There is an old and simple trick for showing the equality of two quantities. We learn it in high school algebra, 1st or 2nd year most of us. It goes like this. If two separate quantities are both equal to a third quantity, then they are also equal to each other. Put another way ; if A = C and B = C, then A = B.
Readers should have a basic understanding of the new action metrology as outlined by David Stanwood and printed in previous Journal articles, and a knowledge of basic leverage principles.
Using the sum of Balance Weight and Front Weight (BW + FW) I intend
to show that the Stanwood Strike Weight Ratio can be expressed as the product
of the appropriate individual ratios of the key stick, whippen and hammer
shank. That is to say, that there exists a set of points (Effort
force, Fulcrum, and Load force) on each of the three levers in the action
that corresponds too and correctly describes the Strike Weight Ratio. This
quantity then, can be arrived at by measuring the appropriate lengths of
the arms of each of these levers much in the same fashion as the overall
action ratio traditionally has been calculated.**
A quick look at Archimedes’ Law of the lever reminds us that for a lever to be in balance, then d1 x W1 = d2 x W2, (W being weight, and d being the distance that weight is from the fulcrum) The quantities of distance and weight are inextricably connected. Levers and leverage in their simplest conception are all about weight, distance and velocity.
We know that the key stick is an example of a simple lever. Its’ ratio is determined by the distance from the top front of the key to the fulcrum divided by the distance from the top of the capstan to the fulcrum. If the ratio is 2:1, then this applies to weight, distance, and speed. 10 mm down movement at the key front will give 5 mm up at the capstan. 10 grams of weight at the capstan will balance 5 grams of weight at the key front. The speed the front of the key travels is twice the speed of capstan travel. The addition of one or more new levers to construct a multi-levered system such as our piano action does not alter this consistency
When referring to the piano keystick it is customary to refer to the ratio in the inverse of the above discussion of simple levers. Our example above has then a Key Ratio of 0.5.
Whatever weight is at the capstan is halved at the key front. In the grand piano action, that weight is balanced by the amount of mass in the key forward of the balance rail pin (Front Weight) added to the quantity referred to as Touchweight or Balance Weight. The key thing to understand here is that Balance Weight is the exact weight needed to add to the existing mass in the front half of the key (FW) to bring the action weight on the capstan into some condition of equilibrium. Hence the term Balance Weight.
We have then that the sum of BW and FW is equal to the weight of the action at the capstan times the key stick ratio. BW + FW = Action Weight x KR. This much is Archimedes all over again ! Nothing more and nothing less.
One way of demonstrating that the Stanwood Strike Weight Ratio reduces to the product of the individual lever ratios is seen in the following example. A second more formal proof is found at the end of this article in which the attentive reader will see one way in which the Balance Equation can be arrived.
Given the following action parameters;
Key Ratio (KR) 0.5
Whippen Ratio (WR) 1.5
Hammer Shank Ratio** (HR) 7
Strike Weight * (SW) 10 grams
Whippen Radius Weight * (WW) 18 grams
Calculating the action ratio the old way we have:
R = KR x WR x HR
R = 0,5 x 1.5 x 7
R = 5.25
Using the weight parameters above with the ratios of the individual levers we can solve for the quantity BW + FW. The Strike Weight times the appropriate Hammer Shank Ratio is felt at the Jack top. This quantity times the whippen ratio added to the whippen weight, is felt at the capstan as the Action Weight. We have at the capstan :
Action Weight = (SW x HR x WR) + WW
Action Weight = (10 x 7 x 1.5) + 18 = 123 grams
When multiplied by the key ratio, we have the weight that BW + FW will balance.
Action Weight x KR = FW + BW
0.5 x 123 = 61.5 grams = FW + BW
Since the Stanwood formula also uses FW + BW …
BW + FW = (SW x R) + (KR x WW)
… we can use our 61.5 grams along with the KR, SW, and WW to see if the resulting ratio using the Stanwood equation is the same as the ratio above.
