David, I think I see where you are trying to go with this, and I think I can see you frustration, but I am a bit uncertain. You begin with: >Let's assume that the left side of the equation produces an action ratio of 5.5-a fairly standard target. What EQ? Is it Pfeiffer's W / S = the products of the RAs / EAs? I assume are you referring to: (keyout/key in) x (wippen out/wippen in) x (shank out/shank in) = (blow distance - letoff) / (key dip-aftertouch) >Then if we look at the right side of the equation and target a key dip of 10mm, say. By your analysis the denominator would be 7.5 (.75 x 10) making the numerator 39.7mm representing blow distance minus let-off??? I assume now we are referring to W / S = the products of the RAs / EAs. How did you arrive at 7.5? In order for the ratio of W / S to = your 5.5, and given that S = your 7.5, then W (i.e., hammer blow - AT) would have to = 41.25 since 41.25 / 7.5 = 5.5. Given your 10 mm total dip, aftertouch (the moment that the jack tender makes contact with the let-off button) would be 2.5 mm. Still, if hammer blow is actually 46, then let off would have to occur at 4.75 mm. Things are not adding up here and I will address it later. For now, and given your 10 mm total dip, and a 5.5 Action ratio*: W = 44 (not 39.7 or even 41.25) S = 8 (not 7.5) At = 2 *NOTE all of the Kawai action values below are retained, but in order to yield David's required 5.5 ratio rather than the 5.58 Kawai original, the short shank lever arm was increased by 0.25mm. >That doesn't seem to bear any resemblance to what one would expect from an action ratio of 5.5 in practice. The EQ W / S = the products of the RAs / EAs resolves to a perfect balance of the EQs in that W is to S as the RAs are to the EAs, which is the point if one wishes to approach the subject via these EQs. >Clearly the dip-aftertouch number is at issue but the assignment of the AT number seems somewhat arbitrary in order to make the formula work. The AT number is neither arbitrary nor assigned; it is required to make the relationships balance. Note that the W value relates to all the RAs on the right side the EQ, and that the S value relates to all the EAs. >What, then, is the point of the right side of the equation at all? Again, if the right side of the EQ is according to Pfeiffer: "The ratio W / S.is of much less importance for us than the right side of the equation, which permits us to survey at a glance not only the relationship between our lever arms, but also the effect which changing one of the lever arms has on the others, or on the stroke of the key or hammer blow distance." Page 110 - 111 If the right side of the EQ is (blow distance - letoff) / (key dip-aftertouch), then I see your frustration in trying to assign a total key dip. More later. Nick From: pianotech-bounces at ptg.org [mailto:pianotech-bounces at ptg.org] On Behalf Of Nick Gravagne Sent: Tuesday, January 05, 2010 8:24 PM To: pianotech at ptg.org; joegarrett at earthlink.net Subject: Re: [pianotech] Action Ratios Any confusion with the formulas as shown in emails below exists with the (key dip - aftertouch). You cannot arbitrarily assign a number of your choosing to this value, although from a practical regulating standpoint this is often done. The idea of the overall transmission (or action) ratio is that the component ratio of (hammer travel - let off) to (key dip - aftertouch) is that of the product of the (output levers) to that of the (input levers). My studies assign input levers as effort arms and output levers as resistance arms. So, if we designate: W = (hammer travel - let off) S = (key dip - aftertouch) H = rear key lever arm resistance (key out) Rs = whippen lever arm resistance (whippen out) K = hammer lever long arm resistance (hammer out) V = front key lever effort (key in) Ra = whippen lever arm effort (whippen in) N = hammer lever short arm effort (hammer in) The relationship, then, of W / S should be that of the product of Resistance Arms / Effort Arms. But note that S (key dip - aftertouch) implies a calculated value, not an arbitrary one. In order to isolate S the formula works out thus: S = (product of Effort Arms times W) / (product of Resistance Arms) Once the theoretical value of S is isolated; the Action ratio can be calculated. So, using some values from a Kawai action model: W = 46 mm - 2 mm = 44 mm) S = (10.5 - aftertouch) H = 126 V = 245 Rs = 94 Ra = 67 K = 141 N = 18.25 (Jack to knuckle contact taken at half stroke) S = (V x Ra x N x W) / (H x Rs x K) = (245 x 67 x 18.25 x 44) / (126 x 94 x 141) = 7.89 mm which is the theoretical key dip minus aftertouch. Thus aftertouch = 2.61 mm. Said another way; dip prior to aftertouch is 75% of the key stroke, while aftertouch accounts for the remaining 25%. Given this, the ratio of W to S is equal to the ratio of lever arms thus: W / S = 44 / 7.89 = 5.58 and the ratio of the products of the RAs / EAs = 5.575. The ratios not only agree, but they define the Action Ratio at half stroke. Now, the so-called aftertouch value of 2.61 mm seems odd, but it is important to realize that the measurement for this (if we can call it that) begins the exact moment that the jack tender makes contact with the let-off button in a well regulated action, and continues to the a solid bottom at full key dip. In addition, if the aftertouch value is far off from the theoretical remember that In any case, measurements aside, this is how the ratios interact. For more, see Pfeiffer's The Piano Hammer pages 110 and 111. It is necessary to work the formulas and read a few things between the lines as some of the info references his other book The Piano Key and Whippen. Nick Gravagne, RPT Piano Technicians Guild Member Society Manufacturing Engineers Voice Mail 928-476-4143 _____ -------------- next part -------------- An HTML attachment was scrubbed... URL: <http://ptg.org/pipermail/pianotech.php/attachments/20100106/f9aef3f1/attachment-0001.htm>
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