Richard Feynman’s exercise on stiff strings (Physics Today, December 2009, page 46) illustrates well his many interests. He and author John Bryner might have saved themselves a lot of trouble (and maybe missed out on some fun) by examining the substantial literature on the subject. The fact that the stiffness of a string stretches the frequencies of the harmonics has long been known. Lord Rayleigh in 1894 treated the case in which the string is free to tilt at the bridge. 1 He also calculated the effect of movement of the end pins, of which Feynman says, “It is too hard for me to figure how big these effects would be.” In 1939 Robert Shankland and I gave formulas for that and for the case in which the string is clamped at the end, 2 and he presented experimental evidence that the piano string acted with a combination of these. Altogether, piano tuning by ear has always had the effect of taking into account the stiffness of the strings.
[Editors’ note: With sadness, we have learned that John Coltman passed away on 10 February 2010.]