61.5 = (10 x R) + (0.5 x 18)
61.5 = (10 x R) + 9
52.5 = 10 x R
R = 5.25
It follows then that
because ? Action Weight x KR = BW + FW = (SW x R) + (KR x WW),
that ? Action Weight x KR = (SW x R) + (KR x WW)
The Strike Weight Ratio can be calculated as the product of the appropriate ratios of key stick, hammer shank, and whippen. The resulting R can be then put directly into the Stanwood formula and used in whatever connections are valid for said.
Front Weights can be solved for in similar fashion, by using the formula shown in figure 2. For those capable of making the relevant action design decisions, awareness these relationships should be a must. Those who do not understand these basic action mechanics well enough to make such decisions, are encouraged to use the consulting services and resources of Touchweight experts in the field.
In addition, because the overall ratio is the same regardless of whether it is taken from distance measurements, weight measurements, or speed measurements, any convention of measurement that does not result in this consistency is either in error, or is a measure of some other relationship. This is important to understand as it explains why the Strike Weight Ratio and other conventions for measuring the action ratio do not yield the same results. The Stanwood Strike Weight Ratio is based on the particular quantity of Strike Weight. It is the vertical weight, including the shank itself, that is felt at the end of the hammer shank with the shank in horizontal position and resting directly under the center pin. This must be clearly understood in order to arrive at the set of points that describe the hammer shank ratio appropriately. Those unfamiliar with this or other metrology mentioned herein should refer to Stanwood documentation for a closer explanation.
It means that one can bypass the use of the Stanwood formula all together and simply use BW + FW = ((SW x HR x WR) + WW ) X KR instead. There are certain advantages to doing exactly this due to the fact that all three lever ratios and their respective weight quantities are visible in the equation. Any change resulting from the manipulation of any single operator is then easier to see and understand. This perspective also begins to open, if ever so slightly, the door to a greater understanding of the dynamics of the action, something the present action ratio concept or the Balance Equation can not describe to any significant degree.
The fact that the Strike Weight Ratio is simply the product of the individual ratios of the three levers also serves greatly to justify Stanwood’s whole approach. The idea that the action ratio describes and allows some degree of purposeful alterations of the mass distribution in the action should be wholeheartedly accepted now without further debate. Such alteration can be motivated from both tonal and touch related concerns. This in no way is meant to imply that the Stanwood ratio describes more then a small part of grand action dynamics. Yet that fact does not detract from the real possibilities the Strike Weight Ratio does provide.
The equivalency of these two formulas should also dispel and clear up argumentations raised against the Stanwood method that speak to assumed conflicts between traditional action geometrical concerns and Touchweight Design concerns.
One of these has been criticisms relating to the need to change capstan
/ heel or knuckle positions that sometimes occurs in altering the ratio
of an action. It should be clear to anyone understanding action design
basics that there exists a range of useable action ratios that are consistent
with action geometrical concerns in any given piano, though a thorough
justification for this claim is beyond the scope of this article. However,
having first assured optimal geometric configuration, the subsequent choice
and execution of a smooth strike weight curve balanced by appropriate front
weights allows the action technician an ability to provide unprecedented
consistency and control in action performance and Touchweight.
*measured as per Stanwood protocols.
** It must be noted that there are several conventions for measuring
hammer shank ratio and it has been long debated which of these is most
Regardless of which convention is used at any given time, it is important to agree on which set of parameters are being discussed so as to assure a common ground for meaningful discussion.
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A second proof of equality.
((SW x HR x WR) + WW) x KR = (SW x R) + (KR x WW)
(SW x HR x WR) + WW = ((SW x R) + (KR x WW)) / KR … divide both
sides by KR
= ((SW x R) / KR) + WW …take KR into both expressions on right side
SW x HR x WR = (SW x R ) / KR ……subtract
WW from both sides.
SW x HR x WR x KR = SW x R …… multiply both sides by KR
HR X WR X KR = R ……. Dividing both sides by SW.
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