| Abstract: | The acoustic spectrum of the kalimba (African thumb piano) is measured and analyzed for tonal structure. The frequency f(1) of the fundamental tone of each tine (key) is investigated in relation to the frequencies of its two dominant overtones, f(2) and f(3). These frequencies are identified as the first three modes of transverse vibration of a beam of rectangular cross section. As is typical for vibrating-beam instruments, the overtone sequence is inharmonic, that is, the sequence f(1), f(2), f(3),[ellipsis (horizontal)] is unevenly spaced and the frequency ratios f(2)/f(1) and f(3)/f(1) are not integers. The kalimba tines are modeled by applying the Euler-Bernoulli beam equation with one end clamped, the other end free, and an intermediate point (the bridge) simply supported. Unlike the cases of free-free and clamped-free beams, it is found that the clamped-supported-free frequency ratios f(2)/f(1) and f(3)/f(1) are not fixed values, but depend uniquely upon where the bridge supports and subdivides the tine. The model solution is more thoroughly investigated analytically for the special case in which the beam segment ratio is unity, which has some analytic solutions. Numerically computed mode frequencies agree well with acoustic measurements, validating the model. Mode shapes are computed for the first three modes of a typical tine. |
