuckle position (SL):
      RA (resistance arm) = √ (bore² + hang distance²)
      RA = √ (2.0² + 5.25²) = 5.62

      EA (effort arm) = √ (knuckle distance² + knuckle height²)

18mm knuckle-to-center (.71):
      EA = √ (.71² + .51²) = .87
      SL = RA ÷ EA

      SL = 5.62 ÷ .87 = 6.46

17mm knuckle-to-center (.67):
      EA = √ (.67² + .51²) = .84
      SL = RA ÷ EA

      SL = 5.62 ÷ .84 = 6.69

15.5mm knuckle-to-center (.61):
      EA = √ (.61² + .51²) = .80
      SL = RA ÷ EA

      SL = 5.62 ÷ .80 = 7.03

A greater knuckle-to-center distance reduces effective hammer weight but increases distance needed at key dip. A smaller knuckle-to-center distance increases effective hammer weight and reduces the amount of dip needed.

The same set of proportions will work for the distance formula, but we need to use what can be observed and tested. When the minimum initiating force of downweight is applied, key, whippen, and hammer will only travel to the point where jack tender and rep lever meet letoff button and dropscrew (the beginning of letoff, drop, and aftertouch). Since we can’t see or measure what happens beyond that point, we need to subtract the unreached parts of key travel and hammer travel from our calculations.

Hammer travel (HT) to key travel (KT):
       HT = BD (blow distance) – LO (letoff)
      BD = 1.75
      LO = .09 (averaged)
      HT = 1.75 - .09 = 1.69
      KT = KD (key dip) - AT (aftertouch)
      AT = .06     
      HT = KT • KL • WL • HL

      KT = HT ÷ SL ÷ WL ÷ KL

18mm knuckle-to-center (.71):
      KT = 1.69 ÷ 6.46 ÷ 1.44 ÷ .53 = .343
      KD = KT + AT = .343 + .06

      KD = .403

17mm knuckle-to-center (.67):
      KT = 1.69 ÷ 6.69 ÷ 1.44 ÷ .53 = .331
      KD = .331 + .06

      KD = .391

15.5mm knuckle-to-center (.61):
      KT = 1.69 ÷ 7.03 ÷ 1.44 ÷ .53 = .315
      KD = .315 + .06

      KD = .375

I find these numbers a little frustrating. Proportionately, they seem about right but the actual key dips needed are deeper. Add about .025" to each. "About right" works for anticipating what will happen when the knuckles are moved, but the math falls short of full accountability. I think this has to do with the change of direction distance takes at letoff. And arcs are longer than straight lines. But for getting the idea of what will happen when one or more ingredients are changed, the models have utility.

If we remove the weight-specific and distance-specific elements from the weight and distance formulas and start in each case from the hammer (it's weight and blow distance givens), we find the same proportions inverted: 

Weight: SL • WL • KL

Distance: SL ÷ WL ÷ KL

This is the nub: weight and distance trade symptoms for any change in lever arms. Only a balance of the two interests will optimize playability for the player. If blow distance, letoff, and aftertouch are fixed, geometric changes will change key dip. If weights of hammer, shank, whippen, and key are fixed, geometric changes will change downweight.

Dip must be not too shallow or too deep and must coordinate with desirable blow distance and aftertouch. And downweight must be not too heavy or too light, with acceptable upweight and inertia. These constraints generally lead to one very specific combination of elements that work best, with any deviations diminishing the quality of results.

Part 1 "Why Weigh Off?" 

Part 2 "My Strategy" 

Part 3 "Method and Style"

Part 3a "Regulation Strategy Review" 

Part 4 "Pragmatics"

Part 5 "Trials and Errors"

Part 6 "Levers"

Part 7 "Playing with Proportions"

Part 9 "Plan by Number?"

Part 10 "Best Balance for Best Art"