Ric and Ed, I'll try to respond to you both. Ed, you say you never knew me to be wrong. You just have not checked closely enough. I'm wrong on occasions just like everyone else, but when I see that I'm wrong, I try to correct it ASAP. Ric, Ed, as for as a longitudinal vibration zipping right past the bridge termination: It all depends upon what you mean. In one respect it does. In another respect, it does not. What is a fact is that the bridge presents an ambiguous termination to the longitudinal vibration. Compared to the normal transverse vibration, the longitudinal is high impedance, i.e., high pressure, very little motion. The bridge is not solid enough to completely terminate the longitudinal wave, but it does not just go zipping past either. If you know the longitudinal wave velocity of the wire, and you calculate the longitudinal frequency by using the standard formula, your answer will come out too high if you use the measured speaking length of the string. But if you use the total length of the string, or even the string length from agraffe to hitch pin, your answer will come out too low. The correct answer will be somewhere in between, and it will depend upon everything else around it, bridge, adjacent strings, tail lengths, combined tail lengths, etc. There is just no single correction factor you can plug in for all cases. It will be different for every make and model piano made, and it will vary over different parts of the bridge. I think that notion about longitudinal modes in the string tails comes from Steinway's ORIGINAL patent for the duplex scale - not the patents that followed later on. He believed he was harmonizing the longitudinal vibrations of the tails with those of the speaking lengths. Some believe this is a mis-translation in the patent. Others believe it is not - that it is literal. Be that as it may, that's not what Steinway was doing at all, even though he might have thought it was. The question remains: Do the transverse modes of individual string tails couple across the bridge to those of individual string speaking lengths? I say they don't, unless the bridge termination is faulty. They DO couple to be sure, but over whole sections of the bridge - the closer the tails to the speaking lengths, the more the coupling. If anyone can send me hard measured data to prove me wrong - not just aural impressions - I'll change my stand in a heart-beat, but until then, I'll maintain it. I say a few more words about longitudinal modes in piano strings on my web site, www.claviertech.com, and the PTG Foundation has just published my book on the subject. I don't know if they have listed it yet, but they do have it now. I don't know if they have decided on the price. Anyone who wants one will need to contact the PTG Home Office. Sincerely, Jim Ellis
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