Re: Neville H. Fletcher, Thomas D. Rossing (auth.) - The Physics of Musical Instruments-Springer-Verlag New York (1998)
Here are a couple of extracts that show it is not as simple as might be expected;
From page 95;
“An orchestral chime or tubular bell, on the other hand, is essentially a long narrow pipe, as also is the common wind-chime. The dimensions are such that this cylindrical shell can best be considered as a form of bar, with a radius of gyration as defined in Fig. 2.18. The mode frequencies for simple transverse vibrations are then given by Eq. (2.63) and vary with the longitudinal mode number n approximately as ( n + ½ )² . There are, of course, higher modes to be considered, particularly those with m > 0 associated with distortions of the tube cross section. There are also corrections to the simple formula (2.63) for the transverse mode frequencies to allow for coupling to distortions of the cross-section, rotary inertia, and other minor effects (Flugge, 1962). The effect of these corrections, broadly, is to slightly lower the frequencies of the higher modes relative to those predicted by the thin-bar formula.”
From page 641;
“One of the interesting characteristics of chimes is that there is no mode of vibration with a frequency at, or even near, the pitch of the strike tone one hears. This is an example of a subjective tone created in the human auditory system. Modes 4, 5, and 6 appear to determine the strike tone.
This can be understood by noting that these modes for a free bar have frequencies in the ratios 9²:11²:13², or 81:121:169, which are close enough to the ratios 2:3:4 for the ear to consider them nearly harmonic and to use them as a basis for establishing a pitch. The largest near-common factor in the numbers 81,121, and 169 is 41.”
